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Academic literature on the topic 'Toric algebra'

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Published: 4 June 2021

Last updated: 4 February 2022

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Journal articles on the topic "Toric algebra"

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Herzog, Jürgen, Raheleh Jafari, and Abbas Nasrollah Nejad. "On the Gauss algebra of toric algebras." Journal of Algebraic Combinatorics 51, no. 1 (January 2, 2019): 1–17. http://dx.doi.org/10.1007/s10801-018-0865-8.

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Petrović, Sonja, and Despina Stasi. "Toric algebra of hypergraphs." Journal of Algebraic Combinatorics 39, no. 1 (April 20, 2013): 187–208. http://dx.doi.org/10.1007/s10801-013-0444-y.

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Petrović, Sonja, Apostolos Thoma, and Marius Vladoiu. "Bouquet algebra of toric ideals." Journal of Algebra 512 (October 2018): 493–525. http://dx.doi.org/10.1016/j.jalgebra.2018.05.016.

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Chirivì, Rocco. "On some properties of LS algebras." Communications in Contemporary Mathematics 22, no. 02 (December 3, 2018): 1850085. http://dx.doi.org/10.1142/s0219199718500852.

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The discrete LS algebra over a totally ordered set is the homogeneous coordinate ring of an irreducible projective (normal) toric variety. We prove that this algebra is the ring of invariants of a finite abelian group containing no pseudo-reflection acting on a polynomial ring. This is used to study the Gorenstein property for LS algebras. Further we show that any LS algebra is Koszul.

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Kim, Jin Hong. "On the integral cohomology of toric varieties." Journal of Algebra and Its Applications 15, no. 02 (October 6, 2015): 1650032. http://dx.doi.org/10.1142/s0219498816500328.

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It is known that the integral cohomology algebra of any smooth compact toric variety XΣ associated to a complete regular fan Σ is isomorphic to the Stanley–Reisner algebra ℤ[Σ] modulo the ideal JΣ generated by linear relations determined by Σ. The aim of this paper is to show how to determine the integral cohomology algebra of a toric variety (in particular, a projective toric variety) associated to a certain simplicial fan. As a consequence, we confirm our expectation that for a certain simplicial fan the integral cohomology algebra is also given by the same formula as in a complete regular fan.

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Katsabekis, Anargyros, and Apostolos Thoma. "Toric sets and orbits on toric varieties." Journal of Pure and Applied Algebra 181, no. 1 (June 2003): 75–83. http://dx.doi.org/10.1016/s0022-4049(02)00305-5.

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Kaveh, Kiumars. "Vector Fields and the Cohomology Ring of Toric Varieties." Canadian Mathematical Bulletin 48, no. 3 (September 1, 2005): 414–27. http://dx.doi.org/10.4153/cmb-2005-039-1.

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AbstractLetXbe a smooth complex projective variety with a holomorphic vector field with isolated zero setZ. From the results of Carrell and Lieberman there exists a filtrationF0⊂F1⊂ · · · ofA(Z), the ring of ℂ-valued functions onZ, such thatas graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra ofX.

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Geiger, Dan, Christopher Meek, and Bernd Sturmfels. "On the toric algebra of graphical models." Annals of Statistics 34, no. 3 (June 2006): 1463–92. http://dx.doi.org/10.1214/009053606000000263.

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Klyachko, A. A. "Toric bundles and problems of linear algebra." Functional Analysis and Its Applications 23, no. 2 (1989): 135–37. http://dx.doi.org/10.1007/bf01078785.

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Callegaro, Filippo, and Emanuele Delucchi. "The integer cohomology algebra of toric arrangements." Advances in Mathematics 313 (June 2017): 746–802. http://dx.doi.org/10.1016/j.aim.2017.04.017.

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Dissertations / Theses on the topic "Toric algebra"

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Solus, Liam. "Normal and Δ-Normal Configurations in Toric Algebra." Oberlin College Honors Theses / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=oberlin1308243895.

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Petrovic, Sonja. "ALGEBRAIC AND COMBINATORIAL PROPERTIES OF CERTAIN TORIC IDEALS IN THEORY AND APPLICATIONS." UKnowledge, 2008. http://uknowledge.uky.edu/gradschool_diss/606.

