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Letteratura scientifica selezionata sul tema "Permutations. Mathematics"

Autore: Grafiati

Pubblicato: 4 giugno 2021

Ultima modifica: 13 luglio 2024

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Articoli di riviste sul tema "Permutations. Mathematics"

1

Brualdi, Richard A., e Geir Dahl. "Permutation Matrices, Their Discrete Derivatives and Extremal Properties". Vietnam Journal of Mathematics 48, n. 4 (24 marzo 2020): 719–40. http://dx.doi.org/10.1007/s10013-020-00392-5.

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AbstractFor a permutation π, and the corresponding permutation matrix, we introduce the notion of discrete derivative, obtained by taking differences of successive entries in π. We characterize the possible derivatives of permutations, and consider questions for permutations with certain properties satisfied by the derivative. For instance, we consider permutations with distinct derivatives, and the relationship to so-called Costas arrays.

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2

Drakakis, Konstantinos. "On the Measurement of the (Non)linearity of Costas Permutations". Journal of Applied Mathematics 2010 (2010): 1–14. http://dx.doi.org/10.1155/2010/149658.

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We study several criteria for the (non)linearity of Costas permutations, with or without the imposition of additional algebraic structure in the domain and the range of the permutation, aiming to find one that successfully identifies Costas permutations as more nonlinear than randomly chosen permutations of the same order.

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3

Steingrı́msson, Einar. "Permutation Statistics of Indexed Permutations". European Journal of Combinatorics 15, n. 2 (marzo 1994): 187–205. http://dx.doi.org/10.1006/eujc.1994.1021.

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4

Hajnal, Péter. "A short note on Layman permutations". Acta Universitatis Sapientiae, Mathematica 14, n. 2 (1 dicembre 2022): 231–38. http://dx.doi.org/10.2478/ausm-2022-0015.

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Abstract A permutation p of [k] = {1, 2, 3, …, k} is called Layman permutation iff i + p(i) is a Fibonacci number for 1 ≤ i ≤ k. This concept is introduced by Layman in the A097082 entry of the Encyclopedia of Integers Sequences, that is the number of Layman permutations of [n]. In this paper, we will study Layman permutations. We introduce the notion of the Fibonacci complement of a natural number, that plays a crucial role in our investigation. Using this notion we prove some results on the number of Layman permutations, related to a conjecture of Layman that is implicit in the A097083 entry of OEIS.

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5

Choi, Wonseok, Jooyoung Lee e Yeongmin Lee. "Building PRFs from TPRPs: Beyond the Block and the Tweak Length Bounds". IACR Transactions on Symmetric Cryptology 2024, n. 1 (1 marzo 2024): 35–70. http://dx.doi.org/10.46586/tosc.v2024.i1.35-70.

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A secure n-bit tweakable block cipher (TBC) using t-bit tweaks can be modeled as a tweakable uniform random permutation, where each tweak defines an independent random n-bit permutation. When an input to this tweakable permutation is fixed, it can be viewed as a perfectly secure t-bit random function. On the other hand, when a tweak is fixed, it can be viewed as a perfectly secure n-bit random permutation, and it is well known that the sum of two random permutations is pseudorandom up to 2n queries.A natural question is whether one can construct a pseudorandom function (PRF) beyond the block and the tweak length bounds using a small number of calls to the underlying tweakable permutations. A straightforward way of constructing a PRF from tweakable permutations is to xor the outputs from two tweakable permutations with c bits of the input to each permutation fixed. Using the multi-user security of the sum of two permutations, one can prove that the (t + n − c)-to-n bit PRF is secure up to 2n+c queries.In this paper, we propose a family of PRF constructions based on tweakable permutations, dubbed XoTPc, achieving stronger security than the straightforward construction. XoTPc is parameterized by c, giving a (t + n − c)-to-n bit PRF. When t < 3n and c = t/3 , XoTPt/3 becomes an (n + 2t/3 )-to-n bit pseudorandom function, which is secure up to 2n+2t/3 queries. It provides security beyond the block and the tweak length bounds, making two calls to the underlying tweakable permutations. In order to prove the security of XoTPc, we extend Mirror theory to q ≫ 2n, where q is the number of equations. From a practical point of view, our construction can be used to construct TBC-based MAC finalization functions and CTR-type encryption modes with stronger provable security compared to existing schemes.

