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Topological and Metric Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
The space (X, d) is said to be separable if there exists a countable set E dense in X. Topological Equivalence. Two metric spaces X=(X,d) $ X = (X,d) $ and Y=(Y,ρ) $ Y = (Y,\rho ) $ are topologically equivalent if X and Y, with topologies induced by metrics d and ρ $ \rho $ , are homeomorphic.
Linear systems
Published in Simon Serovajsky, Optimization and Differentiation, 2017
If the space is separable, then each point can be approximated by elements of a countable set that can be interpreted as a sequence. We use this property for proving of existence theorems for the boundary problems that describe control systems.
Functional characterisation of mechanical joints to facilitate its selection during the design of open architecture products
Published in International Journal of Production Research, 2018
Jaime A. Mesa, Danny Illera, Iván Esparragoza, Heriberto Maury, Humberto Gómez
Mechanical joining methods can be classified according to two main schemes: physical and functional. A physical classification scheme is related to ‘working-principle’ behind the joining method itself. As an example, Groche and Türk classify joining methods into three categories related to joining mechanisms: positive-locking, friction-locking and material-locking connections (Groche and Türk 2011). Similarly, Messler states that there are only three forces involved: mechanical (mechanical joining), chemical (adhesive joining) and physical (welding; Messler 2004). However, from the designer point of view, a function-based classification scheme is more practical, particularly in early design steps. In this regard, Bazrov offers different classification schemes (Bazrov 2010). In term of structural features, the joints could be classified into the following groups: (I) immobile and separable, (II) immobile and inseparable, (III) mobile and separable and (IV) mobile and inseparable. The terms mobile/immobile describe whether or not, there is relative motion between mating components or modules. A separable joint is understood as that which allows to disassemble it without ‘damaging’ or to modify the properties of any of the components of the joint (i.e. could be assembled again using the same initial components without loss of integrity). The classification as mentioned in the above scheme is especially practical at the conceptual design stage, where the functional/physical structure of the product is defined, and a different set of alternatives are to be identified and evaluated. Particularly, recognising that the flux between components/modules may involve energy transfer, a fifth classification is proposed: (V) separable, mobile and mechanical power transmission capable. Table 4 summarises the five joint types, as described before, along with a set of general joint types belonging to each class. The table is expected to grow as more and more joining types (particularly non-conventional) are added, providing information to expand the universe of joining options for the designer.