# Keyword index R

**R-core**- 28/2011 - Relations Between Graphs

**R-retraction**- 28/2011 - Relations Between Graphs

**Rademacher theorem**- 78/2014 - On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

**radial basis functions****radial variation**- 115/2002 - A Rough Lipschitz Function

**radiosity**- 28/2007 - Wavelet radiosity

**radon-nikodym decomposition theorem**- 12/2003 - -quasiconvexity: weak-star convergence and the gap

**radon-nikodyn theorem**- 8/1998 - 3D-2D asymptotic analysis of an optimal design problem for thin films

**Ramanujan differential equation**- 87/2007 - Ramanujan Eisenstein Series, Fa\'a di Bruno Polynomials and Integrable Systems

**Ramanujan's differential equation**- 49/2008 - Riccati Chain, Ramanujan's Differential Equations For Eisenstein Series and Chazy Flows

**ramsey theory**- 1/2006 - Discrepancy of Symmetric Products of Hypergraphs

**random attractor**- 45/2018 - Stochastic FitzHugh-Nagumo neuron model in excitable regime embeds a leaky integrate-and-fire model

**random diffeomorphisms**- 49/2012 - Representation of Markov chains by random maps: existence and regularity conditions

**random dynamical systems**- 26/2011 - Topological bifurcations of minimal invariant sets for set-valued dynamical systems

**random dynamics**- 49/2012 - Representation of Markov chains by random maps: existence and regularity conditions

**random fields****Random functionals**- 14/2007 - Sharp-interface limit of a mesoscopic free energy with a random external field

**random genetic drift****random graphs**- 61/2011 - Random Graphs with Motifs

**random maps****random matrices**- 94/2005 - Probability of local bifurcation type from a fixed point: a random matrix perspective

**Random matrix theory****random media**- 11/2015 - Correlation structure of the corrector in stochastic homogenization

**random neural activity**- 55/2006 - Cortical network modeling: analytical methods for firing rates and some properties of networks of LIF neurons

**random perturbation****random perturbation of Nonhyperbolic dynamics**- 58/2010 - Random fluctuation leads to forbidden escape of particles

**random quantum measurement**- 106/2019 - Incompatibility probability of random quantum measurements

**random quantum state**- 107/2019 - Duistermaat-Heckman measure and the mixture of quantum states

**random quantum states**- 99/2019 - Average skew information-based coherence and its typicality for random quantum states

**Random sampling**- 79/2020 - Distributed Learning via Filtered Hyperinterpolation on Manifolds

**random time change**- 71/2000 - Mesoscopic limit for non-isothermal phase transition

**random walk**- 25/2011 - Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator

**random walk in a random environment**- 33/2009 - Bounds on the speed and on regeneration times for certain processes on regular trees

**Randomly perturbed dynamics**- 62/2010 - Escape from attracting sets in randomly perturbed systems

**range condition**- 42/2013 (
*Lecture note*) - Large Deviations for Stochastic Processes

- 42/2013 (
**range of derivative**- 64/2002 - The set of gradients of a bump

**range-separated tensor format**- 12/2019 - Computing electrostatic potentials using regularization based on the range-separated tensor format

**range-separated tensor formats****range-separated Tucker tensor format**- 39/2016 - Range-separated tensor formats for numerical modeling of many-particle interaction potentials

**rank $3$ simple matroids**- 88/2020 - On the Generation of Rank 3 Simple Matroids with an Application to Terao’s Freeness Conjecture

**rank structured tensor approximation****rank two geometry**- 40/2019 - Injection dimensions of projective varieties

**rank-1 convexity****rank-metric codes**- 10/2021 - Valued rank-metric codes

**rank-one and quasiconvexity**- 34/2000 - Stability of quasiconvex hulls and deformations with finitely many gradients

**rank-one connection**- 35/2000 - Rigidity for the four gradient problem

**rank-one connection and rigidity**- 16/2003 (
*Lecture note*) - Rigidity and Geometry of Microstructures

- 16/2003 (
**rank-one convex hull**- 94/2003 - Algebraic detection of T
_{k}-configurations

