Relevant bibliographies by topics / Connections (Mathematics) Lattice theory

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Connections (Mathematics) Lattice theory'

Author: Grafiati
Published: 4 June 2021
Last updated: 7 February 2022

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Journal articles on the topic "Connections (Mathematics) Lattice theory"

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Bullivant, Alex, Marcos Calçada, Zoltán Kádár, João Faria Martins, and Paul Martin. "Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry." Reviews in Mathematical Physics 32, no. 04 (November 4, 2019): 2050011. http://dx.doi.org/10.1142/s0129055x20500117.

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Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in [Formula: see text] dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.
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Longstaff, W. E., J. B. Nation, and Oreste Panaia. "Abstract reflexive sublattices and completely distributive collapsibility." Bulletin of the Australian Mathematical Society 58, no. 2 (October 1998): 245–60. http://dx.doi.org/10.1017/s0004972700032226.

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There is a natural Galois connection between subspace lattices and operator algebras on a Banach space which arises from the notion of invariance. If a subspace lattice ℒ is completely distributive, then ℒ is reflexive. In this paper we study the more general situation of complete lattices for which the least complete congruence δ on ℒ such that ℒ/δ is completely distributive is well-behaved. Our results are purely lattice theoretic, but the motivation comes from operator theory.
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DAVEY, BRIAN A., JANE G. PITKETHLY, and ROSS WILLARD. "THE LATTICE OF ALTER EGOS." International Journal of Algebra and Computation 22, no. 01 (February 2012): 1250007. http://dx.doi.org/10.1142/s021819671100673x.

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We introduce a new Galois connection for partial operations on a finite set, which induces a natural quasi-order on the collection of all partial algebras on this set. The quasi-order is compatible with the basic concepts of natural duality theory, and we use it to turn the set of all alter egos of a given finite algebra into a doubly algebraic lattice. The Galois connection provides a framework for us to develop further the theory of natural dualities for partial algebras. The development unifies several fundamental concepts from duality theory and reveals a new understanding of full dualities, particularly at the finite level.
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Chen, Mo, Lei Jin, Xiangyang Gong, Xiaojuan Wang, and Wenhua Sun. "Analysis of the spatial cascading effect in networks." International Journal of Modern Physics C 31, no. 04 (February 13, 2020): 2050055. http://dx.doi.org/10.1142/s0129183120500552.

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Reality networks such as power grids and social networks can be spatially embedded. In this paper, we focus on the spatial cascading effect in such networks. The spatial cascading effect is that the failure of one node may cause other nodes that are close to it in space to fail. The phenomenon is very common, such that a person is more likely to have an impact on his neighbors even if he is not connected with his neighbors via social networks. Based on this, we construct a spatial cascading model to simulate the spatial cascading effect. In addition, we apply the exponential distribution [Formula: see text] to fit the real link distances. The networks are generated by two-dimensional lattices. We define two kinds of connections, namely actual spatial connections. The actual connections are links generated by the exponential distribution. The spatial connections are links in the lattice. Simulations show that the spatial embeddedness makes networks more robust in our model, which is different from previous research results. We put forward an algorithm to alter the link distances in the networks without changing node degree values. Using the algorithm verifies our conclusion that if nodes tend to connect with local nodes, networks will be robust to the spatial cascading effect. We further extend our model to a more general form. The nodes embedded in lattice can be sparse, which means that the existing probability of nodes in the lattice is not always 1. The networks in the extension model are more vulnerable compared to those in the original model.
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Valverde-Albacete, Francisco José, and Carmen Peláez-Moreno. "Four-Fold Formal Concept Analysis Based on Complete Idempotent Semifields." Mathematics 9, no. 2 (January 15, 2021): 173. http://dx.doi.org/10.3390/math9020173.

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Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is K¯-four-fold Formal Concept Analysis (K¯-4FCA) where K¯ is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in dually-ordered pairs—e.g., the complete max-plus and min-plus semirings—the basic construction shows dual-order-, row–column- and Galois-connection-induced dualities that appear simultaneously a number of times to provide the full spectrum of variability. Our results lead to a fundamental theorem of K¯-four-fold Formal Concept Analysis that properly defines quadrilattices as 4-tuples of (order-dually) isomorphic lattices of vectors and discuss its relevance vis-à-vis previous formal conceptual analyses and some affordances of their results.
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Djouadi, Yassine, and Henri Prade. "Interval-Valued Fuzzy Galois Connections: Algebraic Requirements and Concept Lattice Construction." Fundamenta Informaticae 99, no. 2 (2010): 169–86. http://dx.doi.org/10.3233/fi-2010-244.