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This work focuses on commutative algebra, its combinatorial and computational aspects, and its interactions with statistics. The main objects of interest are projective varieties in Pn, algebraic properties of their coordinate rings, and the combinatorial invariants, such as Hilbert series and Gröbner fans, of their defining ideals. Specifically, the ideals in this work are all toric ideals, and they come in three flavors: they are defining ideals of a family of classical varieties called rational normal scrolls, cut ideals that can be associated to a graph, and phylogenetic ideals arising in a new and increasingly popular area of algebraic statistics.

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Runge, Piotr. "A Comparison Theorem for the Topological and Algebraic Classification of Quaternionic Toric 8-Manifolds." DigitalCommons@USU, 2009. https://digitalcommons.usu.edu/etd/501.

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In order to discuss topological properties of quaternionic toric 8-manifolds, we introduce the notion of an algebraic morphism in the category of toric spaces. We show that the classification of quaternionic toric 8-manifolds with respect to an algebraic isomorphism is finer than the oriented topological classification. We construct infinite families of quaternionic toric 8-manifolds in the same oriented homeomorphism type but algebraically distinct. To prove that the elements within each family are of the same oriented homeomorphism type, and that we have representatives of all such types of a quaternionic toric 8-manifold, we present and use a method of evaluating the first Pontrjagin class for an arbitrary quaternionic toric 8-manifold.

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Agosti, Claudia. "Cohomology of hyperplane and toric arrangements." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/19510/.

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L'algebra di coomologia del complementare di un arrangiamento torico è più complicata di quella del complementare di un arrangiamento di iperpiani, in quanto il toro complesso ha già di per sè una coomologia non banale e perchè l'intersezione di due sottotori in generale non è connessa. Nel 2005, De Concini e Procesi si sono concentrati sullo studio dell'algebra di coomologia del complementare degli arrangiamenti torici nel quale le intersezioni di sottotori sono sempre connesse (arrangiamenti torici unimodulari) ottenendone una presentazione sullo stile di quella data da Orlik e Solomon per gli arrangiamenti di iperpiani. Nel 2018, Callegaro, D'Adderio, Delucchi, Migliorini e Pagaria hanno generalizzato il lavoro di De Concini e Procesi fornendo una presentazione, sempre sullo stile di quella data da Orlik e Solomon, dell'algebra di coomologia di un generico arrangiamento torico. In questa tesi descriviamo tali presentazioni dell'algebra di coomologia, soffermandoci in particolare su alcuni esempi.

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Lienkaemper, Caitlin. "Toric Ideals, Polytopes, and Convex Neural Codes." Scholarship @ Claremont, 2017. http://scholarship.claremont.edu/hmc_theses/106.

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How does the brain encode the spatial structure of the external world? A partial answer comes through place cells, hippocampal neurons which become associated to approximately convex regions of the world known as their place fields. When an organism is in the place field of some place cell, that cell will fire at an increased rate. A neural code describes the set of firing patterns observed in a set of neurons in terms of which subsets fire together and which do not. If the neurons the code describes are place cells, then the neural code gives some information about the relationships between the place fields–for instance, two place fields intersect if and only if their associated place cells fire together. Since place fields are convex, we are interested in determining which neural codes can be realized with convex sets and in finding convex sets which generate a given neural code when taken as place fields. To this end, we study algebraic invariants associated to neural codes, such as neural ideals and toric ideals. We work with a special class of convex codes, known as inductively pierced codes, and seek to identify these codes through the Gröbner bases of their toric ideals.

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Lin, Matthew. "Graph Cohomology." Scholarship @ Claremont, 2016. https://scholarship.claremont.edu/hmc_theses/82.

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What is the cohomology of a graph? Cohomology is a topological invariant and encodes such information as genus and euler characteristic. Graphs are combinatorial objects which may not a priori admit a natural and isomorphism invariant cohomology ring. In this project, given any finite graph G, we constructively define a cohomology ring H*(G) of G. Our method uses graph associahedra and toric varieties. Given a graph, there is a canonically associated convex polytope, called the graph associahedron, constructed from G. In turn, a convex polytope uniquely determines a toric variety. We synthesize these results, and describe the cohomology of the associated variety directly in terms of the graph G itself.

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French, Josephine. "Toric chiral algebras." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:e16ac71b-7634-48e4-971e-cda490c95f07.

