Publications

Ay, N.
; Jost, J.
; Lê, H.
V.
and L.
J. Schwachhöfer:
*Parametrized measure models*.
Bernoulli,
24 (2018) 3, p. 1692-1725 MIS-Preprint: 69/2015 [DOI] [ARXIV]

Felice, D.
and N. Ay:
*Towards a canonical divergence within information geometry*.
Information geometry,
Vol. not yet known, pp. not yet known

Ghazi-Zahedi, K.
; Langer, C.
and N. Ay:
*Morphological computation : synergy of body and brain*.
Entropy,
19 (2017) 9, 456 [DOI]

Kanwal, M.
S. ; Grochow, J.
A. and N. Ay:
*Comparing information-theoretic measures of complexity in Boltzmann machines*.
Entropy,
19 (2017) 7, 310 [DOI] [ARXIV]

Ghazi-Zahedi, K.
; Haeufle, D.
F. B. ; Montúfar, G.
; Schmitt, S.
and N. Ay:
*Evaluating morphological computation in muscle and DC-motor driven models of hopping
movements*.
Frontiers in robotics and AI,
3 (2016), 42 [DOI] [ARXIV]

Perrone, P.
and N. Ay:
*Hierarchical quantification of synergy in channels*.
Frontiers in robotics and AI,
2 (2016), 35 MIS-Preprint: 86/2015 [DOI] [ARXIV]

Sato, Y.
and N. Ay:
*Information flow in learning a coin-tossing game*.
Nonlinear theory and its applications,
7 (2016) 2, p. 118-125 [DOI]

Ay, N.
:
*Information geometry on complexity and stochastic interaction*.
Entropy,
17 (2015) 4, p. 2432-2458 MIS-Preprint: 95/2001 [DOI]

Ay, N.
:
*Geometric design principles for brains of embodied agents*.
Künstliche Intelligenz : KI,
29 (2015) 4, p. 389-399 [DOI]

Ay, N.
and S.
Amari:
*A novel approach to canonical divergences within information geometry*.
Entropy,
17 (2015) 12, p. 8111-8129 MIS-Preprint: 73/2015 [DOI]

Ay, N.
; Jost, J.
; Lê, H.
V.
and L.
J. Schwachhöfer:
*Information geometry and sufficient statistics*.
Probability theory and related fields,
162 (2015) 1-2, p. 327-364 MIS-Preprint: 65/2012 [DOI] [ARXIV]

Ay, N.
and W.
Löhr:
*The Umwelt of an embodied agent : a measure-theoretic definition*.
Theory in biosciences,
134 (2015) 3-4, p. 105-116 MIS-Preprint: 74/2015 [DOI] [ARXIV]

Montúfar, G.
; Ay, N.
and K.
Ghazi-Zahedi:
*Geometry and expressive power of conditional restricted Boltzmann machines*.
Journal of machine learning research,
16 (2015), p. 2405-2436 MIS-Preprint: 16/2014 [ARXIV] [FREELINK]

Montúfar, G.
; Ghazi-Zahedi, K.
and N. Ay:
*A theory of cheap control in embodied systems*.
PLoS computational biology,
11 (2015) 9, e1004427 MIS-Preprint: 70/2014 [DOI] [ARXIV]

Pfante, O.
and N. Ay:
*Operator-theoretic identification of closed sub-systems of dynamical systems*.
An interdisciplinary journal of discontinuity, nonlinearity, and complexity,
4 (2015) 1, p. 91-109 MIS-Preprint: 4/2015 [DOI]

Salge, C.
; Ay, N.
; Polani, D.
and M.
Prokopenko:
*Zipf's law : balancing signal usage cost and communication efficiency*.
PLoS one,
10 (2015) 10, e0139475 [DOI] [FREELINK]

Steudel, B.
and N. Ay:
*Information-theoretic inference of common ancestors*.
Entropy,
17 (2015) 4, p. 2304-2327 [DOI] [ARXIV]

Weis, S.
; Knauf, A.
; Ay, N.
and M.
Zhao:
*Maximizing the divergence from a hierarchical model of quantum states*.
Open systems and information dynamics,
22 (2015) 1, 1550006 MIS-Preprint: 58/2014 [DOI] [ARXIV]