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Burov, Dmitry A. "Subgroups of direct products of groups invariant under the action of permutations on factors". Discrete Mathematics and Applications 30, n. 4 (26 agosto 2020): 243–55. http://dx.doi.org/10.1515/dma-2020-0021.

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AbstractWe study subgroups of the direct product of two groups invariant under the action of permutations on factors. An invariance criterion for the subdirect product of two groups under the action of permutations on factors is put forward. Under certain additional constraints on permutations, we describe the subgroups of the direct product of a finite number of groups that are invariant under the action of permutations on factors. We describe the subgroups of the additive group of vector space over a finite field of characteristic 2 which are invariant under the coordinatewise action of inversion permutation of nonzero elements of the field.

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7

WANG, LI-YUAN, e HAI-LIANG WU. "APPLICATIONS OF LERCH’S THEOREM TO PERMUTATIONS OF QUADRATIC RESIDUES". Bulletin of the Australian Mathematical Society 100, n. 3 (10 luglio 2019): 362–71. http://dx.doi.org/10.1017/s000497271900073x.

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Let $n$ be a positive integer and $a$ an integer prime to $n$. Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$th power residues modulo $p$ and primitive roots modulo a power of $p$.

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8

BLOCK, LOUIS, ALEXANDER M. BLOKH e ETHAN M. COVEN. "ZERO ENTROPY PERMUTATIONS". International Journal of Bifurcation and Chaos 05, n. 05 (ottobre 1995): 1331–37. http://dx.doi.org/10.1142/s0218127495001009.

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The entropy of a permutation is the (topological) entropy of the "connect-the-dots" map determined by it. We give matrix- and graph-theoretic, geometric, and dynamical characterizations of zero entropy permutations, as well as a procedure for constructing all of them. We also include some information about the number of zero entropy permutations.

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9

ZHOU, YINGCHUN, e MURAD S. TAQQU. "APPLYING BUCKET RANDOM PERMUTATIONS TO STATIONARY SEQUENCES WITH LONG-RANGE DEPENDENCE". Fractals 15, n. 02 (giugno 2007): 105–26. http://dx.doi.org/10.1142/s0218348x07003526.

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Bucket random permutations (shuffling) are used to modify the dependence structure of a time series, and this may destroy long-range dependence, when it is present. Three types of bucket permutations are considered here: external, internal and two-level permutations. It is commonly believed that (1) an external random permutation destroys the long-range dependence and keeps the short-range dependence, (2) an internal permutation destroys the short-range dependence and keeps the long-range dependence, and (3) a two-level permutation distorts the medium-range dependence while keeping both the long-range and short-range dependence. This paper provides a theoretical basis for investigating these claims. It extends the study started in Ref. 1 and analyze the effects that these random permutations have on a long-range dependent finite variance stationary sequence both in the time domain and in the frequency domain.

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10

Cioni, Lapo, e Luca Ferrari. "Sorting with a popqueue". RAIRO - Theoretical Informatics and Applications 58 (2024): 13. http://dx.doi.org/10.1051/ita/2024010.

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We introduce a new sorting device for permutations, which we call popqueue. It consists of a special queue, having the property that any time one wants to extract elements from the queue, actually all the elements currently in the queue are poured into the output. We illustrate two distinct optimal algorithms, called Min and Cons, to sort a permutation using such a device, which allow us also to characterize sortable permutations in terms of pattern avoidance. We next investigate what happens by making two passes through a popqueue, showing that the set of sortable permutations is not a class for Min, whereas it is for Cons. In the latter case we also explicitly find the basis of the class of sortable permutations. Finally, we study preimages under Cons (by means of an equivalent version of the algorithm), and find a characterization of the set of preimages of a given permutation. We also give some enumerative results concerning the number of permutations having k preimages, for k = 1, 2, 3, and we conclude by observing that there exist permutations having k preimages for any value of k ≥ 0.

