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Journal articles on the topic 'Algebraic schemes'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Pan, Yang. "Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes." Forum Mathematicum 30, no. 2 (March 1, 2018): 479–95. http://dx.doi.org/10.1515/forum-2017-0007.

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AbstractWe investigate the saturation rank of a finite group scheme defined over an algebraically closed field{\Bbbk}of positive characteristicp. We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group{\operatorname{SL}_{n}}.

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12

Blundo, Carlo, Sebastià Martín, Barbara Masucci, and CarlEs Padró. "A Linear Algebraic Approach to Metering Schemes." Designs, Codes and Cryptography 33, no. 3 (November 2004): 241–60. http://dx.doi.org/10.1023/b:desi.0000036249.86262.d5.

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13

Ascher, Uri. "On Symmetric Schemes and Differential-Algebraic Equations." SIAM Journal on Scientific and Statistical Computing 10, no. 5 (September 1989): 937–49. http://dx.doi.org/10.1137/0910054.

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14

Banerjee, Abhishek. "Noetherian schemes over abelian symmetric monoidal categories." International Journal of Mathematics 28, no. 07 (May 23, 2017): 1750051. http://dx.doi.org/10.1142/s0129167x17500513.

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In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let [Formula: see text] be a commutative monoid object in an abelian symmetric monoidal category [Formula: see text] satisfying certain conditions and let [Formula: see text]. If the subobjects of [Formula: see text] satisfy a certain compactness property, we say that [Formula: see text] is Noetherian. We study the localization of [Formula: see text] with respect to any [Formula: see text] and define the quotient [Formula: see text] of [Formula: see text] with respect to any ideal [Formula: see text]. We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc.) for schemes over [Formula: see text]. Our notion of a scheme over a symmetric monoidal category [Formula: see text] is that of Toën and Vaquié.

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15

Gross, Philipp. "The resolution property of algebraic surfaces." Compositio Mathematica 148, no. 1 (November 9, 2011): 209–26. http://dx.doi.org/10.1112/s0010437x11005628.

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AbstractWe prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective nor embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over an excellent ring.

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16

Ying, Yu. "The symbolic problems associated with Runge-Kutta methods and their solving in Sage." Discrete and Continuous Models and Applied Computational Science 27, no. 1 (December 15, 2019): 33–41. http://dx.doi.org/10.22363/2658-4670-2019-27-1-33-41.

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Runge-Kutta schemes play a very important role in solving ordinary differential equations numerically. At first we want to present the Sage routine for calculation of Butcher matrix, we call it an rk package. We tested our Sage routine in several numerical experiments with standard and symplectic schemes and verified our result by corporation with results of the calculations made by hand.Second, in Sage there are the excellent tools for investigation of algebraic sets, based on Gröbner basis technique. As we all known, the choice of parameters in Runge- Kutta scheme is free. By the help of these tools we study the algebraic properties of the manifolds in affine space, coordinates of whose are Butcher coefficients in Runge-Kutta scheme. Results are given both for explicit Runge-Kutta scheme and implicit Runge-Kutta scheme by using our rk package. Examples are carried out to justify our results. All calculation are executed in the computer algebra system Sage.

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17

Zhang, Xiaojing, Vladimir Gerdt, and Yury Blinkov. "Algebraic Construction of a Strongly Consistent, Permutationally Symmetric and Conservative Difference Scheme for 3D Steady Stokes Flow." Symmetry 11, no. 2 (February 20, 2019): 269. http://dx.doi.org/10.3390/sym11020269.

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By using symbolic algebraic computation, we construct a strongly-consistent second-order finite difference scheme for steady three-dimensional Stokes flow and a Cartesian solution grid. The scheme has the second order of accuracy and incorporates the pressure Poisson equation. This equation is the integrability condition for the discrete momentum and continuity equations. Our algebraic approach to the construction of difference schemes suggested by the second and the third authors combines the finite volume method, numerical integration, and difference elimination. We make use of the techniques of the differential and difference Janet/Gröbner bases for performing related computations. To prove the strong consistency of the generated scheme, we use these bases to correlate the differential ideal generated by the polynomials in the Stokes equations with the difference ideal generated by the polynomials in the constructed difference scheme. As this takes place, our difference scheme is conservative and inherits permutation symmetry of the differential Stokes flow. For the obtained scheme, we compute the modified differential system and use it to analyze the scheme’s accuracy.

