Relevant bibliographies by topics / Finite topological spaces

Academic literature on the topic 'Finite topological spaces'

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Published: 17 February 2024

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Journal articles on the topic "Finite topological spaces"

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Benoumhani, Moussa, and Ali Jaballah. "Finite fuzzy topological spaces." Fuzzy Sets and Systems 321 (August 2017): 101–14. http://dx.doi.org/10.1016/j.fss.2016.11.003.

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OSAKI, Takao. "Reduction of Finite Topological Spaces." Interdisciplinary Information Sciences 5, no. 2 (1999): 149–55. http://dx.doi.org/10.4036/iis.1999.149.

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Chae, Hi-joon. "FINITE TOPOLOGICAL SPACES AND GRAPHS." Communications of the Korean Mathematical Society 32, no. 1 (January 31, 2017): 183–91. http://dx.doi.org/10.4134/ckms.c160004.

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Bagchi, Susmit. "Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes." Symmetry 14, no. 2 (February 20, 2022): 422. http://dx.doi.org/10.3390/sym14020422.

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The interplay between topological hyperconvex spaces and sigma-finite measures in such spaces gives rise to a set of analytical observations. This paper introduces the Noetherian class of k-finite k-hyperconvex topological subspaces (NHCs) admitting countable finite covers. A sigma-finite measure is constructed in a sigma-semiring in a NHC under a topological ordering of NHCs. The topological ordering relation maintains the irreflexive and anti-symmetric algebraic properties while retaining the homeomorphism of NHCs. The monotonic measure sequence in a NHC determines the convexity and compactness of topological subspaces. Interestingly, the topological ordering in NHCs in two isomorphic topological spaces induces the corresponding ordering of measures in sigma-semirings. Moreover, the uniform topological measure spaces of NHCs need not always preserve the pushforward measures, and a NHC semiring is functionally separable by a set of inner-measurable functions.
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Edelsbrunner, Herbert, and Nimish R. Shah. "Triangulating Topological Spaces." International Journal of Computational Geometry & Applications 07, no. 04 (August 1997): 365–78. http://dx.doi.org/10.1142/s0218195997000223.

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Given a subspace [Formula: see text] and a finite set S⊆ℝd, we introduce the Delaunay complex, [Formula: see text], restricted by [Formula: see text]. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets [Formula: see text] in a non-empty set. By the nerve theorem, [Formula: see text] and [Formula: see text] are homotopy equivalent if all such sets are contractible. This paper proves a sufficient condition for [Formula: see text] and [Formula: see text] be homeomorphic.
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Clader, Emily. "Inverse limits of finite topological spaces." Homology, Homotopy and Applications 11, no. 2 (2009): 223–27. http://dx.doi.org/10.4310/hha.2009.v11.n2.a11.

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Nakasho, Kazuhisa, Hiroyuki Okazaki, and Yasunari Shidama. "Finite Dimensional Real Normed Spaces are Proper Metric Spaces." Formalized Mathematics 29, no. 4 (December 1, 2021): 175–84. http://dx.doi.org/10.2478/forma-2021-0017.

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Summary In this article, we formalize in Mizar [1], [2] the topological properties of finite-dimensional real normed spaces. In the first section, we formalize the Bolzano-Weierstrass theorem, which states that a bounded sequence of points in an n-dimensional Euclidean space has a certain subsequence that converges to a point. As a corollary, it is also shown the equivalence between a subset of an n-dimensional Euclidean space being compact and being closed and bounded. In the next section, we formalize the definitions of L1-norm (Manhattan Norm) and maximum norm and show their topological equivalence in n-dimensional Euclidean spaces and finite-dimensional real linear spaces. In the last section, we formalize the linear isometries and their topological properties. Namely, it is shown that a linear isometry between real normed spaces preserves properties such as continuity, the convergence of a sequence, openness, closeness, and compactness of subsets. Finally, it is shown that finite-dimensional real normed spaces are proper metric spaces. We referred to [5], [9], and [7] in the formalization.
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K. K, Bushra Beevi, and Baby Chacko. "PARACOMPACTNESS IN GENERALIZED TOPOLOGICAL SPACES." South East Asian J. of Mathematics and Mathematical Sciences 19, no. 01 (April 30, 2023): 287–300. http://dx.doi.org/10.56827/seajmms.2023.1901.24.

