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Academic literature on the topic 'Finite fields (Algebra)'

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Published: 4 June 2021
Last updated: 30 July 2024

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Journal articles on the topic "Finite fields (Algebra)"

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KORNYAK, V. V. "COMPUTATION OF COHOMOLOGY OF LIE SUPERALGEBRAS OF VECTOR FIELDS." International Journal of Modern Physics C 11, no. 02 (March 2000): 397–413. http://dx.doi.org/10.1142/s0129183100000353.

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The cohomology of Lie (super)algebras has many important applications in mathematics and physics. It carries most fundamental ("topological") information about algebra under consideration. At present, because of the need for very tedious algebraic computation, the explicitly computed cohomology for different classes of Lie (super)algebras is known only in a few cases. That is why application of computer algebra methods is important for this problem. We describe here an algorithm and its C implementation for computing the cohomology of Lie algebras and superalgebras. The program can proceed finite-dimensional algebras and infinite-dimensional graded algebras with finite-dimensional homogeneous components. Among the last algebras, Lie algebras and superalgebras of formal vector fields are most important. We present some results of computation of cohomology for Lie superalgebras of Buttin vector fields and related algebras. These algebras being super-analogs of Poisson and Hamiltonian algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics.
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Dadarlat, Marius. "Fiberwise KK-equivalence of continuous fields of C*-algebras." Journal of K-Theory 3, no. 2 (May 28, 2008): 205–19. http://dx.doi.org/10.1017/is008001012jkt041.

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AbstractLet A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.
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Iovanov, Miodrag Cristian, and Alexander Harris Sistko. "Maximal subalgebras of finite-dimensional algebras." Forum Mathematicum 31, no. 5 (September 1, 2019): 1283–304. http://dx.doi.org/10.1515/forum-2019-0033.

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AbstractWe study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.
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Mounirh, Karim. "Nicely semiramified division algebras over Henselian fields." International Journal of Mathematics and Mathematical Sciences 2005, no. 4 (2005): 571–77. http://dx.doi.org/10.1155/ijmms.2005.571.

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This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian fieldEwith an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.
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Fratila, Dragos. "Cusp eigenforms and the hall algebra of an elliptic curve." Compositio Mathematica 149, no. 6 (March 4, 2013): 914–58. http://dx.doi.org/10.1112/s0010437x12000784.

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AbstractWe give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function fields and ${\mathrm{GL} }_{n} $. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by Burban and Schiffmann [On the Hall algebra of an elliptic curve I, Preprint (2005), arXiv:math/0505148 [math.AG]] and Schiffmann and Vasserot [The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011), 188–234]).
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GORAZD, TOMASZ A. "FAST ISOMORPHISM TESTING IN ARITHMETICAL VARIETIES." International Journal of Algebra and Computation 13, no. 04 (August 2003): 499–506. http://dx.doi.org/10.1142/s0218196703001572.

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Let [Formula: see text] be a finitely generated, arithmetical variety such that all subdirectly irreducible algebras from [Formula: see text] have linearly ordered congruences. We show that there is a polynomial time algorithm that tests the existing of an isomorphism between any two finite algebras from [Formula: see text]. This includes the following classical structures in algebra: • Boolean algebras. • Varieties of rings generated by finitely many finite fields. • Varieties of Heyting algebras generated by an n–element chain.
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BILLIG, YULY. "MODULES FOR A SHEAF OF LIE ALGEBRAS ON LOOP MANIFOLDS." International Journal of Mathematics 23, no. 08 (July 10, 2012): 1250079. http://dx.doi.org/10.1142/s0129167x12500796.

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We consider a semidirect product of the sheaf of vector fields on a manifold ℂ* × X with a central extension of the sheaf of Lie algebras of maps from ℂ* × X into a finite-dimensional simple Lie algebra, viewed as sheaves on X. Using vertex algebra methods we construct sheaves of modules for this sheaf of Lie algebras. Our results extend the work of Malikov–Schechtman–Vaintrob on the chiral de Rham complex.
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MAYR, PETER. "THE SUBPOWER MEMBERSHIP PROBLEM FOR MAL'CEV ALGEBRAS." International Journal of Algebra and Computation 22, no. 07 (November 2012): 1250075. http://dx.doi.org/10.1142/s0218196712500750.

