Search

This work reports a novel method to estimate the geometrical parameters of a 2-(3-RRPS) parallel robot intended for manufacturing tasks. The method uses camera calibration techniques, and it is based on the concept of vertex space. The advantage of this technique is that the system does not require complex electronic instrumentation, and only uses a CCD camera as a main sensor and planar patterns, which makes it portable, accurate and low cost. To ensure the quality of the measurements, a methodology for characterization of the measurement system is included. The applicability and the advantages of using the proposed method are shown by means of the estimation of the geometrical dimensions of a spatial parallel manipulator with a relatively complex kinematic architecture. Experiments are conducted and show a significant improvement in manipulator accuracy when the parameters estimated with this technique are used.

The current trend in end-user devices' advancements in computing and communication capabilities makes edge computing an attractive solution to pave the way for the coveted ultra-low latency services. The success of the edge computing networking paradigm depends on the proper orchestration of the edge servers. Several Edge applications and services are intolerant to latency, especially in 5G and beyond networks, such as intelligent video surveillance, E-health, Internet of Vehicles, and augmented reality applications. The edge devices underwent rapid growth in both capabilities and size to cope with the service demands. Orchestrating it on the cloud was a prominent trend during the past decade. However, the increasing number of edge devices poses a significant burden on the orchestration delay. In addition to the growth in edge devices, the high mobility of users renders traditional orchestration schemes impractical for contemporary edge networks. Proper segmentation of the edge space becomes necessary to adapt these schemes to address these challenges. In this paper, we introduce a segmentation technique employing lax clustering and segregated mobility-based clustering. We then apply latency mapping to these clusters. The proposed scheme's main objective is to create subspaces (segments) that enable light and efficient edge orchestration by reducing the processing time and the core cloud communication overhead. A bench-marking simulation is conducted with the results showing decreased mobility-related failures and reduced orchestration delay., Comment: accepted at ISNCC 2021

International audience; A novel two-stage approach is proposed for image manipulation and generation. User-interactive image deformation is performed through editing of contours. This is performed in the latent edge space with both color and gradient information. The output of editing is then fed into a multi-scale representation of the image to recover quality output. The model is flexible in terms of transferability and training efficiency.

Edge space is not only the main place for residents’ public activities, but also the key component of urban public space. It promotes people’s spontaneous activities and social activities. Reasonable urban fringe space design can create the possibility for people to transform the city from bottom to top. While preserving the cultural background, Tanhualin historical life block has formed a rich and varied space form. This paper analyses the composition of its marginal space to explore the importance of edge space to urban public activities.

Urban ecological land planning is an important foundation and premise of urban ecosystem protection. Based on the structure of geosynclinal fold belt, the spatial allocation model of ecological land in the urban fringe of Xi’an city is designed. Taking the ecological land in the urban fringe of Xi’an city as an example, after analyzing the structure of geosynclinal fold belt in this area, the comprehensive index model of regional ecological effect based on the structure of geosynclinal fold belt is adopted to calculate the comprehensive index of ecological effect in the urban fringe of Xi’an city and obtain the ecological number in the urban fringe of Xi’an city. According to the ecological data, the paper constructs the spatial planning model of urban ecological land based on ant colony optimization algorithm and realizes the spatial allocation model design of ecological land in the urban fringe of Xi’an city based on geosynclinal fold belt structure. The results show that the edge space of Xi’an city based on geosynclinal fold belt structure is in good agreement with each other after the spatial allocation of ecological land in urban fringe area of Xi’an city by using the designed model, and the ecosystem service value is high, which provides some help for improving the urban ecosystem protection.

Edge detection is one of today’s hottest computer vision issues with widely applications. It is beneficial for improving the capability of many vision systems, such as semantic segmentation, salient object detection and object recognition. Deep convolution neural networks (CNNs) recently have been employed to extract robust features, and have achieved a definite improvement. However, there is still a long run to study this hotspot with the main reason that CNNs-based approaches may cause the edges thicker. To address this problem, a novel semantic edge detection algorithm using multi-scale features is proposed. Our model is deep symmetrical metric learning network, which includes 3 key parts. Firstly, the deep detail layer, as a preprocessing layer and a guide module, is employed to remove some low-frequency information and still maintain the edge. Secondly, the deep encoder-decoder networks extract multi-scale features of original image, integrated for complementing information among each level feature. Finally, metric learning is introduced to generate a metric space used to predict edge result. It is easy to distinguish different categories, such as edge space and object space. Simulations and comparisons on benchmark datasets demonstrate the proposed algorithm is superior to the others through visual and quantitative evaluation, and specifically, the score of ODS reachs 0.788.