- 94/2003 - Algebraic detection of T
**rank-one convexity****Rank-Two Mixed States**- 67/2007 - Local Equivalence of Rank-Two Quantum Mixed States

**rankine-hugoniot conditions**- 4/2002 - The Bifurcation Analysis of the MHD Rankine-Hugoniot Equations for a Perfect Gas

**rate functional**- 5/2005 - Instability of interface under forced displacements

**Rate-independent problems**- 101/2007 - On the inviscid limit of a model for crack propagation

**rate-independent processes**- 14/2009 - Quasistatic delamination problem

**rational functions**- 3/2009 - On the Efficient Computation of High-Dimensional Integrals and the Approximation by Exponential Sums

**rational invariants**- 45/2017 - Rational invariants of ternary forms under the orthogonal group

**rational maps**- 74/2003 - Equivariant Rational maps and Configurations: spherical equidistribution and SO(N,1) contraction

**rational points**- 8/2002 - Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity

**rattling****Rayleigh-Benard convection**- 6/2011 - Rayleigh--B\'enard convection: Improved bounds on the Nusselt number

**Rayleigh-Bénard convection**- 57/2016 - Limitations of the background field method applied to Rayleigh-Bénard convection

**Rayleigh-B\'enard convection**- 116/2014 - Upper bounds on Nusselt number at finite Prandtl number

**reaction diffusion****reaction-diffusion****reaction-diffusion equation**- 35/2015 - Recent Advances in Reaction-Diffusion Equations with Non-Ideal Relays

**reaction-diffusion equations****reaction-diffusion system****reaction-diffusion-system**- 21/2003 - Travelling wave solutions in a model for filtration combustion

**reaction-diffution equation**- 1/2004 - Traveling wave speeds of nonlocally perturbed reaction diffusion equations

**reaction-diﬀusion equations**- 94/2013 - Systems of reaction-diffusion equations with spatially distributed hysteresis

**real algebraic curve****Real Algebraic Curves**- 23/2017 - Sixty-Four Curves of Degree Six

**real connective k theory**- 83/2002 - The eta invariant and the real connective K-theory of the classifying space for quaternion groups

**Real projective space**- 95/2019 - A new family of triangulations of ℝP
_{d}

- 95/2019 - A new family of triangulations of ℝP
**Real rank**- 9/2017 - Spaces of Sums of Powers and Real Rank Boundaries

**real roots**- 59/2018 - Arithmetic aspects of symmetric edge polytopes

**receptive field reconstruction****receptive fields**- 10/2001 - Nonlinear analysis of spatio-temporal receptive fields:

I. Dynamic approximation method

- 10/2001 - Nonlinear analysis of spatio-temporal receptive fields:
**reciprocal spaces**- 108/2020 - Reciprocal maximum likelihood degrees of diagonal linear concentration models

**reciprocity**- 90/2003 - Efficient Computation of Lead Field Bases and Influence Matrix for the FEM-based EEG and MEG Inverse Problem. Part I: Complexity Considerations

**rectiable currents**- 58/2013 - The Steiner tree problem revisited through rectifiable G-currents

**rectifiability**- 61/2016 - Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies
- 95/2002 - Structure of entropy solutions for multi--dimensional scalar conservation laws
- 47/2002 - Structure of entropy solutions: applications to variational problems
- 29/2002 - Stationary measures and rectifiability

**rectifiablity**- 116/2002 - On the rectifiability of defect measures arising in a micromagnetics model

**rectifiablity, stationarity**- 56/2002 - Regularity and blow-up analysis for J-holomorphic maps

**rectifier**- 73/2014 - On the Number of Linear Regions of Deep Neural Networks

**rectifier unit**- 72/2014 - On the number of response regions of deep feedforward networks with piecewise linear activations

**recursive iterator**- 88/2020 - On the Generation of Rank 3 Simple Matroids with an Application to Terao’s Freeness Conjecture

**reduced basis**- 32/2015 - A reduced basis approach for calculation of the Bethe-Salpeter excitation energies using low-rank tensor factorizations

