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Feynman integrals and difference equations [article]

S. Moch, C. Schneider
2007 arXiv   pre-print
The solution to these difference equations in terms of harmonic sums can be constructed algorithmically over difference fields, the so-called Pi-Sigma-fields.  ...  We report on the calculation of multi-loop Feynman integrals for single-scale problems by means of difference equations in Mellin space.  ...  Next, we compute a degree bound b ∈ N 0 for these polynomial solutions Reduction III (polynomial degree reduction).  ... 
arXiv:0709.1769v1 fatcat:bgqhyqsq4nah5f2cxiiqqfhrnm

Solving parameterized linear difference equations in terms of indefinite nested sums and products

Carsten Schneider
2005 Journal of difference equations and applications (Print)  
The described algorithms enable one to find all solutions of parameterized linear difference equations within ΠΣ-fields, a very general class of difference fields.  ...  These algorithms can be applied to a very general class of multisums, for instance, for proving identities and simplifying expressions.  ...  I would like to thank the anonymous referee for his valuable comments.  ... 
doi:10.1080/10236190500138262 fatcat:xuy2urqcozh2fep7itao3vpidu

Transport Map Accelerated Markov Chain Monte Carlo

Matthew D. Parno, Youssef M. Marzouk
2018 SIAM/ASA Journal on Uncertainty Quantification  
Our approach adaptively constructs a lower triangular transport map-an approximation of the Knothe-Rosenblatt rearrangement-using information from previous MCMC states, via the solution of an optimization  ...  The core idea is to use continuous transportation to transform typical Metropolis proposal mechanisms (e.g., random walks, Langevin methods) into non-Gaussian proposal distributions that can more effectively  ...  The authors would also like to thank F. Augustin, B. Calderhead, T. Cui, M. Girolami, T. Moselhy, A. Solonen, and A. Spantini for many helpful comments and suggestions.  ... 
doi:10.1137/17m1134640 fatcat:qrxxa3cfkjfmhltgmrne3fu3py

A Streamlined Difference Ring Theory: Indefinite Nested Sums, the Alternating Sign, and the Parameterized Telescoping Problem

Carsten Schneider
2014 2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing  
The underlying machinery relies on algorithms that compute all solutions of a given parameterized telescoping equation.  ...  In order to accomplish this task, parts of Karr's difference field theory have been extended to a ring theory in which also the alternating sign can be expressed.  ...  First, we bound the degree of the polynomial solutions.  ... 
doi:10.1109/synasc.2014.12 dblp:conf/synasc/Schneider14 fatcat:2meaio7v7vbwxkahh6imgdyusa

Bethe ansatz in stringy sigma models

T Klose, K Zarembo
2006 Journal of Statistical Mechanics: Theory and Experiment  
We compute the exact S-matrix and give the Bethe ansatz solution for three sigma-models which arise as subsectors of string theory in AdS(5)xS(5): Landau-Lifshitz model (non-relativistic sigma-model on  ...  S(2)), Alday-Arutyunov-Frolov model (fermionic sigma-model with su(1|1) symmetry), and Faddeev-Reshetikhin model (string sigma-model on S(3)xR).  ...  K.Z. would like to thank Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, where this work was completed, for kind hospitality. This work was supported in part by  ... 
doi:10.1088/1742-5468/2006/05/p05006 fatcat:vqp6zeghlvhqviuwhzmg7azyza

Weak Power-Counting Theorem for the Renormalization of the Nonlinear Sigma Model in Four Dimensions

Ruggero Ferrari, Andrea Quadri
2006 International Journal of Theoretical Physics  
The formulation of the non-linear sigma model in terms of flat connection allows the construction of a perturbative solution of a local functional equation encoding the underlying gauge symmetry.  ...  The proof uses the linearized functional equation of which we provide the general solution in terms of local functionals.  ...  In particular one has to find a technique to implement the idea of field redefinition.  ... 
doi:10.1007/s10773-006-9217-x fatcat:pqrj5uv2lvfwhk24yremwjqvla

Elliptic genera and real Jacobi forms

Sujay K. Ashok, Jan Troost
2014 Journal of High Energy Physics  
We decompose the real Jacobi form into a mock modular form and a term arising from the continuous spectrum of the conformal field theory.  ...  They arise in a family labeled by two integers N and k which determine the central charge of the infrared fixed point through the formula c=3N(1+ 2N/k).  ...  This work was supported in part by the ANR grant ANR-09-BLAN-0157-02.  ... 
doi:10.1007/jhep01(2014)082 fatcat:o2kv3vbaqjf6hgueyctk476pmy

Wronskian solution for AdS/CFT Y-system

Nikolay Gromov, Vladimir Kazakov, Sebastien Leurent, Zengo Tsuboi
2011 Journal of High Energy Physics  
Using the discrete Hirota integrability we find the general solution of the full quantum Y-system for the spectrum of anomalous dimensions of operators in the planar AdS5/CFT4 correspondence in terms of  ...  We consider it as a useful step towards the construction of a finite system of non-linear integral equations (FiNLIE) for the full spectrum.  ...  The work of ZT was supported by Nishina Memorial Foundation and by Grant-in-Aid for Young Scientists, B #19740244 from The Ministry of Education, Culture, Sports, Science and Technology in Japan.  ... 
doi:10.1007/jhep01(2011)155 fatcat:uhraomuz4fdk7pstom3ui3izsa

