Relevant bibliographies by topics / Algebraic schemes

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

Academic literature on the topic 'Algebraic schemes'

Author: Grafiati
Published: 9 March 2023
Last updated: 10 March 2023

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Journal articles on the topic "Algebraic schemes"

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Yokura, Shoji. "Oriented bivariant theory, II: Algebraic cobordism of S-schemes." International Journal of Mathematics 30, no. 06 (June 2019): 1950031. http://dx.doi.org/10.1142/s0129167x19500319.

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This is a sequel to our previous paper “Oriented bivariant theory, I”. In 2001, Levine and Morel constructed algebraic cobordism for (reduced) schemes [Formula: see text] of finite type over a base field [Formula: see text] in an abstract way and later Levine and Pandharipande reconstructed it more geometrically, using “double point degeneration”. In this paper in a similar manner to the construction of Levine–Morel, we construct an algebraic cobordism for a scheme [Formula: see text] over a fixed scheme [Formula: see text] in such a way that if the target scheme [Formula: see text] is the point [Formula: see text], then our algebraic cobordism is isomorphic to Levine–Morel’s algebraic cobordism. Our algebraic cobordism can be interpreted as “a family of algebraic cobordism” parametrized by the base scheme [Formula: see text].
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Cailotto, Maurizio. "Algebraic connections on logarithmic schemes." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 333, no. 12 (December 2001): 1089–94. http://dx.doi.org/10.1016/s0764-4442(01)02189-9.

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B. Benson, David, and Irène Guessarian. "Algebraic solutions to recursion schemes." Journal of Computer and System Sciences 35, no. 3 (December 1987): 365–400. http://dx.doi.org/10.1016/0022-0000(87)90020-1.

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Chipalkatti, Jaydeep. "Apolar Schemes of Algebraic Forms." Canadian Journal of Mathematics 58, no. 3 (June 1, 2006): 476–91. http://dx.doi.org/10.4153/cjm-2006-020-3.

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AbstractThis is a note on the classical Waring's problem for algebraic forms. Fix integers (n, d, r, s), and let ∧ be a general r-dimensional subspace of degree d homogeneous polynomials in n+1 variables. Let denote the variety of s-sided polar polyhedra of ∧. We carry out a case-by-case study of the structure of for several specific values of (n, d, r, s). In the first batch of examples, is shown to be a rational variety. In the second batch, is a finite set of which we calculate the cardinality.
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Partala, Juha. "Algebraic generalization of Diffie–Hellman key exchange." Journal of Mathematical Cryptology 12, no. 1 (March 1, 2018): 1–21. http://dx.doi.org/10.1515/jmc-2017-0015.

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AbstractThe Diffie–Hellman key exchange scheme is one of the earliest and most widely used public-key primitives. Its underlying algebraic structure is a cyclic group and its security is based on the discrete logarithm problem (DLP). The DLP can be solved in polynomial time for any cyclic group in the quantum computation model. Therefore, new key exchange schemes have been sought to prepare for the time when quantum computing becomes a reality. Algebraically, these schemes need to provide some sort of commutativity to enable Alice and Bob to derive a common key on a public channel while keeping it computationally difficult for the adversary to deduce the derived key. We suggest an algebraically generalized Diffie–Hellman scheme (AGDH) that, in general, enables the application of any algebra as the platform for key exchange. We formulate the underlying computational problems in the framework of average-case complexity and show that the scheme is secure if the problem of computing images under an unknown homomorphism is infeasible. We also show that a symmetric encryption scheme possessing homomorphic properties over some algebraic operation can be turned into a public-key primitive with the AGDH, provided that the operation is complex enough. In addition, we present a brief survey on the algebraic properties of existing key exchange schemes and identify the source of commutativity and the family of underlying algebraic structures for each scheme.
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Căzănescu, Virgil Emil, and Gheorghe Ştefănescu. "Towards a New Algebraic Foundation of Flowchart Scheme Theory." Fundamenta Informaticae 13, no. 2 (April 1, 1990): 171–210. http://dx.doi.org/10.3233/fi-1990-13204.