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In this thesis, we investigate lattice chiral algebras as defined by Beilinson and Drinfeld. Given a factorisation monoid satisfying specific conditions and a super extension of this, Beilinson and Drinfeld show that one can push forward this line bundle (super extension) to give a factorisation algebra. Specifically, they describe this in the case of the factorisation monoid formed by taking Г-valued divisors set-theoretically supported over each divisor, for Г a lattice, as a method of constructing these lattice chiral algebras. In this work, we show that their definitions of such divisors, and of line bundles with factorisation on these, generalise to a wider class of objects given by taking coefficients in any cone, C, in a lattice. We show that, in this more general case, the functors of C-valued divisors with settheoretic pullback contained in S are ind-schemes, and, from this, that they form a factorisation monoid. Further, we show that super line bundles with factorisation exist on this factorisation monoid, and that if we have a super line bundle with factorisation on the factorisation monoid of C-valued divisors, we can push forward such a line bundle to get a chiral (factorisation) algebra as for lattices. Hence, we obtain a new class of chiral algebras via this procedure, which we call toric chiral algebras.

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Prabhu-Naik, Nathan. "Tilting bundles and toric Fano varieties." Thesis, University of Bath, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.690721.

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This thesis constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth toric Fano fourfolds. The tilting bundles lead to a large class of explicit Calabi-Yau-5 algebras, obtained as the corresponding rolled-up helix algebra. We provide two different methods to show that a collection of line bundles is full, whilst the strong exceptional condition is checked using the package QuiversToricVarieties for the computer algebra system Macaulay2, written by the author. A database of the full strong exceptional collections can also be found in this package.

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Alexander, Nicholas Charles. "Algebraic Tori in Cryptography." Thesis, University of Waterloo, 2005. http://hdl.handle.net/10012/1154.

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Communicating bits over a network is expensive. Therefore, cryptosystems that transmit as little data as possible are valuable. This thesis studies several cryptosystems that require significantly less bandwidth than conventional analogues. The systems we study, called torus-based cryptosystems, were analyzed by Karl Rubin and Alice Silverberg in 2003 [RS03]. They interpreted the XTR [LV00] and LUC [SL93] cryptosystems in terms of quotients of algebraic tori and birational parameterizations, and they also presented CEILIDH, a new torus-based cryptosystem. This thesis introduces the geometry of algebraic tori, uses it to explain the XTR, LUC, and CEILIDH cryptosystems, and presents torus-based extensions of van Dijk, Woodruff, et al. [vDW04, vDGP⁺05] that require even less bandwidth. In addition, a new algorithm of Granger and Vercauteren [GV05] that attacks the security of torus-based cryptosystems is presented. Finally, we list some open research problems.

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Bouchat, Rachelle R. "ALGEBRAIC PROPERTIES OF EDGE IDEALS." UKnowledge, 2008. http://uknowledge.uky.edu/gradschool_diss/618.

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Given a simple graph G, the corresponding edge ideal IG is the ideal generated by the edges of G. In 2007, Ha and Van Tuyl demonstrated an inductive procedure to construct the minimal free resolution of certain classes of edge ideals. We will provide a simplified proof of this inductive method for the class of trees. Furthermore, we will provide a comprehensive description of the finely graded Betti numbers occurring in the minimal free resolution of the edge ideal of a tree. For specific subclasses of trees, we will generate more precise information including explicit formulas for the projective dimensions of the quotient rings of the edge ideals. In the second half of this thesis, we will consider the class of simple bipartite graphs known as Ferrers graphs. In particular, we will study a class of monomial ideals that arise as initial ideals of the defining ideals of the toric rings associated to Ferrers graphs. The toric rings were studied by Corso and Nagel in 2007, and by studying the initial ideals of the defining ideals of the toric rings we are able to show that in certain cases the toric rings of Ferrers graphs are level.

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Books on the topic "Toric algebra"

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1975-, Panov Taras E., ed. Toric topology. Providence, Rhode Island: American Mathematical Society, 2015.

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B, Little John, and Schenck Henry K. 1963-, eds. Toric varieties. Providence, R.I: American Mathematical Society, 2011.

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William, Fulton. Introduction to toric varieties. Princeton, N.J: Princeton University Press, 1993.

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Jorge, Donoso Pacheco, ed. Tomic testimonios. [Santiago, Chile]: Centro Latinoamericano Simón Bolivar, 1988.

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Dimer models and Calabi-Yau algebras. Providence, R.I: American Mathematical Society, 2012.