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Tesi sul tema "Permutations. Mathematics"

1

Steingrímsson, Einar. "Permutations statistics of indexed and poset permutations". Thesis, Massachusetts Institute of Technology, 1992. http://hdl.handle.net/1721.1/35952.

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2

West, Julian 1964. "Permutations with forbidden subsequences, and, stack-sortable permutations". Thesis, Massachusetts Institute of Technology, 1990. http://hdl.handle.net/1721.1/13641.

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3

Elizalde, Sergi 1979. "Statistics on pattern-avoiding permutations". Thesis, Massachusetts Institute of Technology, 2004. http://hdl.handle.net/1721.1/16629.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
Includes bibliographical references (p. 111-116).
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
This thesis concerns the enumeration of pattern-avoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics 'number of fixed points' and 'number of excedances' is the same in 321-avoiding as in 132-avoiding permutations. This generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof. The key ideas are to introduce a new class of statistics on Dyck paths, based on what we call a tunnel, and to use a new technique involving diagonals of non-rational generating functions. Next we present a new statistic-preserving family of bijections from the set of Dyck paths to itself. They map statistics that appear in the study of pattern-avoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. In particular, this gives a simple bijective proof of the equidistribution of fixed points in the above two sets of restricted permutations.
(cont.) Then we introduce a bijection between 321- and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. A part of our bijection is based on the Robinson-Schensted-Knuth correspondence. We also show that our bijection preserves additional parameters. Next, motivated by these results, we study the distribution of fixed points and excedances in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions which enumerate them. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statistics in involutions avoiding any subset of patterns of length 3. The main technique consists in using bijections between pattern-avoiding permutations and certain kinds of Dyck paths, in such a way that the statistics in permutations that we consider correspond to statistics on Dyck paths which are easier to enumerate. Finally, we study another kind of restricted permutations, counted by the Motzkin numbers. By constructing a bijection into Motzkin paths, we enumerate them with respect to some parameters, including the length of the longest increasing and decreasing subsequences and the number of ascents.
by Sergi Elizalde.
Ph.D.

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4

Mao, Cheng Ph D. Massachusetts Institute of Technology. "Matrix estimation with latent permutations". Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/117863.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 151-167).
Motivated by various applications such as seriation, network alignment and ranking from pairwise comparisons, we study the problem of estimating a structured matrix with rows and columns shuffled by latent permutations, given noisy and incomplete observations of its entries. This problem is at the intersection of shape constrained estimation which has a long history in statistics, and latent permutation learning which has driven a recent surge of interest in the machine learning community. Shape constraints on matrices, such as monotonicity and smoothness, are generally more robust than parametric assumptions, and often allow for adaptive and efficient estimation in high dimensions. On the other hand, latent permutations underlie many graph matching and assignment problems that are computationally intractable in the worst-case and not yet well-understood in the average-case. Therefore, it is of significant interest to both develop statistical approaches and design efficient algorithms for problems where shape constraints meet latent permutations. In this work, we consider three specific models: the statistical seriation model, the noisy sorting model and the strong stochastic transitivity model. First, statistical seriation consists in permuting the rows of a noisy matrix in such a way that all its columns are approximately monotone, or more generally, unimodal. We study both global and adaptive rates of estimation for this model, and introduce an efficient algorithm for the monotone case. Next, we move on to ranking from pairwise comparisons, and consider the noisy sorting model. We establish the minimax rates of estimation for noisy sorting, and propose a near-linear time multistage algorithm that achieves a near-optimal rate. Finally, we study the strong stochastic transitivity model that significantly generalizes the noisy sorting model for estimation from pairwise comparisons. Our efficient algorithm achieves the rate (n- 3 /4 ), narrowing a gap between the statistically optimal rate Õ(n-1 ) and the state-of-the-art computationally efficient rate [Theta] (n- 1/ 2 ). In addition, we consider the scenario where a fixed subset of pairwise comparisons is given. A dichotomy exists between the worst-case design, where consistent estimation is often impossible, and an average-case design, where we show that the optimal rate of estimation depends on the degree sequence of the comparison topology.
by Cheng Mao.
Ph. D.