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Cherukuri, Aswani Kumar, Radhika Shivhare, Ajith Abraham, Jinhai Li, and Annapurna Jonnalagadda. "A Pragmatic Approach to Understand Hebbian Cell Assembly." International Journal of Cognitive Informatics and Natural Intelligence 15, no. 2 (April 2021): 60–82. http://dx.doi.org/10.4018/ijcini.20210401.oa6.

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Formed at the cerebral cortex, neuron cell assemblies are regarded as basic units in cortical representation. Proposed by Hebb, these cell assemblies are regarded as the distributed neural representation of relevant objects, concepts or constellations. Each cell assembly contains a group of neurons having strong mutual excitatory connections. During a stimulus, these cells get activated. This activation either performs a given action or represent a given percept or concept in brain. This theory is in the strongest connection of the problem of concept forming in the brain. The challenge is to model coordinated activity among neurons in brain mathematically. The need of modelling it mathematically enables this paper to give clear view of functionality of Hebbian cell assembly. Therefore this paper proposes a pragmatic approach to Hebbian cell assemblies using mathematical model grounded in lattice based formalism that utilizes Galois connections. During this proposal, the authors also show the connections of the proposal to cognitive model of memory in particularly long-term memory (LTM).
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Alm, Sven Erick. "Upper Bounds for the Connective Constant of Self-Avoiding Walks." Combinatorics, Probability and Computing 2, no. 2 (June 1993): 115–36. http://dx.doi.org/10.1017/s0963548300000547.

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We present a method for obtaining upper bounds for the connective constant of self-avoiding walks. The method works for a large class of lattices, including all that have been studied in connection with self-avoiding walks. The bound is obtained as the largest eigenvalue of a certain matrix. Numerical application of the method has given improved bounds for all lattices studied, e.g. μ < 2.696 for the square lattice, μ < 4.278 for the triangular lattice and μ < 4.756 for the simple cubic lattice.
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QUIETA, MARIE THERESE ROBLES, and SHENG-UEI GUAN. "OPTIMIZATION OF 2D LATTICE CELLULAR AUTOMATA FOR PSEUDORANDOM NUMBER GENERATION." International Journal of Modern Physics C 16, no. 03 (March 2005): 479–500. http://dx.doi.org/10.1142/s0129183105007303.

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This paper proposes a generalized approach to 2D CA PRNGs — the 2D lattice CA PRNG — by introducing vertical connections to arrays of 1D CA. The structure of a 2D lattice CA PRNG lies in between that of 1D CA and 2D CA grid PRNGs. With the generalized approach, 2D lattice CA PRNG offers more 2D CA PRNG variations. It is found that they can do better than the conventional 2D CA grid PRNGs. In this paper, the structure and properties of 2D lattice CA are explored by varying the number and location of vertical connections, and by searching for different 2D array settings that can give good randomness based on Diehard test. To get the most out of 2D lattice CA PRNGs, genetic algorithm is employed in searching for good neighborhood characteristics. By adopting an evolutionary approach, the randomness quality of 2D lattice CA PRNGs is optimized. In this paper, a new metric, #rn is introduced as a way of finding a 2D lattice CA PRNG with the least number of cells required to pass Diehard test. Following the introduction of the new metric #rn, a cropping technique is presented to further boost the CA PRNG performance. The cost and efficiency of 2D lattice CA PRNG is compared with past works on CA PRNGs.
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Vahala, George, Pavol Pavlo, Linda Vahala, and Nicos S. Martys. "Thermal Lattice-Boltzmann Models (TLBM) for Compressible Flows." International Journal of Modern Physics C 09, no. 08 (December 1998): 1247–61. http://dx.doi.org/10.1142/s0129183198001126.

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The progress and challenges in thermal lattice-Boltzmann modeling are discussed. In particular, momentum and energy closures schemes are contrasted. Higher order symmetric (but no longer space filling) velocity lattices are constructed for both 2D and 3D flows and shown to have superior stability properties to the standard (but lower) symmetry lattices. While this decouples the velocity lattice from the spatial grid, the interpolation required following free-streaming is just 1D. The connection between fixed lattice vectors and temperature-dependent lattice vectors (obtained in the Gauss–Hermite quadrature approach) is discussed. Some (compressible) Rayleigh–Benard simulations on the 2D octagonal lattice are presented for extended BGK collision operators that allow for arbitrary Prandtl numbers.
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Dissertations / Theses on the topic "Connections (Mathematics) Lattice theory"

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Konecny, Jan. "Isotone fuzzy Galois connections and their applications in formal concept analysis." Diss., Online access via UMI:, 2009.

