43 research outputs found
Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order
We employ computer algebra algorithms to prove a collection of identities
involving Bessel functions with half-integer orders and other special
functions. These identities appear in the famous Handbook of Mathematical
Functions, as well as in its successor, the DLMF, but their proofs were lost.
We use generating functions and symbolic summation techniques to produce new
proofs for them.Comment: Final version, some typos were corrected. 21 pages, uses svmult.cl
Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices
We study the face-centered cubic lattice (fcc) in up to six dimensions. In
particular, we are concerned with lattice Green's functions (LGF) and return
probabilities. Computer algebra techniques, such as the method of creative
telescoping, are used for deriving an ODE for a given LGF. For the four- and
five-dimensional fcc lattices, we give rigorous proofs of the ODEs that were
conjectured by Guttmann and Broadhurst. Additionally, we find the ODE of the
LGF of the six-dimensional fcc lattice, a result that was not believed to be
achievable with current computer hardware.Comment: 16 pages, final versio
From Moments to Functions in Quantum Chromodynamics
Single-scale quantities, like the QCD anomalous dimensions and Wilson
coefficients, obey difference equations. Therefore their analytic form can be
determined from a finite number of moments. We demonstrate this in an explicit
calculation by establishing and solving large scale recursions by means of
computer algebra for the anomalous dimensions and Wilson coefficients in
unpolarized deeply inelastic scattering from their Mellin moments to 3-loop
order.Comment: 7 pages, 2 subsidiary file
On the functions counting walks with small steps in the quarter plane
Models of spatially homogeneous walks in the quarter plane
with steps taken from a subset of the set of jumps to the eight
nearest neighbors are considered. The generating function of the numbers of such walks starting at the origin and
ending at after steps is studied. For all
non-singular models of walks, the functions and are continued as multi-valued functions on having
infinitely many meromorphic branches, of which the set of poles is identified.
The nature of these functions is derived from this result: namely, for all the
51 walks which admit a certain infinite group of birational transformations of
, the interval of variation of splits into
two dense subsets such that the functions and are shown to be holonomic for any from the one of them and
non-holonomic for any from the other. This entails the non-holonomy of
, and therefore proves a conjecture of
Bousquet-M\'elou and Mishna.Comment: 40 pages, 17 figure
Iterated Binomial Sums and their Associated Iterated Integrals
We consider finite iterated generalized harmonic sums weighted by the
binomial in numerators and denominators. A large class of these
functions emerges in the calculation of massive Feynman diagrams with local
operator insertions starting at 3-loop order in the coupling constant and
extends the classes of the nested harmonic, generalized harmonic and cyclotomic
sums. The binomially weighted sums are associated by the Mellin transform to
iterated integrals over square-root valued alphabets. The values of the sums
for and the iterated integrals at lead to new
constants, extending the set of special numbers given by the multiple zeta
values, the cyclotomic zeta values and special constants which emerge in the
limit of generalized harmonic sums. We develop
algorithms to obtain the Mellin representations of these sums in a systematic
way. They are of importance for the derivation of the asymptotic expansion of
these sums and their analytic continuation to . The
associated convolution relations are derived for real parameters and can
therefore be used in a wider context, as e.g. for multi-scale processes. We
also derive algorithms to transform iterated integrals over root-valued
alphabets into binomial sums. Using generating functions we study a few aspects
of infinite (inverse) binomial sums.Comment: 62 pages Latex, 1 style fil
Non-geometric flux vacua, S-duality and algebraic geometry
The four dimensional gauged supergravities descending from non-geometric
string compactifications involve a wide class of flux objects which are needed
to make the theory invariant under duality transformations at the effective
level. Additionally, complex algebraic conditions involving these fluxes arise
from Bianchi identities and tadpole cancellations in the effective theory. In
this work we study a simple T and S-duality invariant gauged supergravity, that
of a type IIB string compactified on a orientifold with
O3/O7-planes. We build upon the results of recent works and develop a
systematic method for solving all the flux constraints based on the algebra
structure underlying the fluxes. Starting with the T-duality invariant
supergravity, we find that the fluxes needed to restore S-duality can be simply
implemented as linear deformations of the gauge subalgebra by an element of its
second cohomology class. Algebraic geometry techniques are extensively used to
solve these constraints and supersymmetric vacua, centering our attention on
Minkowski solutions, become systematically computable and are also provided to
clarify the methods.Comment: 47 pages, 10 tables, typos corrected, Accepted for Publication in
Journal of High Energy Physic
Algebraic Comparison of Partial Lists in Bioinformatics
The outcome of a functional genomics pipeline is usually a partial list of
genomic features, ranked by their relevance in modelling biological phenotype
in terms of a classification or regression model. Due to resampling protocols
or just within a meta-analysis comparison, instead of one list it is often the
case that sets of alternative feature lists (possibly of different lengths) are
obtained. Here we introduce a method, based on the algebraic theory of
symmetric groups, for studying the variability between lists ("list stability")
in the case of lists of unequal length. We provide algorithms evaluating
stability for lists embedded in the full feature set or just limited to the
features occurring in the partial lists. The method is demonstrated first on
synthetic data in a gene filtering task and then for finding gene profiles on a
recent prostate cancer dataset
SQCD: A Geometric Apercu
We take new algebraic and geometric perspectives on the old subject of SQCD.
We count chiral gauge invariant operators using generating functions, or
Hilbert series, derived from the plethystic programme and the Molien-Weyl
formula. Using the character expansion technique, we also see how the global
symmetries are encoded in the generating functions. Equipped with these methods
and techniques of algorithmic algebraic geometry, we obtain the character
expansions for theories with arbitrary numbers of colours and flavours.
Moreover, computational algebraic geometry allows us to systematically study
the classical vacuum moduli space of SQCD and investigate such structures as
its irreducible components, degree and syzygies. We find the vacuum manifolds
of SQCD to be affine Calabi-Yau cones over weighted projective varieties.Comment: 49 pages, 1 figur
Simplifying Multiple Sums in Difference Fields
In this survey article we present difference field algorithms for symbolic
summation. Special emphasize is put on new aspects in how the summation
problems are rephrased in terms of difference fields, how the problems are
solved there, and how the derived results in the given difference field can be
reinterpreted as solutions of the input problem. The algorithms are illustrated
with the Mathematica package \SigmaP\ by discovering and proving new harmonic
number identities extending those from (Paule and Schneider, 2003). In
addition, the newly developed package \texttt{EvaluateMultiSums} is introduced
that combines the presented tools. In this way, large scale summation problems
for the evaluation of Feynman diagrams in QCD (Quantum ChromoDynamics) can be
solved completely automatically.Comment: Uses svmult.cls, to appear as contribution in the book "Computer
Algebra in Quantum Field Theory: Integration, Summation and Special
Functions" (www.Springer.com