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5

Yun, Taedong. "Diagrams of affine permutations and their labellings". Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/83702.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 63-64).
We study affine permutation diagrams and their labellings with positive integers. Balanced labellings of a Rothe diagram of a finite permutation were defined by Fomin- Greene-Reiner-Shimozono, and we extend this notion to affine permutations. The balanced labellings give a natural encoding of the reduced decompositions of affine permutations. We show that the sum of weight monomials of the column-strict balanced labellings is the affine Stanley symmetric function which plays an important role in the geometry of the affine Grassmannian. Furthermore, we define set-valued balanced labellings in which the labels are sets of positive integers, and we investigate the relations between set-valued balanced labellings and nilHecke words in the nilHecke algebra. A signed generating function of column-strict set-valued balanced labellings is shown to coincide with the affine stable Grothendieck polynomial which is related to the K-theory of the affine Grassmannian. Moreover, for finite permutations, we show that the usual Grothendieck polynomial of Lascoux-Schiitzenberger can be obtained by flagged column-strict set-valued balanced labellings. Using the theory of balanced labellings, we give a necessary and sufficient condition for a diagram to be a permutation diagram. An affine diagram is an affine permutation diagram if and only if it is North-West and admits a special content map. We also characterize and enumerate the patterns of permutation diagrams.
by Taedong Yun.
Ph.D.

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6

Boberg, Jonas. "Counting Double-Descents and Double-Inversions in Permutations". Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-54431.

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In this paper, new variations of some well-known permutation statistics are introduced and studied. Firstly, a double-descent of a permutation π is defined as a position i where πi ≥ 2πi+1. By proofs by induction and direct proofs, recursive and explicit expressions for the number of n-permutations with k double-descents are presented. Also, an expression for the total number of double-descents in all n-permutations is presented. Secondly, a double-inversion of a permutation π is defined as a pair (πi,πj) where i<j but πi ≥ 2πj. The total number of double-inversions in all n-permutations is presented.

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Stadler, Jonathan. "Schur functions, juggling, and statistics on shuffled permutations /". The Ohio State University, 1997. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487947501135397.

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8

Bóna, Miklós. "Exact and asymptotic enumeration of permutations with subsequence conditions". Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/42691.

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Hammett, Adam Joseph. "On comparability of random permutations". Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1172592365.

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10

Marcus, Adam Wade. "New combinatorial techniques for nonlinear orders". Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24685.

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Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008.
Committee Chair: Prasad Tetali; Committee Member: Dana Randall; Committee Member: Robin Thomas; Committee Member: Vijay Vazirani; Committee Member: William T. Trotter

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Libri sul tema "Permutations. Mathematics"

1

Education, Alberta Alberta. Mathematics 30: Permutations and combinations. Edmonton: Alberta Education, 1991.

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2

Combinatorics of permutations. Boca Raton: Chapman & Hall/CRC, 2004.

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Combinatorics of permutations. 2^a ed. Boca Raton, FL: Chapman and Hall/CRC, 2012.

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4

Introduction to pure mathematics: Permutations : Group theory block. Milton Keynes: Open University Press, 1991.

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service), SpringerLink (Online, a cura di. Patterns in Permutations and Words. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 2011.

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Alexandre, Borovik, e Myasnikov Alexei G. 1955-, a cura di. Computational and experimental group theory: AMS-ASL joint special session, Interactions between logic, group theory, and computer science : January 15-16, 2003, Baltimore, Maryland. Providence, R.I: American Mathematical Society, 2004.