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Thesis (Ph. D.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Systems Science and Industrial Engineering, 2009.
Includes bibliographical references.
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Jenkinson, Justin. "Convex Geometric Connections to Information Theory." Case Western Reserve University School of Graduate Studies / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=case1365179413.

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Romanovski, Iakov. "Connections between descriptive set theory and HF-logic." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/MQ37160.pdf.

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Buckle, John Francis. "Computational aspects of lattice theory." Thesis, University of Warwick, 1989. http://wrap.warwick.ac.uk/106446/.

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The use of computers to produce a user-friendly safe environment is an important area of research in computer science. This dissertation investigates how computers can be used to create an interactive environment for lattice theory. The dissertation is divided into three parts. Chapters two and three discuss mathematical aspects of lattice theory, chapter four describes methods of representing and displaying distributive lattices and chapters five, six and seven describe a definitive based environment for lattice theory. Chapter two investigates lattice congruences and pre-orders and demonstrates that any lattice congruence or pre-order can be determined by sets of join-irreducibles. By this correspondence it is shown that lattice operations in a quotient lattice can be calculated by set operations on the join-irreducibles that determine the congruence. This alternative characterisation is used in chapter three to obtain closed forms for all replacements of the form "h can replace g when computing an element f", and hence extends the results of Beynon and Dunne into general lattices. Chapter four investigates methods of representing and displaying distributive lattices. Techniques for generating Hasse diagrams of distributive lattices are discussed and two methods for performing calculations on free distributive lattices and their respective advantages are given. Chapters five and six compare procedural and functional based notations with computer environments based on definitive notations for creating an interactive environment for studying set theory. Chapter seven introduces a definitive based language called Pecan for creating an interactive environment for lattice theory. The results of chapters two and three are applied so that quotients, congruences and homomorphic images of lattices can be calculated efficiently.
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Menix, Jacob Scott. "Properties of Functionally Alexandroff Topologies and Their Lattice." TopSCHOLAR®, 2019. https://digitalcommons.wku.edu/theses/3147.

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This thesis explores functionally Alexandroff topologies and the order theory asso- ciated when considering the collection of such topologies on some set X. We present several theorems about the properties of these topologies as well as their partially ordered set. The first chapter introduces functionally Alexandroff topologies and motivates why this work is of interest to topologists. This chapter explains the historical context of this relatively new type of topology and how this work relates to previous work in topology. Chapter 2 presents several theorems describing properties of functionally Alexandroff topologies ad presents a characterization for the functionally Alexandroff topologies on a finite set X. The third and fourth chapters present facts about the lattice of functionally Alexandroff topologies, with Chapter 4 being dedicated to an algorithm which generates a complement in this lattice.
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Walters, J. L. "Uniform sigma frames and the cozero part of uniform frames." Master's thesis, University of Cape Town, 1989. http://hdl.handle.net/11427/18467.

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In this thesis some general results on uniform frames are established and then, after defining a 'uniform sigma frame', the correspondence between the two is explored via the 'uniform cozero part' of a uniform frame. It is shown that the Lindelof uniform frames and the uniform sigma frames are in fact equivalent as categories, and that properties of, and constructions using separable uniform frames can be obtained by considering the uniform cozero part. For example, the Samuel compactification of a separable uniform frame can be obtained via the Samuel compactification (in the sigma frame sense) of the underlying cozero part of the uniform frame. Throughout the thesis, choice principles such as the axioms of choice and countably dependent choice, are used, and generally without mention.
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Radu, Ion. "Stone's representation theorem." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3087.

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The thesis analyzes some aspects of the theory of distributive lattices, particularly two representation theorems: Birkhoff's representation theorem for finite distributive lattices and Stone's representation theorem for infinite distributive lattices.
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Krohne, Edward. "Continuous Combinatorics of a Lattice Graph in the Cantor Space." Thesis, University of North Texas, 2016. https://digital.library.unt.edu/ark:/67531/metadc849680/.

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We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2ᴳ) from the Cantor space, where the group G is the additive group of integer pairs ℤ². That is, X is the set of aperiodic {0,1} labelings of the two-dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two-coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications.
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Unwin, James. "On connections between dark matter and the baryon asymmetry." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:5d7d6d06-5ef8-4921-8d4f-9ab19e21a031.