**reduced higher order SVD**- 29/2012 - Tensor-structured factorized calculation of two-electron integrals in a general basis

**reduced model**- 33/2012 - A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types.

**reduction****referential choice**- 50/2002 - Referential Choice and Activation Factors: A Neural Network Approach

**Refined grid**- 53/2009 - On the efficient convolution with the Newton potential

**regeneration times**- 33/2009 - Bounds on the speed and on regeneration times for certain processes on regular trees

**regular lagrangian system**- 55/2003 - The Morse index theorem for regular Lagrangian systems

**regular variation**- 62/2003 - Approach to self-similarity in Smoluchowski's coagulation equations

**regularity**- 6/2017 - Coarse Regularity of Solutions to a Nonlinear Sigma-model with L
^{p}gravitino - 69/2011 - The boundary value problem for Dirac-harmonic maps
- 32/2008 - Regularity for weakly Dirac-harmonic maps to hypersurfaces
- 106/2007 - Regularity of domain walls in magnetic nanowires
- 42/2007 - Nonlinear Dirac equations on Riemann surfaces
- 120/2006 - C
^{1,α}Regularity for Solutions to the p-Harmonic thin Obstacle Problem. - 4/2004 - Regularity properties of isometric immersions
- 68/2003 - On the regularity of critical points of polyconvex functionals
- 11/2003 - Parabolic systems with nowhere smooth solutions
- 107/2002 - An ϵ-regularity result for generalized harmonic maps into spheres
- 106/2002 - A short proof of the self-improving regularity of quasiregular mappings.
- 26/2002 - Partial regularity for the Landau-Lifshitz equation in small dimensions

- 6/2017 - Coarse Regularity of Solutions to a Nonlinear Sigma-model with L
**regularity for energy minimzing maps**- 3/2016 - On the number of singular points for planar multivalued harmonic functions

**Regularity of Minimal Surfaces**- 2/2016 - Almgren's type regularity for semicalibrated currents

**regularity of solutions****Regularity of the free boundary**- 16/2015 - An epiperimetric inequality for the thin obstacle problem

**regularization**- 9/2002 - Local stress regularity in scalar non-convex variational problems

**regularization of nonconvex problems**- 43/2002 - Application of a result of Ioffe/Tichomirov to multidimensional control problems of Dieudonn\'e-Rashevsky type

**regularization of potentials**- 53/2006 - About solutions of Poisson's equation with transition condition in non-smooth domains

**regularized Newton method**- 78/2010 - A Regularized Newton method for the Efficient Approximation of Tensors Represented in the Canonical Tensor Format

**Reilly's formula**- 54/2014 - A generalization of Reilly's formula and its applications to a new Heintze-Karcher type inequality

**Reinforcement Learning****relative entropy**- 73/2015 - A Novel Approach to Canonical Divergences within Information Geometry
- 75/2011 - Information topologies on non-commutative state spaces
- 72/2011 - Entropy Distance: New Quantum Phenomena
- 82/2009 - Finding the Maximizers of the Information Divergence from an Exponential Family
- 39/2008 - Mathematical strategies in the coarse-graining of extensive systems: error quantification and adaptivity
- 71/2007 - C
^{*}-algebraic approach to the Bose-Hubbard model - 75/2006 - Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems

**relative entropy of coherence**- 53/2018 - Ordering states with various coherence measures

**relative Hamiltonian**- 71/2007 - C
^{*}-algebraic approach to the Bose-Hubbard model

- 71/2007 - C
**relative morse index**- 55/2003 - The Morse index theorem for regular Lagrangian systems

**Relative nullity foliation**- 38/2011 - Screen conformal lightlike geometry in indefinite Kenmotsu space forms

**relativistic klein-gordon equation**- 24/2005 (
*Lecture note*) - Lectures on Global Attractors of Hamilton Nonlinear Wave Equations