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We develope a formalism for the algebraic study of flowchart schemes and their behaviours, based on a new axiomatic looping operation, called feedback. This formalism is based on certain flownomial expressions. Such an expression is built up from two types of atomic schemes (i.e., elements in a double-ranked set X considered as unknown computation processes, and elements in a “theory” T considered as known computation processes) by using three operations: sum, composition, and feedback. Flownomial expressions are subject to certain rules of identification. The axiomatization of flowchart schemes is based on the fact that a flowchart scheme may be identified with a class of isomorphic flownomial expressions in normal form. The corresponding algebra for flowchart schemes is called biflow. This axiomatization is extended to certain types of behaviour. We present axiomatizations for accessible flowchart schemes, reduced flowchart schemes, minimal flowchart schemes with respect to the input behaviour, minimal flowchart schemes with respect to the input-output behaviour etc. Some results are new, others are simple translations in terms of feedback of previous results obtained by using Elgot’s iteration or Kleene’s repetition. The paper also contains some historical comments.
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Lowrey, Parker E., and Timo Schürg. "DERIVED ALGEBRAIC COBORDISM." Journal of the Institute of Mathematics of Jussieu 15, no. 2 (October 30, 2014): 407–43. http://dx.doi.org/10.1017/s1474748014000334.

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We construct a cohomology theory using quasi-smooth derived schemes as generators and an analog of the bordism relation using derived fiber products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that, in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.
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Barrenechea, Gabriel R., Volker John, and Petr Knobloch. "Analysis of Algebraic Flux Correction Schemes." SIAM Journal on Numerical Analysis 54, no. 4 (January 2016): 2427–51. http://dx.doi.org/10.1137/15m1018216.

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Jun, Jaiung. "Hyperstructures of affine algebraic group schemes." Journal of Number Theory 167 (October 2016): 336–52. http://dx.doi.org/10.1016/j.jnt.2016.03.016.

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Ferguson, Pamela A., and Alexandre Turull. "Algebraic decompositions of commutative association schemes." Journal of Algebra 96, no. 1 (September 1985): 211–29. http://dx.doi.org/10.1016/0021-8693(85)90047-x.

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Dissertations / Theses on the topic "Algebraic schemes"

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Abou-Rached, John. "Sheaves and schemes: an introduction to algebraic geometry." Kansas State University, 2016. http://hdl.handle.net/2097/32608.

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Master of Science
Department of Mathematics
Roman Fedorov
The purpose of this report is to serve as an introduction to the language of sheaves and schemes via algebraic geometry. The main objective is to use examples from algebraic geometry to motivate the utility of the perspective from sheaf and scheme theory. Basic facts and definitions will be provided, and a categorical approach will be frequently incorporated when appropriate.
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Bechtold, Benjamin [Verfasser], and Jürgen [Akademischer Betreuer] Hausen. "Cox sheaves on graded schemes, algebraic actions and F1-schemes / Benjamin Bechtold ; Betreuer: Jürgen Hausen." Tübingen : Universitätsbibliothek Tübingen, 2018. http://d-nb.info/1168803829/34.

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Goward, Russell A. "A simple algorithm for principalization of monomial ideals /." free to MU campus, to others for purchase, 2001. http://wwwlib.umi.com/cr/mo/fullcit?p3012972.

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Cliff, Emily Rose. "Universal D-modules, and factorisation structures on Hilbert schemes of points." Thesis, University of Oxford, 2015. https://ora.ox.ac.uk/objects/uuid:9edee0a0-f30a-4a54-baf5-c833222303ca.

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This thesis concerns the study of chiral algebras over schemes of arbitrary dimension n. In Chapter I, we construct a chiral algebra over each smooth variety X of dimension n. We do this via the Hilbert scheme of points of X, which we use to build a factorisation space over X. Linearising this space produces a factorisation algebra over X, and hence, by Koszul duality, the desired chiral algebra. We begin the chapter with an overview of the theory of factorisation and chiral algebras, before introducing our main constructions. We compute the chiral homology of our factorisation algebra, and show that the D-modules underlying the corresponding chiral algebras form a universal D-module of dimension n. In Chapter II, we discuss the theory of universal D-modules and OO- modules more generally. We show that universal modules are equivalent to sheaves on certain stacks of étale germs of n-dimensional varieties. Furthermore, we identify these stacks with the classifying stacks of groups of automorphisms of the n-dimensional disc, and hence obtain an equivalence between the categories of universal modules and the representation categories of these groups. We also define categories of convergent universal modules and study them from the perspectives of the stacks of étale germs and the representation theory of the automorphism groups.
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Lee, Hwa Young. "The flag Hilbert scheme of points on nodal curves and the punctual Hilbert scheme of points of the cusp curve." Diss., UC access only, 2009. http://proquest.umi.com/pqdweb?did=1907270841&sid=1&Fmt=7&clientId=48051&RQT=309&VName=PQD.

