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Détails du magazine
Format
Magazine
eISSN
1788-800X
Première publication
30 Mar 2015
Période de publication
4 fois par an
Langues
Anglais

Chercher

Volume 27 (2018): Edition 1 (July 2018)

Détails du magazine
Format
Magazine
eISSN
1788-800X
Première publication
30 Mar 2015
Période de publication
4 fois par an
Langues
Anglais

Chercher

5 Articles
access type Accès libre

On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras

Publié en ligne: 06 Aug 2018
Pages: 1 - 17

Résumé

Abstract

The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let ℛ be a 2-torsion free semiprime *-ring. In this paper it has been shown that, if admits an additive mapping D : ℛ→ℛsatisfying either D(xyx) = D(xy)x*+ xyD(x) for all x,y, or D(xyx) = D(x)y*x*+ xD(yx) for all pairs x, y, then D is a *-derivation. Moreover this result makes it possible to prove that if satis es 2D(xn) = D(xn−1)x* + xn−1D(x) + D(x)(x*)n−1 + xD(xn−1) for all x and some xed integer n ≥ 2, then D is a Jordan *-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras 𝒜(). In particular, we prove that if be a real or complex Hilbert space, with dim() > 1, admitting a linear mapping D : 𝒜() → ℬ() (where () stands for the bounded linear operators) such that

2D(An)=D(An1)A*+An1D(A)+D(A)(A*)n1+AD(An1)$$2D\left( {A^n } \right) = D\left( {A^{n - 1} } \right)A^* + A^{n - 1} D\left( A \right) + D\left( A \right)\left( {A^* } \right)^{n - 1} + AD\left( {A^{n - 1} } \right)$$

for all A𝒜(). Then D is of the form D(A) = AB−BA* for all A𝒜() and some fixed B(), which means that D is Jordan *-derivation.

Mots clés

  • prime ring
  • semiprime ring
  • standard operator algebra
  • Jordan *derivation
  • Jordan triple *-derivation

MSC 2010

  • 16N60
  • 16W10
  • 46K15
  • 16W25
access type Accès libre

Forests and pattern-avoiding permutations modulo pure descents

Publié en ligne: 06 Aug 2018
Pages: 18 - 31

Résumé

Abstract

We investigate an equivalence relation on permutations based on the pure descent statistic. Generating functions are given for the number of equivalence classes for the set of all permutations, and the sets of permutations avoiding exactly one pattern of length three. As a byproduct, we exhibit a permutation set in one-to-one correspondence with forests of ordered binary trees, which provides a new combinatorial class enumerated by the single-source directed animals on the square lattice. Furthermore, bivariate generating functions for these sets are given according to various statistics.

Mots clés

  • permutation
  • equivalence class
  • pure descent
  • pattern
  • Catalan and Motzkin numbers
  • forest
  • directed animal

MSC 2010

  • 05A05
  • 05A15
  • 05A19
access type Accès libre

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

Publié en ligne: 06 Aug 2018
Pages: 32 - 61

Résumé

Abstract

Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.

Mots clés

  • pattern avoidance
  • generating function
  • generating tree
  • kernel method

MSC 2010

  • 05A05
  • 05A15
access type Accès libre

Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns

Publié en ligne: 06 Aug 2018
Pages: 62 - 97

Résumé

Abstract

This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is shown to be algebraic of degree 3. We use standard methods, usually involving detailed consideration of the left to right maxima, and sometimes the initial letters, to obtain an algebraic or functional equation for the generating function.

Mots clés

  • pattern avoidance
  • Wilf-equivalence
  • generating function
  • kernel method

MSC 2010

  • 05A05
  • 05A15
access type Accès libre

A generalization of André-Jeannin’s symmetric identity

Publié en ligne: 06 Aug 2018
Pages: 98 - 118

Résumé

Abstract

In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers Wn, defined by a three-term recurrence Wn+2 = P Wn+1− QWn with constant coefficients. In this paper, we extend this identity to sequences {an}n∈ℕ satisfying a three-term recurrence an+2 = pn+1an+1 + qn+1an with arbitrary coefficients. Then, we specialize such an identity to several q-polynomials of combinatorial interest, such as the q-Fibonacci, q-Lucas, q-Pell, q-Jacobsthal, q-Chebyshev and q-Morgan-Voyce polynomials.