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18

JOSHUA, ROY. "Algebraic K-theory and higher Chow groups of linear varieties." Mathematical Proceedings of the Cambridge Philosophical Society 130, no. 1 (January 2001): 37–60. http://dx.doi.org/10.1017/s030500410000476x.

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The main focus in this paper is the algebraic K-theory and higher Chow groups of linear varieties and schemes. We provide Kunneth spectral sequences for the higher algebraic K-theory of linear schemes flat over a base scheme and for the motivic cohomology of linear varieties defined over a field. The latter provides a Kunneth formula for the usual Chow groups of linear varieties originally obtained by different means by Totaro. We also obtain a general condition under which the higher cycle maps of Bloch from mod-lv higher Chow groups to mod-lv étale cohomology are isomorphisms for projective nonsingular varieties defined over an algebraically closed field of arbitrary characteristic p [ges ] 0 with l ≠ p. It is observed that the Kunneth formula for the Chow groups implies this condition for linear varieties and we compute the mod-lv motivic cohomology and mod-lv algebraic K-theory of projective nonsingular linear varieties to be free ℤ/lv-modules.

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19

Malykh, Mikhail, and Leonid Sevastianov. "Finite Difference Schemes as Algebraic Correspondences between Layers." EPJ Web of Conferences 173 (2018): 03016. http://dx.doi.org/10.1051/epjconf/201817303016.

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For some differential equations, especially for Riccati equation, new finite difference schemes are suggested. These schemes define protective correspondences between the layers. Calculation using these schemes can be extended to the area beyond movable singularities of exact solution without any error accumulation.

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20

Shen, Y. F., D. H. Yuan, and S. Z. Yang. "Polynomial Reproduction of Vector Subdivision Schemes." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/104840.

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We discuss the polynomial reproduction of vector subdivision schemes with general integer dilationm≥2. We first present a simple algebraic condition for polynomial reproduction of such schemes with standard subdivision symbol. We then extend it to general subdivision symbol satisfying certain order of sum rules. We also illustrate our results with several examples. Our results show that such kind of scheme can produce exactly the same scalar polynomial from which the data is sampled by convolving with a finite nonzero sequence of vectors.

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21

Cortiñas, Guillermo, Christian Haesemeyer, Mark E. Walker, and Charles Weibel. "Toric varieties, monoid schemes and cdh descent." Journal für die reine und angewandte Mathematik (Crelles Journal) 2015, no. 698 (January 1, 2015): 1–54. http://dx.doi.org/10.1515/crelle-2012-0123.

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22

Toën, Bertrand, and Michel Vaquié. "Au-dessous de Specℤ." Journal of K-theory 3, no. 3 (September 4, 2008): 437–500. http://dx.doi.org/10.1017/is008004027jkt048.

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AbstractIn this article we use the theories of relative algebraic geometry and of homotopical algebraic geometry (cf. [HAGII]) to construct some categories of schemes defined under Specℤ. We define the categories of ℕ-schemes, 1-schemes, -schemes, +-schemes and 1-schemes, where (from an intuitive point of view) ℕ is the semi-ring of natural numbers, 1 is the field with one element, is the ring spectra of integers, + is the semi-ring spectra of natural numbers and 1 is the ring spectra with one element. These categories of schemes are linked together by base change functors, and all of them have a base change functor to the category of ℤ-schemes. We show that the linear group Gln and the toric varieties can be defined as objects in these categories.

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23

Kreuzer, Martin, Tran N. K. Linh, and Le Ngoc Long. "The Dedekind different of a Cayley–Bacharach scheme." Journal of Algebra and Its Applications 18, no. 02 (February 2019): 1950027. http://dx.doi.org/10.1142/s0219498819500270.