The edge connectivity is a kind of classic measure of fault tolerance of networks. It is well known that the edge-connectivity of a simple, connected, vertex transitive graph attains its regular degree. It is then natural to consider the relationship between the edge connectivity and the number of orbits of its automorphism group. The double-orbit graphs with two orbits of the same size is a generalization of vertex transitive networks, which contains several classic network models. In this note, we obtain a sufficient condition for such double-orbit graphs to be super- λ ′ .

Vertex models are generally powerful tools for exploring biological insights into multicellular dynamics. In these models, a multicellular structure is represented by a network, which is dynamically rearranged using topological operations. Remarkably, the topological dynamics of the network are important in guaranteeing the results from the models and their biological implications. However, it remains unclear whether the entire pattern of multicellular topological dynamics can be accurately expressed by a set of operators in the models. Surprisingly, vertex models have been empirically used for several decades without any mathematical verification. In this study, we propose a rigorous two-/three-dimensional (2D/3D) vertex model to describe multicellular topological dynamics. To do this, we classify several types of vertex models from a graph-theoretic perspective. Based on the classification, mathematical analyses reveal several conditions that enable us to apply the operators accurately without topological errors. Under these conditions, the operators can completely express the entire pattern of multicellular topological dynamics. From these results, we newly propose rigorous 2D/3D vertex models that can be applied to general multicellular dynamics, and we clarify several points to verify the results obtained from previous models.

The second author showed how Katsura's construction of the C*-algebra of a topological graph E may be twisted by a Hermitian line bundle L over the edge space E. The correspondence defining the algebra is obtained as the completion of the compactly supported continuous sections of L. We prove that the resulting C*-algebra is isomorphic to a twisted groupoid C*-algebra where the underlying groupoid is the Renault-Deaconu groupoid of the topological graph with Yeend's boundary path space as its unit space.

Given a topological graph $E$, we give a complete description of tracial states on the $\text{C}^{\ast }$-algebra $\text{C}^{\ast }(E)$ which are invariant under the gauge action; there is an affine homeomorphism between the space of gauge invariant tracial states on $\text{C}^{\ast }(E)$ and Radon probability measures on the vertex space $E^{0}$ which are, in a suitable sense, invariant under the action of the edge space $E^{1}$. It is shown that if $E$ has no cycles, then every tracial state on $\text{C}^{\ast }(E)$ is gauge invariant. When $E^{0}$ is totally disconnected, the gauge invariant tracial states on $\text{C}^{\ast }(E)$ are in bijection with the states on $\text{K}_{0}(\text{C}^{\ast }(E))$.

Given an integer K and a simple edge-weighted undirected graph G = ( V , E ) , the Distance Geometry Problem questions the existence of a vertex realization function V → R K such that each vertex pair adjacent to an edge is placed at a distance which is equal to the edge weight. This problem has many applications to science and engineering, and many methods have been proposed to solve it. We propose some new formulation-based methods.

The edge space lies between the “black” and “white” part of the city, associates the vagueness of the space with “ambiguity” and are the places to inspire people to interact with each other. Edge space is also becoming more and more important in increasingly alienated cities due to its heterogeneity and integration, but the vitality of edge space in cities is uneven. This paper taking Tan Hualin edge space configuration as the research unit, according to the chart of survey data, conducts a comparative analysis of the spatial “field” and human “behaviors” of the edge space between Tan Hualin commercial district and Tan Hualin residential district, so as to explore the reasons why the edge space of urban residential district is more dynamic and to enrich the types of environment construction of edge space in urban district .

The reconstruction of historic district is a hot topic in urban design in recent years. Weighing the interests of all parties, improving the space quality and meeting the use needs has become a century problem. In this paper, the Deshengqiao historic block is taken as an example to investigate by observing and recording the form of the inside and outside of the street and the edge space, people's behavior state, so that we can explore the use and future development of the edge space. According to the demand of the Deshengqiao residents who “want people to come in”, this paper puts forward a solution to present the Deshengqiao bolck’s most characteristic and life-rich articles and buildings in an artistic form, so as to create an attractive Street entrance and a comfortable communication place and spiritual space, enhancing the image of the city and giving play to the functions of edge spaces.

