Search results for: Hilbert spaces

Authors: Salvatore Triolo

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We discuss the possibility of extending the well-known Gelfand-Naimark-Segal representation to modules over a C*algebra. We focus our attention on the case of Hilbert modules. We consider, in particular, the problem of the existence of a faithful representation. Non-commutative Lᵖ-spaces are shown to constitute examples of a class of CQ*-algebras. Finally, we have shown that any semisimple proper CQ*-algebra (X, A#), with A# a W*-algebra can be represented as a CQ*-algebra of measurable operators in Segal’s sense.

Keywords: Gelfand-Naimark-Segal representation, CQ*-algebras, faithful representation, non-commutative Lᵖ-spaces, operator in Hilbert spaces

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Authors: Jin Liang, Guanghua Shi

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In this paper, we are concerned with the further refinements of the Heinz operator inequalities in Hilbert spaces. Our purpose is to derive several new Heinz-type operator inequalities. First, with the help of the Taylor series of some hyperbolic functions, we obtain some refinements of the ordering relations among Heinz means defined by Bhatia with different parameters, which would be more suitable in obtaining the corresponding operator inequalities. Second, we present some generalizations of Heinz operator inequalities. Finally, we give a matrix version of the Heinz inequality for the Hilbert-Schmidt norm.

Keywords: Hilbert space, means inequality, norm inequality, positive linear operator

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Authors: Mehdi Rashidi-Kouchi, Asghar Rahimi

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Sun introduced a generalization of frames and showed that this includes more other cases of generalizations of frame concept and proved that many basic properties can be derived within this more general context. Another generalization of frames is frames in Hilbert C*-module. It has been proved that every g-frame in Hilbert space H respect to Hilbert space K is a frame for B(H;K) as Hilbert C*-module. We show that every g-Riesz basis for Hilbert space H respect to K by add a condition is a Riesz basis for Hilbert B(K)-module B(H;K). Also, we investigate similar result for g-orthonormal and orthogonal bases.

Keywords: frame, g-frame, Riesz basis, g-Riesz basis, Hilbert C*-module

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Authors: Francis O. Nwawuru, Jeremiah N. Ezeora

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In this paper, a new algorithm for solving the system of split equilibrium and fixed point problems in real Hilbert spaces is considered. The equilibrium bifunction involves a nite family of pseudo-monotone mappings, which is an improvement over monotone operators. More so, it turns out that the solution of the finite family of nonexpansive mappings. The regularized parameters do not depend on Lipschitz constants. Also, the computations of the stepsize, which plays a crucial role in the convergence analysis of the proposed method, do require prior knowledge of the norm of the involved bounded linear map. Furthermore, to speed up the rate of convergence, an inertial term technique is introduced in the proposed method. Under standard assumptions on the operators and the control sequences, using a modified Halpern iteration method, we establish strong convergence, a desired result in applications. Finally, the proposed scheme is applied to solve some optimization problems. The result obtained improves numerous results announced earlier in this direction.

Keywords: equilibrium, Hilbert spaces, fixed point, nonexpansive mapping, extragradient method, regularized equilibrium

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Authors: Francis O. Nwawuru

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The convergence analysis of split monotone inclusion problems and fixed point problems of certain nonlinear mappings are investigated in the setting of real Hilbert spaces. Inertial extrapolation term in the spirit of Polyak is incorporated to speed up the rate of convergence. Under standard assumptions, a strong convergence of the proposed algorithm is established without computing the resolvent operator or involving Yosida approximation method. The stepsize involved in the algorithm does not depend on the spectral radius of the linear operator. Furthermore, applications of the proposed algorithm in solving some related optimization problems are also considered. Our result complements and extends numerous results in the literature.

Keywords: fixedpoint, hilbertspace, monotonemapping, resolventoperators

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Authors: A. Bhandari, S. Mukherjee

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Two frames in a Hilbert space are called woven or weaving if all possible merge combinations between them generate frames of the Hilbert space with uniform frame bounds. Weaving frames are powerful tools in wireless sensor networks which require distributed data processing. Considering the practical applications, this article deals with finite woven frames. We provide methods of constructing finite woven frames, in particular, bounded linear operators are used to construct woven frames from a given frame. Several examples are discussed. We also introduce the notion of woven frame sequences and characterize them through the concepts of gaps and angles between spaces.