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In this paper we introduce the concepts G - locally finite, σG - locally finite and G - paracompactness. Also discuss about some properties of these concepts. Here we investigate that some properties in topological spaces and generalized topological spaces (GTS) are coincides if we replace open sets by generalized open sets (G - open sets ). Also, we provide some examples to show some results are invalid in the case of GTS.
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Kang, Jeong, Sang-Eon Han, and Sik Lee. "The Fixed Point Property of Non-Retractable Topological Spaces." Mathematics 7, no. 10 (September 21, 2019): 879. http://dx.doi.org/10.3390/math7100879.

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Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for Khalimsky (K-, for short) topological spaces, the present paper studies the product property of the FPP for K-topological spaces. Furthermore, the paper investigates the FPP of various types of connected K-topological spaces such as non-K-retractable spaces and some points deleted K-topological (finite) planes, and so on. To be specific, after proving that not every one point deleted subspace of a finite K-topological plane X is a K-retract of X, we study the FPP of a non-retractable topological space Y, such as one point deleted space Y ∖ { p } .
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Nogin, Maria, and Bing Xu. "Modal Logic Axioms Valid in Quotient Spaces of Finite CW-Complexes and Other Families of Topological Spaces." International Journal of Mathematics and Mathematical Sciences 2016 (2016): 1–3. http://dx.doi.org/10.1155/2016/9163014.

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In this paper we consider the topological interpretations of L□, the classical logic extended by a “box” operator □ interpreted as interior. We present extensions of S4 that are sound over some families of topological spaces, including particular point topological spaces, excluded point topological spaces, and quotient spaces of finite CW-complexes.
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Dissertations / Theses on the topic "Finite topological spaces"

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Lesser, Alice. "Optimal and Hereditarily Optimal Realizations of Metric Spaces." Doctoral thesis, Uppsala University, Department of Mathematics, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-8297.

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This PhD thesis, consisting of an introduction, four papers, and some supplementary results, studies the problem of finding an optimal realization of a given finite metric space: a weighted graph which preserves the metric's distances and has minimal total edge weight. This problem is known to be NP-hard, and solutions are not necessarily unique.

It has been conjectured that extremally weighted optimal realizations may be found as subgraphs of the hereditarily optimal realization Γd, a graph which in general has a higher total edge weight than the optimal realization but has the advantages of being unique, and possible to construct explicitly via the tight span of the metric.

In Paper I, we prove that the graph Γd is equivalent to the 1-skeleton of the tight span precisely when the metric considered is totally split-decomposable. For the subset of totally split-decomposable metrics known as consistent metrics this implies that Γd is isomorphic to the easily constructed Buneman graph.

In Paper II, we show that for any metric on at most five points, any optimal realization can be found as a subgraph of Γd.

In Paper III we provide a series of counterexamples; metrics for which there exist extremally weighted optimal realizations which are not subgraphs of Γd. However, for these examples there also exists at least one optimal realization which is a subgraph.

Finally, Paper IV examines a weakened conjecture suggested by the above counterexamples: can we always find some optimal realization as a subgraph in Γd? Defining extremal optimal realizations as those having the maximum possible number of shortest paths, we prove that any embedding of the vertices of an extremal optimal realization into Γd is injective. Moreover, we prove that this weakened conjecture holds for the subset of consistent metrics which have a 2-dimensional tight span

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Tamburini, Caterina. "The isomorphism problem for directed acyclic graphs: an application to multivector fields." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/15793/.