We present an edge-based framework for the study of geometric elastic network models to model mechanical interactions in physical systems. We use a formulation in the edge space, instead of the usual node-centric approach, to characterise edge fluctuations of geometric networks defined in d-dimensional space and define the edge mechanical embeddedness, an edge mechanical susceptibility measuring the force felt on each edge given a force applied on the whole system. We further show that this formulation can be directly related to the infinitesimal rigidity of the network, which additionally permits three- and four-centre forces to be included in the network description. We exemplify the approach in protein systems, at both the residue and atomistic levels of description.

Given a directed graph E, we construct for each real number l a quiver whose vertex space is the topological realisation of E, and whose edges are directed paths of length l in the vertex space. These quivers are not topological graphs in the sense of Katsura, nor topological quivers in the sense of Muhly and Tomforde. We prove that when l = 1 and E is finite, the infinite-path space of the associated quiver is homeomorphic to the suspension of the one-sided shift of E. We call this quiver the suspension of E. We associate both a Toeplitz algebra and a Cuntz-Krieger algebra to each of the quivers we have constructed, and show that when l = 1 the Cuntz-Krieger algebra admits a natural faithful representation on the \ell^2-space of the suspension of the one-sided shift of E. For graphs E in which sufficiently many vertices both emit and receive at least two edges, and for rational values of l, we show that the Toeplitz algebra and the Cuntz-Krieger algebra of the associated quiver are homotopy equivalent to the Toeplitz algebra and Cuntz-Krieger algebra respectively of a graph that can be regarded as encoding the l-th higher shift associated to the one-sided shift space of E., 41 pages

A Compton camera has been suggested for use in single photon emission computed tomography because a conventional gamma camera has low efficiency. Here we consider a cone transform brought about by a Compton camera with line detectors. A cone transform takes a given function on the 3-dimensional space and assigns to it the surface integral of the function over cones determined by the 1-dimensional vertex space, the 1-dimensional central axis, and the 1-dimensional opening angle. We generalize this cone transform to $n$-dimensional space and provide an inversion formula. Also, numerical simulations are presented to demonstrate our suggested algorithm in three dimensions.

Prostor na rubovima gradova, shvaćen u najopćenitijem smislu, kao i naselja koja su dio toga prostora specifičan su socioprostorni laboratorij današnjice. Grad prodire u svoju okolicu na nebrojene načine, od kojih su neki vidljivi oku, a neki počivaju na simboličkoj razini, pa možemo govoriti o skupu odnosa koji prožimaju prostor i stvaraju nove slojeve. Definiranje suburbanizacije i srodnih procesa iz socioprostornog aspekta zahtijeva slojevit interdisciplinarni pristup s obzirom na važnost povijesnoga, geografskoga i društvenoga konteksta, čije međudjelovanje stvara specifične socijalne i prostorne konstrukte. Polazeći od tvrdnje da smo svjedoci planetarnoga suburbanizma, analizirat ćemo proces suburbanizacije fokusirajući se na trendove u suvremenom razvoju rubnih naselja. Proučavanje (sub)urbanoga prostora zahtijeva izmicanje iz strogih definicija grada i njegove okolice, odnosno prihvaćanje da su granice grada permeabilne, stoga se u ovom radu istražuje suburbanizacija kroz široku lepezu teorijskih i istraživačkih praksi. Promišljajući proces suburbanizacije kroz brojne aspekte, kao što su dinamika procesa, prostorna i socijalna dinamika te kroz istraživanje konceptualnoga i semantičkoga bogatstva koje se razvijalo u zadnjim desetljećima, ovaj rad teži objasniti glavne pokretače i rezultate suvremene suburbanizacije., Space on the urban edge as well as settlements that are part of that space are considered to be specific contemporary socio-spatial laboratories. Edge settlements, towns or villages, are affected by the central city which penetrates into its surroundings in multiple ways, some of which are visible, spatial transformations, but many are manifested at symbolic level and therefore represent a set of relationships that define space. The definition of suburbanization and all other related processes is based on the analysis of the key concepts such as urbanization, space and place and the consequences for the urban and suburban everyday life that emerge from the reconceptualization of those concepts. Starting from the presumption that we are all witnessing planetary suburbanism, this paper analyzes the process of suburbanization and focuses on contemporary spatial trends in edge settlements. Suburbanization processes are studied from a sociological perspective, which implies the expansion of the urban way of life and departure from strict definitions of cities and their surroundings by stressing the permeability of the urban boundaries in contemporary spatial classification. In this paper, the author attempts to explain the main agents of suburbanization and its socio-spatial results by rethinking the process of suburbanization through process dynamics, spatial and social dynamics as well as through research of the conceptual and semantic richness that has emerged in recent decades.