Keywords: frames, woven frames, gap, angle

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Authors: Ti-Jun Xiao, Zhe Xu

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Controllability is one of the fundamental issues in control systems. In this paper, we study the controllability of second order evolutional control systems in Hilbert spaces with memory and boundary controls, which model dynamic behaviors of some viscoelastic materials. Transferring the control problem into a moment problem and showing the Riesz property of a family of functions related to Cauchy problems for some integrodifferential equations, we obtain a general boundary controllability theorem for these second order evolutional control systems. This controllability theorem is applicable to various concrete 1D viscoelastic systems and recovers some previous related results. It is worth noting that Riesz sequences can be used for numerical computations of the control functions and the identification of new Riesz sequence is of independent interest for the basis-function theory. Moreover, using the Riesz sequences, we obtain the existence and uniqueness of (weak) solutions to these second order evolutional control systems in Hilbert spaces. Finally, we derive the exact boundary controllability of a viscoelastic beam equation, as an application of our abstract theorem.

Keywords: evolutional control system, controllability, boundary control, existence and uniqueness

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Authors: Apirak Sombat, Teerapol Saleewong, Poom Kumam, Parin Chaipunya, Wiyada Kumam, Anantachai Padcharoen, Yeol Je Cho, Thana Sutthibutpong

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This research is aimed to study a two-step iteration process defined over a finite family of σ-asymptotically quasi-nonexpansive nonself-mappings. The strong convergence is guaranteed under the framework of Banach spaces with some additional structural properties including strict and uniform convexity, reflexivity, and smoothness assumptions. With similar projection technique for nonself-mapping in Hilbert spaces, we hereby use the generalized projection to construct a point within the corresponding domain. Moreover, we have to introduce the use of duality mapping and its inverse to overcome the unavailability of duality representation that is exploit by Hilbert space theorists. We then apply our results for σ-asymptotically quasi-nonexpansive nonself-mappings to solve for ideal efficiency of vector optimization problems composed of finitely many objective functions. We also showed that the obtained solution from our process is the closest to the origin. Moreover, we also give an illustrative numerical example to support our results.

Keywords: asymptotically quasi-nonexpansive nonself-mapping, strong convergence, fixed point, uniformly convex and uniformly smooth Banach space

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Authors: Sakshi Dhruve, Ar. Sarang Barbarwar

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Public spaces are the locus of activity and interaction in any urban area. Such spaces provide identity to cities, towns or neighborhoods and define the people and culture over there. Inclusiveness is one of the core aspects of public or community spaces. With its humongous population and rapidly expanding urban areas, India needs more inclusivity in public spaces to attain true equitable development. The aim of the paper is to discuss the sensitivity of public spaces in India to the transgender community. The study shows how this community was legally included as ‘Third Gender’ in country’s legislation yet lacks social acceptance and security. It shows the challenges and issues faced by them at public spaces. The community was studied on ethnographic basis to understand their culture, lifestyle, requirements, etc. The findings have indicated towards a social stigma from people and insensitivity in designing of civic spaces. The larger objective of the study is also to provide recommendations on the design aspects and interventions in public places to increase their inclusiveness towards the transgender society.

Keywords: community spaces, ethnographic, stigma, Third Gender community

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Authors: U. M. Swamy

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A compact Hausdorff and totally disconnected topological space are known as Boolean space in view of the stone duality between Boolean algebras and such topological spaces. A sheaf over X is a triple (S, p, X) where S and X are topological spaces and p is a local homeomorphism of S onto X (that is, for each element s in S, there exist open sets U and G containing s and p(s) in S and X respectively such that the restriction of p to U is a homeomorphism of U onto G). Here we mainly concern on sheaves over Boolean spaces. From a given sheaf over a Boolean space, we obtain an algebraic structure in such a way that there is a one-to-one correspondence between these algebraic structures and sheaves over Boolean spaces.