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This thesis is based on a project developed by a group of researchers at the Faculty of Mathematics and Computer Science at the Jagiellonian University of Krakow. They study sampled dynamics using combinatorial multivector fields. Applying a decomposition into strongly connected components, it is possible to create a directed acyclic graph, called Morse graph, which is a description of the multivector field's global dynamics. Therefore the purpose of this thesis is to compare directed acyclic graphs. In the first chapter we describe the creation process of a Morse graph and an algorithm to study the graph isomorphism problem. The second chapter is dedicated to our personal work, so we describe four Python tests we developed to establish whether two directed acyclic graphs are definitely not isomorphic. In the third chapter we sum up many examples. The last chapter aims to present a possible way for the future work, that is to treat a combinatorial multivector filed as a finite topological space.
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Ameen, Zanyar. "Finitely additive measures on topological spaces and Boolean algebras." Thesis, University of East Anglia, 2015. https://ueaeprints.uea.ac.uk/56864/.

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The thesis studies some problems in measure theory. In particular, a possible generalization corresponding to Maharam Theorem for �nitely additive measures (charges). In the �rst Chapter, we give some de�nitions and results on di�erent areas of Mathematics that will be used during this work. In Chapter two, we recall the de�nitions of nonatomic, continuous and Darboux charges, and show their relations tWe hereby con�rm that all work without reference are our work excluding the chapter one. We hereby con�rm that all work without reference are our work excluding the chapter one. o each other. The relation between charges on Boolean algebras and the induced measures on their Stone spaces is mentioned in this chapter. We also show that for any charge algebra, there exists a compact zero-dimensional space such that its charge algebra is isomorphic to the given charge algebra. In Chapter three, we give the de�nition of Jordan measure and some of its outcomes. We de�ne another measure on an algebra of subsets of some set called Jordanian measure, and investigate it. Then we de�ne the Jordan algebras and Jordanian algebras, and study some of their properties. Chapter four is mostly devoted to the investigation of uniformly regular measures and charges (on both Boolean algebras and topological spaces). We show how the properties of a charge on a Boolean algebra can be transferred to the induced measure on its Stone space. We give a di�erent proof to a result by Mercourakis in [36, Remark 1.10]. In 2013, Borodulin-Nadzieja and Dºamonja [10, Theorem 4.1] proved the countable version of Maharam Theorem for charges using uniform regularity. We show that 3 4 this result can be proved under weaker assumption and further extended. The �nal Chapter is concerned with the higher versions of uniform regularity which are called uniform �-regularity. We study these types of measures and obtain several results and characterizations. The major contribution to this work is that we show we cannot hope for a higher analogue of Maharam Theorem for charges using uniform �-regularity. In particular, Theorem 4.1 in [10] and Remark 1.10 in [36] cannot be extended for all cardinals. We prove that a higher version of Theorem 4.1 in [10] (resp. Remark 1.10 in [36]) can be proved only for charges on free algebras on � many generators (resp. measures on a product of compact metric spaces). We also generalize Proposition 2.10 in [26].
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Ayadi, Mohamed. "Propriétés algébriques et combinatoires des espaces topologiques finis." Electronic Thesis or Diss., Université Clermont Auvergne (2021-...), 2022. http://www.theses.fr/2022UCFAC106.

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Ibrahim, Caroline Maher Boulis Heil Wolfgang. "Finite abelian group actions on orientable circle bundles over surfaces." 2004. http://etd.lib.fsu.edu/theses/available/etd-07122004-135529.

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Thesis (Ph. D.)--Florida State University, 2004.
Advisor: Dr. Wolfgang Heil, Florida State University, College of Arts and Sciences, Dept. of Mathematics. Title and description from dissertation home page (viewed Sept. 28, 2004). Includes bibliographical references.
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Jasinski, Jakub. "Hrushovski and Ramsey Properties of Classes of Finite Inner Product Structures, Finite Euclidean Metric Spaces, and Boron Trees." Thesis, 2011. http://hdl.handle.net/1807/29762.

