We consider the sequence of integers whose
n
th term has base-
p
expansion given by the
n
th row of Pascal’s triangle modulo
p
(where
p
is a prime number). We first present and generalize well-known ...relations concerning this sequence. Then, with the great help of Sloane’s On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as a subsequence of a 2-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involving alternating sum-of-digits modulo
p
. This note ends with a discussion about Pascal’s pyramid built with trinomial coefficients.
On tortkara triple systems Bremner, Murray
Communications in algebra,
06/2018, Letnik:
46, Številka:
6
Journal Article
Recenzirano
The commutator a,b = ab−ba in a free Zinbiel algebra (dual Leibniz algebra) is an anticommutative operation which satisfies no new relations in arity 3. Dzhumadildaev discovered a relation T(a,b,c,d) ...which he called the tortkara identity and showed that it implies every relation satisfied by the Zinbiel commutator in arity 4. Kolesnikov constructed examples of anticommutative algebras satisfying T(a,b,c,d) which cannot be embedded into the commutator algebra of a Zinbiel algebra. We consider the tortkara triple product a,b,c = a,b,c in a free Zinbiel algebra and use computer algebra to construct a relation TT(a,b,c,d,e) which implies every relation satisfied by a,b,c in arity 5. Thus, although tortkara algebras are defined by a cubic binary operad (with no Koszul dual), the corresponding triple systems are defined by a quadratic ternary operad (with a Koszul dual). We use computer algebra to construct a relation in arity 7 satisfied by a,b,c which does not follow from the relations of lower arity. It remains an open problem to determine whether there are further new identities in arity n≥9.
In the present paper we investigate minimal supervarieties of given superexponent over fields of characteristic zero. We show that any minimal supervariety of finite basic rank is generated by one of ...the minimal superalgebras, introduced by Giambruno and Zaicev in 2003. Furthermore it is proved that any minimal superalgebra, whose graded simple components of the semisimple part are simple, generates a minimal supervariety. Finally we state that the same conclusion holds when the semisimple part of a minimal superalgebra has exactly two arbitrary graded simple components.
The third part of our survey series concerning the evaluation of matrix elements (MEs) of the unitary group generators and of their products in the electronic Gel’fand–Tsetlin basis of the two-column ...irreps of U(
n
)—which are essential in the unitary group approach (UGA) to the many-electron correlation problem as handled by the configuration interaction (CI) and the coupled cluster (CC) approaches—relies on what we refer to as the
Green-Gould
(G-G)
approach
. In addition to the CI and CC methods, the G-G formalism proved to be very helpful in a number of other tasks, particularly in handling of the spin-dependent operators, the density matrices, or partitioned basis sets adapted to a chosen group chain.
Let F be an algebraically closed field of characteristic zero and let G be a finite group. In this paper we will show that the asymptotics of cnG(A), the G-graded codimension sequence of a finite ...dimensional G-simple F-algebra A, has the form αn1−dimFAe2(dimFA)n (as conjectured by E. Aljadeff, D. Haile and M. Natapov), where α is some positive real number and Ae denotes the identity component of A. In the special case where A is the algebra of m×m matrices with an arbitrary elementary G-grading we succeeded in calculating the constant α explicitly.
Let Z〈X〉 be the free unital associative ring freely generated by an infinite countable set X={x1,x2,…}. Define a left-normed commutator x1,x2,…,xn by a,b=ab−ba, a,b,c=a,b,c. For n⩾2, let T(n) be the ...two-sided ideal in Z〈X〉 generated by all commutators a1,a2,…,an(ai∈Z〈X〉). It can be easily seen that the additive group of the quotient ring Z〈X〉/T(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z〈X〉/T(3) is also free abelian. In the present note we show that this is not the case for Z〈X〉/T(4). More precisely, let T(3,2) be the ideal in Z〈X〉 generated by T(4) together with all elements a1,a2,a3a4,a5(ai∈Z〈X〉). We prove that T(3,2)/T(4) is a non-trivial elementary abelian 3-group and the additive group of Z〈X〉/T(3,2) is free abelian.
Let
$F$
be an algebraically closed field of characteristic zero, and let
$A$
be an associative unitary
$F$
-algebra graded by a group of prime order. We prove that if
$A$
is finite dimensional then ...the graded exponent of
$A$
exists and is an integer.
Let
F be an infinite field of characteristic
p
>
2
and let
E be the Grassmann algebra generated by an infinite dimensional vector space
V over
F. In this paper, we describe the
T
2
-ideal of the
Z
2
...-graded polynomial identities of the Grassmann algebra
E for any
Z
2
-grading such that
V is homogeneous in the grading. In particular, we give a description of the
T
2
-ideal of the graded identities of
E in the case there is a finite number of homogeneous elements of the linear basis of
E belonging to one of the homogenous components of
E
.
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