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Given tuples a1, …, ak and b in An for some algebraic structure A, the subpower membership problem asks whether b is in the subalgebra of An that is generated by a1, …, ak. For A a finite group, there is a folklore algorithm which decides this problem in time polynomial in n and k. We show that the subpower membership problem for any finite Mal'cev algebra is in NP and give a polynomial time algorithm for any finite Mal'cev algebra with finite signature and prime power size that has a nilpotent reduct. In particular, this yields a polynomial algorithm for finite rings, vector spaces, algebras over fields, Lie rings and for nilpotent loops of prime power order.
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Regev, Amitai. "Grassmann algebras over finite fields." Communications in Algebra 19, no. 6 (January 1991): 1829–49. http://dx.doi.org/10.1080/00927879108824231.

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PÉREZ, EFRÉN. "ON SEMIGENERIC TAMENESS AND BASE FIELD EXTENSION." Glasgow Mathematical Journal 58, no. 1 (July 21, 2015): 39–53. http://dx.doi.org/10.1017/s0017089515000051.

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AbstractThe notions of central endolength and semigeneric tameness are introduced, and their behaviour under base field extension for finite-dimensional algebras over perfect fields are analysed. Forka perfect field,Kan algebraic closure and Λ a finite-dimensionalk-algebra, here there is a proof that Λ is semigenerically tame if and only if Λ ⊗kKis tame.
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Dissertations / Theses on the topic "Finite fields (Algebra)"

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Rovi, Carmen. "Algebraic Curves over Finite Fields." Thesis, Linköping University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-56761.

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This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of Nq(g) is now known.

At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.

 

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Pizzato, Marco. "Some Problems Concerning Polynomials over Finite Fields, or Algebraic Divertissements." Doctoral thesis, Università degli studi di Trento, 2013. https://hdl.handle.net/11572/367913.

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In this thesis we consider some problems concerning polynomials over finite fields. The first topic is the action of some groups on irreducible polynomials. We describe orbits and stabilizers. Next, we consider transformations of irreducible polynomials by quadratic and cubic maps and study the irreducibility of the polynomials obtained. Finally, starting from PN functions and monomials, we generalize this concept, introducing k-PN monomials and classifying them for small values of k and for fields of order p, p^2 and p^4.
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Prešern, Mateja. "Existence problems of primitive polynomials over finite fields." Connect to e-thesis. Move to record for print version, 2007. http://theses.gla.ac.uk/50/.

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Thesis (Ph.D.) - University of Glasgow, 2007.
Ph.D. thesis submitted to the Department of Mathematics, Faculty of Information and Mathematical Sciences, University of Glasgow, 2007. Includes bibliographical references.
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GOMEZ-CALDERON, JAVIER. "POLYNOMIALS WITH SMALL VALUE SET OVER FINITE FIELDS." Diss., The University of Arizona, 1986. http://hdl.handle.net/10150/183933.