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This thesis is dedicated to the study of a prominent class of dark matter (DM) models, in which the DM relic density is linked to the baryon asymmetry, often referred to as Asymmetric Dark Matter (ADM) theories. In ADM the relic density is set by a particle-antiparticle asymmetry, in direct analogue to the baryons. This is partly motivated by the observed proximity of the baryon and DM relic densities Ω_{DM} ≈ 5 Ω_{B}, as this can be explained if the DM and baryon asymmetries are linked. A general requisite of models of ADM is that the vast majority of the symmetric component of the DM number density, the DM-antiDM pairs, must be removed for the asymmetry to set the DM relic density and thus to explain the coincidence of Ω_{DM} and Ω_{B}. However we shall argue that demanding the efficient annihilation of the symmetric component leads to a tension with experimental constraints in a large class of models. In order to satisfy the limits coming from direct detection and colliders searches, it is almost certainly required that the DM be part of a richer hidden sector of interacting states. Subsequently, examples of such extended hidden sectors are constructed and studied, in particular we highlight that the presence of light pseudoscalars can greatly aid in alleviating the experimental bounds and are well motivated from a theoretical stance. Finally, we highlight that self-conjugate DM can be generated from hidden sector particle asymmetries, which can lead to distinct phenomenology. Further, this variant on the ADM scenario can circumvent some of the leading constraints.
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Kanaan, Mona N. "Cross-spectral analysis for spatial point-lattice processes." Thesis, [n.p.], 2000. http://dart.open.ac.uk/abstracts/page.php?thesisid=94.

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Books on the topic "Connections (Mathematics) Lattice theory"

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service), SpringerLink (Online, ed. Lattice Theory: Foundation. Basel: Springer Basel AG, 2011.

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Metaharmonic lattice point theory. Boca Raton: Taylor & Francis, 2011.

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Călugăreanu, Grigore. Lattice Concepts of Module Theory. Dordrecht: Springer Netherlands, 2000.

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Oitmaa, Jaan. Series expansion methods for strongly interacting lattice models. Cambridge: Cambridge University Press, 2010.

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Mycielski, Jan. A lattice of chapters of mathematics: Interpretations between theorems. Providence, R.I., USA: American Mathematical Society, 1990.

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Mycielski, Jan. A lattice of chapters of mathematics: Interpretations between theorems. Providence, R.I., USA: American Mathematical Society, 1990.

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The theory of quantaloids. Harlow: Longman, 1996.

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Group theory: Classes, representation and connections, and applications. New York: Nova Science Publishers, 2010.

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Sardanashvili, G. A. (Gennadiĭ Aleksandrovich), ed. Connections in classical and quantum field theory. Singapore: World Scientific, 2000.

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service), SpringerLink (Online, ed. Lattices and Codes: A Course Partially Based on Lectures by Friedrich Hirzebruch. 3rd ed. Wiesbaden: Springer Fachmedien Wiesbaden, 2013.

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Book chapters on the topic "Connections (Mathematics) Lattice theory"

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Sabbah, Claude. "Good Meromorphic Connections (Formal Theory)." In Lecture Notes in Mathematics, 159–75. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-31695-1_11.

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Paugam, Frédéric. "Connections and Curvature." In Towards the Mathematics of Quantum Field Theory, 159–66. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-04564-1_7.

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Haiman, Mark. "Linear lattice proof theory: An overview." In Lecture Notes in Mathematics, 129–41. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0098460.

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Grobler, Jacobus J. "101 Years of Vector Lattice Theory: A Vector Lattice-Valued Daniell Integral." In Trends in Mathematics, 173–92. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-70974-7_8.

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Kaburlasos, Vassilis G. "Connections with Established Paradigms." In Towards a Unified Modeling and Knowledge-Representation based on Lattice Theory, 141–72. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. http://dx.doi.org/10.1007/978-3-540-34170-3_10.

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Gross, Lconard. "Lattice gauge theory; Heuristics and convergence." In Stochastic Processes — Mathematics and Physics, 130–40. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/bfb0080213.

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Daverman, Robert J. "The intimate connections among decomposition theory, embedding theory, and manifold structure theory." In Lecture Notes in Mathematics, 43–47. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/bfb0081417.

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Entezari, Alireza, Ramsay Dyer, and Torsten Möller. "From Sphere Packing to the Theory of Optimal Lattice Sampling." In Mathematics and Visualization, 227–55. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/b106657_12.

