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We study the Laplacian on family preserving metric graphs. These are graphs that have a certain symmetry that, as we show, allows for a decomposition into a direct sum of one-dimensional operators whose properties are explicitly related to the structure of the graph. Such decompositions have been extremely useful in the study of Schrödinger operators on metric trees. We show that the tree structure is not essential, and moreover, obtain a direct and simple correspondence between such decompositions in the discrete and continuum case.operators whose structure is directly related to the structure of the tree and to the boundary conditions at the vertices. A similar method exists in the discrete case, where the graph is considered as a combinatorial object and the operator studied is the discrete Laplacian or the adjacency matrix (see, e.g., [3,9,10,12]). While the similarity between the decomposition in the continuous and discrete case is clear and lies in the exploitation of the symmetry properties of the graph, it is important to note there are essential technical differences. Whereas the discrete case involves studying cyclic subspaces generated by specially chosen functions, the continuum case (as presented in [28,32]) involves defining the relevant invariant subspaces directly and relies heavily on the structure of the tree.It has recently been realized in [13] that in the discrete case, the tree structure is not essential for this decomposition. It is in fact possible to carry out this procedure for a more general class of graphs that we call 'family preserving' and whose definition we give below (see Definition 2.2) 1 . This class contains radially symmetric trees, antitrees (see Section 5 for the definition), and various other graphs.We refer the reader to [13] for examples and some graphics.The objective of this work is to extend this decomposition for family preserving graphs to the continuum case of metric graphs as well. Since, as remarked above, the standard decomposition technique for metric trees relies heavily on the tree structure, one needs to take a different approach. A natural approach to this task, and the one that we shall take, is to try and obtain a direct translation of the discrete decomposition to the decomposition in the metric case. Such an approach has also the added bonus of making explicit the connection between the combinatorial and continuum decompositions, and as we shall show, can recover the original Naimark-Solomyak procedure from the procedure in the discrete case.While employing the discrete decomposition scheme is indeed natural for getting a decomposition in the metric case, the results in this paper should by no means be viewed as a direct extension of those in [7]. First, as the functions in the metric case live on edges, whereas those in the discrete case live on vertices, it is a non-trivial task to adapt one procedure to the other. Second, as is evident in Section 4 (where we describe the decomposition algorithm) the discrete decomposition is only one of several steps...

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The standard requirements for transverse steel reinforcement (TSR) confinement in reinforced-concrete (RC) columns are mainly to provide the following: ductile behavior, minimum axial load capacity of the column’s core, and prevention of longitudinal bars buckling. It is well-known that the passive confinement due the TSR action is less effective in high-strength concrete (HSC) compared to normal-strength concrete (NSC). Therefore, the TSR amounts required by the standards for HSC columns are high, and in some cases, especially in the lower stories columns of high-rise buildings, are impractical. This paper presents a new construction method using textile-reinforced concrete (TRC) as internal confinement together with reduced TSR amounts. Moreover, comparison of the proposed method with RC columns casted in fiber-reinforced polymer (FRP) stay-in-place forms as additional external confinement, is presented. Eleven large-scale column specimens were tested under axial compression. The results give an insight on the application feasibility of the proposed construction method. It is shown that the TRC-TSR dual internal confinement action can be an option to reduce the standard required TSR amounts while maintaining similar levels of ductile behavior.

We study Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a connection between asymptotic properties of solutions to the eigenvalue equations and continuity properties of the spectral measure with respect to the Lebesgue measure. We also use this theory in order to derive results regarding the multiplicity of the singular spectrum.