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7

C++ math class library: Permutations, partitions, calculators, and gaming. New York: J. Wiley, 1994.

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Cameron, Peter J. Oligomorphic permutation groups. Cambridge: Cambridge University Press, 1990.

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9

OmeGA: A competent genetic algorithm for solving permutation and scheduling problems. Boston: Kluwer Academic Publishers, 2002.

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10

Gerard, Scott N. C[plus plus] math class library: Permutations, partitions, calculators, and gaming. New York: Wiley, 1994.

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Capitoli di libri sul tema "Permutations. Mathematics"

1

Armstrong, M. A. "Permutations". In Undergraduate Texts in Mathematics, 26–31. New York, NY: Springer New York, 1988. http://dx.doi.org/10.1007/978-1-4757-4034-9_6.

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2

Sane, Sharad S. "Permutations". In Texts and Readings in Mathematics, 39–56. Gurgaon: Hindustan Book Agency, 2013. http://dx.doi.org/10.1007/978-93-86279-55-2_3.

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3

Effinger, Gove, e Gary L. Mullen. "Permutations". In An Elementary Transition to Abstract Mathematics, 57–63. Boca Raton : CRC Press, Taylor … Francis Group, 2020.: CRC Press, 2019. http://dx.doi.org/10.1201/9780429324819-10.

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4

Roman, Steven. "Catalan Numbers and Permutations". In Compact Textbooks in Mathematics, 61–71. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-22144-1_10.

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5

Gil, Juan B., e Michael D. Weiner. "On Pattern-Avoiding Fishburn Permutations". In Trends in Mathematics, 431–46. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-57050-7_25.

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6

Corteel, Sylvie, Guy Louchard e Robin Pemantle. "Common Intervals of Permutations". In Mathematics and Computer Science III, 3–14. Basel: Birkhäuser Basel, 2004. http://dx.doi.org/10.1007/978-3-0348-7915-6_1.

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Rotman, Joseph J. "Permutations and the Mathieu Groups". In Graduate Texts in Mathematics, 247–306. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-4176-8_9.

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Müller, Paul F. X. "Permutations of the Haar system". In Lecture Notes in Mathematics, 125–26. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/bfb0089218.

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9

Arboretti, Rosa, Eleonora Carrozzo e Luigi Salmaso. "Permutation Tests for Multivariate Stratified Data: Synchronized or Unsynchronized Permutations?" In Springer Proceedings in Mathematics & Statistics, 47–56. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-57306-5_5.

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10

Milosz, Robin, e Sylvie Hamel. "Medians of Permutations: Building Constraints". In Algorithms and Discrete Applied Mathematics, 264–76. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-29221-2_23.

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Atti di convegni sul tema "Permutations. Mathematics"

1

Ryabov, Vladimir Gennadievich. "On number of substitutions of vector space over finite field with affine approximations with given accuracy". In Academician O.B. Lupanov 14th International Scientific Seminar "Discrete Mathematics and Its Applications". Keldysh Institute of Applied Mathematics, 2022. http://dx.doi.org/10.20948/dms-2022-88.

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Nonlinear mapping of a vector space over a finite field in vector space over the same field and, in particular, the non-linearity the permutation space is defined in terms of the Hamming distance to sets of affine mappings. For permutations of an n-dimensional vector space over a field of q elements, an upper bound for the number permutations with non-linearity, not higher than a given value r, where 0 ≤ r < q<sup>n</sup> - q<sup>n-1</sup>, and for r < (q<sup>n</sup> - q<sup>n-1</sup>)/2, the exact the value of the specified number, which allows us to draw conclusions about distribution of nonlinearity on the set of substitutions.

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2

Makarov, Vladimir Vladimirovich. "On homeomorphisms of groups of automata permutations". In Academician O.B. Lupanov 14th International Scientific Seminar "Discrete Mathematics and Its Applications". Keldysh Institute of Applied Mathematics, 2022. http://dx.doi.org/10.20948/dms-2022-35.