- 24/2005 (
**relaxation**- 68/2008 - Quasiconvex relaxation of multidimensional control problems with integrands f(t,ξ,v)
- 111/2007 - Relaxation of three solenoidal wells and characterization of three-phase H-measures
- 6/2005 - From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence
- 72/2000 - Nonconvex potentials and microstructures in finite-strain plasticity
- 40/1999 - Weierstrass-type maximum principle for microstructure in micromagnetics
- 24/1998 - Regularity and optimal design results for elastic membranes
- 8/1998 - 3D-2D asymptotic analysis of an optimal design problem for thin films

**relaxation oscillations****relaxed energy**- 50/1998 - Relaxation of some multi-well problems

**relaxed problem**- 9/2002 - Local stress regularity in scalar non-convex variational problems

**Remez algorithm**- 80/2017 - Computation of Best L
^{∞}Exponential Sums for 1∕x by Remez’ Algorithm

- 80/2017 - Computation of Best L
**renewal theorem**- 31/2004 - A notion of Euler characteristic for fractals

**renormalization**- 10/2006 - Renormalization : A number theoretical model

**renormalization group**- 9/2004 - Scaling algebras for charged fields and short-distance analysis for localizable and topological charges

**renormalization group map**- 75/2006 - Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems

**renormalization, general covariance**- 12/2004 - The current status of quantum fields in curved spacetime (Transparencies of a talk given at DPG meeting, Ulm, 17 March 2004)

**renormalized solutions**- 67/2003 - Well--posedness for a class of hyperbolic systems of conservation laws in several space dimensions

**reparametrization**- 43/2014 - An example of a vector field with the oriented shadowing property

**replication****representation**- 129/2006 - Explicit realization of induced and coinduced modules over Lie superalgebras by differential operators

**Representation class**- 75/2007 - Representation Class and Geometrical Invariants of Quantum States under Local Unitary Transformations

**representation formula**- 85/2003 - A representation formula for the inverse harmonic mean curvature flow

**Representation of measures**- 69/2016 - On the regular representation of measures

**representation rings****representation theory****representation, symmetry group, lagrangian theory**- 25/2005 (
*Lecture note*) - Lectures on Quantum Mechanics (nonlinear PDEs point of view)

- 25/2005 (
**Representational Power**- 106/2014 - Discrete Restricted Boltzmann Machines
- 74/2014 - Universal Approximation Depth and Errors of Narrow Belief Networks with Discrete Units
- 72/2014 - On the number of response regions of deep feedforward networks with piecewise linear activations
- 27/2011 - Expressive Power and Approximation Errors of Restricted Boltzmann Machines

**Representative Volume Element**- 47/2017 - A numerical primer in 2D stochastic homogenization: CLT scaling in the Representative Volume Element

**reproducing kernel Hilbert space**- 152/2006 - Deterministic Error Analysis of Support Vector Regression and Related Regularized Kernel Methods

**reproducing kernel Hilbert spaces****residual resultant**- 65/2019 - Implicitization of tensor product surfaces via virtual projective resolutions

**residually finite-dimensional C*-algebras**- 50/2016 - Quantum logic is undecidable

**resolvent****resolvents**- 11/2007 - On the Robustness of Elliptic Resolvents Computed by means of the Technique of Hierarchical Matrices

**resonance arrangement,**- 86/2020 - The universality of the resonance arrangement and its Betti numbers

**resource theory**- 40/2016 - A Resource Theory for Work and Heat

**resting cell**- 99/2006 - Actin Dynamics and Cell Motility: Minimal Models for the Initiation of Cell Movement

**restricted Boltzmann machine****restricted word problem**- 50/2016 - Quantum logic is undecidable

**resultants**- 65/2019 - Implicitization of tensor product surfaces via virtual projective resolutions

**resurgence**- 34/2009 - Shatalov-Sternin's construction of complex WKB solutions -- a comment on geometric issues.

**retarded potential integral equation**- 93/2013 - Adaptive Time Discretization for Retarded Potentials

**retarded potentials**- 42/2011 - Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation

**Riccati and Chazy equations**- 143/2006 - Riccati Chain, Higher Order Painlev\'e Type Equations and Stabilizer Set of Virasoro Orbit