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Given a 0-dimensional scheme [Formula: see text] in a projective space [Formula: see text] over a field [Formula: see text], we characterize the Cayley–Bacharach property (CBP) of [Formula: see text] in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley–Bacharach schemes by Dedekind’s formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes.

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24

Lu, Jianguang, Yong Feng, Xiaolin Qin, and Juan Tang. "Group Preserving Correction Methods for Differential Algebraic Equations." Journal of Computational and Theoretical Nanoscience 13, no. 10 (October 1, 2016): 7719–25. http://dx.doi.org/10.1166/jctn.2016.4508.

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The group preserving methods proposed by Liu [Int. J. Non-Linear Mech., 2001 and CMES-Comp. Model. Eng., 2006] for ordinary differential equations or differential algebraic equations (DAEs) adopted the Cayley transform or exponential mapping to formulate the Lie group from its Lie algebra. In this paper, we combine the Euler scheme with the group preserving methods to obtain the high accuracy group preserving techniques. We propose a group preserving correction scheme (GPCS) via exponential mapping and a modified group preserving correction scheme (MGPCS) by considering constraint. The two schemes provide single-step explicit time integrators for systems of DAEs. Some numerical examples are examined, showing that the GPCS and MGPCS work very well and have good computational efficiency and high accuracy.

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25

Podlovchenko, R. I., and A. E. Molchanov. "About Algebraic Program Models with Procedures." Modeling and Analysis of Information Systems 19, no. 5 (March 4, 2015): 100–114. http://dx.doi.org/10.18255/1818-1015-2012-5-100-114.

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Algebraic program models with procedures are designed to analyze program semantic properties on their models called program schemes. The concepts that give foundation to the theory of such models are stated along with a description of their implementation. The key point of the theory is the equivalence of program schemes that belong to a particular model. A class of special algebraic models with procedures, called gateway models, is studied. Necessary and sufficient conditions of the equivalence problem decidability in such models are proposed.

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26

Iwanari, Isamu. "Tannakization in derived algebraic geometry." Journal of K-theory 14, no. 3 (December 2014): 642–700. http://dx.doi.org/10.1017/is014008019jkt278.

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AbstractIn this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

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27

Calkavur, Selda. "Some algebraic (n,n)-secret image sharing schemes." Applied Mathematical Sciences 11 (2017): 2807–15. http://dx.doi.org/10.12988/ams.2017.710309.

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28

Ghorbel, M., and T. Huillet. "On two fragmentation schemes with algebraic splitting probability." Applicationes Mathematicae 33, no. 1 (2006): 95–110. http://dx.doi.org/10.4064/am33-1-8.

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29

Karcanias, Nicos, and John Leventides. "DESIGN OF DECENTRALISED CONTROL SCHEMES : AN ALGEBRAIC APPROACH." IFAC Proceedings Volumes 38, no. 1 (2005): 922–27. http://dx.doi.org/10.3182/20050703-6-cz-1902.00554.

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30

Takeda, Yuichiro. "Lefschetz-Riemann-Roch theorem for smooth algebraic schemes." Mathematische Zeitschrift 211, no. 1 (December 1992): 643–56. http://dx.doi.org/10.1007/bf02571452.

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31

Dai, Shouxin. "Algebraic cobordism and Grothendieck groups over singular schemes." Homology, Homotopy and Applications 12, no. 1 (2010): 93–110. http://dx.doi.org/10.4310/hha.2010.v12.n1.a8.

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32

Bulatov, Mikhail V., and Liubov S. Solovarova. "Collocation-variation difference schemes for differential-algebraic equations." Mathematical Methods in the Applied Sciences 41, no. 18 (April 20, 2018): 9048–56. http://dx.doi.org/10.1002/mma.4884.

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33

Hanaki, Akihide, and Katsuhiro Uno. "Algebraic structure of association schemes of prime order." Journal of Algebraic Combinatorics 23, no. 2 (March 2006): 189–95. http://dx.doi.org/10.1007/s10801-006-6923-7.