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We investigate two combinatorial properties of classes of finite structures, as well as related applications to topological dynamics. Using the Hrushovski property of classes of finite structures -- a finite extension property of homomorphisms -- we can show the existence of ample generics. For example, Solecki proved the existence of ample generics in the context of finite metric spaces that do indeed possess this extension property. Furthermore, Kechris, Pestov and Todorcevic have shown that the Ramsey property of Fraisse classes of finite structures implies that the automorphism group of the corresponding Fraisse limit is extremely amenable, i.e., it possesses a very strong fixed point property. Gromov and Milman had shown that the unitary group of the infinite-dimensional separable Hilbert space is extremely amenable using non-combinatorial methods. This result encourages a deeper look into structural Euclidean Ramsey theory, i.e., Euclidean Ramsey theory in which we colour more than just points. In particular, we look at complete finite labeled graphs whose vertex sets are subsets of the Hilbert space and whose labels correspond to the inner products. We prove "Ramsey-type" and "Hrushovski-type" theorems for linearly ordered metric subspaces of "sufficiently" orthogonal sets. In particular, the latter is used to show a "Hrushovski version" of the Ramsey-type Matousek-Rodl theorem for simplices. It is known that the square root of the metric induced by the distance between vertices in graphs produces a metric space embeddable in a Euclidean space if and only if the graph is a metric subgraph of the Cartesian product of three types of graphs. These three are the half-cube graphs, the so-called cocktail party graphs, and the Gosset graph. We show that the class of metric spaces related to half-cube graphs -- metric spaces on sets with the symmetric difference metric -- satisfies the Hrushovski property up to 3 points, but not more. Moreover, the amalgamation in this class can be too restrictive to permit the Ramsey Property. Finally, following the work of Fouche, we compute the Ramsey degrees of structures induced by the leaf sets of boron trees. Also, we briefly show that this class does not satisfy the full Hrushovski property. Fouche's trees are in fact related to ultrametric spaces, as was observed by Lionel Nguyen van The. We augment Fouche's concept of orientation so that it applies to these boron tree structures. The upper bound computation of the Ramsey degree in this case, turns out to be an "asymmetric" version of the Graham-Rothschild theorem. Finally, we extend these structures to "oriented" ones, yielding a Ramsey class and a corresponding Fraisse limit whose automorphism group is extremely amenable.
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Books on the topic "Finite topological spaces"

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Barmak, Jonathan A. Algebraic Topology of Finite Topological Spaces and Applications. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22003-6.

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Barmak, Jonathan A. Algebraic topology of finite topological spaces and applications. Heidelberg: Springer, 2011.

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Engelking, Ryszard. Theory of dimensions, finite and infinite. Lemgo, Germany: Heldermann, 1995.

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Talsi, Jussi. Imbeddings of equivariant complexes into representation spaces. Helsinki: Suomalainen Tiedeakatemia, 1994.

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Spaces of constant curvature. 6th ed. Providence, R.I: AMS Chelsea Pub., 2011.

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Jaco, William H. Topology and geometry in dimension three: Triangulations, invariants, and geometric structures : conference in honor of William Jaco's 70th birthday, June 4-6, 2010, Oklahoma State University, Stillwater, OK. Providence, R.I: American Mathematical Society, 2011.

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Stanford Symposium on Algebraic Topology: Applications and New Directions (2012 : Stanford, Calif.), ed. Algebraic topology: Applications and new directions : Stanford Symposium on Algebraic Topology: Applications and New Directions, July 23--27, 2012, Stanford University, Stanford, CA. Providence, Rhode Island: American Mathematical Society, 2014.

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1953-, Campillo Antonio, ed. Zeta functions in algebra and geometry: Second International Workshop on Zeta Functions in Algebra and Geometry, May 3-7, 2010, Universitat de Les Illes Balears, Palma de Mallorca, Spain. Providence, R.I: American Mathematical Society, 2012.

