36,354 research outputs found
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
Words and polynomial invariants of finite groups in non-commutative variables
Let V be a complex vector space with basis {x_1,x_2,...,x_n} and G be a
finite subgroup of GL(V). The tensor algebra T(V) over the complex is
isomorphic to the polynomials in the non-commutative variables x_1, x_2,...,
x_n with complex coefficients. We want to give a combinatorial interpretation
for the decomposition of T(V) into simple G-modules. In particular, we want to
study the graded space of invariants in T(V) with respect to the action of G.
We give a general method for decomposing the space T(V) into simple modules in
terms of words in a Cayley graph of the group G. To apply the method to a
particular group, we require a homomorphism from a subalgebra of the group
algebra into the character algebra. In the case of G as the symmetric group, we
give an example of this homomorphism from the descent algebra. When G is the
dihedral group, we have a realization of the character algebra as a subalgebra
of the group algebra. In those two cases, we have an interpretation for the
graded dimensions of the invariant space in term of those words
Sylvester: Ushering in the Modern Era of Research on Odd Perfect Numbers
In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester\u27s bound stood as the best demonstrated until Gradstein improved it by one in 1925. Today, we know that the number of distinct prime divisors that an odd perfect number can have is at least eight. This was demonstrated by Chein in 1979 in his doctoral thesis. However, he published nothing of it. A complete proof consisting of almost 200 manuscript pages was given independently by Hagis. An outline of it appeared in 1980.
What motivated Sylvester\u27s sudden interest in odd perfect numbers? Moreover, we also ask what prompted this mathematician who was primarily noted for his work in algebra to periodically direct his attention to famous unsolved problems in number theory? The objective of this paper is to formulate a response to these questions, as well as to substantiate the assertion that much of the modern work done on the subject of odd perfect numbers has as it roots, the series of papers produced by Sylvester in 1888
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