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Let K(q) be the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coefficients in K(q). Let C(f) denote the number of distinct values of f(x) as x ranges over K(q). It is easy to show that C(f) ≤ [|(q - 1)/d|] + 1. Now, there is a characterization of polynomials of degree d < √q for which C(f) = [|(q - 1)/d|] +1. The main object of this work is to give a characterization for polynomials of degree d < ⁴√q for which C(f) < 2q/d. Using two well known theorems: Hurwitz genus formula and Andre Weil's theorem, the Riemann Hypothesis for Algebraic Function Fields, it is shown that if d < ⁴√q and C(f) < 2q/d then f(x) - f(y) factors into at least d/2 absolutely irreducible factors and f(x) has one of the following forms: (UNFORMATTED TABLE FOLLOWS) f(x - λ) = D(d,a)(x) + c, d|(q² - 1), f(x - λ) = D(r,a)(∙ ∙ ∙ ((x²+b₁)²+b₂)²+ ∙ ∙ ∙ +b(m)), d|(q² - 1), d=2ᵐ∙r, and (2,r) = 1 f(x - λ) = (x² + a)ᵈ/² + b, d/2|(q - 1), f(x - λ) = (∙ ∙ ∙((x²+b₁)²+b₂)² + ∙ ∙ ∙ +b(m))ʳ+c, d|(q - 1), d=2ᵐ∙r, f(x - λ) = xᵈ + a, d|(q - 1), f(x - λ) = x(x³ + ax + b) + c, f(x - λ) = x(x³ + ax + b) (x² + a) + e, f(x - λ) = D₃,ₐ(x² + c), c² ≠ 4a, f(x - λ) = (x³ + a)ⁱ + b, i = 1, 2, 3, or 4, f(x - λ) = x³(x³ + a)³ +b, f(x - λ) = x⁴(x⁴ + a)² +b or f(x - λ) = (x⁴ + a) ⁱ + b, i = 1,2 or 3, where D(d,a)(x) denotes the Dickson’s polynomial of degree d. Finally to show other polynomials with small value set, the following equation is obtained C((fᵐ + b)ⁿ) = αq/d + O(√q) where α = (1 – (1 – 1/m)ⁿ)m and the constant implied in O(√q) is independent of q.
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Pizzato, Marco. "Some Problems Concerning Polynomials over Finite Fields, or Algebraic Divertissements." Doctoral thesis, University of Trento, 2013. http://eprints-phd.biblio.unitn.it/1121/1/PizzatoPhDThesisbis.pdf.

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In this thesis we consider some problems concerning polynomials over finite fields. The first topic is the action of some groups on irreducible polynomials. We describe orbits and stabilizers. Next, we consider transformations of irreducible polynomials by quadratic and cubic maps and study the irreducibility of the polynomials obtained. Finally, starting from PN functions and monomials, we generalize this concept, introducing k-PN monomials and classifying them for small values of k and for fields of order p, p^2 and p^4.
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Akleylek, Sedat. "On The Representation Of Finite Fields." Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12612727/index.pdf.

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The representation of field elements has a great impact on the performance of the finite field arithmetic. In this thesis, we give modified version of redundant representation which works for any finite fields of arbitrary characteristics to design arithmetic circuits with small complexity. Using our modified redundant representation, we improve many of the complexity values. We then propose new representations as an alternative way to represent finite fields of characteristic two by using Charlier and Hermite polynomials. We show that multiplication in these representations can be achieved with subquadratic space complexity. Charlier and Hermite representations enable us to find binomial, trinomial or quadranomial irreducible polynomials which allows us faster modular reduction over binary fields when there is no desirable such low weight irreducible polynomial in other representations. These representations are very interesting for the NIST and SEC recommended binary fields GF(2^{283}) and GF(2^{571}) since there is no optimal normal basis (ONB) for the corresponding extensions. It is also shown that in some cases the proposed representations have better space complexity even if there exists an ONB for the corresponding extension.
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Jogia, Danesh Michael Mathematics &amp Statistics Faculty of Science UNSW. "Algebraic aspects of integrability and reversibility in maps." Publisher:University of New South Wales. Mathematics & Statistics, 2008. http://handle.unsw.edu.au/1959.4/40947.

