Journal articles: 'Algorithm algebra'

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Bancerek, Grzegorz. "Analysis of Algorithms: An Example of a Sort Algorithm." Formalized Mathematics 21, no. 1 (January 1, 2013): 1–23. http://dx.doi.org/10.2478/forma-2013-0001.

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INAMI, T., Y. MATSUO, and I. YAMANAKA. "EXTENDED CONFORMAL ALGEBRA WITH N=2 SUPERSYMMETRY." International Journal of Modern Physics A 05, no. 23 (December 10, 1990): 4441–67. http://dx.doi.org/10.1142/s0217751x90001860.

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We construct an N=2 supersymmetric extension of Zamolodchikov’s W algebra. The generators of this extended conformal algebra consist of the stress-tensor superfield and a pair of chiral currents [Formula: see text] of integer or half-integer spin Δ and opposite U(I) charges ±τ. The algorithm for deriving the operator product algebra [Formula: see text] is given for general Δ and the algebra is worked out explicitly for Δ=3/2. The N=2 super-W algebra has an interesting feature not shared by other conformal algebras, i.e. it has two types of algebra—short and long algebras.

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Vasyluk, Andrii, and Taras Basyuk. "Synthesis System Оf Algebra Algorithms Formulas." Vìsnik Nacìonalʹnogo unìversitetu "Lʹvìvsʹka polìtehnìka". Serìâ Ìnformacìjnì sistemi ta merežì 9 (June 10, 2021): 11–22. http://dx.doi.org/10.23939/sisn2021.09.011.

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In the article the authors have developed a mathematical support for the process of generating subject unitherms of formulas of algebra of algorithms. The analysis of features of construction of formulas of algebra of algorithms as a result of which it was found out, that today, subsystems with realization of processes of generation of subject unitherms on the basis of abstract unitherms with the subsequent adaptation of formulas are not realized in known systems that served as stimulus to intellectual analysis formulas of algebra of algorithms. It is described that the synthesis of algebra formulas of algorithms, and especially the generation of subject unitherms on the basis of abstract ones is an extremely complex and laborious process. Since all elements of the formula are interconnected, all changes in the algorithm’s formula affect its structure. Therefore, this is the main reason for the complexity of the described processes. One aspect of the synthesis of the formulas of the algebra of algorithms is the process of generating subject unitherms based on abstract unitherms. The signs of operations of the algebra of algorithms are briefly described. Mathematical support of the process of synthesis of algorithm algebra formulas is developed, which takes into account vertical and horizontal orientation and type of algorithm algebra formula: text unitherm, sequencing operation, elimination operation, parallel operation and corresponding cyclic sequencing operations, elimination and parallelization, as well as geometric parameters. The process of generating subject unitherms on the basis of abstract ones is previously described. The list of necessary eliminations and sequences for the synthesis of the corresponding formulas is determined. According to the properties of the signs of operations of the algebra of algorithms, the synthesized formulas of the algorithms are minimized by the number of unitherms. Also, in accordance with the properties of the formulas of the algorithms of algebra, the corresponding unitherms are taken out as signs of operations, as a result of which the formula of the algorithm for the synthesis of algorithm formulas is obtained taking into account the generation of subject unitherms based on abstract unitherms.

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Mei, Xu Shi. "Research on Vector Algebra Algorithm of Network Coding." Applied Mechanics and Materials 416-417 (September 2013): 1614–18. http://dx.doi.org/10.4028/www.scientific.net/amm.416-417.1614.

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The vector algebra algorithm has been conducted in - depth study, through the vector algorithm to adjust the conversion of vector data and raster data, it can finally realize the quantization scheme of vector algebra algorithms that are applied to network coding, vector algebra algorithm will be conducive to the network and retrieval analysis, the graphical display also has good quality and high precision. At the same time, data structure is simple and easy to spatial analysis and surface modeling.

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FLAUT, CRISTINA, and DIANA SAVIN. "Some examples of division symbol algebras of degree 3 and 5." Carpathian Journal of Mathematics 31, no. 2 (2015): 197–204. http://dx.doi.org/10.37193/cjm.2015.02.07.

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In this paper we provide an algorithm to compute the product between two elements in a symbol algebra of degree n and we find an octonion algebra (in general, without division) in a symbol algebra of degree three. Moreover, using MAGMA software, we will provide some examples of division symbol algebras of degree 3 and of degree 5.

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Casas, J. M., M. Ladra, B. A. Omirov, and U. A. Rozikov. "On Evolution Algebras." Algebra Colloquium 21, no. 02 (April 11, 2014): 331–42. http://dx.doi.org/10.1142/s1005386714000285.