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Thesis (Ph. D.)--University of California, Riverside, 2009.
Includes abstract. Includes bibliographical references (leaf 71). Issued in print and online. Available via ProQuest Digital Dissertations.
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BORGHESI, SIMONE. "Algebraic Morava K-theories and the higher degree formula." Doctoral thesis, Northwestern University, 2000. http://hdl.handle.net/10281/39205.

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This manuscript consists of two parts. In the first, a cohomology theory on the category of algebraic schemes over a field of characteristic zero is provided. This theory shares several properties with the topological Morava K-theories, hence the name. The second part contains a proof of Voevodsky and Rost conjectured degree formulae. The proof uses algebraic Morava K-theories.
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Heinze, Aiso. "Applications of Schur rings in algebraic combinatorics graphs, partial difference sets and cyclotomic schemes /." [S.l. : s.n.], 2001. http://deposit.ddb.de/cgi-bin/dokserv?idn=962888532.

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Kuang, Yu Rang. "Algebraic coupled-state calculation of positron-hydrogen collisions at low energy, using large coupling schemes." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/nq23106.pdf.

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Carissimi, Nicola. "Reconstruction of schemes via the tensor triangulated category of perfect complexes." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23343/.

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This elaborate consists of a detailed presentation of the construction introduced for the first time by Paul Balmer and aimed to define a locally ringed space associated to a given tensor triangulated category, the so called spectrum of the category. The focus of this thesis is the case of the tensor triangulated category of perfect complexes on a noetherian scheme X, the full triangulated subcategory of the derived category of sheaves of modules consisting of complexes of sheaves locally quasi-isomorphic to complexes of locally free sheaves. This category inherits the structure of derived tensor product of complexes of sheaves of modules, becoming a tensor triangulated category. Its spectrum is a scheme isomorphic to X, providing a powerful reconstruction result.
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ALMashrafi, Mufeed Juma. "Analysis of stream cipher based authenticated encryption schemes." Thesis, Queensland University of Technology, 2012. https://eprints.qut.edu.au/60916/1/Mufeed_ALMashrafi_Thesis.pdf.