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We study the cause of the signature over finite fields of integrability in two dimensional discrete dynamical systems by using theory from algebraic geometry. In particular the theory of elliptic curves is used to prove the major result of the thesis: that all birational maps that preserve an elliptic curve necessarily act on that elliptic curve as addition under the associated group law. Our result generalises special cases previously given in the literature. We apply this theorem to the specific cases when the ground fields are finite fields of prime order and the function field $mathbb{C}(t)$. In the former case, this yields an explanation of the aforementioned signature over finite fields of integrability. In the latter case we arrive at an analogue of the Arnol'd-Liouville theorem. Other results that are related to this approach to integrability are also proven and their consequences considered in examples. Of particular importance are two separate items: (i) we define a generalization of integrability called mixing and examine its relation to integrability; and (ii) we use the concept of rotation number to study differences and similarities between birational integrable maps that preserve the same foliation. The final chapter is given over to considering the existence of the signature of reversibility in higher (three and four) dimensional maps. A conjecture regarding the distribution of periodic orbits generated by such maps when considered over finite fields is given along with numerical evidence to support the conjecture.
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Park, Hong Goo. "Polynomial Isomorphisms of Cayley Objects Over a Finite Field." Thesis, University of North Texas, 1989. https://digital.library.unt.edu/ark:/67531/metadc331144/.

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In this dissertation the Bays-Lambossy theorem is generalized to GF(pn). The Bays-Lambossy theorem states that if two Cayley objects each based on GF(p) are isomorphic then they are isomorphic by a multiplier map. We use this characterization to show that under certain conditions two isomorphic Cayley objects over GF(pn) must be isomorphic by a function on GF(pn) of a particular type.
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Baktir, Selcuk. "Efficient algorithms for finite fields, with applications in elliptic curve cryptography." Link to electronic thesis, 2003. http://www.wpi.edu/Pubs/ETD/Available/etd-0501103-132249.

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Thesis (M.S.)--Worcester Polytechnic Institute.
Keywords: multiplication; OTF; optimal extension fields; finite fields; optimal tower fields; cryptography; OEF; inversion; finite field arithmetic; elliptic curve cryptography. Includes bibliographical references (p. 50-52).
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Veliz-Cuba, Alan A. "The Algebra of Systems Biology." Diss., Virginia Tech, 2010. http://hdl.handle.net/10919/28240.

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In order to understand biochemical networks we need to know not only how their parts work but also how they interact with each other. The goal of systems biology is to look at biological systems as a whole to understand how interactions of the parts can give rise to complex dynamics. In order to do this efficiently, new techniques have to be developed. This work shows how tools from mathematics are suitable to study problems in systems biology such as modeling, dynamics prediction, reverse engineering and many others. The advantage of using mathematical tools is that there is a large number of theory, algorithms and software available. This work focuses on how algebra can contribute to answer questions arising from systems biology.
Ph. D.
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Books on the topic "Finite fields (Algebra)"

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Lidl, Rudolf, and Harald Niederreiter. Finite Fields. Cambridge: Cambridge University Press, 1996.

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Moreno, Carlos. Algebraic curvesover finite fields. Cambridge: Cambridge University Press, 1991.

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Jacobson, Nathan. Finite-dimensional division algebras over fields. Berlin: Springer, 1996.

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Charles, Small. Arithmetic of finite fields. New York: M. Dekker, 1991.

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Lidl, Rudolf. Finite fields. 2nd ed. Cambridge: Cambridge University Press, 1997.

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Lidl, Rudolf. Introductionto finite fields and their applications. Cambridge: Cambridge University Press, 1986.

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Shparlinski, Igor E. Finite fields: Theory and computation : the meeting point of number theory, computer science, coding theory, and cryptography. Dordrecht: Kluwer Academic Publishers, 1999.

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Lidl, Rudolf. Introduction to finite fields and their applications. Cambridge: Cambridge University Press, 1994.

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International Conference on Finite Fields : Theory, Applications, and Algorithms (2nd 1993 Las Vegas, Nev.). Finite fields: Theory, applications, and algorithms. Edited by Mullen Gary L and Shiue Peter Jau-Shyong 1941-. Providence, R.I: American Mathematical Society, 1994.

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Gockenbach, Mark S. Finite-dimensional linear algebra. Boca Raton: Chapman & Hall/CRC, 2010.

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Book chapters on the topic "Finite fields (Algebra)"

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Lang, Serge. "Finite Fields." In Undergraduate Algebra, 291–307. New York, NY: Springer New York, 1990. http://dx.doi.org/10.1007/978-1-4757-6898-5_8.