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The structural constants of an evolution algebra are given by a quadratic matrix. In this work we establish an equivalence between nil, right nilpotent evolution algebras and evolution algebras defined by upper triangular matrices. The classification of 2-dimensional complex evolution algebras is obtained. For an evolution algebra with a special form of the matrix, we describe all its isomorphisms and their compositions. We construct an algorithm running under Mathematica which decides if two finite dimensional evolution algebras are isomorphic.

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RIZELL, GEORGIOS DIMITROGLOU. "Nontriviality results for the characteristic algebra of a DGA." Mathematical Proceedings of the Cambridge Philosophical Society 162, no. 3 (July 28, 2016): 419–33. http://dx.doi.org/10.1017/s0305004116000645.

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AbstractAssume that we are given a semifree noncommutative differential graded algebra (DGA for short) whose differential respects an action filtration. We show that the canonical unital algebra map from the homology of the DGA to its characteristic algebra, i.e. the quotient of the underlying algebra by the two-sided ideal generated by the boundaries, is a monomorphism. The main tool that we use is the weak division algorithm in free noncommutative algebras due to P. Cohn.

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NURAKUNOV, ANVAR M., and MICHAŁ M. STRONKOWSKI. "PROFINITENESS IN FINITELY GENERATED VARIETIES IS UNDECIDABLE." Journal of Symbolic Logic 83, no. 04 (December 2018): 1566–78. http://dx.doi.org/10.1017/jsl.2017.89.

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AbstractProfinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety ${\cal V}$ is standard if every Boolean topological algebra with the algebraic reduct in ${\cal V}$ is profinite.We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety $V\left( {\bf{A}} \right)$ generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson.We accomplish this by combining two results. The first one is Moore’s theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether $V\left( {\bf{A}} \right)$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.

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Chen, Yuqun. "Gröbner–Shirshov Bases for Extensions of Algebras." Algebra Colloquium 16, no. 02 (June 2009): 283–92. http://dx.doi.org/10.1142/s1005386709000285.

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An algebra [Formula: see text] is called an extension of the algebra M by B if M2 = 0, M is an ideal of [Formula: see text] and [Formula: see text] as algebras. In this paper, by using Gröbner–Shirshov bases, we characterize completely the extensions of M by B. An algorithm to find the conditions of an algebra A to be an extension of M by B is obtained.

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Vejdemo-Johansson, Mikael. "Blackbox computation of A ∞-algebras." gmj 17, no. 2 (June 2010): 391–404. http://dx.doi.org/10.1515/gmj.2010.005.

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Abstract Kadeishvili's proof of theminimality theorem [T. Kadeishvili, On the homology theory of fiber spaces, Russ. Math. Surv. 35:3 (1980), 231–238] induces an algorithm for the inductive computation of an A ∞-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete A ∞-algebra structure after a finite amount of computational work.

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Zhilina, Svetlana. "Orthogonality graphs of real Cayley–Dickson algebras. Part II: The subgraph on pairs of basis elements." International Journal of Algebra and Computation 31, no. 04 (May 12, 2021): 691–725. http://dx.doi.org/10.1142/s0218196721500338.

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We consider zero divisors of an arbitrary real Cayley–Dickson algebra such that their components are both standard basis elements. We construct inductively the orthogonality graph on these elements. Then we show that, if we restrict our attention to at least [Formula: see text]-dimensional algebras, two algebras are isomorphic if and only if their graphs are isomorphic. We also provide an algorithm to retrieve the Cayley–Dickson parameters of an algebra from its graph.

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Feng, Youyang, Qing Wang, and Hao Zhang. "Total Least-Squares Iterative Closest Point Algorithm Based on Lie Algebra." Applied Sciences 9, no. 24 (December 7, 2019): 5352. http://dx.doi.org/10.3390/app9245352.

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In geodetic surveying, input data from two coordinates are needed to compute rigid transformations. A common solution is a least-squares algorithm based on a Gauss–Markov model, called iterative closest point (ICP). However, the error in the ICP algorithm only exists in target coordinates, and the algorithm does not consider the source model’s error. A total least-squares (TLS) algorithm based on an errors-in-variables (EIV) model is proposed to solve this problem. Previous total least-squares ICP algorithms used a Euler angle parameterization method, which is easily affected by a gimbal lock problem. Lie algebra is more suitable than the Euler angle for interpolation during an iterative optimization process. In this paper, Lie algebra is used to parameterize the rotation matrix, and we re-derive the TLS algorithm based on a GHM (Gauss–Helmert model) using Lie algebra. We present two TLS-ICP models based on Lie algebra. Our method is more robust than previous TLS algorithms, and it suits all kinds of transformation matrices.