Privacy is one of the most important issues in social social network data sharing. Structure anonymization is a effective method to protect user from being reidentfied through graph modifications. The data utility of the distorted graph structure after the anonymization is a really severe problem. Reducing the utility loss is a new measurement while k-anonymity as a criterion to guarantee privacy protection. The existing anonymization algorithms that use vertex's degree modification usually introduce a large amount of distortion to the original social network graph. In this paper, we present a $$k$$k-degree anonymity with vertex and edge modification algorithm which includes two phase: first, finding the optimal target degree of each vertex; second, deciding the candidates to increase the vertex degree and adding the edges between vertices to satisfy the requirement. The community structure factors of the social network and the path length between vertices are used to evaluated the anonymization methods. Experimental results on real world datasets show that the average relative performance between anonymized data and original data is the best with our approach.

A través de informes eclesiásticos, entrevistas a maestros, agrimensores y campesinos, cartografía, encuestas y otras fuentes complementarias, buscamos reconstruir la trayectoria de la práctica educativa, recrear la historia institucional de la escuela y recuperar el significado de acceder a la misma en el paraje rural Chos Malal, del centro de Argentina. Utilizamos la categoría “espacio de borde” para referirnos a esos lugares menos insertos en la dinámica del capital, con dependencia y dominación de otros espacios, pero que tienen un margen de autonomía en sus lógicas territoriales y prácticas. Educar en la frontera supone enfrentar diversos obstáculos para poder acceder y permanecer en la escuela. En este camino avanzamos en el proceso de territorialización de este paraje rural situándolo espacial y temporalmente, para luego plantear cuatro momentos significativos en la práctica educativa del lugar. Through ecclesiastical reports, interviews with teachers, surveyors and peasants, cartography, surveys and other complementary sources, we seek to reconstruct the trajectory of educational practice, recreate the institutional history of the school and recover the meaning of access to it in the rural place called Chos Malal, central Argentina. We use the category “edge space” to refer to those places less embedded in the dynamics of capital, with dependence and domination of other spaces, but which have a margin of autonomy in their territorial and practical logics. To educate in the border supposes to face diverse obstacles to be able to accede and to remain in the school. In this way we advance in the process of territorialization of this rural place situating it spatially and temporally, and then raise four significant moments in the educational practice of the place.

Let G be a connected and simple graph with vertex set V ( G ) and edge set E ( G ). A total labeling f : V ∪ E → {1, 2,. . ., k }is called a vertex irregular total k-labeling of G if every two distinct vertices x and y in V ( G ) satisfy wf ( x ) ≠ wf ( y ), where . The total vertex irregularity strength of G , denoted by tvs ( G ), is the minimum k for which G has a vertex irregular total k -labeling. In this paper, we provide an upper bound on the total vertex irregularity strength of the Cartesian product of P 2 and an arbitrary regular graph G .

Depth images have recently attracted much attention in computer vision and high-quality 3D content for 3DTV and 3D movies. In this paper, we present a new semi self-taught learning application framework for enhancing resolution of depth maps without making use of ancillary color images data at the target resolution, or multiple aligned depth maps. Our framework consists of cascade random forests reaching from coarse to fine results. We learn the surface information and structure transformations both from a small high-quality depth exemplars and the input depth map itself across different scales. Considering that edge plays an important role in depth map quality, we optimize an effective regularized objective that calculates on output image space and input edge space in random forests. Experiments show the effectiveness and superiority of our method against other techniques with or without applying aligned RGB information

This paper addresses the problem of optimally modifying the edge lengths such that a prespecified vertex becomes the furthest vertex from a given fixed vertex in the perturbed network. We call this problem the inverse eccentric vertex problem. We show that the problem is \(NP\)-complete even on cactus graphs. However, if the underlying graph is a cycle or a tree, we develop efficient algorithms with linear time complexity.