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Lang, Serge. "Finite Fields." In Undergraduate Algebra, 184–98. New York, NY: Springer New York, 1987. http://dx.doi.org/10.1007/978-1-4684-9234-7_8.

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Lang, Serge. "Finite Fields." In Undergraduate Algebra, 291–307. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-59275-1_8.

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Hibbard, Allen C., and Kenneth M. Levasseur. "Finite Fields." In Exploring Abstract Algebra With Mathematica®, 227–33. New York, NY: Springer New York, 1999. http://dx.doi.org/10.1007/978-1-4612-1530-1_27.

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Rose, Harvey E. "Interlude on Finite Fields." In Linear Algebra, 109–24. Basel: Birkhäuser Basel, 2002. http://dx.doi.org/10.1007/978-3-0348-8189-0_5.

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Meijer, Alko R. "Properties of Finite Fields." In Algebra for Cryptologists, 105–22. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30396-3_6.

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Gårding, Lars, and Torbjörn Tambour. "Polynomial rings, algebraic fields, finite fields." In Algebra for Computer Science, 107–25. New York, NY: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4613-8797-8_7.

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Childs, Lindsay N. "Classifying Finite Fields." In A Concrete Introduction to Higher Algebra, 464–82. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4419-8702-0_30.

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Mignotte, Maurice. "Polynomials Over Finite Fields." In Mathematics for Computer Algebra, 229–88. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4613-9171-5_6.

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Caruso, Fabrizio, Jacopo D’Aurizio, and Alasdair McAndrew. "Efficient Finite Fields in the Maxima Computer Algebra System." In Arithmetic of Finite Fields, 62–76. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-69499-1_6.

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Conference papers on the topic "Finite fields (Algebra)"

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Coquereaux, R., and G. E. Schieber. "Action of finite quantum group on the algebra of complex N×N matrices." In Particles, fields and gravitation. AIP, 1998. http://dx.doi.org/10.1063/1.57119.

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Nerode, Anil, and J. B. Remmel. "Complexity Theoretic Algebra I: Vector Spaces over Finite Fields." In Proceeding Structure in Complexity Theory. IEEE, 1987. http://dx.doi.org/10.1109/psct.1987.10319273.

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Muchtadi-Alamsyah, I., F. Yuliawan, and A. Muchlis. "Finite Field Basis Conversion and Normal Basis in Characteristic Three." In The International Conference on Algebra 2010 - Advances in Algebraic Structures. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814366311_0034.

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Dumas, Jean Guillaume, Thierry Gautier, and Clément Pernet. "Finite field linear algebra subroutines." In the 2002 international symposium. New York, New York, USA: ACM Press, 2002. http://dx.doi.org/10.1145/780506.780515.

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Elsayed, Mostafa S. A., and Damiano Pasini. "Characterization and Performance Optimization of 2D Lattice Materials With Hexagonal Bravais Lattice Symmetry." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-87540.