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Goodwin, Simon M., Gerhard Röhrle, and Glenn Ubly. "On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type." LMS Journal of Computation and Mathematics 13 (September 2, 2010): 357–69. http://dx.doi.org/10.1112/s1461157009000205.

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AbstractWe consider the finiteW-algebraU(𝔤,e) associated to a nilpotent elemente∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem forU(𝔤,e), we verify a conjecture of Premet, thatU(𝔤,e) always has a 1-dimensional representation when 𝔤 is of typeG2,F4,E6orE7. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal inU(𝔤) whose associated variety is the coadjoint orbit corresponding to e.

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Karaman, Sertac, Tal Shima, and Emilio Frazzoli. "A Process Algebra Genetic Algorithm." IEEE Transactions on Evolutionary Computation 16, no. 4 (August 2012): 489–503. http://dx.doi.org/10.1109/tevc.2011.2160400.

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Chen, Chin Chun, Yuan Horng Lin, Jeng Ming Yih, and Sue Fen Huang. "Construct Knowledge Structure of Linear Algebra." Advanced Materials Research 211-212 (February 2011): 793–97. http://dx.doi.org/10.4028/www.scientific.net/amr.211-212.793.

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Apply interpretive structural modeling to construct knowledge structure of linear algebra. New fuzzy clustering algorithms improved fuzzy c-means algorithm based on Mahalanobis distance has better performance than fuzzy c-means algorithm. Each cluster of data can easily describe features of knowledge structures individually. The results show that there are six clusters and each cluster has its own cognitive characteristics. The methodology can improve knowledge management in classroom more feasible.

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Moldovyan, Nikolay, Dmitriy Moldovyan, and Alexandr Moldovyan. "A Novel Method for Developing Post-quantum Digital Signature Algorithms on Non-commutative Associative Algebras." Information and Control Systems, no. 1 (March 2, 2022): 44–53. http://dx.doi.org/10.31799/1684-8853-2022-1-44-53.

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Introduction: Development of practical post-quantum signature algorithms is a current challenge in the area of cryptography. Recently, several candidates on post-quantum signature schemes, in which the exponentiation operations in a hidden commutative group contained in a non-commutative algebra is used, were proposed. Search for new mechanisms of using a hidden group, while developing signature schemes resistant to quantum attacks, is of significant practical interest. Purpose: Development of a new method for designing post-quantum signature algorithms on finite non-commutative associative algebras. Results: A novel method for developing digital signature algorithms on non-commutative algebras. A new four-dimensional finite non-commutative associative algebra set over the ground field GF(p) have been proposed as algebraic support of the signature algorithms. To provide a higher performance of the algorithm, in the introduced algebra the vector multiplication is defined by a sparse basis vector multiplication table. Study of the algebra structure has shown that it can be represented as a set of commutative subalgebras of three different types, which intersect exactly in the set of scalar vectors. Using the proposed method and introduced algebra, a new post-quantum signature scheme has been designed. The introduced method is characterized in using one of the elements of the signature (e, S) in form of the four-dimensional vector S that is computed as a masked product of two exponentiated elements G and H of a hidden commutative group: S = B-1GnHmC-1, where non-permutable vectors B and C are masking multipliers; the natural numbers n and m are calculated depending on the signed document M and public key. The pair <G, H> composes a minimum generator systems of the hidden group. The signature verification equation has the form R = (Y1SZ1)e(Y2SZ2)e2, where pairwise non-permutable vectors Y1, Z1, Y2, and Z2 are element of the public key and natural number e that is computed depending on the value M and the vector R. Practical relevance: Due to sufficiently small size of public key and signature and high, the developed digital signature scheme represents interest as a practical post-quantum signature algorithm. The introduced method is very attractive to develop a post-quantum digital signature standard.

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Wang, Rui, Yue Wang, Yanping Li, Wenming Cao, and Yi Yan. "Geometric Algebra-Based ESPRIT Algorithm for DOA Estimation." Sensors 21, no. 17 (September 3, 2021): 5933. http://dx.doi.org/10.3390/s21175933.