Through to the street corner space of the urban residents' daily life experience of social investigation, put forward the guiding principles for the location of corner space landscape design can identification, space versatility, use safety, environment suitability principle. Conclusion from the location awareness, zoning, personality construction and processing in the detail design method. For waterscape design, plant configuration, pavement and recreation facilities such as the design of the landscape elements to improve the edge space landscape quality present situation, the cultural connotation and aesthetic value, improve environment conducive to the development of spiritual civilization.

A feedback vertex set in an undirected graph is a subset of vertices removal of which leaves a graph with no cycles. Razgon (in: Proceedings of the 10th Scandinavian workshop on algorithm theory (SWAT 2006), pp. 160---171, 2006) gave a $$1.8899^n n^{O(1)}$$1.8899nnO(1)-time algorithm for finding a minimum feedback vertex set in an $$n$$n-vertex undirected graph, which is the first exact algorithm for the problem that breaks the trivial barrier of $$2^n$$2n. Later, Fomin et al. (Algorithmica 52:293---307, 2008) improved the result to $$1.7548^n n^{O(1)}$$1.7548nnO(1). In this paper, we further improve the result to $$1.7266^n n^{O(1)}$$1.7266nnO(1). Our algorithm is analyzed by the measure-and-conquer method. We get the improvement by designing new reductions based on biconnectivity of instances and introducing a new measure scheme on the structure of reduced graphs.

An edge coloring c^':E->{1,2,...,t} of a graph G=(V,E) is an edge t-ranking if for any two edges of the same color, every path between them contains an intermediate edge with a larger color. The edge ranking number @g"r^'(G) is the smallest value of t such that G has an edge t-ranking. In this paper, we introduce a relation between edge ranking number and vertex partitions. By using the proposed recurrence formula, we show that the edge ranking number of the Sierpinski graph @g"r^'(S(n,k))[email protected]"r^'(K"k) for any n,k>=2 where K"k denotes a complete graph of k vertices.

A proper edge-coloring of a graph defines at each vertex the set of colors of its incident edges. This set is called the palette of the vertex. In this paper we are interested in the minimum number of palettes taken over all possible proper colorings of a graph.

We introduce a vector space representation of concepts using Wikipedia graph structure to calculate semantic relatedness. The proposed method starts from the neighborhood graph of a concept as the primary form and transfers this graph into a vector space to obtain the final representation. The proposed method achieves state of the art results on various relatedness datasets.

We consider in this paper the edge classification problem in networks, which is defined as follows. Given a graph-structured network G(N, A), where N is a set of vertices and A ⊆ N ×N is a set of edges, in which a subset Al ⊆ A of edges are properly labeled a priori, determine for those edges in Au = A\Al the edge labels which are unknown. The edge classification problem has numerous applications in graph mining and social network analysis, such as relationship discovery, categorization, and recommendation. Although the vertex classification problem has been well known and extensively explored in networks, edge classification is relatively unknown and in an urgent need for careful studies. In this paper, we present a series of efficient, neighborhood-based algorithms to perform edge classification in networks. To make the proposed algorithms scalable in large-scale networks, which can be either disk-resident or streamlike, we further devise efficient, cost-effective probabilistic edge classification methods without a significant compromise to the classification accuracy. We carry out experimental studies in a series of real-world networks, and the experimental results demonstrate both the effectiveness and efficiency of the proposed methods for edge classification in large networks.

We establish existence of the eta-invariant as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the Gauss-Bonnet and the spin Dirac operator. We derive an analogue of the Atiyah-Patodi-Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments employ microlocal analysis of the heat kernel asymptotics on incomplete edge spaces and the classical argument of Atiyah-Patodi-Singer. As an application, we discuss stability results for the two rho-invariants we have defined., 65 pages, 2 figures

This paper analyzes ( r | p ) - centroid problems on networks with vertex and edge demand under a binary choice rule. Bilevel programming models are presented for the discrete problem class. Furthermore, NP-hardness proofs for the discrete and continuous ( 1 | p ) - centroid problem on general networks with edge demand only are provided. Nevertheless, an efficient algorithm to determine a discrete ( 1 | p ) - centroid of a tree network with vertex and edge demand can be derived.