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1980-, Blazquez-Sanz David, Morales Ruiz, Juan J. (Juan José), 1953-, and Lombardero Jesus Rodriguez 1961-, eds. Symmetries and related topics in differential and difference equations: Jairo Charris Seminar 2009, Escuela de Matematicas, Universidad Sergio Arboleda, Bogotá, Colombia. Providence, R.I: American Mathematical Society, 2011.

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Richmond, Thomas Alan. Finite-point order compactifications. 1986.

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Book chapters on the topic "Finite topological spaces"

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Kono, Susumu, and Fumihiro Ushitaki. "Geometry of Finite Topological Spaces and Equivariant Finite Topological Spaces." In K-Monographs in Mathematics, 53–63. Dordrecht: Springer Netherlands, 2002. http://dx.doi.org/10.1007/978-94-009-0003-5_4.

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Tikhomirov, V. M. "Finite Coverings of Topological Spaces." In Selected Works of A. N. Kolmogorov, 221–25. Dordrecht: Springer Netherlands, 1991. http://dx.doi.org/10.1007/978-94-011-3030-1_31.

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Barmak, Jonathan A. "Basic Topological Properties of Finite Spaces." In Algebraic Topology of Finite Topological Spaces and Applications, 19–35. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22003-6_2.

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Barmak, Jonathan A. "Minimal Finite Models." In Algebraic Topology of Finite Topological Spaces and Applications, 37–47. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22003-6_3.

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Barmak, Jonathan A. "Simple Homotopy Types and Finite Spaces." In Algebraic Topology of Finite Topological Spaces and Applications, 49–72. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22003-6_4.

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Barmak, Jonathan A. "Preliminaries." In Algebraic Topology of Finite Topological Spaces and Applications, 1–18. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22003-6_1.

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Barmak, Jonathan A. "Fixed Points and the Lefschetz Number." In Algebraic Topology of Finite Topological Spaces and Applications, 129–35. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22003-6_10.

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Barmak, Jonathan A. "The Andrews–Curtis Conjecture." In Algebraic Topology of Finite Topological Spaces and Applications, 137–50. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22003-6_11.

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Barmak, Jonathan A. "Strong Homotopy Types." In Algebraic Topology of Finite Topological Spaces and Applications, 73–84. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22003-6_5.

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Barmak, Jonathan A. "Methods of Reduction." In Algebraic Topology of Finite Topological Spaces and Applications, 85–91. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22003-6_6.

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Conference papers on the topic "Finite topological spaces"

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Muradov, Firudin Kh. "Ternary semigroups of topological transformations of open sets of finite-dimensional Euclidean spaces." In FOURTH INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2020). AIP Publishing, 2021. http://dx.doi.org/10.1063/5.0042197.

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Stimpfl, Franz, Josef Weinbub, René Heinzl, Philipp Schwaha, Siegfried Selberherr, Theodore E. Simos, George Psihoyios, and Ch Tsitouras. "A Unified Topological Layer for Finite Element Space Discretization." In ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010. AIP, 2010. http://dx.doi.org/10.1063/1.3498151.

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Tasolamprou, A. C., M. Kafesaki, C. M. Soukoulis, E. N. Economou, and Th Koschny. "Topological surface states at the free space termination of uncorrugated finite square photonic crystals." In 2021 Fifteenth International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials). IEEE, 2021. http://dx.doi.org/10.1109/metamaterials52332.2021.9577199.

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Rashid, Mark M., Mili Selimotic, and Tarig Dinar. "General Polyhedral Finite Elements for Rapid Nonlinear Analysis." In ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/detc2008-49248.