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34

Ying, Yu, and Mikhail D. Malykh. "On conjugate difference schemes: the midpoint scheme and the trapezoidal scheme." Discrete and Continuous Models and Applied Computational Science 29, no. 1 (December 15, 2021): 63–72. http://dx.doi.org/10.22363/2658-4670-2021-29-1-63-72.

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The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations x=f(x), found by the trapezoidal scheme, is investigated. For this purpose, a relation has been established between the trapezoidal scheme and the midpoint scheme, which preserves all quadratic integrals of motion by virtue of Coopers theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Coopers theorem by analogy with the duality principle in projective geometry. It is proved that on the approximate solution found by the trapezoidal scheme, not the quadratic integral itself is preserved, but a more complicated expression, which turns into an integral in the limit as t0.Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. In both cases, expressions preserved by the trapezoidal scheme are presented explicitly.

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35

Nesmelova, Olga. "Nonlinear boundary value problems for nondegenerate differential-algebraic systems." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 32 (December 28, 2018): 78–91. http://dx.doi.org/10.37069/1683-4720-2018-32-9.

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The article proposes original solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem. And we use the matrix pseudo-inversion technique of Moore-Penrose. The posed problem in the article continues the study of conditions of solvability and schemes for finding solutions of the nonlinear Noetherian boundary-value problems given in the monographs by A. Poincare, A.M. Lyapunov, I.G. Malkin, J. Hale, Yu.A. Ryabov, A.M. Samoylenko, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina and A.A. Boychuk. We studied a general case, when a linear bounded operator corresponding to the homogeneous part of the linear Noetherian differential-algebraic boundary value problem has no inverse. Sufficient conditions for reducibility of the differential algebraic equation to the system uniting a differential and algebraic equation are found. Thus, the differential-algebraic boundary value problem is reduced to the nonlinear Noetherian boundary value problem for the system of ordinary differential equations. We studied the case of the presence of simple roots of the equation for generating amplitudes. Constructive necessary and sufficient conditions of existence were obtained to find solutions to the problem in the critical case, and the converging iterative scheme was constructed. The proposed solvability conditions, and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem are illustrated in detail by the example from the nonlinear Noetherian differential-algebraic boundary value problem for Duffing type equations. For control of the rate of the iterative scheme convergence to the exact solution of the differential-algebraic boundary value problem for the Duffing type equation, we used the residuals of the obtained approximations in the Duffing type equation in the space of continuous functions.

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36

Gilmer, Patrick M., and Stepan Yu Orevkov. "Signatures of real algebraic curves via plumbing diagrams." Journal of Knot Theory and Its Ramifications 27, no. 03 (March 2018): 1840003. http://dx.doi.org/10.1142/s0218216518400035.

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We define and calculate signature and nullity invariants for complex schemes for curves in [Formula: see text]. We use an analog of the Murasugi–Tristram inequality to prohibit certain schemes from being realized by real algebraic curves. We give new formulas for Casson–Gordon invariants of graph manifolds, and signatures of graph links.

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37

Berman, J., G. Grätzer, and C. R. Platt. "Extending Algebras to Model Congruence Schemes." Canadian Journal of Mathematics 38, no. 2 (April 1, 1986): 257–76. http://dx.doi.org/10.4153/cjm-1986-012-8.

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This paper is concerned with the description of principal congruence relations. Given elements a and b of a universal algebra , let θ(a, b) denote the smallest congruence relation on containing the pair 〈a, b〉. One of the earliest characterizations of θ(a, b) is Mal'cev's well-known result [5, Theorem 1.10.3], which says that c ≡ d(θ(a, b)) if and only if there exists a sequence z0, z1, …, zn of elements of and a sequence f1, f2, …, fn of unary algebraic functions such that c = z0, d = zn, and for each i = 1, …, n,Although this describes θ(a, b) in terms of a set of unary algebraic functions, it is not possible to predict the number or complexity of the unary functions used independently of the choice of a, b, c and d. Several recent papers ([1], [2], [3], [4], [6]) investigate classes of algebras in which principal congruences are simpler.

