5,485 research outputs found
Elementary Evaluation of Convolution Sums involving the Sum of Divisors Function for a Class of positive Integers
We discuss an elementary method for the evaluation of the convolution sums
for those for which
and , where
and is a finite product of distinct odd primes.
Modular forms are used to achieve this result. We also generalize the
extraction of the convolution sum to all natural numbers. Formulae for the
number of representations of a positive integer by octonary quadratic forms
using convolution sums belonging to this class are then determined when
or . To achieve this
application, we first discuss a method to compute all pairs
necessary for the determination of such formulae
for the number of representations of a positive integer by octonary
quadratic forms when has the above form and or . We illustrate our approach by
explicitly evaluating the convolution sum for and , and by
revisiting the evaluation of the convolution sums for , ,
, , . We then apply these convolution sums to determine formulae
for the number of representations of a positive integer by octonary
quadratic forms. In addition, we determine formulae for the number of
representations of a positive integer when , , ,
.Comment: 29 pages, 8 table
Representations of simple noncommutative Jordan superalgebras I
In this article we begin the study of representations of simple
finite-dimensional noncommutative Jordan superalgebras. In the case of degree
we show that any finite-dimensional representation is completely
reducible and, depending on the superalgebra, quasiassociative or Jordan. Then
we study representations of superalgebras and
and prove the Kronecker factorization theorem for
superalgebras . In the last section we use a new
approach to study noncommutative Jordan representations of simple Jordan
superalgebras
Evaluation of Convolution Sums entailing mixed Divisor Functions for a Class of Levels
Let be such that .
We carry out the evaluation of the convolution sums and for all levels , by
using in particular modular forms. We next apply convolution sums belonging to
this class of levels to determine formulae for the number of representations of
a positive integer by the quadratic forms in twelve variables
when the level , and when the level .
Our approach is then illustrated by explicitly evaluating the convolution sum
for , , , , , , , , , , ,
, , , . These convolution sums are then applied to determine
explicit formulae for the number of representations of a positive integer
by quadratic forms in twelve variables.Comment: 51 pages, 9 tables. arXiv admin note: text overlap with
arXiv:1609.01343, arXiv:1607.0108
The Structure Constants of the Exceptional Lie Algebra in the Cartan-Weyl Basis
The purpose of this paper is to answer the question whether it is possible to
realize simultaneously the relations ,
and
by the structure constants of the Lie algebra . We show that
if the structure constants obey the first relation, the three last ones are
violated, and vice versa. Contrary to the second case, the first one uses the
Cartan matrix elements to derive the structure constants in the form of
. The commutation relations corresponding to
the first case are exactly documented in the prior literature. However, as
expected, a Lie algebra isomorphism is established between the Cartan-Weyl
bases obtained in both approaches.Comment: 15 pages, 2 table
Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category
If H is a Hopf algebra with bijective antipode and \alpha, \beta \in
Aut_{Hopf}(H), we introduce a category_H{\cal YD}^H(\alpha, \beta),
generalizing both Yetter-Drinfeld and anti-Yetter-Drinfeld modules. We
construct a braided T-category {\cal YD}(H) having all these categories as
components, which if H is finite dimensional coincides with the representations
of a certain quasitriangular T-coalgebra DT(H) that we construct. We also prove
that if (\alpha, \beta) admits a so-called pair in involution, then_H{\cal
YD}^H(\alpha, \beta) is isomorphic to the category of usual Yetter-Drinfeld
modules_H{\cal YD}^H.Comment: 12 pages, Latex, no figure
Representation of the Gauss hypergeometric function by multiple polylogarithms and relations of multiple zeta values
We describe a solution of the Gauss hypergeometric equation,
by power series in paramaters
whose coefficients are linear combinations of multiple polylogarithms. And
using the representation and connection reletions of solutions of the
hypergeometric equation, we show some relations of multiple zeta values
Grey Brownian motion local time: Existence and weak-approximation
In this paper we investigate the class of grey Brownian motions
(, ). We show that grey Brownian
motion admits different representations in terms of certain known processes,
such as fractional Brownian motion, multivariate elliptical distribution or as
a subordination. The weak convergence of the increments of
in , -variables are studied. Using the Berman criterium we show that
admits a -square integrable local time
almost surely ( Lebesgue measure).
Moreover, we prove that this local time can be weak-approximated by the number
of crossings , of level , of the
convolution approximation of grey Brownian
motion.Comment: 20 page
C*-algebras associated with topological group quivers II: K-groups
Topological quivers generalize the notion of directed graphs in which the
sets of vertices and edges are locally compact (second countable) Hausdorff
spaces. Associated to a topological quiver is a -correspondence, and
in turn, a Cuntz-Pimsner algebra Given a locally compact
group and and endomorphisms on one may construct a
topological quiver with vertex set and
edge set \Omega_{\alpha,\beta}(\Gamma)= \{(x,y)\in\Gamma\times\Gamma\st
\alpha(y)=\beta(x)\}. In \cite{Mc1}, the author examined the Cuntz-Pimsner
algebra \cO_{\alpha,\beta}(\Gamma):=C^*(Q_{\alpha,\beta}(\Gamma)) and found
generators (and their relations) of \cO_{\alpha,\beta}(\Gamma). In this
paper, the author uses this information to create a six term exact sequence in
order to calculate the -groups of \cO_{\alpha,\beta}(\Gamma).Comment: 27 page
Homomorphisms from Specht Modules to Signed Young Permutation Modules
We construct a class of homomorphisms from a Specht
module to a signed permutation module
which generalises James's construction of
homomorphisms whose codomain is a Young permutation module. We show that any
lies in the -span of
, a subset of corresponding to
semistandard -tableaux of type . We also study the
conditions for which - a subset of
induced by - is linearly independent, and show that it
is a basis for
when is semisimple
The structure of simple noncommutative Jordan superalgebras
In this paper we describe all subalgebras and automorphisms of simple
noncommutative Jordan superalgebras and
and compute the derivations of the nontrivial simple
finite-dimensional noncommutative Jordan superalgebras
- …