Keywords: Boolean algebra, Boolean space, sheaf, stone duality

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Authors: Melike Orhan

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Along with impetuous physical change, configuration and increase in the density of cities, urban public spaces have started to become a transition area rather than spaces to inhabit. The insufficiency of public spaces, one of the most significant components of a city, where communal life is maintained and the decrease in the quality of urban spaces have led to an increase in the use of semi-public spaces as urban space. Semi-public spaces are those that ensure transition between private and public spaces and can be seen, observed, reached and used by urban-dwellers. Humans are in a constant relation to their surroundings and care for integration as part of their surroundings. Semi-public spaces providing balance for the individual between private spaces (structures) and urban-public spaces make this integration easier. Spaces with a semi-public characteristic serve for a particular neighboring unit and the user (i.e. common use areas in residential spaces and dwellings, common outdoor areas situated between office buildings, and etc.) These spaces, whose density of usage is increased with distinctive functions and activities, gain different attributions according to the characteristics of the urban space they are located in (commercial, residential, touristic, and etc.) and to the functions of the structures with which they are in relation. At the same time, they begin to serve other neighboring units along with an increase in public usage. As a result, the interaction between environment-space-structure-humans changes, which directly affects the urban space quality. The aim of this study is to determine how and depending on what characteristics the public usage density of semi-public spaces change and to put forth the effects of this change on the urban environment it is located in and to designate its role in terms of 'urban space quality'. In conclusion, within the scope of this study, semi-public spaces located in urban spaces with distinctive functions will be explored through examples, and the effects of these spaces with altered public usage and density on urban space and quality of life will be put forward. Accordingly, applicable criteria will be determined by means of semi-public spaces oriented at increasing and sustaining the quality of urban space.

Keywords: semi-public spaces, urban public spaces, urban space quality, public usage

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Authors: Chaithanya Pothuganti

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Open and semi-open spaces in educational buildings like corridors, mid landings, seating spaces, lobby, courtyards are traditionally have been the places of social communion and interaction which helps in promoting the knowledge, performance, activeness, and motivation in students. Factors like availability of land, commercialization, of educational facilities, especially in e-techno and smart schools, led to closed classrooms to accommodate students thereby lack quality open and semi-open spaces. This insufficient attention towards open space design which is a means of informal learning misses an opportunity to encourage the student’s skill development, behavior and learning skills. The core objective of this paper is to find the level of impact on student learning behavior and to identify the suitable proportions and configuration of spaces that shape the schools. In order to achieve this, different types of open spaces in schools and their impact on student’s performance in various existing models are analysed using case studies to draw some design principles. The study is limited to indoor open spaces like corridors, break out spaces and courtyards. The expected outcome of the paper is to suggest better design considerations for the development of semi-open and open spaces which functions as an element for informal learnings. Its focus is to provide further thinking on designing and development of open spaces in educational buildings.

Keywords: configuration of spaces and proportions, informal learning, open spaces, schools, student’s behavior

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Authors: Shams Alyusof

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This work is to enrich the studies of the frames due to their prominent role in pure mathematics as well as in applied mathematics and many applications in computer science and engineering. Recently, there are remarkable studies of operators that preserve frames on some spaces, and this research could be considered as an extension of such studies. Indeed, this paper is to we characterize weighted composition operators that preserve frames in weighted Hardy spaces on the open unit disk. Moreover, it shows that this characterization does not apply to generalized weighted composition operators on such spaces. Nevertheless, this study could be extended to provide more specific characterizations.

Keywords: frames, generalized weighted composition operators, weighted Hardy spaces, analytic functions

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Authors: Ruichong Zhang

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This study proposes a recording-based approach to characterize and quantify earthquake-induced site nonlinearity, exemplified as soil nonlinearity and/or liquefaction. Alternative to Fourier spectral analysis (FSA), the paper introduces time-frequency analysis of earthquake ground motion recordings with the aid of so-called Hilbert-Huang transform (HHT), and offers justification for the HHT in addressing the nonlinear features shown in the recordings. With the use of the 2001 Nisqually earthquake recordings, this study shows that the proposed approach is effective in characterizing site nonlinearity and quantifying the influences in seismic ground responses.