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38

Conrad, Brian, Max Lieblich, and Martin Olsson. "Nagata compactification for algebraic spaces." Journal of the Institute of Mathematics of Jussieu 11, no. 4 (July 13, 2012): 747–814. http://dx.doi.org/10.1017/s1474748011000223.

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AbstractWe prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.

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39

Wheeler, Mary L. "Check-Digit Schemes." Mathematics Teacher 87, no. 4 (April 1994): 228–30. http://dx.doi.org/10.5951/mt.87.4.0228.

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Identification codes have been used for many years. We've seen a large increase in the use of scanners and computers in the last two decades. As a result, the need for identification codes increased tremendously. These codes are an assignment of numbers to an item for identification of information about that item. They are very useful with computers. information can be stored in a condensed format, and with the use of algebraic algorithms the accuracy of that information can be checked by using what is called a check-digit scheme. Identification codes are also very useful for secwity information. The use of a simple or complicated mathematical algorithm allows the information to be disguised.

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40

Shentu, Junchao, and Dong Wang. "Notes on algebraic log stack." International Journal of Mathematics 27, no. 10 (September 2016): 1650081. http://dx.doi.org/10.1142/s0129167x16500816.

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Let [Formula: see text] be a stack over the category of fine log schemes. If [Formula: see text] has a representable fppf covering, then, it has enough compatible minimal objects. As a consequence, we prove the equivalence between two notions of log moduli stacks which appear in literatures. Also, we obtain several fundamental results of algebraic log stacks.

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41

Gerdt, Vladimir P., Mikhail D. Malykh, Leonid A. Sevastianov, and Yu Ying. "On the properties of numerical solutions of dynamical systems obtained using the midpoint method." Discrete and Continuous Models and Applied Computational Science 27, no. 3 (December 15, 2019): 242–62. http://dx.doi.org/10.22363/2658-4670-2019-27-3-242-262.

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The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

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42

Bannenberg, M. W. F. M., A. Ciccazzo, and M. Günther. "Reduced order multirate schemes for coupled differential-algebraic systems." Applied Numerical Mathematics 168 (October 2021): 104–14. http://dx.doi.org/10.1016/j.apnum.2021.05.023.

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Giraud, X., E. Boutillon, and J. C. Belfiore. "Algebraic tools to build modulation schemes for fading channels." IEEE Transactions on Information Theory 43, no. 3 (May 1997): 938–52. http://dx.doi.org/10.1109/18.568703.

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Källström, Rolf. "Liftable derivations for generically separably algebraic morphisms of schemes." Transactions of the American Mathematical Society 361, no. 01 (June 26, 2008): 495–523. http://dx.doi.org/10.1090/s0002-9947-08-04534-0.

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Hashimoto, Mitsuyasu. "“Geometric quotients are algebraic schemes” based on Fogarty’s idea." Journal of Mathematics of Kyoto University 43, no. 4 (2003): 807–14. http://dx.doi.org/10.1215/kjm/1250281736.

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Manetti, Marco, and Francesco Meazzini. "Deformations of algebraic schemes via Reedy–Palamodov cofibrant resolutions." Indagationes Mathematicae 31, no. 1 (January 2020): 7–32. http://dx.doi.org/10.1016/j.indag.2019.08.007.

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Chu, Peter C., and Chenwu Fan. "Accuracy Progressive Calculation of Lagrangian Trajectories from a Gridded Velocity Field." Journal of Atmospheric and Oceanic Technology 31, no. 7 (July 1, 2014): 1615–27. http://dx.doi.org/10.1175/jtech-d-13-00204.1.