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Direction-of-arrival (DOA) estimation plays an important role in array signal processing, and the Estimating Signal Parameter via Rotational Invariance Techniques (ESPRIT) algorithm is one of the typical super resolution algorithms for direction finding in an electromagnetic vector-sensor (EMVS) array; however, existing ESPRIT algorithms treat the output of the EMVS array either as a “long vector”, which will inevitably lead to loss of the orthogonality of the signal components, or a quaternion matrix, which may result in some missing information. In this paper, we propose a novel ESPRIT algorithm based on Geometric Algebra (GA-ESPRIT) to estimate 2D-DOA with double parallel uniform linear arrays. The algorithm combines GA with the principle of ESPRIT, which models the multi-dimensional signals in a holistic way, and then the direction angles can be calculated by different GA matrix operations to keep the correlations among multiple components of the EMVS. Experimental results demonstrate that the proposed GA-ESPRIT algorithm is robust to model errors and achieves less time complexity and smaller memory requirements.

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Kang, Seok-Jin, and Duncan J. Melville. "Rank 2 symmetric hyperbolic Kac-Moody algebras." Nagoya Mathematical Journal 140 (December 1995): 41–75. http://dx.doi.org/10.1017/s0027763000005419.

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Affine Kac-Moody algebras represent a well-trodden and well-understood littoral beyond which stretches the vast, chaotic, and poorly-understood ocean of indefinite Kac-Moody algebras. The simplest indefinite Kac-Moody algebras are the rank 2 Kac-Moody algebras (a) (a ≥ 3) with symmetric Cartan matrix , which form part of the class known as hyperbolic Kac-Moody algebras. In this paper, we probe deeply into the structure of those algebras (a), the e. coli of indefinite Kac-Moody algebras. Using Berman-Moody’s formula ([BM]), we derive a purely combinatorial closed form formula for the root multiplicities of the algebra (a), and illustrate some of the rich relationships that exist among root multiplicities, both within a single algebra and between different algebras in the class. We also give an explicit description of the root system of the algebra (a). As a by-product, we obtain a simple algorithm to find the integral points on certain hyperbolas.

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Khan, Muhammad, Junaid Alam Khan, and Muhammad Binyamin. "SAGBI Bases in G-Algebras." Symmetry 11, no. 2 (February 13, 2019): 221. http://dx.doi.org/10.3390/sym11020221.

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In this article, we develop the theory of SAGBI bases in G-algebras and create a criterion through which we can check if a set of polynomials in a G-algebra is a SAGBI basis or not. Moreover, we will construct an algorithm to compute SAGBI bases from a subset of polynomials contained in a subalgebra of a G-algebra.

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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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Mikhalev, Alexander A., and Andrej A. Zolotykh. "Standard Gröbner-Shirshov Bases of Free Algebras Over Rings, I." International Journal of Algebra and Computation 08, no. 06 (December 1998): 689–726. http://dx.doi.org/10.1142/s021819679800034x.

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We consider standard bases of ideals of free associative algebras over rings. The main result of the article is a criterion for a subset of a free associative algebra to be a standard basis of the ideal it generates. Based on this result, we present an infinite algorithm to construct the reduced standard basis of an ideal. A generalization in case of some semigroup algebras is presented. We also describe a way to construct weak standard bases and reduced standard bases of ideals of a free associative algebra over an arbitrary finitely generated ring (over a finitely generated algebra over a field). Some examples of constructions of standard bases and of solutions of the equality problem are included.

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Thiel, U. "Champ: a Cherednik algebraMagmapackage." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 266–307. http://dx.doi.org/10.1112/s1461157015000054.

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We present a computer algebra package based onMagmafor performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded$G$-module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups.Supplementary materials are available with this article.

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La Scala, Roberto. "Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms." International Journal of Algebra and Computation 24, no. 08 (December 2014): 1157–82. http://dx.doi.org/10.1142/s0218196714500519.

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Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

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García-Sánchez, Pedro A., Christopher O’Neill, and Gautam Webb. "The computation of factorization invariants for affine semigroups." Journal of Algebra and Its Applications 18, no. 01 (January 2019): 1950019. http://dx.doi.org/10.1142/s0219498819500191.

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We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of computing the tame degree of an affine semigroup, and (iii) a dynamic algorithm to compute catenary degrees of affine semigroup elements. Our algorithms rely on theoretical results from combinatorial commutative algebra involving Gröbner bases, Hilbert bases, and other standard techniques. Implementation in the computer algebra system GAP is discussed.

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Magaard, K., and R. A. Wilson. "Algorithmic construction of Chevalley bases." LMS Journal of Computation and Mathematics 15 (December 1, 2012): 436–43. http://dx.doi.org/10.1112/s1461157012001180.

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AbstractWe present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic p⩾5, our algorithm has complexity involving the seventh power of the Lie rank, which is likely to be close to best possible.

