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Feynman integrals and difference equations
[article]

2007
*
arXiv
*
pre-print

The

arXiv:0709.1769v1
fatcat:bgqhyqsq4nah5f2cxiiqqfhrnm
*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). ...##
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Solving parameterized linear difference equations in terms of indefinite nested sums and products

2005
*
Journal of difference equations and applications (Print)
*

The described algorithms enable one

doi:10.1080/10236190500138262
fatcat:xuy2urqcozh2fep7itao3vpidu
*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. ...##
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Transport Map Accelerated Markov Chain Monte Carlo

2018
*
SIAM/ASA Journal on Uncertainty Quantification
*

Our approach adaptively constructs a lower triangular transport map-an approximation

doi:10.1137/17m1134640
fatcat:qrxxa3cfkjfmhltgmrne3fu3py
*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. ...##
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A Streamlined Difference Ring Theory: Indefinite Nested Sums, the Alternating Sign, and the Parameterized Telescoping Problem

2014
*
2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
*

The underlying machinery relies on algorithms that compute all

doi:10.1109/synasc.2014.12
dblp:conf/synasc/Schneider14
fatcat:2meaio7v7vbwxkahh6imgdyusa
*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*. ...##
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Bethe ansatz in stringy sigma models

2006
*
Journal of Statistical Mechanics: Theory and Experiment
*

We compute the exact S-matrix and give the Bethe ansatz

doi:10.1088/1742-5468/2006/05/p05006
fatcat:vqp6zeghlvhqviuwhzmg7azyza
*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 ...##
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Weak Power-Counting Theorem for the Renormalization of the Nonlinear Sigma Model in Four Dimensions

2006
*
International Journal of Theoretical Physics
*

The formulation

doi:10.1007/s10773-006-9217-x
fatcat:pqrj5uv2lvfwhk24yremwjqvla
*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. ...##
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Elliptic genera and real Jacobi forms

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

doi:10.1007/jhep01(2014)082
fatcat:o2kv3vbaqjf6hgueyctk476pmy
*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. ...##
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Wronskian solution for AdS/CFT Y-system

2011
*
Journal of High Energy Physics
*

Using the discrete Hirota integrability we

doi:10.1007/jhep01(2011)155
fatcat:uhraomuz4fdk7pstom3ui3izsa
*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. ...##
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The Hierarchy Principle and the Large Mass Limit of the Linear Sigma Model

2007
*
International Journal of Theoretical Physics
*

While the chiral limit is straightforward, the matching

doi:10.1007/s10773-007-9371-9
fatcat:jfliqggsifbg3fgvdkx6z37awq
*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. ...##
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Generalized Discriminant Analysis Using a Kernel Approach

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. ...

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

2020
*
SIAM journal on computing (Print)
*

So, it does not matter whether the running time is

doi:10.1137/18m1166869
fatcat:cfjriedfenae7escnsgsb3xvuu
*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 ...##
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Finite volume spectrum of 2D field theories from Hirota dynamics

2009
*
Journal of High Energy Physics
*

It is close

doi:10.1088/1126-6708/2009/12/060
fatcat:t7qmhlew7rculamxolbpnj4yhm
*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. ...##
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Strong-Interaction Matter under Extreme Conditions from Chiral Quark Models with Nonlocal Separable Interactions

2021
*
Symmetry
*

, the effects produced by strong external magnetic

doi:10.3390/sym13010121
fatcat:ohd5q7jfpnfa3jwuissmdtl5uu
*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. ...##
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Quantum adiabatic algorithms and large spin tunnelling

2003
*
Physical Review A. Atomic, Molecular, and Optical Physics
*

We show that

doi:10.1103/physreva.68.062321
fatcat:mfy26nf4yng2vkds5n2aosgzc4
*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. ...##
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Off-shell renormalization in the presence of dimension 6 derivative operators. Part I. General theory

2019
*
Journal of High Energy Physics
*

This allows one

doi:10.1007/jhep09(2019)032
fatcat:u5e75u6lkncobpu2ujyib7swym
*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 ...
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