Keywords: site nonlinearity, site amplification, site damping, Hilbert-Huang Transform (HHT), liquefaction, 2001 Nisqually Earthquake

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Authors: Piyush Hajela

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Public spaces within Indian markets are vibrant, colorful and contain dimensions that make them attractive and therefore act as popular gathering spaces. Most of these public spaces emerge as squares, plazas of varied shapes and sizes spread at different locations within the market. These public spaces grow organically and are discovered by the people themselves as they respond positively to the collective human senses. On the other hand, there are the planned and designed public spaces as well that are less active. This research evaluates both the planned and the organically transformed public spaces in Indian markets from an Urban Design point of view. The purpose of such research is to provide a basis for design solutions to ensure the success of designed public spaces. The evaluation is done for identified Attributes, namely Comfort, Protection, Familiarity, Activities, Form, Legibility, Engagement, Safety, Accessibility, Environment and Transformations by which a Public Space attains its recognition. The evaluation is based on a rating done for forty-four parameters falling under eleven attributes of public space. An opinion survey of professionals is conducted for their priorities of attributes while designing Public spaces. A comparison is made to rank these attributes between Planned and Organically transformed Public spaces and, opinion of the professionals. After dues analysis, the research suggests the learning from the organically transformed Public spaces for ensuring the success of designed public spaces. The suggestions may be in the form of Design decisions or administrative regulations, or both for achieving the desirables.

Keywords: assessment, attributes, engagement, interaction

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Authors: Mehdi Jafari Matehkolaee

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In this paper, we have obtained the adjoint of an arbitrary operator (linear and nonlinear) in Hilbert space by introducing an n-dimensional Riemannian manifold. This general formalism covers every linear operator (non – differential) in Hilbert space. In fact, our approach shows that instead of using the adjoint definition of an operator directly, it can be obtained directly by relying on a suitable generalized space according to the action of the operator in question. For the case of nonlinear operators, we have to change the definition of the linear operator adjoint. But here, we have obtained an adjoint of these operators with respect to the definition of the derivative of the operator. As a matter of fact, we have shown one of the straight applications of the ''Frechet derivative'' in the algebra of the operators.

Keywords: adjoint operator, non-linear operator, differentiable operator, manifold

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Authors: Lina Wu, Ye Li

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An equivalent relation between a harmonic form and a closed co-closed form is established on a complete non-compact manifold. This equivalence has been generalized for a differential k-form ω from Lq spaces to non-Lq spaces when q=2 in the context of p-balanced growth where p=2. Especially for a simple differential k-form on a complete non-compact manifold, the equivalent relation has been verified with the extended scope of q for from finite q-energy in Lq spaces to infinite q-energy in non-Lq spaces when with 2-balanced growth. Generalized Hadamard Theorem, Cauchy-Schwarz Inequality, and Calculus skills including Integration by Parts as well as Convergent Series have been applied as estimation techniques to evaluate growth rates for a differential form. In particular, energy growth rates as indicated by an appropriate power range in a selected test function lead to a balance between a harmonic differential form and a closed co-closed differential form. Research ideas and computational methods in this paper could provide an innovative way in the study of broadening Lq spaces to non-Lq spaces with a wide variety of infinite energy growth for a differential form.

Keywords: closed forms, co-closed forms, harmonic forms, L^q spaces, p-balanced growth, simple differential k-forms

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Authors: Edson Ephraim Sanga

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This paper stems to explore possible places used for recreation in unplanned settlements in order to avail knowledge on how to create and shape urban spaces essential for recreation and leisure. The context of unplanned settlements is spatially characterized compactness and congestions of buildings developed by residents without professional inputs. These characteristics surpass greenery landscapes such as parks and squares essential for health, happiness and wellbeing. The lack of recreational greenery landscape arises a question on how possible can recreation take places in the settlements? This study used qualitative methods mainly observation and in-depth interview to explore the recreational situation in Manzese informal settlements as an instrumental case and found that ordinary spaces are re-configured into recreational spaces and used as ‘parks’ and ‘squares’ in the settlements. The spaces are diverse and complex as they possess different spatial characteristics based on their physical attributes and the way they are used and interpreted by respective users. This paper argues that the re-configuration processes of ordinary spaces should not be taken for granted because they portray the appropriation of spaces from quotidian dimensions in a particular context.

