In this paper, we consider Barnes’ type multiple degenerate Bernoulli and Euler polynomials and numbers which are derived from multivariate bosonic (or fermionic) p-adic integral on Zp, and we ...investigate some properties and identities of those polynomials.
A multivariable generalization of Faulhaber's formula is demonstrated and, using Barnes zeta functions, a multivariable generalization of an asymptotic formula for sums of non-integral powers of ...integers is shown. We also give an asymptotic formula for Barnes zeta functions.
Three loop ladder and V-topology diagrams contributing to the massive operator matrix element AQg are calculated. The corresponding objects can all be expressed in terms of nested sums and ...recurrences depending on the Mellin variable N and the dimensional parameter ε. Given these representations, the desired Laurent series expansions in ε can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin–Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist–Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product–sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of N are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of V-topologies.
We present an algorithm for generating approximations for the logarithm of Barnes G-function in the half-plane Re(z)≥3/2. These approximations involve only elementary functions and are easy to ...implement. The algorithm is based on a two-point Padé approximation and we use it to provide two approximations to ln(G(z)), accurate to 3×10−16 and 3×10−31 in the half-plane Re(z)≥3/2; a reflection formula is then used to compute Barnes G-function in the entire complex plane. A by-product of our algorithm is that it also produces accurate approximations to the gamma function.
The Modernist Anthropocene examines how modernist writers forged new and innovative ways of responding to rapidly changing planetary conditions and emergent ideas about nonhuman life, environmental ...change and the human species. Drawing on ecocritical analysis, posthumanist theory, archival research and environmental history, this book resituates key works of modernist fiction within the ecological moment of the early twentieth century, a period in which new configurations of the relationship between human life and the natural world were migrating between the sciences, philosophy and literary culture. The author makes the case that the early twentieth century is pivotal in our understanding of the Anthropocene both as a planetary epoch and a critical concept. In doing so, he positions James Joyce, Djuna Barnes and Virginia Woolf as theorists of the modernist Anthropocene, showing how their oeuvres are shaped by, and actively respond to, changing ideas about the nonhuman that continue to reverberate today.
This paper describe a package written in
MATHEMATICA that automatizes typical operations performed during evaluation of Feynman graphs with Mellin–Barnes (MB) techniques. The main procedure allows to ...analytically continue a MB integral in a given parameter without any intervention from the user and thus to resolve the singularity structure in this parameter. The package can also perform numerical integrations at specified kinematic points, as long as the integrands have satisfactory convergence properties. It is demonstrated that, at least in the case of massive graphs in the physical region, the convergence may turn out to be poor, making naïve numerical integration of MB integrals unusable. Possible solutions to this problem are presented, but full automatization in such cases may not be achievable.
Title of program: MB
Program summary URL:
http://cpc.cs.qub.ac.uk/summaries/ADYG_v1_0
Catalogue identifier: ADYG_v1_0
Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland
Computers: All
Operating systems: All
Programming language used:
MATHEMATICA,
Fortran 77 for numerical evaluation
Memory required to execute with typical data: Sufficient for a typical installation of
MATHEMATICA.
No. of lines in distributed program, including test data, etc.: 12 013
No. of bytes in distributed program, including test data, etc.: 231 899
Distribution format: tar.gz
Libraries used:
CUBA T. Hahn, Comput. Phys. Commun. 168 (2005) 78 for numerical evaluation of multidimensional integrals and
CERNlib CERN Program Library, obtainable from:
http://cernlib.web.cern.ch/cernlib/ for the implementation of Γ and
ψ functions in
Fortran.
Nature of physical problem: Analytic continuation of Mellin–Barnes integrals in a parameter and subsequent numerical evaluation. This is necessary for evaluation of Feynman integrals from Mellin–Barnes representations.
Method of solution: Recursive accumulation of residue terms occurring when singularities cross integration contours. Numerical integration of multidimensional integrals with the help of the
CUBA library.
Restrictions on the complexity of the problem: Limited by the size of the available storage space.
Typical running time: Depending on the problem. Usually seconds for moderate dimensionality integrals.
The Mathematica toolkit
AMBRE derives Mellin–Barnes (MB) representations for Feynman integrals in
d
=
4
−
2
ε
dimensions. It may be applied for tadpoles as well as for multi-leg multi-loop scalar and ...tensor integrals. The package uses a loop-by-loop approach and aims at lowest dimensions of the final MB representations. The present version works fine for planar Feynman diagrams. The output may be further processed by the package
MB for the determination of its singularity structure in
ε. The
AMBRE package contains various sample applications for Feynman integrals with up to six external particles and up to four loops.
Program title:AMBRE
Catalogue identifier:ADZR_v1_0
Program summary URL:
http://cpc.cs.qub.ac.uk/summaries/ADZR_v1_0.html
Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland
Licensing provisions:standard CPC licence,
http://cpc.cs.qub.ac.uk/licence/licence.html
No. of lines in distributed program, including test data, etc.:21 387
No. of bytes in distributed program, including test data, etc.:100 004
Distribution format:tar.gz
Programming language:MATHEMATICA v.5.0 and later versions
Computer:all
Operating system:all
RAM:sufficient for a typical installation of MATHEMATICA
Classification:5; 11.1
External routines:The examples in the package use:
–
MB.m M. Czakon, Comput. Phys. Commun. 175 (2006) 559 (CPC Cat. Id. ADYG), for expansions in
ε;
–
CUBA T. Hahn, Comput. Phys. Commun. 168 (2005) 78 (CPC Cat. Id. ADVH), for numerical evaluation of multidimensional integrals;
–
CERNlib CERN Program Library,
http://cernlib.web.cern.ch/cernlib/, for the implementation of
Γ and
Ψ functions in Fortran.
Nature of problem:Derivation of a representation for a Feynman diagram with
L loops and
N internal lines in
d dimensions by Mellin–Barnes integrals; the subsequent evaluation, after an analytical continuation in
ε
=
(
4
−
d
)
/
2
, has to be done with other packages.
Solution method:Introduction of
N Feynman parameters
x
i
, integration over the loop momenta, and subsequent integration over
x, introducing thereby representations of sums of monomials in
x by Mellin–Barnes integrals.
Restrictions:Limited by the size of the available storage space.
Running time:Depending on the problem; usually seconds.
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