**riccati equation**- 49/2008 - Riccati Chain, Ramanujan's Differential Equations For Eisenstein Series and Chazy Flows
- 87/2007 - Ramanujan Eisenstein Series, Fa\'a di Bruno Polynomials and Integrable Systems
- 84/2005 - Nonlinear multigrid for the solution of large scale Riccati equations in low-rank and -matrix format
- 62/2002 - Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices
- 18/2002 - Introduction to Hierarchical Matrices with Applications

**Ricci blowup**- 123/2006 - Completeness, Ricci blowup, the Osserman and the conformal Osserman condition for Walker signature (2,2) manifolds.

**ricci curvature**- 73/2002 - Some properties of the Schouten tensor and applications to conformal geometry

**Ricci flow****Ricci operator**- 73/2007 - Examples of signature (2,2) manifolds with commuting curvature operators

**Ricci soliton**- 67/2014 - Omori-Yau maximum principles, V-harmonic maps and their geometric applications

**ricci tensor**- 43/2005 - Conformal deformations of the smallest eigenvalue of the Ricci tensor

**Richardson extrapolation**- 36/2008 - Fast and Accurate Tensor Approximation of Multivariate Convolution with Linear Scaling in Dimension

**Riemann tensor**- 81/2006 - The nonholonomic Riemann and Weyl tensors for flag manifolds

**riemann-hilbert factorization**- 13/2005 - On the Method of Steepest Descent

**riemann-hilbert problems**- 14/2005 - Semiclassical Limit of the Focusing Nonlinear Schr\"odinger Equation under Barrier Initial Data

**Riemannian geometries**- 103/2018 - Dynamical Systems induced by Canonical Divergence in dually flat manifolds

**Riemannian submersion**- 116/2006 - The spectral geometry of the canonical Riemannian submersion of a compact Lie Group

**riesz transform**- 29/1999 - Rank-one convexity implies quasiconvexity on diagonal matrices

**rigid-perfect plasticity**- 49/2004 - Examples of nonlinear homogenization in plane strain involving degenerate energies

**rigidity**- 104/2020 - Epsilon local rigidity and numerical algebraic geometry
- 85/2019 - Nonlinear algebra via tensegrity structures
- 33/2004 - Geometric Rigidity of Conformal Matrices
- 4/2004 - Regularity properties of isometric immersions
- 53/2003 - Rigidity for the hyperbolic Monge-Ampere equation
- 47/2003 - Sharp rigidity estimates for nearly umbilical surfaces
- 35/2000 - Rigidity for the four gradient problem

**rigidity of isometric immersions**- 41/2003 - On the Sobolev space of isometric immersions

**rigorous numerics**- 39/2005 - Rigorous Numerics for Dissipative PDEs III. An effective algorithm for rigorous computation of trajectories and Poincaré maps

**risk**- 101/2019 - How signatures affect expected return and volatility: a rough model under transaction cost

**Rivière regularity**- 68/2016 - Regularity of Solutions of the Nonlinear Sigma Model with Gravitino

**robin function****robustness**- 14/2006 - Geometric robustness theory and biological networks

**rod theory****rotating drops**- 16/2006 - On toroidal rotating drops

**rough differential equations**- 101/2019 - How signatures affect expected return and volatility: a rough model under transaction cost

**rough path**- 91/2018 - The rough Veronese variety

**rough path theory**- 101/2019 - How signatures affect expected return and volatility: a rough model under transaction cost

**routes to chaos**- 92/2005 - Routes to chaos in high-dimensional dynamical systems: a qualitative numerical study

**rubber material**- 40/2001 - Soft elastic response of stretched sheets of nematic elastomers: a numerical study

**Runge-Kutta convolution quadrature****Runge-Kutta methods**- 72/2012 - Sparsity of Runge-Kutta convolution weights for three-dimensional wave equation
- 59/2010 - Runge-Kutta convolution quadrature for operators arising in wave propagation
- 27/2010 - An error analysis of Runge-Kutta convolution quadrature
- 58/2009 - Multistep and multistage boundary integral methods for the wave equation