Keywords: ordinary spaces, recreation, unplanned settlement, urban spaces

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Authors: A. N. Paithane, D. S. Bormane, S. D. Shirbahadurkar

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It has been seen that emotion recognition is an important research topic in the field of Human and computer interface. A novel technique for Feature Extraction (FE) has been presented here, further a new method has been used for human emotion recognition which is based on HHT method. This method is feasible for analyzing the nonlinear and non-stationary signals. Each signal has been decomposed into the IMF using the EMD. These functions are used to extract the features using fission and fusion process. The decomposition technique which we adopt is a new technique for adaptively decomposing signals. In this perspective, we have reported here potential usefulness of EMD based techniques.We evaluated the algorithm on Augsburg University Database; the manually annotated database.

Keywords: intrinsic mode function (IMF), Hilbert-Huang transform (HHT), empirical mode decomposition (EMD), emotion detection, electrocardiogram (ECG)

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Authors: Nazneen Khan, Vakeel A. Khan

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In this article we introduce the Paranorm Zweier I-convergent sequence spaces, for a sequence of positive real numbers. We study some topological properties, prove the decomposition theorem and study some inclusion relations on these spaces.

Keywords: ideal, filter, I-convergence, I-nullity, paranorm

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Authors: Jorge Duarte de Sá

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The appearance of the new media technologies has opened new fields of intervention in architecture creating a new dynamic communication in the relationship between public and space, where are present technological devices that enable a new sensory experience, aesthetic and even spiritual. This connection makes relevant the idea of rehabilitate architectonic spaces with new media technologies such as sacred spaces. This research aims to create a media project integrated in sacred spaces that combine Architecture, Art and New Technologies, exploring new perspectives and different dynamics in space.

Keywords: media, architecture, religious spaces, projections, contemplation

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Authors: Bibi Somayeh Aliakbari, Niknaz Kachooei, Fatemeh Amiri Najafabadi

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According to the results of contemporary urbanism, social living moved into buildings and the quality of urban space has been declining. But still, there are life in open public space and it is one of reason attendance and activities of people in open public spaces.The purpose of this research is finding reason creation friendly public space in urban spaces and also use these in new urban spaces.The research methodology consisted of a qualitative model based on observation and graphical analysis. In this paper case study is public space historical, moderns in urban scales and local scales in Iran.This paper shows that Existence of friendly public space in cities cause is attendance and activities of people in open public spaces that it is reason the revitalization of public open spaces in cities.

Keywords: public space, public open space, friendly public space, Iran

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Authors: Jasim Azhar, Morten Gjerde

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A city’s space is comprehended as a phenomenon that emerges from the ongoing negotiation between the constructed environment, urban processes, and bodily experience. Many spaces do not represent a static notion but are continually challenged and reconstituted. The ability to recognize those leftover spaces in the urban context is an integral part of an urban redevelopment process, where structured and layered approaches become useful in understanding to transform these spaces into places. Contemporary urban leftover spaces exist as a result of several factors and are present in every major city that often disrupts the flow of districts by creating visually unappealing places. These spaces can be designed, transformed and integrated so as to achieve environmental gains and social preferences. The paper explores how those small changes in visual quality of an urban leftover spaces in Wellington city influence a person’s experience significantly and its potential usage. These spaces can be seen as a catalyst for a change through an ecological sustainability’s framework. A creative and flexible design would lead to psychologically healthy places by improving the image of a city from within. The qualitative research is undertaken through the visual preference studies which will inform the planning initiatives by knowing what people feel about those visual changes in these leftover spaces. Those visual preferences can guide behavior and the emotional responses of different users for the redesign of those spaces with the meaningful attributes. The research is driven by the hypothesis that if the attributes are made visible, the likelihood of stimulating the interest of users should increase.