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Corr, Brian P., Tomasz Popiel, and Cheryl E. Praeger. "Nilpotent-independent sets and estimation in matrix algebras." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 404–18. http://dx.doi.org/10.1112/s146115701500008x.

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Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for such algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families $N$ of elements in the algebra of all $d\times d$ matrices over a field of order $q$, where membership of a matrix in $N$ depends only on its ‘invertible part’. The method is based on the availability of estimates for proportions of certain non-singular matrices depending on $N$, so that existing estimation techniques for non-singular matrices can be used to deal with families containing singular matrices. As an application, we investigate primary cyclic matrices, which are used in the Holt–Rees MEATAXE algorithm for testing irreducibility of matrix algebras.

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Glad, S. T. "Implementing Ritt’s Algorithm of Differential Algebra." IFAC Proceedings Volumes 25, no. 13 (June 1992): 219–23. http://dx.doi.org/10.1016/s1474-6670(17)52285-5.

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Alves, Rafael, and Carlile Lavor. "Clifford Algebra Applied to Grover’s Algorithm." Advances in Applied Clifford Algebras 20, no. 3-4 (March 9, 2010): 477–88. http://dx.doi.org/10.1007/s00006-010-0206-z.

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Braker, J. G., and G. J. Olsder. "The power algorithm in max algebra." Linear Algebra and its Applications 182 (March 1993): 67–89. http://dx.doi.org/10.1016/0024-3795(93)90492-7.

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Chen, Chin Chun, Yuan Horng Lin, and Jeng Ming Yih. "Management of Abstract Algebra Concepts Based on Knowledge Structure." Applied Mechanics and Materials 284-287 (January 2013): 3537–42. http://dx.doi.org/10.4028/www.scientific.net/amm.284-287.3537.

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Knowledge Management of Mathematics Concepts was essential in educational environment. The purpose of this study is to provide an integrated method of fuzzy theory basis for individualized concept structure analysis. This method integrates Fuzzy Logic Model of Perception (FLMP) and Interpretive Structural Modeling (ISM). The combined algorithm could analyze individualized concepts structure based on the comparisons with concept structure of expert. Fuzzy clustering algorithms are based on Euclidean distance function, which can only be used to detect spherical structural clusters. A Fuzzy C-Means algorithm based on Mahalanobis distance (FCM-M) was proposed to improve those limitations of GG and GK algorithms, but it is not stable enough when some of its covariance matrices are not equal. A new improved Fuzzy C-Means algorithm based on a Normalized Mahalanobis distance (FCM-NM) is proposed. Use the best performance of clustering Algorithm FCM-NM in data analysis and interpretation. Each cluster of data can easily describe features of knowledge structures. Manage the knowledge structures of Mathematics Concepts to construct the model of features in the pattern recognition completely. This procedure will also useful for cognition diagnosis. To sum up, this integrated algorithm could improve the assessment methodology of cognition diagnosis and manage the knowledge structures of Mathematics Concepts easily.

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HART, WILLIAM B. "A ONE LINE FACTORING ALGORITHM." Journal of the Australian Mathematical Society 92, no. 1 (February 2012): 61–69. http://dx.doi.org/10.1017/s1446788712000146.

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AbstractWe describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm described and the tuned assortment of factoring algorithms in the Pari/GP computer algebra package.

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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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BIRMAJER, DANIEL. "CONSTRUCTING FULL BLOCK TRIANGULAR REPRESENTATIONS OF ALGEBRAS." Journal of Algebra and Its Applications 06, no. 02 (April 2007): 259–65. http://dx.doi.org/10.1142/s021949880700217x.

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Every finite dimensional representation of an algebra is equivalent to a finite direct sum of indecomposable representations. Hence, the classification of indecomposable representations of algebras is a relevant (and usually complicated) task. In this note we study the existence of full block triangular representations, an interesting example of indecomposable representations, from a computational perspective. We describe an algorithm for determining whether or not an associative finitely presented k-algebra R has a full block triangular representation over [Formula: see text].

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Wires, Alexander. "On finite Taylor algebras." International Journal of Algebra and Computation 26, no. 08 (December 2016): 1547–71. http://dx.doi.org/10.1142/s0218196716500685.

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We establish a new hereditary characterization of finite idempotent Taylor algebras. This generalizes the algebraic constructions which figured in the recent successful characterization of the correctness of the bounded width algorithm for constraint satisfaction problems by finite idempotent algebras generating congruence meet-semidistributive varieties. We introduce the cyclic reduct of a finite Taylor algebra and prove a collapse of Mal’cev conditions.

