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Journal of algebra, ISSN 0021-8693, 2017, Volume 481, pp. 293 - 326
Quasi-shuffle products, introduced by the first author, have been useful in studying multiple zeta values and some of their analogues and generalizations... 
Quasi-shuffle product | Multiple zeta values | Infinitesimal Hopf algebra | Hopf algebra | MATHEMATICS | MULTIPLE HARMONIC SUMS | ZETA-STAR VALUES | Algebra
Journal Article
Taiwanese Journal of Mathematics, ISSN 1027-5487, 6/2018, Volume 22, Issue 3, pp. 529 - 543
Using the combinatorial descriptions of stuffle product, we obtain recursive formulas for the stuffle product of multiple zeta values and of multiple zeta-star values... 
Multiple zeta values | Stuffle product | Multiple zeta-star values | MATHEMATICS | DECOMPOSITION | multiple zeta-star values | Stuffel product | multiple zeta values | SHUFFLE PRODUCT
Journal Article
Journal of Number Theory, ISSN 0022-314X, 02/2017, Volume 171, pp. 79 - 111
Using the combinatorial description of shuffle product, we prove or reformulate several shuffle product formulas of multiple zeta values, including a general formula of the shuffle product of two... 
Multiple zeta values | Shuffle product | MATHEMATICS | DECOMPOSITION | SERIES | DIRICHLET FUNCTIONS
Journal Article
Discrete Applied Mathematics, ISSN 0166-218X, 12/2017, Volume 233, pp. 1 - 18
We define and investigate a family of permutations matrices, called shuffling matrices, acting on a set of N=n1⋯nm elements, where m≥2 and ni≥2 for any... 
Kronecker product | Shuffling matrix | Rooted tree | Perfect shuffle | Discrete Fourier Matrix | MATHEMATICS, APPLIED | CUTOFF PHENOMENON | Mathematics - Combinatorics
Journal Article
SIAM Journal on Scientific Computing, ISSN 1064-8275, 2015, Volume 37, Issue 5, pp. S526 - S543
The infinitesimal generator matrix underlying a multidimensional Markov chain can be represented compactly by using sums of Kronecker products of small rectangular matrices... 
Markov chain | Shuffle algorithm | Vector-Kronecker product multiplication | Kronecker representation | MATHEMATICS, APPLIED | MODELS | MARKOVIAN REPRESENTATIONS | PARALLEL SYSTEMS | vector-Kronecker product multiplication | STOCHASTIC AUTOMATA NETWORKS | BLOCK SOR | MULTILEVEL METHODS | shuffle algorithm | SERVERS
Journal Article
Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, 11/2019, Volume 42, Issue 6, pp. 3047 - 3072
.... We construct free commutative modified Rota–Baxter algebras by a variation of the shuffle product and describe the structure both recursively and explicitly... 
Rota–Baxter algebra | 16S10 | Bialgebra | 16T99 | Modified Rota–Baxter algebra | Mathematics, general | Mathematics | Applications of Mathematics | Hopf algebra | Shuffle product | 16W99 | MATHEMATICS | ASSOCIATIVE ALGEBRAS | Modified Rota-Baxter algebra | Rota-Baxter algebra | O-OPERATORS | Algebra
Journal Article
Information Processing Letters, ISSN 0020-0190, 05/2019, Volume 145, pp. 68 - 73
We show that the shuffle L⊔⊔F of a piecewise-testable language L and a finite language F is piecewise-testable. The proof relies on a classic but little-used... 
Piecewise-testable languages | Subwords and subsequences | Dot-depth-one and star-free languages | Shuffle product | Formal languages | COMPUTER SCIENCE, INFORMATION SYSTEMS | Computer Science - Formal Languages and Automata Theory | Computer Science
Journal Article
Taiwanese Journal of Mathematics, ISSN 1027-5487, 2/2016, Volume 20, Issue 1, pp. 13 - 24
The classical Euler decomposition theorem expressed a product of two Riemann zeta values... 