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Authenticated Encryption (AE) is the cryptographic process of providing simultaneous confidentiality and integrity protection to messages. This approach is more efficient than applying a two-step process of providing confidentiality for a message by encrypting the message, and in a separate pass providing integrity protection by generating a Message Authentication Code (MAC). AE using symmetric ciphers can be provided by either stream ciphers with built in authentication mechanisms or block ciphers using appropriate modes of operation. However, stream ciphers have the potential for higher performance and smaller footprint in hardware and/or software than block ciphers. This property makes stream ciphers suitable for resource constrained environments, where storage and computational power are limited. There have been several recent stream cipher proposals that claim to provide AE. These ciphers can be analysed using existing techniques that consider confidentiality or integrity separately; however currently there is no existing framework for the analysis of AE stream ciphers that analyses these two properties simultaneously. This thesis introduces a novel framework for the analysis of AE using stream cipher algorithms. This thesis analyzes the mechanisms for providing confidentiality and for providing integrity in AE algorithms using stream ciphers. There is a greater emphasis on the analysis of the integrity mechanisms, as there is little in the public literature on this, in the context of authenticated encryption. The thesis has four main contributions as follows. The first contribution is the design of a framework that can be used to classify AE stream ciphers based on three characteristics. The first classification applies Bellare and Namprempre's work on the the order in which encryption and authentication processes take place. The second classification is based on the method used for accumulating the input message (either directly or indirectly) into the into the internal states of the cipher to generate a MAC. The third classification is based on whether the sequence that is used to provide encryption and authentication is generated using a single key and initial vector, or two keys and two initial vectors. The second contribution is the application of an existing algebraic method to analyse the confidentiality algorithms of two AE stream ciphers; namely SSS and ZUC. The algebraic method is based on considering the nonlinear filter (NLF) of these ciphers as a combiner with memory. This method enables us to construct equations for the NLF that relate the (inputs, outputs and memory of the combiner) to the output keystream. We show that both of these ciphers are secure from this type of algebraic attack. We conclude that using a keydependent SBox in the NLF twice, and using two different SBoxes in the NLF of ZUC, prevents this type of algebraic attack. The third contribution is a new general matrix based model for MAC generation where the input message is injected directly into the internal state. This model describes the accumulation process when the input message is injected directly into the internal state of a nonlinear filter generator. We show that three recently proposed AE stream ciphers can be considered as instances of this model; namely SSS, NLSv2 and SOBER-128. Our model is more general than a previous investigations into direct injection. Possible forgery attacks against this model are investigated. It is shown that using a nonlinear filter in the accumulation process of the input message when either the input message or the initial states of the register is unknown prevents forgery attacks based on collisions. The last contribution is a new general matrix based model for MAC generation where the input message is injected indirectly into the internal state. This model uses the input message as a controller to accumulate a keystream sequence into an accumulation register. We show that three current AE stream ciphers can be considered as instances of this model; namely ZUC, Grain-128a and Sfinks. We establish the conditions under which the model is susceptible to forgery and side-channel attacks.
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Books on the topic "Algebraic schemes"

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1950-, Shokurov Vyacheslav V., ed. Algebraic curves, algebraic manifolds, and schemes. Berlin: Springer, 1998.

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Görtz, Ulrich, and Torsten Wedhorn. Algebraic Geometry I: Schemes. Wiesbaden: Springer Fachmedien Wiesbaden, 2020. http://dx.doi.org/10.1007/978-3-658-30733-2.

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Oystaeyen, F. van. Algebraic geometry for associative algebras. New York: M. Dekker, 2000.

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Eisenbud, David. The geometry of schemes. New York: Springer, 2000.

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Eisenbud, David. Schemes: The language of modern algebraic geometry. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, 1992.

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1951-, Harris Joe, ed. Schemes: The language of modern algebraic geometry. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, 1992.

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Zieschang, Paul-Hermann. An Algebraic Approach to Association Schemes. Berlin, Heidelberg: Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/bfb0097032.

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Zieschang, Paul-Hermann. An algebraic approach to association schemes. Berlin: Springer, 1996.

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An algebraic approach to association schemes. Berlin: Springer, 1996.

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author, Pianzola Arturo 1955, ed. Torsors, reductive group schemes and extended affine lie algebras. Providence, Rhode Island: American Mathematical Society, 2013.

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Book chapters on the topic "Algebraic schemes"

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Shafarevich, I. R. "Schemes." In Algebraic Geometry I, 280–93. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/978-3-642-57878-6_9.

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Görtz, Ulrich, and Torsten Wedhorn. "Schemes." In Algebraic Geometry I, 66–92. Wiesbaden: Vieweg+Teubner, 2010. http://dx.doi.org/10.1007/978-3-8348-9722-0_4.

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Shafarevich, Igor R. "Schemes." In Basic Algebraic Geometry 2, 3–48. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/978-3-642-57956-1_1.

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Jantzen, Jens. "Schemes." In Representations of Algebraic Groups, 3–18. Providence, Rhode Island: American Mathematical Society, 2007. http://dx.doi.org/10.1090/surv/107/01.

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Shafarevich, Igor R. "Schemes." In Basic Algebraic Geometry 2, 3–47. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-38010-5_1.

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Kriz, Igor, and Sophie Kriz. "Schemes." In Introduction to Algebraic Geometry, 57–100. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-62644-0_2.

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Zieschang, Paul-Hermann. "Algebraic prerequisites." In An Algebraic Approach to Association Schemes, 65–96. Berlin, Heidelberg: Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/bfb0097035.