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The current paper examines the static performance of 2D infinite lattice materials with hexagonal Bravais lattice symmetry. Two novel microscopic cell topologies are proposed. The first topology is a semi-regular lattice that has the modified Schafli symbol 34.6, which describes the type of regular polygons surrounding the joints of the lattice. Here, 34.6 indicates four (4) regular triangles (3) successively surrounding a node followed by a regular hexagon (6). The second topology is an irregular lattice that is referred here as Double Hexagonal Triangulation (DHT). The lattice material is considered as a pin-jointed micro-truss where determinacy analysis of the material micro structure is used to distinguish between bending dominated and stretching dominated behaviours. The finite structural performance of unit cells of the proposed topologies is assessed by the matrix methods of linear algebra. The Dummy Node Hypothesis is developed to generalize the analysis to tackle any lattice topology. Collapse mechanisms and states of self-stress are deduced from the four fundamental subspaces of the kinematic and the equilibrium matrices of the finite unit cell structures, respectively. The generated finite structural matrices are employed to analyze the infinite structural performance of the lattice using the Bloch’s theorem. To find macroscopic strain fields generated by periodic mechanisms, the Cauchy-Born hypothesis is adopted. An explicit expression of the microscopic cell element deformations in terms of the macroscopic strain field is generated which is employed to derive the strain energy density of the lattice material. Finally, the strain energy density is used to derive the material macroscopic stiffness properties. The results showed that the proposed lattice topologies can support all macroscopic strain fields. Their stiffness properties are compared with those of lattice materials with hexagonal Bravais symmetry available in literature. The comparison showed that the lattice material with 34.6 cell topology has superior isotropic stiffness properties. When compared with the Kagome’ lattice, the 34.6 lattice generates isotropic stiffness properties, with additional stiffness to mass ratio of 18.5% and 93.2% in the direct and the coupled direct stiffness, respectively. However, it generates reduced shear stiffness to mass ratio by 18.8%.
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Zuckerman, Neil, and Jin Fang. "Two Carrier Heat Transfer Modeling for Heat Assisted Magnetic Recording." In ASME 2013 Heat Transfer Summer Conference collocated with the ASME 2013 7th International Conference on Energy Sustainability and the ASME 2013 11th International Conference on Fuel Cell Science, Engineering and Technology. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/ht2013-17235.

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In this work we show the structure and application of a two carrier thermal model applied to a near field transducer, representative of that used in Heat Assisted Magnetic Recording (HAMR). As part of the HAMR device operation, high energy non-thermalized electrons are initially excited by laser incidence on a gold nanostructure. The high energy electrons can travel in a ballistic fashion over longer distances than the optical thickness of gold, resulting in a spreading of the local heat. During their travel the hot electrons collide with lower-energy electrons, thermalizing the hot electrons via inelastic scattering. The thermalized electrons then transfer energy to the lattice due to electron-phonon coupling, as captured in the two carrier model. Starting with an electromagnetic solution for local heating in a sub-micron-scale microfabricated gold structure, the chosen modeling technique applies physical effects of unique interest at the nanometer scale, including brief ballistic transport of hot electrons, experimentally-verified interface thermal resistance, and electron-phonon temperature mismatch. By design, the model is built to use far-field boundary conditions from conventional one-carrier FEMs as well as lubrication-flow computational fluid dynamics. The fundamental governing equations of the two carrier model are two versions of Poisson’s Equation for heat diffusion, coupled by empirically determined terms. These equations are combined with equations for interfacial discontinuities in the temperature fields, yielding a third degree of freedom. The continuous fields are discretized using the finite difference method, and solved using algorithms developed for linear algebra, such as Gaussian Elimination, or non-direct iterative methods. Through use of the model we explore effects of ballistic electron transport length, electron-phonon coupling, as well as interfacial thermal resistance between gold and neighboring ceramics. The model results show the relative impact of the nanoscale heat transfer phenomena in a nanometer scale metal-ceramic structure, allowing us to identify the relative importance of design features and compare candidate designs.
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7

Voight, John. "Curves over finite fields with many points: an introduction." In Computational Aspects of Algebraic Curves. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812701640_0010.

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8

De Feo, Luca, Hugues Randriam, and Édouard Rousseau. "Standard Lattices of Compatibly Embedded Finite Fields." In ISSAC '19: International Symposium on Symbolic and Algebraic Computation. New York, NY, USA: ACM, 2019. http://dx.doi.org/10.1145/3326229.3326251.

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9

Schost, Eric. "Algorithms for Finite Field Arithmetic." In ISSAC'15: International Symposium on Symbolic and Algebraic Computation. New York, NY, USA: ACM, 2015. http://dx.doi.org/10.1145/2755996.2756637.

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Villard, Gilles. "Elimination ideal and bivariate resultant over finite fields." In ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023. New York, NY, USA: ACM, 2023. http://dx.doi.org/10.1145/3597066.3597100.

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