Integers | Mathematical theorems | Value theorems | Multiple zeta values | Shuffle product | Euler decomposition theorem | MATHEMATICS | SUM FORMULAS
Journal Article
Journal of Algebraic Combinatorics, ISSN 0925-9899, 1/2000, Volume 11, Issue 1, pp. 49 - 68
Journal Article
Journal of Number Theory, ISSN 0022-314X, 11/2014, Volume 144, pp. 219 - 233
In this paper we obtain a recursive formula for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values (MZVs... 
Multiple zeta values | Euler's decomposition formula | Restricted decomposition formula | Shuffle product | MIXED TATE MOTIVES | POLYLOGARITHMS | MATHEMATICS | HARMONIC SERIES | Computer science
Journal Article
Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 02/2013, Volume 398, Issue 1, pp. 392 - 402
.... in terms of MCD copulas and the ∗-product discovered by Darsow, Nguyen and Olsen. Since any shuffle of a copula is the copula of the corresponding shuffle of the two... 
Shuffles of copulas | Shuffles of Min | Measure-preserving | Copulas | Sobolev norm [formula omitted]-product | Measure of dependence | Sobolev norm -product | MATHEMATICS | MATHEMATICS, APPLIED | PRODUCT
Journal Article
Kybernetika, ISSN 0023-5954, 2007, Volume 43, Issue 2, pp. 235 - 244
.... In particular, these constructions are generalizations of the *-product and the *-product for copulas introduced by Darsow, Nguyen and Olsen in 1992... 
Ordinal sum | Shuffle of Min | Copula | Concordance | shuffle of Min | copula | ordinal sum | COMPUTER SCIENCE, CYBERNETICS | concordance
Journal Article
Theoretical Computer Science, ISSN 0304-3975, 2017
.... A word u∈A* is a square for the shuffle product if it is the shuffle of two identical words (i.e., u∈v[U+29E2]v for some v... 
Combinatorics on words | Complexity | Shuffle operator | Data Structures and Algorithms | Computer Science
Journal Article
Journal of Pure and Applied Algebra, ISSN 0022-4049, 06/2020, Volume 224, Issue 6, p. 106260
For a prime number p and a free profinite group S on the basis X, let S(n,p), n=1,2,…, be the lower p-central filtration of S. For p>n, we give a combinatorial... 
Massey products | Lower p-central sequence | Absolute Galois groups | Shuffle algebra | Unipotent upper-triangular representations | Galois cohomology | MATHEMATICS | MATHEMATICS, APPLIED | QUOTIENTS | representations | TRIPLE MASSEY PRODUCTS | Unipotent upper-triangular
Journal Article
Journal of Pure and Applied Algebra, ISSN 0022-4049, 2005, Volume 200, Issue 3, pp. 293 - 317
.... We consider these constructions as operads and their products and duals, in terms of generators and relations, with the goal to clarify and simplify the process of obtaining new algebra structures... 
SHUFFLE PRODUCTS | MATHEMATICS | MATHEMATICS, APPLIED | HOPF-ALGEBRAS | TREES | THEOREM | QUANTUM-FIELD THEORY | BAXTER ALGEBRAS | HOMOLOGY | RENORMALIZATION | DIALGEBRAS
Journal Article
Journal of Algebra and its Applications, ISSN 0219-4988, 08/2014, Volume 13, Issue 5, pp. 1350160 - 1-1350160-38
Journal Article
Ramanujan Journal, ISSN 1382-4090, 2019, Volume 49, Issue 1, pp. 215 - 230
In this paper we give two idelic representations of the multiple zeta valuesone using iterated integrals over the finite ideles and the other using iterated... 
Multiple zeta values | Adeles | Shuffle product | Iterated integrals | MATHEMATICS | Computer science | Community colleges
Journal Article
International Journal of Algebra and Computation, ISSN 0218-1967, 06/2017, Volume 27, Issue 4, pp. 421 - 454
Journal Article
Journal of Algebra, ISSN 0021-8693, 11/2017, Volume 489, pp. 552 - 581
Journal Article
Journal of Algebra, ISSN 0021-8693, 02/2020, Volume 543, pp. 111 - 155
.... Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborified zeta values are algebra morphisms for these new products on trees. 
Multiple zeta values | Rooted trees | Shuffle products | Rota-Baxter algebras | MATHEMATICS
Journal Article
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