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1962-, Sturmfels Bernd, ed. Introduction to tropical geometry. Providence, Rhode Island: American Mathematical Society, 2015.

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P, Banks Stephen. Clifford algebras, nonlinear dynamical systems and isochronous tori. Sheffield: University of Sheffield, Dept. of Automatic Control and Systems Engineering, 1997.

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Oda, Tadao. Convex bodies and algebraic geometry: An introduction to the theory of toric varieties. Berlin: Springer-Verlag, 1988.

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Musson, Ian M. Invariants under tori of rings of differential operators and related topics. Providence, R.I: American Mathematical Society, 1998.

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Ewald, Günter. Combinatorial convexity and algebraic geometry. New York: Springer, 1996.

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Book chapters on the topic "Toric algebra"

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Assi, Abdallah, and Margherita Barile. "Toric Modifications of Free Toric Varieties." In Algebra, Arithmetic and Geometry with Applications, 175–82. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-642-18487-1_8.

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Duarte, Daniel, and Daniel Green Tripp. "Nash Modification on Toric Curves." In Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, 191–202. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-96827-8_8.

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Beelen, Peter, and Diego Ruano. "The Order Bound for Toric Codes." In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 1–10. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-02181-7_1.

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Teissier, Bernard. "Two Points of the Boundary of Toric Geometry." In Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, 107–17. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-96827-8_4.

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Dickenstein, Alicia, Ioannis Z. Emiris, and Anna Karasoulou. "Plane Mixed Discriminants and Toric Jacobians." In SAGA – Advances in ShApes, Geometry, and Algebra, 105–21. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-08635-4_6.

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Morales, Marcel. "Simplicial Toric Varieties Which Are Set-Theoretic Complete Intersections." In Commutative Algebra and its Interactions to Algebraic Geometry, 217–56. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-75565-6_4.

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Helmer, Martin. "Computing the Chern–Schwartz–MacPherson Class of Complete Simplical Toric Varieties." In Applications of Computer Algebra, 207–17. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-56932-1_13.

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Steenbrink, J. H. M. "Motivic Milnor Fibre for Nondegenerate Function Germs on Toric Singularities." In Bridging Algebra, Geometry, and Topology, 255–67. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-09186-0_16.

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Cortiñas, G., C. Haesemeyer, M. E. Walker, and C. A. Weibel. "The K-Theory of Toric Schemes Over Regular Rings of Mixed Characteristic." In Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, 455–79. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-96827-8_19.

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Campillo, Antonio, and Carlos Galindo. "Toric Structure of the Graded Algebra Relative to a Valuation." In Algebra, Arithmetic and Geometry with Applications, 219–34. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-642-18487-1_12.

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Conference papers on the topic "Toric algebra"

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Krone, Robert. "Equivariant Gröbner Bases of Symmetric Toric Ideals." In ISSAC '16: International Symposium on Symbolic and Algebraic Computation. New York, NY, USA: ACM, 2016. http://dx.doi.org/10.1145/2930889.2930902.

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RUEDA, SONIA L. "ACTIONS OF TORI AND FINITE FANS." In Proceedings of the International Conference on Algebras, Modules and Rings. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774552_0019.

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Tsiporkova, K. A., G. S. Lukyanova, S. N. Mashnina, L. S. Revkova, N. I. Tsiporkov, and N. A. Lukyanov. "USING PRACTICAL CONTENT TASKS IN THE CLASSROOM ON THE TOPIC "LINEAR ALGEBRA»." In Modern Technologies in Science and Education MTSE-2020. Ryazan State Radio Engineering University, 2020. http://dx.doi.org/10.21667/978-5-6044782-8-8-104-112.

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Rúa Taborda, María Isabel. "Analytical solution using computer algebra of a biosensor for detecting toxic substances in water." In SPIE Sensing Technology + Applications, edited by Tuan Vo-Dinh, Robert A. Lieberman, and Günter G. Gauglitz. SPIE, 2014. http://dx.doi.org/10.1117/12.2061933.

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Fan, Junfeng, Lejla Batina, Kazuo Sakiyama, and Ingrid Verbauwhede. "FPGA Design for Algebraic Tori-Based Public-Key Cryptography." In 2008 Design, Automation and Test in Europe. IEEE, 2008. http://dx.doi.org/10.1109/date.2008.4484857.