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The author previously proved that the automaton permutation group AS<sub>2</sub> generated by elements of infinite order. The question remains open on the generation of AS<sub>2</sub> by elements of the second order. The article provides an example subgroup of AS<sub>2</sub> generated by involutions containing a homomorphic prototype of the entire AS<sub>2</sub>.

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3

Ren, Nianxin. "On the largest minimal suitable sets of permutations". In 2023 3rd International Conference on Applied Mathematics, Modelling and Intelligent Computing (CAMMIC 2023), a cura di Xuebin Chen e Hari Mohan Srivastava. SPIE, 2023. http://dx.doi.org/10.1117/12.2685909.

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4

Antonides, Joseph, e Michael T. Battista. "Two prospective middle school teachers reinvent combinatorial formulas: permutations and arrangements". In 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. PMENA, 2020. http://dx.doi.org/10.51272/pmena.42.2020-177.

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5

Demiralp, Metin. "Relations between the Permutations and the Matrix Norm in Denumerable Infinite Vector Folding to Semi-denumerable Infinite Matrices". In 2014 International Conference on Mathematics and Computers in Sciences and in Industry (MCSI). IEEE, 2014. http://dx.doi.org/10.1109/mcsi.2014.32.

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Kharinov, Mikhail. "Natural non‐group symmetry in modern applications". In V International Scientific Workshop on Modeling, Information Processing and Computing. CEUR-WS.org, 2022. http://dx.doi.org/10.47813/dnit-mip5/2022-3091-38-45.

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Intuitively perceived symmetry is formalized for effective application in physics, mathematics, and engineering. In this regard, several scientific research directions are indicated, which are expressed by three generalizations: a) the concept of symmetry using the example of normalized Hadamard matrices; b) cross vector product for the cases of three arguments and seven-dimensional space, c) Lorentz transformations for doubling the spacetime dimension. To generalize and formalize the concept of symmetry, the preservation of the symmetry of matrices under permutations of rows (columns) is studied. It is shown that the set of symmetrypreserving permutations does not constitute a group. For the development of the octonion toolkit and the best generalization of the vector product, based on symmetry considerations, the decomposition of the triple product of octonions into the sum of a triple anticommutator, a triple commutator (generalized vector product) and an associator is deduced. To begin the generalization of Lorentz transformations Lorentz boost is recorded in terms of quaternions so that the treated expressions retain their meaning in the octonionic space. To speed up the assimilation of the research results, the paper proposes some elementary information on the three listed topics, which it is desirable to place in reference books, as well as bring to the attention of students in general education courses at technical universities.

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Alsalem, Shuker, Abu Firas Al Musawi e Enoch Suleiman. "On Permutation Upper and Transitive Permutation BE-Algebras". In 2022 14th International Conference on Mathematics, Actuarial Science, Computer Science and Statistics (MACS). IEEE, 2022. http://dx.doi.org/10.1109/macs56771.2022.10022454.

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Alsalem, Shuker, Abu Firas Al Musawi e Enoch Suleiman. "On maximal permutation BH—ideals of Permutation BH—Algebras". In 2022 7th International Conference on Mathematics and Computers in Sciences and Industry (MCSI). IEEE, 2022. http://dx.doi.org/10.1109/mcsi55933.2022.00013.

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Alsalem, Shuker, Abu Firas Al Musawi e Enoch Suleiman. "On Permutation B-center and Derived Permutation B-algebras". In 2022 7th International Conference on Mathematics and Computers in Sciences and Industry (MCSI). IEEE, 2022. http://dx.doi.org/10.1109/mcsi55933.2022.00014.

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Alsalem, Shuker, Abu F. Almusawi e Enoch Suleiman. "On permutation G-part in permutation Q-algebras". In International Conference on Mathematical and Statistical Physics, Computational Science, Education, and Communication (ICMSCE 2022), a cura di Lazim Abdullah e Norma bt Alias. SPIE, 2023. http://dx.doi.org/10.1117/12.2674993.

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