Bertschinger, N.
; Rauh, J.
; Olbrich, E.
; Jost, J.
and N. Ay:
*Quantifying unique information*.
Entropy,
16 (2014) 4, p. 2161-2183 MIS-Preprint: 102/2013 [DOI] [ARXIV]

Krakauer, D.
C. ; Bertschinger, N.
; Olbrich, E.
; Ay, N.
and J.
C. Flack:
*The information theory of individuality*.
Theory in biosciences,
Vol. not yet known, pp. not yet known[ARXIV]

Montúfar, G.
; Rauh, J.
and N. Ay:
*On the Fisher metric of conditional probability polytopes*.
Entropy,
16 (2014) 6, p. 3207-3233 MIS-Preprint: 87/2014 [DOI] [ARXIV]

Moritz, P.
; Reichardt, J.
and N. Ay:
*Discriminating between causal structures in Bayesian Networks given partial observations*.
Kybernetika,
50 (2014) 2, p. 284-295 [DOI]

Pfante, O.
; Bertschinger, N.
; Olbrich, E.
; Ay, N.
and J.
Jost:
*Comparison between different methods of level identification*.
Advances in complex systems,
17 (2014) 2, 1450007 [DOI]

Pfante, O.
; Olbrich, E.
; Bertschinger, N.
; Ay, N.
and J.
Jost:
*Closure measures for coarse-graining of the tent map*.
Chaos,
24 (2014), 013136 MIS-Preprint: 108/2013 [DOI]

Rauh, J.
and N. Ay:
*Robustness, canalyzing functions and systems design*.
Theory in biosciences,
133 (2014) 2, p. 63-78 MIS-Preprint: 66/2012 [DOI] [ARXIV]

Ghazi-Zahedi, K.
and N. Ay:
*Quantifying morphological computation*.
Entropy,
15 (2013) 5, p. 1887-1915 MIS-Preprint: 11/2013 [DOI] [ARXIV]

Ghazi-Zahedi, K.
; Martius, G.
and N. Ay:
*Linear combination of one-step predictive information with an external reward in an
episodic policy gradient setting : a critical analysis*.
Frontiers in psychology / Frontiers in cognitive science,
4 (2013), 801 MIS-Preprint: 60/2013 [DOI] [ARXIV]

Lohmann, G.
; Stelzer, J.
; Neumann, J.
; Ay, N.
and R.
Turner:
*'More is different' in functional magnetic resonance imaging : a review of recent
data analysis techniques*.
Brain Connectivity,
3 (2013) 3, p. 223-239 [DOI]

Martius, G.
; Der, R.
and N. Ay:
*Information driven self-organization of complex robotic behaviors*.
PLoS one,
8 (2013) 5, e63400 MIS-Preprint: 15/2013 [DOI] [ARXIV]

Ay, N.
; Bernigau, H.
; Der, R.
and M.
Prokopenko:
*Information-driven self-organization : the dynamical system approach to autonomous
robot behavior*.
Theory in biosciences,
131 (2012) 3, p. 161-179 [DOI]

Ay, N.
; Der, R.
and M.
Prokopenko:
*Guided self-organization : perception-action loops of embodied systems*.
Theory in biosciences,
131 (2012) 3, p. 125-127 [DOI]

Ay, N.
and W.
Wenzel:
*On solution sets of information inequalities*.
Kybernetika,
48 (2012) 5, p. 845-864 MIS-Preprint: 16/2011 [FREELINK]

Löhr, W.
; Szkoła, A.
and N. Ay:
*Process dimension of classical and non-commutative processes*.
Open systems and information dynamics,
19 (2012) 1, 1250007 MIS-Preprint: 52/2011 [DOI] [ARXIV]

Ay, N.
; Müller, M.
and A.
Szkoła:
*Effective complexity of stationary process realizations*.
Entropy,
13 (2011) 6, p. 1200-1211 MIS-Preprint: 2/2010 [DOI] [ARXIV]

Ay, N.
; Olbrich, E.
; Bertschinger, N.
and J.
Jost:
*A geometric approach to complexity*.
Chaos,
21 (2011) 3, 037103 MIS-Preprint: 53/2011 [DOI]

Montúfar, G.
and N. Ay:
*Refinements of universal approximation results for deep belief networks and restricted
Boltzmann machines*.
Neural computation,
23 (2011) 5, p. 1306-1319 MIS-Preprint: 23/2010 [DOI]