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Jantzen, Jens. "Schubert schemes." In Representations of Algebraic Groups, 353–64. Providence, Rhode Island: American Mathematical Society, 2007. http://dx.doi.org/10.1090/surv/107/23.

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Görtz, Ulrich, and Torsten Wedhorn. "One-dimensional schemes." In Algebraic Geometry I, 485–502. Wiesbaden: Vieweg+Teubner, 2010. http://dx.doi.org/10.1007/978-3-8348-9722-0_16.

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Görtz, Ulrich, and Torsten Wedhorn. "Schemes over fields." In Algebraic Geometry I, 118–44. Wiesbaden: Vieweg+Teubner, 2010. http://dx.doi.org/10.1007/978-3-8348-9722-0_6.

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Conference papers on the topic "Algebraic schemes"

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Urivskiy, Alexey. "On Identity-Based-Like Non-Interactive Key Establishment for Key Predistribution Schemes." In 2020 Algebraic and Combinatorial Coding Theory (ACCT). IEEE, 2020. http://dx.doi.org/10.1109/acct51235.2020.9383385.

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Suga, Yuji. "An algebraic interpretation of the XOR-based Secret Sharing Schemes." In 2018 International Symposium on Information Theory and Its Applications (ISITA). IEEE, 2018. http://dx.doi.org/10.23919/isita.2018.8664382.

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Albertini, Francesca, and Domenico D'Alessandro. "Algebraic conditions for indirect controllability in quantum coherent feedback schemes." In 2013 European Control Conference (ECC). IEEE, 2013. http://dx.doi.org/10.23919/ecc.2013.6669189.

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Di Franco, Pierluigi, Giordano Scarciotti, and Alessandro Astolfi. "Discretization schemes for constraint stabilization in nonlinear differential-algebraic systems." In 2018 17th European Control Conference (ECC). IEEE, 2018. http://dx.doi.org/10.23919/ecc.2018.8550189.

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Marin, Rodrigo A., and Placid M. Ferreira. "A Kinematic Approach to the Design and Analysis of Optimal Machining Fixtures." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-33587.

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This paper discusses the design and analysis of machining fixtures from a kinematics perspective. We use screw theory to model the fixture elements, the motions and forces/moments involved. We present linear algebraic methods that are quick and reasonably accurate, as opposed to existing methods. First, we study the determinism of 3-2-1 locator schemes and present simple formal geometric conditions to verify it. They also give an insight into the design issues influencing the robustness of the locator scheme. Next, we discuss the problem of synthesizing optimal clamping schemes that minimize the maximum clamping force. We present linear programs to solve this problem with and without friction. Finally, we focus our attention on the robustness of the fixturing scheme and discuss the use of form closure to provide it. We present linear algebraic methods to compute the regions of the workpart within which the clamps can be placed to produce form closure, both dependently and independently of each other. The method is easily incorporated in the linear program to compute optimal clamping schemes.
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Chang, Yanni, Dezhi Dai, and Albert Y. Tong. "A Comparative Study of Interpolation Schemes in Overset Meshes for the PLIC-VOF Method in Multiphase Flows." In ASME 2020 Fluids Engineering Division Summer Meeting collocated with the ASME 2020 Heat Transfer Summer Conference and the ASME 2020 18th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/fedsm2020-20399.

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Abstract Piecewise Linear Interface Calculation (PLIC) schemes have been extensively employed in the Volume-of-Fluid (VOF) method for interface capturing in numerical simulations of multiphase flows. Dynamic overset meshes can be especially useful in applications involving component motions and complex geometric shapes. The basic idea of the overset mesh is the variable field interpolation within the overlapped region between the background and body meshes. The acceptor cell value is evaluated by a weighted average of its donors. The weighting factors are calculated by different algebraic methods, such as the averageValue, injection and inverseDistance schemes, which are implanted in the foam-extend library. A geometric interpolation scheme of the VOF field in overset meshes for the PLIC-VOF method has been proposed in the present study. The VOF value of an acceptor cell is evaluated geometrically with the reconstructed interfaces from the corresponding donor elements. Test cases of advecting liquid columns of different shapes inside a unit square/cube with a prescribed rotational velocity field have been performed to demonstrate the accuracy of the proposed overset interpolation scheme by comparing it with three algebraic ones. The proposed scheme has been shown to yield higher accuracy.
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Abla, Parhat. "Lattice Based Group Key Exchange Protocol in the Standard Model." In 2nd International Conference on Machine Learning &Trends (MLT 2021). AIRCC Publishing Corporation, 2021. http://dx.doi.org/10.5121/csit.2021.111113.