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Fan, Junfeng, Lejla Batina, Kazuo Sakiyama, and Ingrid Verbauwhede. "FPGA design for algebraic tori-based public-key cryptography." In the conference. New York, New York, USA: ACM Press, 2008. http://dx.doi.org/10.1145/1403375.1403687.

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Brunetto, Domenico, and Ana Moura Santos. "Designing active Maths for undergraduate STEAM students." In Seventh International Conference on Higher Education Advances. Valencia: Universitat Politècnica de València, 2021. http://dx.doi.org/10.4995/head21.2021.13082.

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This work presents a set of student-centred activities that may help undergraduate students understand mathematics in their first year of a STEAM degree. In particular, the authors refer to the difficulties students meet in making connections between syntactic and semantic dimensions in learning mathematics, especially in Linear Algebra topics. The specific goal of this paper is to present and discuss how it can work in the case of linear transformations. This topic stands in the middle of every Linear Algebra standard course and is pivotal in many recent applications, such as computer graphics. The study describes the teaching-learning experience and reports the results of the first pilot study, which involves about 100 undergraduate Architecture students of Politecnico di Milano. One of the peculiarities of this work is its context since the class is composed of heterogeneous group of students, in terms of knowledge background and attitudes towards mathematics. The main findings of this paper are underlining how a student-centred strategy, based on asynchronous activities and synchronous class discussion, allows misconceptions to emerge and be appropriately addressed

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Sanchez, Nestor. "Analytical Reformulation of the Multibody Dynamics of a Satellite." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/vib-8241.

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Abstract The topic of dynamics has been somehow reshaped by computational power. The areas of computer algebra and symbolics now allow us to deal with a more involved analytical manipulation of equations. At the same time, the everyday increasing power of numerics put into our hands new tools to solve old problems. In this case, we reformulate the problem of the dynamics of a three body multibody system by using symbolic manipulation of the Newtonian equations, to produce a set of differential equations that can be solve with standard codes. This treatment should produce not only the same results as the numerical approach, but it allows us to use the analytical equations to expand the analysis into design, control and stability. The paper shows the process to build the symbolic code using Maple language, or any algebraic manipulator. The proper equations will be derived to solve for the unknowns angles {ψ,ϕ,θ}, in terms of the prescribed quantities {α(t),β(t),γ(t)t}, and initial conditions. This procedure gives a good idea about the nonlinear response of the satellite to the control parameters. The size of the equations obtained is large. However, considering the type of analysis that could be done with a set like this and the capacity of large computers, it will pay off the extra effort. The codes that could be used for further analysis would find folds, branch points, period doubling bifurcations, Hopf bifurcations, torus bifurcations, by changing the parameters of the governing equation. A large number of important applications will develop in this area in the near future.

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Sam, Sazilah, and Mohd Faizal Nizam Lee Abdullah. "Development and evaluation of form three mathematics i-Think module (Mi-T3) on algebraic formulae topic." In PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON EDUCATION, MATHEMATICS AND SCIENCE 2016 (ICEMS2016) IN CONJUNCTION WITH 4TH INTERNATIONAL POSTGRADUATE CONFERENCE ON SCIENCE AND MATHEMATICS 2016 (IPCSM2016). Author(s), 2017. http://dx.doi.org/10.1063/1.4983895.

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Müller, Andreas, and Zdravko Terze. "A Constraint Stabilization Method for Time Integration of Constrained Multibody Systems in Lie Group Setting." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-34899.

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The stabilization of geometric constraints is vital for an accurate numerical solution of the differential-algebraic equations (DAE) governing the dynamics of constrained multibody systems (MBS). Although this has been a central topic in numerical MBS dynamics using classical vector space formulations, it has not yet been sufficiently addressed when using Lie group formulations. A straightforward approach is to impose constraints directly on the Lie group elements that represent the MBS motion, which requires additional constraints accounting for the invariants of the Lie group. On the other hand, most numerical Lie group integration schemes introduce local coordinates within the integration step, and it is natural to perform the stabilization in terms of these local coordinates. Such a formulation is presented in this paper for index 1 formulation. The stabilization method is applicable to general coordinate mappings (canonical coordinates, Cayley-Rodriguez, Study) on the MBS configuration space Lie group. The stabilization scheme resembles the well-known vectors space projection and pseudo-inverse method consisting in an iterative procedure. A numerical example is presented and it is shown that the Lie group stabilization scheme converges normally within one iteration step, like the scheme in the vector space formulation.

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