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Magazev, Alexey Anatolievich, and Maria Nikolaevna Boldyreva. "Schrödinger Equations in Electromagnetic Fields: Symmetries and Noncommutative Integration." Symmetry 13, no. 8 (August 19, 2021): 1527. http://dx.doi.org/10.3390/sym13081527.

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We study symmetry properties and the possibility of exact integration of the time-independent Schrödinger equation in an external electromagnetic field. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra central extensions. Based on the well-known classification of the subalgebras of the algebra e(3), we classify all electromagnetic fields for which the corresponding time-independent Schrödinger equations admit first-order symmetry algebras. Moreover, we select the integrable cases, and for physically interesting electromagnetic fields, we reduced the original Schrödinger equation to an ordinary differential equation using the noncommutative integration method developed by Shapovalov and Shirokov.

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36

Noeske, Felix. "Matching simple modules of condensation algebras." LMS Journal of Computation and Mathematics 15 (December 1, 2012): 374–84. http://dx.doi.org/10.1112/s1461157012001118.

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AbstractWe revise the matching algorithm of Noeske (LMS J. Comput. Math. 11 (2008) 213–222) and introduce a new approach via composition series to expedite the calculations. Furthermore, we show how the matching algorithm may be applied in the more general and frequently occurring setting that we are only given subalgebras of the condensed algebras which each contain the separable algebra of one of their Wedderburn–Malcev decompositions.

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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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38

Bokut, L. A., Yuqun Chen, and Zerui Zhang. "Gröbner–Shirshov bases method for Gelfand–Dorfman–Novikov algebras." Journal of Algebra and Its Applications 16, no. 01 (January 2017): 1750001. http://dx.doi.org/10.1142/s0219498817500013.

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We establish Gröbner–Shirshov base theory for Gelfand–Dorfman–Novikov algebras over a field of characteristic [Formula: see text]. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem of Gelfand–Dorfman–Novikov algebras with finite homogeneous relations. We also construct a subalgebra of one generated free Gelfand–Dorfman–Novikov algebra which is not free.

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Deng, FangAn, Shouheng Tuo, Longquan Yong, and Tao Zhou. "Construction Example for Algebra System Using Harmony Search Algorithm." Mathematical Problems in Engineering 2015 (2015): 1–15. http://dx.doi.org/10.1155/2015/836925.

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The construction example of algebra system is to verify the existence of a complex algebra system, and it is a NP-hard problem. In this paper, to solve this kind of problems, firstly, a mathematical optimization model for construction example of algebra system is established. Secondly, an improved harmony search algorithm based on NGHS algorithm (INGHS) is proposed to find as more solutions as possible for the optimization model; in the proposed INGHS algorithm, to achieve the balance between exploration power and exploitation power in the search process, a global best strategy and parameters dynamic adjustment method are present. Finally, nine construction examples of algebra system are used to evaluate the optimization model and performance of INGHS. The experimental results show that the proposed algorithm has strong performance for solving complex construction example problems of algebra system.

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Liu, Hsiang Chuan, Yen Kuei Yu, Jeng Ming Yih, and Chin Chun Chen. "Identifying the Mastery Concepts in Linear Algebra by Using FCM-CM Algorithm." Applied Mechanics and Materials 44-47 (December 2010): 3897–901. http://dx.doi.org/10.4028/www.scientific.net/amm.44-47.3897.

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Euclidean distance function based fuzzy clustering algorithms can only be used to detect spherical structural clusters. Gustafson-Kessel (GK) clustering algorithm and Gath-Geva (GG) clustering algorithm were developed to detect non-spherical structural clusters by employing Mahalanobis distance in objective function, however, both of them need to add some constrains for Mahalanobis distance. In this paper, the authors’ improved Fuzzy C-Means algorithm based on common Mahalanobis distance (FCM-CM) is used to identify the mastery concepts in linear algebra, for comparing the performances with other four partition algorithms; FCM-M, GG, GK, and FCM. The result shows that FCM-CM has better performance than others.

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Bonchi, Filippo, Marcello M. Bonsangue, Helle H. Hansen, Prakash Panangaden, Jan J. M. M. Rutten, and Alexandra Silva. "Algebra-coalgebra duality in brzozowski's minimization algorithm." ACM Transactions on Computational Logic 15, no. 1 (February 2014): 1–29. http://dx.doi.org/10.1145/2490818.

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42

von Sohsten de Medeiros, Airton. "Elementary Linear Algebra and the Division Algorithm." College Mathematics Journal 33, no. 1 (January 2002): 51. http://dx.doi.org/10.2307/1558982.