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Hartung, Tobias, Karl Jansen, Hernan Leövey, and Julia Volmer. "Avoiding the Sign Problem in Lattice Field Theory." In Springer Proceedings in Mathematics & Statistics, 231–49. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43465-6_11.

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Sabbah, Claude. "Good Meromorphic Connections (Analytic Theory) and the Riemann–Hilbert Correspondence." In Lecture Notes in Mathematics, 177–93. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-31695-1_12.

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Conference papers on the topic "Connections (Mathematics) Lattice theory"

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Uragami, Daisuke, and Yurika Suzuki. "Analysis of human body motion by Lattice theory." In INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016). Author(s), 2017. http://dx.doi.org/10.1063/1.4992533.

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KASAMATSU, KENICHI, IKUO ICHINOSE, and TETSUO MATSUI. "ATOMIC QUANTUM SIMULATIONS OF LATTICE GAUGE THEORY: EFFECT OF GAUGE SYMMETRY BREAKING." In Summer Workshop on Physics, Mathematics, and All That Quantum Jazz. WORLD SCIENTIFIC, 2014. http://dx.doi.org/10.1142/9789814602372_0015.

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Hennig, Markus, and Bärbel Mertsching. "Innovative 3D Animations for Teaching Electromagnetic Field Theory and its Mathematics in Undergraduate Engineering." In Third International Conference on Higher Education Advances. Valencia: Universitat Politècnica València, 2017. http://dx.doi.org/10.4995/head17.2017.5327.

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In this work, an innovative approach for the design and structuring of teaching videos systematically using 3D animations is presented. The approach focuses on the quantitative description of electromagnetic fields and the mathematical methods and competencies required for this purpose, exemplarily with regard to an undergraduate electrical engineering course during the initial phase of corresponding degree programs. An essential part of this course is the spatial and time-dependent description of electromagnetic fields. For this purpose, students have to work with multiple integrals in 3D space and in different coordinate systems. Such subjects are typically covered only later in mathematics courses and without a technical context, therefore leading to major difficulties for many students. The videos presented in this work are intended to support students and lecturers to work with these subjects in an instructive fashion. The 3D animations allow for effectively clarifying complex connections between technical and mathematical aspects. The videos and their specific design are discussed with regard to didactic and technical considerations. Additionally, their integration with existing interventions for the course is described.
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Hunt, Emily M., Pamela Lockwood-Cooke, and Paul Fisher. "A Practical Approach for Problem-Based Learning in Engineering." In ASME 2007 International Mechanical Engineering Congress and Exposition. ASMEDC, 2007. http://dx.doi.org/10.1115/imece2007-42088.

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Problem-based Learning (PBL) is a motivating, problem-centered teaching method with exciting potential in engineering education. PBL can be used in engineering education to bridge the gap between theory and practice in a gradual way. The most common problem encountered when attempting to integrate PBL into the undergraduate engineering classroom is the time requirement to complete a significant, useful problem. Because PBL has such potential in engineering, mathematics, and science education, professors from engineering, mathematics, and physics have joined together to solve small pieces of a large engineering problem concurrently in an effort to reduce the time required to solve a complex problem in any one class. This is a pilot project for a National Science Foundation (NSF) supported Science Talent Expansion Program (STEP) grant entitled Increasing Numbers, Connections, and Retention in Science and Engineering (INCRSE) (NSF 0622442). The students involved are undergraduate mechanical engineering students that are co-enrolled in Engineering Statics, Calculus II, and Engineering Physics I. These classes are linked using PBL to increase both student engagement and success. The problem addresses concepts taught in class, reinforces connections among the courses, and provides real-world applications. Student, faculty, and industry assessment of the problem reveals a mutually beneficial experience that provides a link for students between in-class concepts and real-world application. This method of problem-based learning provides a practical application that can be used in engineering curricula.
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Li, Y. T., and Y. X. Wang. "A Mathematical Functional Decomposition Approach Through Granularity Partition Process in Quotient Space." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-86217.