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An analysis system for solid mechanics applications is described in which a new finite element method that can accommodate general polyhedral elements is exploited. The essence of the method is direct polynomial approximation of the shape functions on the physical element, without transformation to a canonical element. The main motive is elimination of the requirement that all elements be similar to a canonical element via the usual isoparametric mapping. It is this topological restriction that largely drives the design of mesh-generation algorithms, and ultimately leads to the considerable human effort required to perform complex analyses. An integrated analysis system is described in which the flexibility of the polyhedral element method is leveraged via a robust computational geometry processor. The role of the latter is to perform rapid Boolean intersection operations between hex meshes and surface representations of the body to be analyzed. A typical procedure is to create a space-filling structured hex mesh that contains the body, and then extract a polyhedral mesh of the body by intersecting the hex mesh and the body’s surface. The result is a mesh that is directly usable in the polyhedral finite element method. Some example applications are: 1) simulation on very complex geometries; 2) rapid geometry modification and re-analysis; and 3) analysis of material-removal process steps following deformation processing. This last class of problems is particularly challenging for the conventional FE methodology, because the element boundaries are, in general, not aligned with the cutting geometry following the deformation (e.g. forging) step.
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Yuksel, Osman, and Cetin Yilmaz. "Size and Topology Optimization of Inertial Amplification Induced Phononic Band Gap Structures." In ASME 2017 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/imece2017-71342.

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In this study, inertial amplification induced phononic band gaps are attained by performing structural optimization on a compliant unit cell mechanism of a one-dimensional periodic structure. First of all, stop band characteristics of the lumped parameter model of the unit cell mechanism is discussed. Next, the distributed parameter model of the compliant unit cell is presented. In order to obtain wide and deep inertial amplification induced stop bands, both size and topology optimization methods are utilized considering the distributed parameter model of the unit cell mechanism. The band gap characteristics of the infinite periodic size and topologically optimized mechanisms are compared. Moreover, vibration transmissibility of the finite periodic size and topologically optimized mechanisms are calculated and the effect of number of unit cells is discussed. Finally, a parametric study is carried out to demonstrate the effect of topology optimization design space volume fraction on the band gap limits.
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Abdel-Malek, K., Walter Seaman, and Harn-Jou Yeh. "An Exact Method for NC Verification of up to 5-Axis Machining." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/dac-8560.

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Abstract The motion of a cutter tool is modeled as a surface undergoing a sweep operation along another geometric entity. A numerically controlled machining verification method is developed based on a formulation for delineating the volume generated by the motion of a cutting tool on the workpiece (stock). Varieties and subvarieties that are subsets of some Eucledian space defined by the zeros of a finite number of analytic functions are computed and are characterized as closed form equations of surface patches of this volume. A topological space describing the swept volume will be built as a stratified manifold with corners. Singularities of the variety are loci of points where the Jacobian of the manifold has lower rank than maximal. It is shown that varieties appearing inside the manifold representing the removed material are due to a lower degree strata of the Jacobian. Some of the varieties are complicated (so as not to confuse with varieties in complex Cn) and will be shown to be reducible because of their parametrization and are addressed. Benefits of this method are evident in its ability to depict the manifold and to compute a value for the volume.
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Sundararaman, Venkatesh, Matthew P. O'Donnell, Isaac V. Chenchiah, and Paul M. Weaver. "Topology Morphing Lattice Structures." In ASME 2021 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/smasis2021-67531.

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Abstract Planar cellular lattice structures subject to axial compression may undergo elastic bending or buckling of the unit cells. If sufficient compression is applied, the columns of adjacent cells make contact. This changes the topology of the lattice by establishing new load paths. This topology change induces a corresponding shift in the effective stiffness characteristics of the lattice — in particular, the shear modulus undergoes a step-change. The ability to embed adaptive stiffness characteristics through a topology change allows structural reconfiguration to meet changing load/operational requirements efficiently. The concept, of topological reconfiguration, can be exploited across a range of length scales, from (meta-)materials to components. Here we focus on macroscopic behaviour presenting results obtained from finite element analysis that shows excellent correlation with the observed response of 3D-printed PLA lattices. Through a parametric study, we explore the role of key geometric and stiffness parameters and identify desirable regions of the design space. The non-linear responses demonstrated by this topology morphing lattice structure may offer designers a route to develop bespoke elastic systems.
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Takacs, Peter Z., and Eugene L. Church. "Surface profiles and scatter from soft-x-ray optics." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1990. http://dx.doi.org/10.1364/oam.1990.tuo1.