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Group key exchange schemes allow group members to agree on a session key. Although there are many works on constructing group key exchange schemes, but most of them are based on algebraic problems which can be solved by quantum algorithms in polynomial time. Even if several works considered lattice based group key exchange schemes, believed to be post-quantum secure, but only in the random oracle model. In this work, we propose a group key exchange scheme based on ring learning with errors problem. On contrast to existing schemes, our scheme is proved to be secure in the standard model. To achieve this, we define and instantiate multi-party key reconciliation mechanism. Furthermore, using known compiler with lattice based signature schemes, we can achieve authenticated group key exchange with postquantum security.
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Zagumennov, Denis, Vladimir Deundyak, Alexander Gufan, and Vyacheslav Mkrtichan. "Algebraic Geometry Codes for Special Broadcast Encryption Schemes in Telecommunication Nets." In 2020 Moscow Workshop on Electronic and Networking Technologies (MWENT). IEEE, 2020. http://dx.doi.org/10.1109/mwent47943.2020.9067411.

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Wu, Jianping, Fukang Yin, and Jinhui Yang. "Two Schemes to Improve the Implementation of the Aggregation Based Algebraic Multigrid Preconditioner." In 2017 International Conference on Applied Mathematics, Modeling and Simulation (AMMS 2017). Paris, France: Atlantis Press, 2017. http://dx.doi.org/10.2991/amms-17.2017.7.

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Woodward, Jerry. "Foundational algebraic reasoning in the schemes of middle school students with learning disabilities." In 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. PMENA, 2020. http://dx.doi.org/10.51272/pmena.42.2020-40.

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Reports on the topic "Algebraic schemes"

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McKenzie, Edwin, and Richard Snodgrass. Scheme Evolution and the Relationship Algebra. Revision. Fort Belvoir, VA: Defense Technical Information Center, May 1988. http://dx.doi.org/10.21236/ada201297.

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Goldberg, Moshe, and Marvin Marcus. Stability Analysis of Finite Difference Schemes for Hyperbolic Systems and Problems in Applied and Computational Linear Algebra. Fort Belvoir, VA: Defense Technical Information Center, July 1988. http://dx.doi.org/10.21236/ada201083.

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Marcus, Marvin, and Moshe Goldberg. Stability Analysis of Finite Difference Schemes for Hyperbolic Systems, and Problems in Applied and Computational Linear Algebra. Fort Belvoir, VA: Defense Technical Information Center, August 1985. http://dx.doi.org/10.21236/ada161092.

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Bond, W., Maria Seale, and Jeffrey Hensley. A dynamic hyperbolic surface model for responsive data mining. Engineer Research and Development Center (U.S.), April 2022. http://dx.doi.org/10.21079/11681/43886.

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Data management systems impose structure on data via a static representation schema or data structure. Information from the data is extracted by executing queries based on predefined operators. This paradigm restricts the searchability of the data to concepts and relationships that are known or assumed to exist among the objects. While this is an effective and efficient means of retrieving simple information, we propose that such a structure severely limits the ability to derive breakthrough knowledge that exists in data under the guise of “unknown unknowns.” A dynamic system will alleviate this dependence, allowing theoretically infinite projections of the data to reveal discoverable relationships that are hidden by traditional use case-driven, static query systems. In this paper, we propose a framework for a data-responsive query algebra based on a dynamic hyperbolic surface model. Such a model could provide more intuitive access to analytics and insights from massive, aggregated datasets than existing methods. This model will significantly alter the means of addressing the underlying data by representing it as an arrangement on a dynamic, hyperbolic plane. Consequently, querying the data can be viewed as a process similar to quantum annealing, in terms of characterizing data representation as an energy minimization problem with numerous minima.
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