This paper investigates the constant-orientation workspace of five-degree-of-freedom parallel mechanisms generating the three translations and two independent rotations and comprising five identical limbs of the P@?RUR type. The general mechanism was proposed recently from the type synthesis performed for 5-DOF parallel mechanisms with identical limb structures. In this study, the emphasis is placed on the determination of the constant-orientation workspace using a geometric interpretation of the so-called vertex space, i.e., the motion generated by a limb for a given orientation. The geometric investigation is carried out using geometric constructive approach, which is implemented in a computer algebra system and in a CAD system. This paper shows that these two approaches are complementary tools to investigate the workspace of parallel mechanisms. The geometric constructive approach proposed in this paper bring insight into the architecture optimization and it can be regarded as a guideline for the workspace analysis of parallel mechanisms whose vertex spaces generate Bohemian dome.

A stochastic network model is developed which describes the 3D morphology of the pore space in fibre-based materials. It has the form of a random geometric graph, where the vertex set is modelled by random point processes and the edges are put using tools from graph theory and Markov chain Monte Carlo simulation. The model parameters are fitted to real image data gained by X-ray synchrotron tomography. In particular, they are specified in such a way that the distributions of vertex degrees and edge lengths, respectively, coincide to a large extent for real and simulated data. Furthermore, the network model is used to introduce a morphology-based notion of pores and their sizes. The model is validated by considering physical characteristics which are relevant for transport processes in the pore space, like geometric tortuosity, i.e., the distribution of shortest path lengths through the material relative to its thickness.

This paper investigates some kinematic properties of a five-degree-of-freedom parallel mechanism generating the 3T2R motion and comprising five identical limbs of the RPUR type. The general mechanism originates from the type synthesis performed for symmetrical 5-DOF parallel mechanism. In this study, two classes of simplified designs are proposed whose forward kinematic problem have either a univariate or a closed-form solution. The principal contributions of this study are the solution of the forward kinematic problem for some simplified designs—which may have more solutions than the FKP of the general 6-DOF Stewart platform with 40 solutions—and the determination of the constant-orientation workspace which is based on the topology of the vertex space (Bohemian dome) and a geometric constructive approach.

Crystal-structure topologies, represented by periodic nets, are described by labelled quotient graphs (or voltage graphs). Because the edge space of a finite graph is the direct sum of its cycle and co-cycle spaces, a Euclidian representation of the derived periodic net is provided by mapping a basis of the cycle and co-cycle spaces to a set of real vectors. The mapping is consistent if every cycle of the basis is mapped on its own net voltage. The sum of all outgoing edges at every vertex may be chosen as a generating set of the co-cycle space. The embedding maps the cycle space onto the lattice L. By analogy, the concept of the co-lattice L* is defined as the image of the generators of the co-cycle space; a co-lattice vector is proportional to the distance vector between an atom and the centre of gravity of its neighbours. The pair (L, L*) forms a complete geometric descriptor of the embedding, generalizing the concept of barycentric embedding. An algebraic expression permits the direct calculation of fractional coordinates. Non-zero co-lattice vectors allow nets with collisions, displacive transitions etc. to be dealt with. The method applies to nets of any periodicity and dimension, be they crystallographic nets or not. Examples are analyzed: α-cristobalite, the seven unstable 3-periodic minimal nets etc.

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces., Comment: 27pgs No figs, v3: final version, incorporating referee's comments, to appear in Geometriae Dedicata

The present paper deals with differential equations with edge degeneration in spaces with asymptotics. We give the definition of an edge space with asymptotics, prove the continuity of operators with edge degeneration in the scale of these spaces, present statements of problems with edge operators, and state conditions providing their Fredholm property.

Among the various interconnection networks, the star graph has been an attractive one. In this paper, we consider the cycle embedding problem in star graphs with conditional edge faults. We show that there exist cycles of all even lengths from 6 to n! in an n-dimensional star graph with = =4.

We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly as related to Primc's vertex Lie algebra and (universal) enveloping vertex algebras., 26 pages