Rauh, J.
; Kahle, T.
and N. Ay:
*Support sets in exponential families and oriented matroid theory*.
International journal of approximate reasoning,
52 (2011) 5, p. 613-626 MIS-Preprint: 28/2009 [DOI] [ARXIV]

Ay, N.
; Müller, M.
and A.
Szkoła:
*Effective complexity and its relation to logical depth*.
IEEE transactions on information theory,
56 (2010) 9, p. 4593-4607 [DOI] [ARXIV]

Ghazi-Zahedi, K.
; Ay, N.
and R.
Der:
*Higher coordination with less control : a result of information maximization in the
sensorimotor loop*.
Adaptive behavior,
18 (2010) 3/4, p. 338-355 [DOI] [ARXIV]

Olbrich, E.
; Kahle, T.
; Bertschinger, N.
; Ay, N.
and J.
Jost:
*Quantifying structure in networks*.
The European physical journal / B,
77 (2010) 2, p. 239-247 MIS-Preprint: 81/2009 [DOI] [ARXIV]

Prokopenko, M.
; Ay, N.
; Obst, O.
and D.
Polani:
*Phase transitions in least-effort communications*.
Journal of statistical mechanics,
2010 (2010) 11, P11025 [DOI]

Ay, N.
:
*A refinement of the common cause principle*.
Discrete applied mathematics,
157 (2009) 10, p. 2439-2457 [DOI]

Kahle, T.
; Olbrich, E.
; Jost, J.
and N. Ay:
*Complexity measures from interaction structures*.
Physical review / E,
79 (2009) 2, pt. 2, 026201 MIS-Preprint: 44/2008 [DOI] [ARXIV]

Kahle, T.
; Wenzel, W.
and N. Ay:
*Hierarchical models, marginal polytopes, and linear codes*.
Kybernetika,
45 (2009) 2, p. 189-207 MIS-Preprint: 30/2008 [ARXIV] [FREELINK]

Löhr, W.
and N. Ay:
*On the generative nature of prediction*.
Advances in complex systems,
12 (2009) 2, p. 169-194 MIS-Preprint: 8/2008 [DOI]

Ay, N.
; Bertschinger, N.
; Der, R.
; Güttler, F.
and E.
Olbrich:
*Predictive information and explorative behavior of autonomous robots*.
The European physical journal / B,
63 (2008) 3, p. 329-339 [DOI]

Ay, N.
and D.
Polani:
*Information flows in causal networks*.
Advances in complex systems,
11 (2008) 1, p. 17-41 MIS-Preprint: 47/2006 [DOI]

Bertschinger, N.
; Olbrich, E.
; Ay, N.
and J.
Jost:
*Autonomy : an information-theoretic perspective*.
Biosystems,
91 (2008) 2, p. 331-345 [DOI]

Olbrich, E.
; Bertschinger, N.
; Ay, N.
and J.
Jost:
*How should complexity scale with system size?*.
The European physical journal / B,
63 (2008) 3, p. 407-415 [DOI]

Ay, N.
; Flack, J.
C. and D.
C. Krakauer:
*Robustness and complexity co-constructed in multimodal signalling networks*.
Philosophical transactions of the Royal Society of London / B,
362 (2007) 1479, p. 441-447 [DOI]

Ay, N.
and D.
C. Krakauer:
*Geometric robustness theory and biological networks*.
Theory in biosciences,
125 (2007) 2, p. 93-121 MIS-Preprint: 14/2006 [DOI]

Jost, J.
; Bertschinger, N.
; Olbrich, E.
; Ay, N.
and S.
Frankel:
*An information theoretic approach to system differentiation on the basis of statistical
dependencies between subsystems*.
Physica / A,
378 (2007) 1, p. 1-10 [DOI]

Wennekers, T.
; Ay, N.
and P.
Andras:
*High-resolution multiple-unit EEG in cat auditory cortex reveals large spatio-temporal
stochastic interactions*.
Biosystems,
89 (2007) 1/3, p. 190-197 [DOI]

Ay, N.
and A.
Knauf:
*Maximizing multi-information*.
Kybernetika,
42 (2006) 5, p. 517-538 MIS-Preprint: 42/2003 [ARXIV]