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Abstract Reduction of computational error is a key issue in computing Lagrangian trajectories using gridded velocities. Computational accuracy enhances from using the first term (constant velocity scheme), the first two terms (linear uncoupled scheme), the first three terms (linear coupled scheme), to using all four terms (nonlinear coupled scheme) of the two-dimensional interpolation. A unified “analytical form” is presented in this study for different truncations. Ordinary differential equations for predicting Lagrangian trajectory are linear using the constant velocity/linear uncoupled schemes (both commonly used in atmospheric and ocean modeling), the linear coupled scheme, and the nonlinear using the nonlinear coupled scheme (both proposed in this paper). The location of the Lagrangian drifter inside the grid cell is determined by two algebraic equations that are solved explicitly with the constant velocity/linear uncoupled schemes, and implicitly using the Newton–Raphson iteration method with the linear/nonlinear coupled schemes. The analytical Stommel ocean model on the f plane is used to illustrate great accuracy improvement from keeping the first term to keeping all the terms of the two-dimensional interpolation.

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Cai, Hongliang, and Dan Tang. "Multi Secret Image Sharing Scheme of General Access Structure with Meaningful Shares." Mathematics 8, no. 9 (September 14, 2020): 1582. http://dx.doi.org/10.3390/math8091582.

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A Multi Secret Image sharing scheme can share several secret images among certain participators securely. Boolean-based secret sharing schemes are one kind of secret sharing method with light-weighted computation compared to the previous complex algebraic-based methods, which can realize the sharing of multi secret images. However, the existing Boolean-based multi secret sharing schemes are mostly restricted to the particular case of (2, n) and (n, n), only few Boolean-based multi secret sharing schemes study the general access structure, and the shares are mostly meaningless. In this paper, a new Boolean-based multi secret sharing scheme with the general access structure is proposed. All the shares are meaningful, which can avoid attracting the attention of adversaries, and the secret images can be recovered in a lossless manner. The feasibility of the scheme is proven, the performance is validated by the experiments on the gray images, and the analysis of the comparison with other methods is also given out.

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Guryanov, D. Yu, D. N. Moldovyan, and A. A. Moldovyan. "Post-quantum digital signature schemes: setting a hidden group with two-dimensional cyclicity." Informatization and communication 4 (November 2020): 75–82. http://dx.doi.org/10.34219/2078-8320-2020-11-4-75-82.

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For the construction of post-quantum digital signature schemes that satisfy the strengthened criterion of resistance to quantum attacks, an algebraic carrier is proposed that allows one to define a hidden commutative group with two-dimensional cyclicity. Formulas are obtained that describe the set of elements that are permutable with a given fixed element. A post-quantum signature scheme based on the considered finite non-commutative associative algebra is described.

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Kandiy, S. O., and G. A. Maleeva. "Analysis of the complexity of attacks on multivariate cryptographic transformations using algebraic field structure." Radiotekhnika, no. 204 (April 9, 2021): 59–65. http://dx.doi.org/10.30837/rt.2021.1.204.06.

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In recent years, interest in cryptosystems based on multidimensional quadratic transformations (MQ transformations) has grown significantly. This is primarily due to the NIST PQC competition [1] and the need for practical electronic signature schemes that are resistant to attacks on quantum computers. Despite the fact that the world community has done a lot of work on cryptanalysis of the presented schemes, many issues need further clarification. NIST specialists are very cautious about the standardization process and urge cryptologists [4] in the next 3 years to conduct a comprehensive analysis of the finalists of the NIST PQC competition before their standardization. One of the finalists is the Rainbow electronic signature scheme [2]. It is a generalization of the UOV (Unbalanced Oil and Vinegar) scheme [3]. Recently, another generalization of this scheme – LUOV (Lifted UOV) [5] was found to attack [6], which in polynomial time is able to recover completely the private key. The peculiarity of this attack is the use of the algebraic structure of the field over which the MQ transformation is given. This line of attack has emerged recently and it is still unclear whether it is possible to use the field structure in the Rainbow scheme. The aim of this work is to systematize the techniques used in attacks using the algebraic field structure for UOV-based cryptosystems and to analyze the obstacles for their generalization to the Rainbow scheme.

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