The Hierarchy Principle and the Large Mass Limit of the Linear Sigma Model

D. Bettinelli, R. Ferrari, A. Quadri
2007 International Journal of Theoretical Physics  
While the chiral limit is straightforward, the matching in the strong coupling limit requires careful use of the normalization conditions of the linear theory, in order to exploit the functional equation  ...  and the complete set of local solutions of its linearized form.  ...  This in turn allows to use the local invariant solutions of the linearized functional equation both for Γ N L andΓ.  ... 
doi:10.1007/s10773-007-9371-9 fatcat:jfliqggsifbg3fgvdkx6z37awq

Generalized Discriminant Analysis Using a Kernel Approach

G. Baudat, F. Anouar
2000 Neural Computation  
In the transformed space, linear properties make it easy to extend and generalize the classical Linear Discriminant Analysis (LDA) to non linear discriminant analysis.  ...  Using a different kernel, one can cover a wide class of nonlinearities. For both simulated data and alternate kernels, we give classification results as well as the shape of the separating function.  ...  We also thank Rodrigo Fernandez (Research Associate at the university of Paris Nord) for accepting to compare results using his own SVM classifier software.  ... 
doi:10.1162/089976600300014980 pmid:11032039 fatcat:zxrzdl2oaffexnsgg2vbi36lvu

From Gap-Exponential Time Hypothesis to Fixed Parameter Tractable Inapproximability: Clique, Dominating Set, and More

Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, Luca Trevisan
2020 SIAM journal on computing (Print)  
So, it does not matter whether the running time is polynomial on the size of the input or depends on OPT. 774 CHALERMSOOK ET AL. assumption that solving quadratic equations over a finite field under a  ...  In this paper we focus on the standard parameterization which parameterizes the optimal solution. 2 In fact, for maximization problems, it can be shown that a problem admits an f (OPT)-FPTapproximation  ...  Pasin would like to thank Igor Shinkar and Daniel Reichman for discussions on a related problem that inspires part of the proof, and Aviad Rubinstein for useful discussions regarding FPT inapproximability  ... 
doi:10.1137/18m1166869 fatcat:cfjriedfenae7escnsgsb3xvuu

Finite volume spectrum of 2D field theories from Hirota dynamics

Nikolay Gromov, Vladimir Kazakov, Pedro Vieira
2009 Journal of High Energy Physics  
It is close in spirit to the Destri-deVega (DdV) equation.  ...  We propose, using the example of the O(4) sigma model, a general method for solving integrable two dimensional relativistic sigma models in a finite size periodic box.  ...  We also would like to thank A.Tsvelik, J.Teschner, V.Schomerus, K.Zarembo for many enlightening discussions, as well as A.Kozak who participated in this project on its the early stage.  ... 
doi:10.1088/1126-6708/2009/12/060 fatcat:t7qmhlew7rculamxolbpnj4yhm

Strong-Interaction Matter under Extreme Conditions from Chiral Quark Models with Nonlocal Separable Interactions

Daniel Gómez Dumm, Juan Pablo Carlomagno, Norberto N. Scoccola
2021 Symmetry  
, the effects produced by strong external magnetic fields and the application to the description of compact stellar objects.  ...  The analysis includes the predictions for various hadron properties in vacuum, as well as the study of the features of deconfinement and chiral restoration phase transitions for systems at finite temperature  ...  Dashed and solid lines correspond to parameterizations PI (Equation ( Table 9 . 9 Numerical results for mean field values of scalar fields and chiral quark condensates.  ... 
doi:10.3390/sym13010121 fatcat:ohd5q7jfpnfa3jwuissmdtl5uu

Quantum adiabatic algorithms and large spin tunnelling

A. Boulatov, V. N. Smelyanskiy
2003 Physical Review A. Atomic, Molecular, and Optical Physics  
We show that different control Hamiltonians can be parameterized by a set of independent parameters that are expansion coefficients of H(\tau) in a certain universal set of operators.  ...  We show that it is possible to select a coefficient for this operator that guarantees a polynomial complexity of the algorithm for all problem instances.  ...  For a successfully designed family of paths the fraction f is bounded from below by a polynomial in l / n which leads to the average polynomial complexity of QAA. III.  ... 
doi:10.1103/physreva.68.062321 fatcat:mfy26nf4yng2vkds5n2aosgzc4

Off-shell renormalization in the presence of dimension 6 derivative operators. Part I. General theory

D. Binosi, A. Quadri
2019 Journal of High Energy Physics  
This allows one to disentangle the gauge-invariant contributions to off-shell 1-PI amplitudes from those associated with the gauge-fixing and (generalized) non-polynomial field redefinitions (that do appear  ...  We solve the Slavnov-Taylor identity to all orders in the loop expansion by homotopy techniques and a suitable choice of invariant field coordinates (named bleached variables) for the linearly realized  ...  This boils down to the solution of a linear system associated with a change of basis on a space of local functionals, bounded by the power counting (operators up to dimension 12 in the one-loop case at  ... 
doi:10.1007/jhep09(2019)032 fatcat:u5e75u6lkncobpu2ujyib7swym
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