Wennekers, T.
and N. Ay:
*A temporal learning rule in recurrent systems supports high spatio-temporal stochastic
interactions*.
Neurocomputing,
69 (2006) 10/12, p. 1199-1202 [DOI]

Ay, N.
and I.
Erb:
*On a notion of linear replicator equations*.
Journal of dynamics and differential equations,
17 (2005) 2, p. 427-451 [DOI]

Wennekers, T.
and N. Ay:
*Finite state automata resulting from temporal information maximization and a temporal
learning rule*.
Neural computation,
17 (2005) 10, p. 2258-2290 [DOI]

Wennekers, T.
and N. Ay:
*Stochastic interaction in associative nets*.
Neurocomputing,
65 (2005), p. 387-392 [DOI]

Erb, I.
and N. Ay:
*Multi-information in the thermodynamic limit*.
Journal of statistical physics,
115 (2004) 3-4, p. 949-976 MIS-Preprint: 58/2003 [DOI]

Ay, N.
and W.
Tuschmann:
*Duality versus dual flatness in quantum information geometry*.
Journal of mathematical physics,
44 (2003) 4, p. 1512-1518 MIS-Preprint: 69/2002 [DOI]

Ay, N.
and T.
Wennekers:
*Temporal infomax leads to almost deterministic dynamical systems*.
Neurocomputing,
52 (2003) 4, p. 461-466 [DOI]

Ay, N.
and T.
Wennekers:
*Dynamical properties of strongly interacting Markov chains*.
Neural networks,
16 (2003) 10, p. 1483-1497 MIS-Preprint: 107/2001 [DOI]

Wennekers, T.
and N. Ay:
*Spatial and temporal stochastic interaction in neuronal assemblies*.
Theory in biosciences,
122 (2003) 1, p. 5-18 [DOI]

Wennekers, T.
and N. Ay:
*Temporal Infomax on Markov chains with input leads to finite state automata*.
Neurocomputing,
52 (2003) 4, p. 431-436 [DOI]

Ay, N.
:
*Locality of global stochastic interaction in directed acyclic networks*.
Neural computation,
14 (2002) 12, p. 2959-2980 MIS-Preprint: 54/2001 [DOI]

Ay, N.
:
*An information-geometric approach to a theory of pragmatic structuring*.
The annals of probability,
30 (2002) 1, p. 416-436 MIS-Preprint: 52/2000 [FREELINK]

Ay, N.
and W.
Tuschmann:
*Dually flat manifolds and global information geometry*.
Open systems and information dynamics,
9 (2002) 2, p. 195-200 MIS-Preprint: 24/2002 [DOI]

Wenzel, W.
; Ay, N.
and F.
Pasemann:
*Hyperplane arrangements separating arbitrary vertex classes in n-cubes*.
Advances in applied mathematics,
25 (2000) 3, p. 284-306 MIS-Preprint: 35/1999 [DOI]

Ay, N.
; Jost, J.
; Lê, H.
V.
and L.
J. Schwachhöfer:
*Parametrized measure models*.
Information geometry / N.
Ay... (eds.). Springer, 2017. - P. 121-184
(Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A series of modern surveys
in mathematics ; 64)
MIS-Preprint: 69/2015 [DOI] [ARXIV]

Amari, S.
and N. Ay:
*Standard divergence in manifold of dual affine connections*.
Geometric science of information : second international conference, GSI 2015, Palaiseau,
France, October 28-30, 2015, proceedings / F.
Nielsen... (eds.). Springer, 2015. - P. 320-325
(Lecture notes in computer science ; 9389)
[DOI]

Perrone, P.
and N. Ay:
*Decomposition of Markov kernels*.
Proceedings of the 10th workshop on uncertainty processing WUPES '15, Moninec, Czech
Republic, September 16-19, 2015 / V.
Kratochvíl (ed.). Oeconomica, 2015. - P. 167-178[FREELINK]

Schwachhöfer, L.
J. ; Ay, N.
; Jost, J.
and H.
V.
Lê:
*Invariant geometric structures on statistical models*.
Geometric science of information : second international conference, GSI 2015, Palaiseau,
France, October 28-30, 2015, proceedings / F.
Nielsen... (eds.). Springer, 2015. - P. 150-158
(Lecture notes in computer science ; 9389)
[DOI]