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Jeremic, Marina, Aleksandar Rakicevic, and Ivana Dragovic. "Interpolative Boolean algebra based multicriteria routing algorithm." Yugoslav Journal of Operations Research 25, no. 3 (2015): 397–412. http://dx.doi.org/10.2298/yjor140430029j.

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In order to improve the quality-of-service of distributed applications, we propose a multi-criteria algorithm based on interpolative Boolean algebra for routing in an overlay network. We use a mesh topology because it can be easily implemented, and it makes addressing of the cores quite simple during routing. In this paper, we consider four criteria: buffer usage, the distance between peers, bandwidth, and remaining battery power. The proposed routing algorithm determines the path which satisfies quality-of service requirements using interpolative Boolean algebra; the decision at each node is made based on the ranking of available options considering multiple constraints. The simulation shows that the proposed approach provides better results than the standard shortest path routing algorithm.

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Gan, Weichao, Zhengming Ma, and Shuyu Liu. "Dimensionality reduction for tensor data based on projection distance minimization and hilbert-schmidt independence criterion maximization1." Journal of Intelligent & Fuzzy Systems 40, no. 5 (April 22, 2021): 10307–22. http://dx.doi.org/10.3233/jifs-202582.

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Tensor data are becoming more and more common in machine learning. Compared with vector data, the curse of dimensionality of tensor data is more serious. The motivation of this paper is to combine Hilbert-Schmidt Independence Criterion (HSIC) and tensor algebra to create a new dimensionality reduction algorithm for tensor data. There are three contributions in this paper. (1) An HSIC-based algorithm is proposed in which the dimension-reduced tensor is determined by maximizing HSIC between the dimension-reduced and high-dimensional tensors. (2) A tensor algebra-based algorithm is proposed, in which the high-dimensional tensor are projected onto a subspace and the projection coordinate is set to be the dimension-reduced tensor. The subspace is determined by minimizing the distance between the high-dimensional tensor data and their projection in the subspace. (3) By combining the above two algorithms, a new dimensionality reduction algorithm, called PDMHSIC, is proposed, in which the dimensionality reduction must satisfy two criteria at the same time: HSIC maximization and subspace projection distance minimization. The proposed algorithm is a new attempt to combine HSIC with other algorithms to create new algorithms and has achieved better experimental results on 8 commonly-used datasets than the other 7 well-known algorithms.

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Demmel, James, Ioana Dumitriu, Olga Holtz, and Plamen Koev. "Accurate and efficient expression evaluation and linear algebra." Acta Numerica 17 (April 25, 2008): 87–145. http://dx.doi.org/10.1017/s0962492906350015.

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We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By ‘accurate’ we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: most of our results will use the so-called traditional model (TM), where the computed result of op(a, b), a binary operation like a+b, is given by op(a, b) * (1+δ) where all we know is that |δ| ≤ ε ≪ 1. Here ε is a constant also known as machine epsilon.We will see a common reason for the following disparate problems to permit accurate and efficient algorithms using only the four basic arithmetic operations: finding the eigenvalues of a suitably discretized scalar elliptic PDE, finding eigenvalues of arbitrary products, inverses, or Schur complements of totally non-negative matrices (such as Cauchy and Vandermonde), and evaluating the Motzkin polynomial. Furthermore, in all these cases the high accuracy is ‘deserved’, i.e., the answer is determined much more accurately by the data than the conventional condition number would suggest.In contrast, we will see that evaluating even the simple polynomial x + y + z accurately is impossible in the TM, using only the basic arithmetic operations. We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high-accuracy algorithm or a proof that none exists.When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as x + y + z, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case.Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.

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Martin, Paul P., and David Woodcock. "Generalized Blob Algebras and Alcove Geometry." LMS Journal of Computation and Mathematics 6 (2003): 249–96. http://dx.doi.org/10.1112/s1461157000000450.

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AbstractA sequence of finite-dimensional quotients of affine Hecke algebras is studied. Each element of the sequence is constructed so as to have a weight space labelling scheme for Specht⁄standard modules. As in the weight space formalism of algebraic Lie theory, there is an action of an affine reflection group on this weight space that fixes the set of labelling weights. A linkage principle is proved in each case. Further, it is shown that the simplest non-trivial example may essentially be identified with the blob algebra (a physically motivated quasihereditary algebra whose representation theory is very well understood by Lie-theory-like methods). An extended role is hence proposed for Soergel's tilting algorithm, away from its algebraic Lie theory underpinning, in determining the simple content of standard modules for these algebras. This role is explicitly verified in the blob algebra case. A tensor space representation of the blob algebra is constructed, as a candidate for a full tilting module (subsequently proven to be so in a paper by Martin and Ryom-Hansen), further evidencing the extended utility of Lie-theoretic methods. Possible generalisations of this representation to other elements of the sequence are discussed.