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The performance of soft-x-ray optics in synchrotron beam lines and for space astronomy use requires detailed knowledge of the topological properties of their surfaces. Instrumentation now exists for the measurement of surface profiles on spherical and aspherical surfaces up to one meter in length with subnanometer height resolution and submicroradian slope resolution. Analysis of surface profile data in terms of the power spectral density function over the entire range of spatial periods—from micrometers to meters—allows one to predict the performance of the component in a unified theoretical approach encompassing both the specular image and scattered light distribution. There is no longer any need for an artificial distinction between surface figure and surface finish. We will discuss numerous examples of actual surface profile measurements on grazing incidence and normal incidence optics, and we will relate them to measured performance at x-ray wavelengths where possible.
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Charlesworth, William W., and David C. Anderson. "Applications of Non-Manifold Topology." In ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium collocated with the ASME 1995 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/cie1995-0737.

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Abstract It is widely recognized that a solid model based on a non-manifold boundary representation can have a more complicated surface topology than one based on a manifold boundary representation, but non-manifold topology has other capabilities that may be more valuable to the application developer. Non-manifold topology can be put to use in existing application areas in ways that differ significantly from the techniques developed for manifold modeling and it can be put to use in new applications that have not been satisfactorily solved by manifold topology. Several applications of non-manifold topology that would be difficult or impossible to implement using a purely manifold geometric modeler are illustrated: automatic formulation of finite element analyses from solid models, automatic generation of machining tool paths for 2½-dimensional pockets, and construction of geometric models using topological constraints. These applications demonstrate how a non-manifold model partitions the entire space in which an object is embedded, preserves elements of the model that would be discarded by conventional schemes, and permits the implementation of a common merge operation. All three applications have been implemented using a two dimensional non-manifold (non-1-manifold) geometric modeler.
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Choi, Haejoon, Adrian Matias Chung Baek, and Namhum Kim. "Design of Non-Periodic Lattice Structures by Allocating Pre-Optimized Building Blocks." In ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/detc2019-98204.

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Abstract The development of lattice design approaches using topology optimization to create complex lattice structures is the focus of ongoing research as the lattice structures show great potential throughout a wider range of research fields. Unfortunately, many of the methods suggested are often very difficult to industrialize due to their complexity and excessive computational costs in the design process. This paper proposes a novel framework of generating non-periodic lattice structures using topologically pre-optimized building blocks to improve the computational efficiency of the optimization process while maintaining the performance of the target structure and to make the process of obtaining optimal design more accessible for people with limited knowledge of topology optimization. The proposed method is comprised of two phases: the generation of a library holding optimized building blocks that can endure various loading conditions and the allocation of these building blocks at the proper location in the design space based on stress information obtained from the finite element analysis. Numerical examples are presented to show that the method is feasible and able to achieve reasonable performances, demonstrating its efficacy.
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Reports on the topic "Finite topological spaces"

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Lutz, Carsten, and Frank Wolter. Modal Logics of Topological Relations. Technische Universität Dresden, 2004. http://dx.doi.org/10.25368/2022.142.

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The eight topological RCC8(or Egenhofer-Franzosa)- relations between spatial regions play a fundamental role in spatial reasoning, spatial and constraint databases, and geographical information systems. In analogy with Halpern and Shoham’s modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the RCC8-relations. The semantics is based on region spaces induced by standard topological spaces, in particular the real plane. We investigate the expressive power and computational complexity of the logics obtained in this way. It turns our that, similar to Halpern and Shoham’s logic, the expressive power is rather natural, but the computational behavior is problematic: topological modal logics are usually undecidable and often not even recursively enumerable. This even holds if we restrict ourselves to classes of finite region spaces or to substructures of region spaces induced by topological spaces. We also analyze modal logics based on the set of RCC5relations, with similar results.
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