Ay, N.
and K.
Ghazi-Zahedi:
*On the causal structure of the sensorimotor loop*.
Guided self-organization : inception / M.
Prokopenko (ed.). Springer, 2014. - P. 261-294
(Emergence, complexity, and computation)
[DOI]

Prokopenko, M.
; Polani, D.
and N. Ay:
*On the cross-disciplinary nature of guided self-organisation*.
Guided self-organization : inception / M.
Prokopenko (ed.). Springer, 2014. - P. 3-15
(Emergence, complexity, and computation)
[DOI]

Ay, N.
and K.
Ghazi-Zahedi:
*Causal effects for prediction and deliberative decision making of embodied systems*.
Advances in cognitive neurodynamics III : proceedings of the 3rd International Conference
on Cognitive Neurodynamics 2011 ; [June 9-13, 2011, Hilton Niseko Village, Hokkaido,
Japan] / Y.
Yamaguchi (ed.). Springer, 2013. - P. 499-506
(Advances in cognitive neurodynamics)
MIS-Preprint: 22/2011 [DOI]

Ay, N.
; Montúfar, G.
and J.
Rauh:
*Selection criteria for neuromanifolds of stochastic dynamics*.
Advances in cognitive neurodynamics III : proceedings of the 3rd International Conference
on Cognitive Neurodynamics 2011 ; [June 9-13, 2011, Hilton Niseko Village, Hokkaido,
Japan] / Y.
Yamaguchi (ed.). Springer, 2013. - P. 147-154
(Advances in cognitive neurodynamics)
MIS-Preprint: 15/2011 [DOI]

Montúfar, G.
; Rauh, J.
and N. Ay:
*Maximal information divergence from statistical models defined by neural networks*.
Geometric science of information : first international conference, GSI 2013, Paris,
France, August 28-30, 2013. Proceedings / F.
Nielsen... (eds.). Springer, 2013. - P. 759-766
(Lecture notes in computer science ; 8085)
MIS-Preprint: 31/2013 [DOI] [ARXIV]

Moritz, P.
; Reichardt, J.
and N. Ay:
*A new common cause principle for bayesian networks*.
9th Workshop on Uncertainty Processing : WUPES'12 ; Marianske Lazne, Czech Republik
; 12-15th September 2012 Academy of Sciences of the Czech Republik / Institute of Information Theory and Automation, 2012. - P. 149-162[FREELINK]

Montúfar, G.
; Rauh, J.
and N. Ay:
*Expressive power and approximation errors of restricted Boltzmann machines*.
Advances in neural information processing systems 24 : 25th Annual Conference on Neural
Information Processing Systems 2011, Granada, Spain December 12th - 15th ; NIPS 2011 / J. Shawe-Taylor (ed.). Neural Information Processing Systems, 2011. - P. 415-423 MIS-Preprint: 27/2011 [ARXIV] [FREELINK]

Krakauer, D.
C. ; Flack, J.
C. and N. Ay:
*Probabilistic design principles for robust multimodal communication networks*.
Modelling perception with artificial neural networks / C.
R. Tosh... (eds.). Cambridge University Press, 2010. - P. 255-268 [DOI]

Löhr, W.
and N. Ay:
*Non-sufficient memories that are sufficient for prediction*.
Complex sciences : first international conference, Complex 2009, Shanghai, China,
February 23 - 25, 2009, revised papers. Pt. 1 / J.
Zhou (ed.). Springer, 2009. - P. 265-276
(Lecture notes of the Institute for Computer Science, Social Informatics and Telecommunications
Engineering ; 4)
[DOI]

Der, R.
; Güttler, F.
and N. Ay:
*Predictive information and emergent cooperativity in a chain of mobile robots*.
Artificial Life XI : Proceedings of the Eleventh International Conference on the Simulation
and Synthesis of Living Systems MIT Press, 2008. - P. 166-172

Ay, N.
; Olbrich, E.
; Bertschinger, N.
and J.
Jost:
*A unifying framework for complexity measures of finite systems*.
ECCS'06 : proceedings of the European Conference on Complex Systems 2006 ; towards
a science of complex systems / J.
Jost... (eds.). European Complex Systems Society, 2006. - P. 80-80[FREELINK]