Keywords: leftover spaces, visual preferences, tactical urbanism, ecological sustainability

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Authors: E. J. Cilliers, Z. Goosen

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The impact that urban green spaces have on sustainability and quality of life is phenomenal. This is also true for the local South African environment. However, in reality green spaces in urban environments are decreasing due to growing populations, increasing urbanization and development pressure. This further impacts on the provision of child-friendly spaces, a concept that is already limited in local context. Child-friendly spaces are described as environments in which people (children) feel intimately connected to, influencing the physical, social, emotional, and ecological health of individuals and communities. The benefits of providing such spaces for the youth are well documented in literature. This research therefore aimed to investigate the concept of child-friendly spaces and its applicability to the South African planning context, in order to guide the planning of such spaces for future communities and use. Child-friendly spaces in the urban environment of the city of Durban, was used as local case study, along with two international case studies namely Mullerpier public playground in Rotterdam, the Netherlands, and Kadidjiny Park in Melville, Australia. The aim was to determine how these spaces were planned and developed and to identify tools that were used to accomplish the goal of providing successful child-friendly green spaces within urban areas. The need and significance of planning for such spaces was portrayed within the international case studies. It is confirmed that minimal provision is made for green space planning within the South African context, when there is reflected on the international examples. As a result international examples and disciples of providing child-friendly green spaces should direct planning guidelines within local context. The research concluded that child-friendly green spaces have a positive impact on the urban environment and assist in a child’s development and interaction with the natural environment. Regrettably, the planning of these child-friendly spaces is not given priority within current spatial plans, despite the proven benefits of such.

Keywords: built environment, child-friendly spaces, green spaces, public places, urban area

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Authors: O. Poleshchuk, E. Komarov

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In this paper, semantic spaces with the properties of completeness and orthogonality (complete orthogonal semantic spaces) were chosen as models of expert evaluations. As the theoretical and practical studies have shown all the properties of complete orthogonal semantic spaces correspond to the thinking activity of experts that is why these semantic spaces were chosen for modeling. Two methods of construction such spaces were proposed. Models of comparative and fuzzy cluster analysis of expert evaluations were developed. The practical application of the developed methods has demonstrated their viability and validity.

Keywords: expert evaluation, comparative analysis, fuzzy cluster analysis, theoretical and practical studies

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Authors: César Garza

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In supervised binary classification and regression problems, it is well-known that learnability is equivalent to a uniform convergence of the hypothesis class, and if a problem is learnable, it is learnable by empirical risk minimization. For the general learning setting of unsupervised learning tasks, there are non-trivial learning problems where uniform convergence does not hold. We present here the task of learning centers of mass with an extra feature that “activates” some of the coordinates over the unit ball in a Hilbert space. We show that the learning problem is learnable under a stable RLM rule. We introduce a family of distributions over the domain space with some mild restrictions for which the sample complexity of uniform convergence for these problems must grow logarithmically with the dimension of the Hilbert space. If we take this dimension to infinity, we obtain a learnable problem for which the uniform convergence property fails for a vast family of distributions.

Keywords: statistical learning theory, learnability, uniform convergence, stability, regularized loss minimization

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Authors: Hazem Abuorf

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From the perspective of viewing urban design as the architecture of public spaces, the latter has many advantages; for example, in achieving attractive and vibrant public spaces, ensuring safety and amenity, supporting a strong sense of place and local character. Besides all advantages, such spaces nevertheless trigger numerous dilemmas, how to design urban spaces preserving the quality of life in the long term while equally achieving cohesion between new urban developments and the already existing urban structure without causing a split in history through the cause of functional and aesthetic degradation. Analysis of this article seeks to propose a methodology deemed essential for assessing and stimulating design criteria of the public spaces when rehabilitating urban sites. The method’s utility is shown by analyzing rehabilitation projects in Gaza Strip, whose design of the public spaces has mainly focused on the physical aspect ignoring the place’s local identity, users’ needs, and history. The proposed methodology serves as a guide for municipal authorities and technical teams to deal with interventions that would rework the role of urban design towards making sense of place.