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BREMNER, MURRAY R., and JUANA SÁNCHEZ-ORTEGA. "LEIBNIZ TRIPLE SYSTEMS." Communications in Contemporary Mathematics 16, no. 01 (January 21, 2014): 1350051. http://dx.doi.org/10.1142/s021919971350051x.

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We define Leibniz triple systems in a functorial manner using the algorithm of Kolesnikov and Pozhidaev which converts identities for algebras into identities for dialgebras; this algorithm is a concrete realization of the white Manin product introduced by Vallette by the permutad Perm introduced by Chapoton. We verify that Leibniz triple systems are natural analogues of Lie triple systems by showing that both the iterated bracket in a Leibniz algebra and the permuted associator in a Jordan dialgebra satisfy the defining identities for Leibniz triple systems. We construct the universal Leibniz envelopes of Leibniz triple systems and prove that every identity satisfied by the iterated bracket in a Leibniz algebra is a consequence of the defining identities for Leibniz triple systems. In the last section, we present some examples of two-dimensional Leibniz triple systems and their universal Leibniz envelopes.

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48

Herman, Allen. "A Constructive Brauer-Witt Theorem for Certain Solvable Groups." Canadian Journal of Mathematics 48, no. 6 (December 1, 1996): 1196–209. http://dx.doi.org/10.4153/cjm-1996-063-1.

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AbstractDivision algebras occurring in simple components of group algebras of finite groups over algebraic number fields are studied. First, well-known restrictions are presented for the structure of a group that arises once no further Clifford Theory reductions are possible. For groups with these properties, a character-theoretic condition is given that forces the p-part of the division algebra part of this simple component to be generated by a predetermined p-quasi-elementary subgroup of the group, for any prime integer p. This is effectively a constructive Brauer-Witt Theorem for groups satisfying this condition. It is then shown that it is possible to constructively compute the Schur index of a simple component of the group algebra of a finite nilpotent-by-abelian group using the above reduction and an algorithm for computing Schur indices of simple algebras generated by finite metabelian groups.

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Chen, Zhongzhe, Jianzhang Xiao, and Guifeng Wang. "An Effective Path Planning of Intelligent Mobile Robot Using Improved Genetic Algorithm." Wireless Communications and Mobile Computing 2022 (May 9, 2022): 1–8. http://dx.doi.org/10.1155/2022/9590367.

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With the rapid development of the robotics industry, the problem of effective and fast path planning for intelligent mobile robots has always been one of the hot spots in the field of robotics research. Intelligent mobile robot path planning is divided into global path planning and local path planning, and its mathematical modeling and adaptive algorithms are different. Therefore, the research of robot path planning based on improved genetic algorithm is of great significance. This paper mainly studies the robot path planning problem based on improved genetic algorithm. Based on the research of the basic genetic algorithm, the improved genetic algorithm is applied to the mobile four-wheel robot to guide the four-wheel robot to complete path planning and other related tasks. Experiments show that the optimization probability and convergence speed of the genetic algorithm can be improved by improving the genetic algorithm. Studies have shown that evolutionary algebra and population size are inversely proportional to the optimal path length, so it is directly proportional to the search ability. However, as the evolutionary algebra and population size increase, the amount of calculation is also increasing, and the calculation time increases. Comprehensive considerations according to various factors, the best value of population size is 60, the best value of mutation probability is 0.09, the best value of crossover probability is 0.8, and the best value of evolutionary algebra is 150 generations.

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Potapov, Viktor S., and Sergei M. Gushansky. "Development of a technique for modeling entangled quantum computations that are applicable in Simon’s quantum algorithm." Informatization and communication, no. 3 (May 5, 2020): 66–70. http://dx.doi.org/10.34219/2078-8320-2020-11-3-66-70.

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This paper describes the basics of developing quantum algorithms and modeling entangled quantum computations applicable in quantum algorithms. Quantum algorithms involve the use of vector and matrix algebra. The basic tasks of the simulation proposed in the work are determined within the framework of the algorithm for executing quantum algorithms, taking into account entanglement. A technique has been developed for modeling entangled quantum calculations applicable in the Simon quantum algorithm, which helps to predict the behavior of the quantum algorithm (or any other computing process that proceeds as part of the work of a quantum computer system) with partial entanglement.

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Journal articles on the topic 'Algorithm algebra'

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