Bertschinger, N.
; Olbrich, E.
; Ay, N.
and J.
Jost:
*Autonomy : an information theoretic perspective*.
Artificial life X : proceedings of the tenth international conference on the synthesis
and simulation of living systems / L.
M.
Rocha... (eds.). MIT Press, 2006. - P. 7-12

Bertschinger, N.
; Olbrich, E.
; Ay, N.
and J.
Jost:
*Information and closure in systems theory*.
Explorations in the complexity of possible life : abstracting and synthesizing the
principles of living systems ; proceedings of the 7th German Workshop on Artificial
Life, July 26 - 28, 2006, Jena, Germany / S. Artmann... (eds.). IOS Press, 2006. - P. 9-19

Kahle, T.
and N. Ay:
*Support sets of distributions with given interaction structure*.
7th Workshop on Uncertainty Processing : WUPES'06 ; Mikulov, Czech Republik ; 16-20th
September 2006 Academy of Sciences of the Czech Republik / Institute of Information Theory and Automation, 2006. - P. 52-61 MIS-Preprint: 94/2006 [FREELINK]

Matus, F.
and N. Ay:
*On maximization of the information divergence from an exponential family*.
Proceedings of 6th workshop on uncertainty processing : Hejnice, September 24-27,
2003 Oeconomica, 2003. - P. 199-204 MIS-Preprint: 46/2003

Schwachhöfer, L.
J. ; Ay, N.
; Jost, J.
and H.
V.
Lê:
*Congruent families and invariant tensors*.
[ARXIV]

Montúfar, G.
; Ghazi-Zahedi, K.
and N. Ay:
*Information theoretically aided reinforcement learning for embodied agents*.
[ARXIV]

Perrone, P.
and N. Ay:
*Iterative scaling algorithm for channels*.
[ARXIV]

Montúfar, G.
; Ghazi-Zahedi, K.
and N. Ay:
*Geometry and determinism of optimal stationary control in partially observable Markov
decision processes*.
MIS-Preprint: 22/2016 [ARXIV]

Ay, N.
and W.
Löhr:
*A measure-theoretic description of the intrinsic view of embodied agents*.
[FREELINK]

Ay, N.
and G.
Struck:
*Stochastic independence of crosstalk variables in the hopfield model*.
MIS-Preprint: 65/2006

Sato, Y.
and N. Ay:
*Adaptive dynamics for interacting Markovian processes*.
[ARXIV]

Wennekers, T.
and N. Ay:
*Information-theoretic grounding of finite automata in neural systems*.
MIS-Preprint: 52/2002

Ay, N.
:
*Aspekte einer Theorie pragmatischer Informationsstrukturierung*.
Dissertation, Universität Leipzig, 2001

Ay, N.
; Krakauer, D.
C. and J.
C. Flack:
*Robustness, causal networks, and experimental design*.
Princeton University Press, 2018

Ay, N.
; Jost, J.
; Lê, H.
V.
and L.
J. Schwachhöfer:
*Information geometry*.
Springer, 2017. - XI, 407 p.
(Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A series of modern surveys
in mathematics ; 64)
ISBN 978-3-319-56477-7 [DOI]

Reports

N. Ay. *Principles of Robustness*

(in German). Annual Report of the Max Planck Society (2006).

(in German). Annual Report of the Max Planck Society (2006).

N. Ay, T. Wennekers. *Complexity of Cognitive Systems*

(in German). Annual Report of the Max Planck Society (2003).

(in German). Annual Report of the Max Planck Society (2003).

Media and Popular Science (refers to work with Ralf Der)

Telegraph (London) August 11, 2008 by Roger Highfield.

BBC News August 6, 2008 by Jason Palmer.

Tagesanzeiger (Zürich).

Die Welt (Hamburg).

Computer Zeitung Online.

Pressemitteilung der Max-Planck-Gesellschaft (MPG, München).

TV: *Eigensinnige Roboter.*

3sat (Newton, 2008).

3sat (Newton, 2008).

Max-Planck-Forschung, Wissenschaftsmagazin der MPG, Heft 1, 2009.

Wissenschaftsvideo der MPG 2009, to appear.

Spektrum der Wissenschaft, to appear in February 2010.

FOCUS Nr. 27, 2009.

Berliner Zeitung, September 5, 2009.