Keywords: urban design, public realm, rehabilitation projects, quality of life

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Authors: Mildred Moreno Villanueva

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Public space is essential to strengthen the social and urban fabric and the social cohesion; there lies the importance of its study. Hence, the aim of this paper is to analyze the quality of public space in the XXI century in both quantitative and qualitative terms. In this article, the concept of public space includes open spaces such as parks, public squares and walking areas. To make this analysis we take Mexico City as the case study. It has a population of nearly 9 million inhabitants and it is composed of sixteen boroughs. For this analysis, we consider both, existing public spaces and the government intervention for building and improvement of new and existent public spaces. Results show that on the one hand, quantitatively there is not an equitable distribution of public spaces because of both, the growth of the city itself, as well as for the absence of political will to create public spaces. Another factor is the evolution of this city, which has been growing merely in a 'patched pattern', where public space has played no role at all with a total absence of urban design. On the other hand, qualitatively, even the boroughs with the most public spaces have not shown interest in making these spaces qualitatively inclusive and open to the general population aiming for integration. Therefore, urban projects that privatize public space seem to be the rule, rather than a rehabilitation effort of the existent public spaces. Hence, state intervention should reinforce its role as an agent of social change acting in the benefit of the majority of the inhabitants with the promotion of more inclusive public spaces.

Keywords: exclusion, inclusion, Mexico City, public space

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Authors: Faris M. Altaf, Abdullah M Elkeshy

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Forty-two albino male rats aged between 30 and 40 days (weighted 200 g to 250 g) were used in the present study. The left sural nerves of 36 rats were subjected to crush injury at 1 to 6 weeks intervals using 6 animals at each interval. The right and left sural nerves of the rest 6 rats were used as a control. After 2 weeks of the crush injury, the endoneurium showed channel-like spaces that were lined by the fibroblast-like cells and collagen bundles. These channels contained degenerated myelin and were connected with the perivascular and subperineurial spaces. Some of the flattened fibroblast-like cells were arranged in several layers in the subperineurial and perivascular spaces, forming barrier-like cellular sheets localizing the endoneurial edema in these spaces. Fibroblast-like cells also wrapped the regenerating nerve fibers by their branching cytoplasmic processes. At the end of the third week, the flattened fibroblasts formed nearly continuous sheets in the subperineurial and perivascular spaces. Macrophages were frequently noticed between these cellular barrier-like sheets and in the subperineurial and perivascular spaces. Conclusion: it could be concluded that the endoneurial fibroblast-like cells form barrier-like cellular sheets that localized the endoneurial edema in the subperineurial and perivascular spaces and create also the endoneurial channel-like spaces containing degenerated myelin and endoneurial edema helping the resolution of such edema.

Keywords: sural nerve, endoneurial fibroblast-like cells, endoneurial edema, barrier-like and channel-like spaces

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Authors: Rania Ajmi, Faiza Allouche Khebour, Aude Nuscia Taibi, Sirine Essasi

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Urban green spaces (UGS) as an important contributor can be a significant part of sustainable development. A spatial method was employed to assess and map the spatial distribution of UGS in five districts in Sousse, Tunisia. Ecological management of UGS is an essential factor for the sustainable development of the city; hence the municipality of Sousse has decided to support the districts according to different green spaces characters. And to implement this policy, (1) a new GIS web application was developed, (2) then the implementation of the various green spaces was carried out, (3) a spatial mapping of UGS using Quantum GIS was realized, and (4) finally a data processing and statistical analysis with RStudio programming language was executed. The intersection of the results of the spatial and statistical analyzes highlighted the presence of an imbalance in terms of the spatial UGS distribution in the study area. The discontinuity between the coast and the city's green spaces was not designed in a spirit of network and connection, hence the lack of a greenway that connects these spaces to the city. Finally, this GIS support will be used to assess and monitor green spaces in the city of Sousse by decision-makers and will contribute to improve the well-being of the local population.

Keywords: distributions, GIS, green space, imbalance, spatial analysis

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