Mots clés

  • combinatorial sums
  • sums of reciprocals
  • three-term recurrences
  • -Fibonacci polynomials
  • -Fibonacci numbers
  • -Lucas polynomials
  • -Lucas numbers
  • -Pell polynomials
  • -Pell numbers
  • -Jacobsthal polynomials
  • -Jacobsthal numbers
  • -Chebyshev polynomials
  • -Morgan-Voyce polynomials

MSC 2010

  • Primary 05A19
  • Secondary 05A30, 11B65
5 Articles
access type Accès libre

On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras

Publié en ligne: 06 Aug 2018
Pages: 1 - 17

Résumé

Abstract

The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let ℛ be a 2-torsion free semiprime *-ring. In this paper it has been shown that, if admits an additive mapping D : ℛ→ℛsatisfying either D(xyx) = D(xy)x*+ xyD(x) for all x,y, or D(xyx) = D(x)y*x*+ xD(yx) for all pairs x, y, then D is a *-derivation. Moreover this result makes it possible to prove that if satis es 2D(xn) = D(xn−1)x* + xn−1D(x) + D(x)(x*)n−1 + xD(xn−1) for all x and some xed integer n ≥ 2, then D is a Jordan *-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras 𝒜(). In particular, we prove that if be a real or complex Hilbert space, with dim() > 1, admitting a linear mapping D : 𝒜() → ℬ() (where () stands for the bounded linear operators) such that

2D(An)=D(An1)A*+An1D(A)+D(A)(A*)n1+AD(An1)$$2D\left( {A^n } \right) = D\left( {A^{n - 1} } \right)A^* + A^{n - 1} D\left( A \right) + D\left( A \right)\left( {A^* } \right)^{n - 1} + AD\left( {A^{n - 1} } \right)$$

for all A𝒜(). Then D is of the form D(A) = AB−BA* for all A𝒜() and some fixed B(), which means that D is Jordan *-derivation.

Mots clés

  • prime ring
  • semiprime ring
  • standard operator algebra
  • Jordan *derivation
  • Jordan triple *-derivation

MSC 2010

  • 16N60
  • 16W10
  • 46K15
  • 16W25
access type Accès libre

Forests and pattern-avoiding permutations modulo pure descents

Publié en ligne: 06 Aug 2018
Pages: 18 - 31

Résumé

Abstract

We investigate an equivalence relation on permutations based on the pure descent statistic. Generating functions are given for the number of equivalence classes for the set of all permutations, and the sets of permutations avoiding exactly one pattern of length three. As a byproduct, we exhibit a permutation set in one-to-one correspondence with forests of ordered binary trees, which provides a new combinatorial class enumerated by the single-source directed animals on the square lattice. Furthermore, bivariate generating functions for these sets are given according to various statistics.

Mots clés

  • permutation
  • equivalence class
  • pure descent
  • pattern
  • Catalan and Motzkin numbers
  • forest
  • directed animal

MSC 2010

  • 05A05
  • 05A15
  • 05A19
access type Accès libre

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

Publié en ligne: 06 Aug 2018
Pages: 32 - 61

Résumé

Abstract

Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.

Mots clés

  • pattern avoidance
  • generating function
  • generating tree
  • kernel method

MSC 2010

  • 05A05
  • 05A15
access type Accès libre

Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns

Publié en ligne: 06 Aug 2018
Pages: 62 - 97

Résumé

Abstract

This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is shown to be algebraic of degree 3. We use standard methods, usually involving detailed consideration of the left to right maxima, and sometimes the initial letters, to obtain an algebraic or functional equation for the generating function.

Mots clés

  • pattern avoidance
  • Wilf-equivalence
  • generating function
  • kernel method

MSC 2010

  • 05A05
  • 05A15
access type Accès libre

A generalization of André-Jeannin’s symmetric identity

Publié en ligne: 06 Aug 2018
Pages: 98 - 118

Résumé

Abstract

In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers Wn, defined by a three-term recurrence Wn+2 = P Wn+1− QWn with constant coefficients. In this paper, we extend this identity to sequences {an}n∈ℕ satisfying a three-term recurrence an+2 = pn+1an+1 + qn+1an with arbitrary coefficients. Then, we specialize such an identity to several q-polynomials of combinatorial interest, such as the q-Fibonacci, q-Lucas, q-Pell, q-Jacobsthal, q-Chebyshev and q-Morgan-Voyce polynomials.

Mots clés

  • combinatorial sums
  • sums of reciprocals
  • three-term recurrences
  • -Fibonacci polynomials
  • -Fibonacci numbers
  • -Lucas polynomials
  • -Lucas numbers
  • -Pell polynomials
  • -Pell numbers
  • -Jacobsthal polynomials
  • -Jacobsthal numbers
  • -Chebyshev polynomials
  • -Morgan-Voyce polynomials

MSC 2010

  • Primary 05A19
  • Secondary 05A30, 11B65

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