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Over the past decades, several methodologies have coalesced around the functional decomposition and partial solution manipulation techniques. These methodologies take designers through steps that help decompose a design problem and build conceptual solutions based on the intended, product functionality. However, this kind of subjective decomposition restricts solutions of conceptual design within designers’ intended the local, rather the whole, solution space. In such cases, the ability for AI-based functional reasoning systems to obtain creative conceptual design solutions is weakened. In this paper, a functional decomposition model based on the domain decomposition theory in quotient space is proposed for carrying out functional decomposition without needing functional reasoning knowledge to support. In this model, the functional decomposition is treated as a granularity partition process in quotient space composed of three variables: the domain granularities, the attribute properties, and the topological structures. The closeness degrees and the attribute properties in fuzzy mathematics are utilized to describe the fuzzy equivalence relations between the granularities in the up-layer and in the lower-layer of the functional hierarchies. According to the order characteristics in the partially sequential quotient space, based on the homomorphism principle, the attribute properties and the topological structures corresponding to the lower-layer of the functional hierarchies are constructed then. Here, the attribute properties are expressed with membership functions pointed to the lower-layer from the up-layer of the functional hierarchies, and the topological structures are expressed with matrixes and the directed function network represent the topological connections among the subfunctions in the lower-layer of the functional hierarchies. Through refining the functional decomposition process step by step, and traversing all tree branches and leaf nodes in the functional decomposition tree, the functional hierarchies are obtained. Since the functional decomposition process not need the user to indicate or manage desired functionality, the model presented in this paper can reduce designers’ prejudices or preconceptions on the functional hierarchies, as well as extend the solution space of conceptual design.
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Brown, Ashland O. "Undergraduate Finite Element Instruction Using Commercial Finite Element Software Tutorials and the Kolb Learning Cycle." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-60756.

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As background, the Kolb learning cycle describes an entire cycle around which a learning experience progresses [1]. The goal, therefore, is to structure learning activities that will proceed completely around this cycle, providing the maximum opportunity for full student comprehension of the course material. This model has been used previously to evaluate and enhance teaching in engineering [2, 3, and 4]. Most college education is geared toward abstract conceptualiztion, but complete learning is enhanced by the use of all four learning stages Abstract Hypothesis and Conceptualization, Active Experimentation, Concrete Experience and Reflective Observation. Some parts of this paper were presented at an earlier conference [13]. The Finite Element (FE) method is a numerical procedure that is widely used to analyze engineering problems accurately and quickly in many corporations. It has become an essential and powerful analytical tool in designing products with ever-shorter development cycles [5, 6, and 7]. The use of commercial finite element software tutorials along with the Kolb model of learning has been used for the past three years to instruct undergraduate students in an introductory FE course. This paper provides outlines of the use of the commercial software tutorials using two Kolb learning cycles, a global learning cycle for the course and a micro learning cycle for the FE tutorials. The commercial FE software tutorials provide an excellent method to reinforce student’s retention of this complex numerical procedure. The software tutorials provide hands-on learning experiences that students need to reinforce the theoretical concepts covered in the lectures. The students are provided “Abstract Hypothesis/Conceptual Theory” that begins with the background of the FE method, fundamental mathematics of FE, move through the concept of “stiffness-analysis,” one-dimensional direct stiffness analysis of various structures, the topology of the various finite elements, error analysis of FE results, and concludes with engineering analysis of a typical engineering problem. These activities are interlaced with the hands-on MSC.Nastran1 software tutorials that begin stating the proposed problem in a manner that is “real-world” in nature then the student is supplied with background theory for the analysis they will attempt. The tutorials provide specific instructions on how to build the FE model of the problem using this commercial FEM code. The tutorial includes a step-by-step outline of the problem modeling with text and illustrations. The student then performs the analysis. Instead of doing this in a blind manner, the tutorial provides a connection to the abstract theory of FE and asks the student to perturb certain parameters in the model to predict the results apriori. This causes the students to make connections between the modeling techniques and the IMECE2004-60756 Undergraduate Finite Element Instruction using Commercial Finite Element Software Tutorials and the Kolb Learning Cycle underlying physics. This focuses in on the “Active Experimentation” part of Kolb’s cycle. After the student performs the analysis, they are asked to attempt to explain the differences between the FEM modeling and theoretical results. This requires students to engage in the “Reflective Observation” portion of Kolb’s cycle. In designing the learning experiences to completely transverse the Kolb cycle, students are fully engaged to understand the fundamentals of FE modeling and maximize the learning experience the tutorials provide. Near the conclusion of this course students are asked to develop prototype models of designs for engineering problems using FE and then asked to conduct experiments to verify their FE analysis. The Kolb model describes an entire cycle around which learning experiences progress Abstract Hypothesis and Conceptualization, Active Experimentation, Concrete Experience and Reflective Observation, and is shown below in Figure 1.
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