Surface acoustic waves (SAWs) on piezoelectric insulators can generate dynamic periodic potentials inside one-dimensional and two-dimensional materials. These periodic potentials have been utilized or proposed for various applications, including acoustoelectric charge pumping. In this study, we investigate acoustoelectric charge pumping in graphene with very low electrostatic disorder. By employing a graphite top gate on boron-nitride-encapsulated graphene, we adjust the graphene carrier concentration over a broad range, enabling us to examine the acoustoelectric signal in both mixed-carrier and single-carrier regimes. We discuss the benefits of hBN-encapsulated graphene for charge pumping applications and introduce a model that describes the acoustoelectric signal across all carrier concentrations, including at the charge neutrality point. This quantitative model will support future SAW-enabled explorations of phenomena in low-dimensional materials and guide the design of novel SAW sensors.
Commensurability phenomena abound in nature and are typically associated with mismatched lengths, as can occur in quasiperiodic systems. However, not all commensuration effects are spatial in nature. In finite-sized Dirac systems, an intriguing example arises in tilted or warped Dirac cones wherein the degeneracy in the speed of right- and left-moving electrons within a given Dirac cone or valley is lifted. Bound states can be purely fast-moving or purely slow-moving, giving rise to incommensurate energy level spacings and a vernier spectrum. In this work, we present evidence for this vernier spectrum in Coulomb blockade measurements of ultraclean suspended carbon nanotube quantum dots. The addition-energy spectrum of the quantum dots reveals an energy-level structure that oscillates between aligned and misaligned energy levels. Our data suggest that the fast- and slow-moving bound states hybridize at certain gate voltages. Thus, gate-voltage tuning can select states with varying degrees of hybridization, suggesting numerous applications based on accessing this isospin-like degree of freedom.
We seek to infer the parameters of an ergodic Markov process from samples taken independently from the steady state. Our focus is on non-equilibrium processes, where the steady state is not described by the Boltzmann measure, but is generally unknown and hard to compute, which prevents the application of established equilibrium inference methods. We propose a quantity we call propagator likelihood, which takes on the role of the likelihood in equilibrium processes. This propagator likelihood is based on fictitious transitions between those configurations of the system which occur in the samples. The propagator likelihood can be derived by minimising the relative entropy between the empirical distribution and a distribution generated by propagating the empirical distribution forward in time. Maximising the propagator likelihood leads to an efficient reconstruction of the parameters of the underlying model in different systems, both with discrete configurations and with continuous configurations. We apply the method to non-equilibrium models from statistical physics and theoretical biology, including the asymmetric simple exclusion process (ASEP), the kinetic Ising model, and replicator dynamics.
Inverse problems in statistical physics are motivated by the challenges of `big data' in different fields, in particular high-throughput experiments in biology. In inverse problems, the usual procedure of statistical physics needs to be reversed: Instead of calculating observables on the basis of model parameters, we seek to infer parameters of a model based on observations. In this review, we focus on the inverse Ising problem and closely related problems, namely how to infer the coupling strengths between spins given observed spin correlations, magnetisations, or other data. We review applications of the inverse Ising problem, including the reconstruction of neural connections, protein structure determination, and the inference of gene regulatory networks. For the inverse Ising problem in equilibrium, a number of controlled and uncontrolled approximate solutions have been developed in the statistical mechanics community. A particularly strong method, pseudolikelihood, stems from statistics. We also review the inverse Ising problem in the non-equilibrium case, where the model parameters must be reconstructed based on non-equilibrium statistics.
The inverse Ising problem seeks to reconstruct the parameters of an Ising Hamiltonian on the basis of spin configurations sampled from the Boltzmann measure. Over the last decade, many applications of the inverse Ising problem have arisen, driven by the advent of large-scale data across different scientific disciplines. Recently, strategies to solve the inverse Ising problem based on convex optimisation have proven to be very successful. These approaches maximise particular objective functions with respect to the model parameters. Examples are the pseudolikelihood method and interaction screening. In this paper, we establish a link between approaches to the inverse Ising problem based on convex optimisation and the statistical physics of disordered systems. We characterise the performance of an arbitrary objective function and calculate the objective function which optimally reconstructs the model parameters. We evaluate the optimal objective function within a replica-symmetric ansatz and compare the results of the optimal objective function with other reconstruction methods. Apart from giving a theoretical underpinning to solving the inverse Ising problem by convex optimisation, the optimal objective function outperforms state-of-the-art methods, albeit by a small margin.
Non-equilibrium systems lack an explicit characterisation of their steady state like the Boltzmann distribution for equilibrium systems. This has drastic consequences for the inference of parameters of a model when its dynamics lacks detailed balance. Such non-equilibrium systems occur naturally in applications like neural networks or gene regulatory networks. Here, we focus on the paradigmatic asymmetric Ising model and show that we can learn its parameters from independent samples of the non-equilibrium steady state. We present both an exact inference algorithm and a computationally more efficient, approximate algorithm for weak interactions based on a systematic expansion around mean-field theory. Obtaining expressions for magnetisations, two- and three-point spin correlations, we establish that these observables are sufficient to infer the model parameters. Further, we discuss the symmetries characterising the different orders of the expansion around the mean field and show how different types of dynamics can be distinguished on the basis of samples from the non-equilibrium steady state.
Thermoelectric effects allow the generation of electrical power from waste heat and the electrical control of cooling and heating. Remarkably, these effects are also highly sensitive to the asymmetry in the density of states around the Fermi energy and can therefore be exploited as probes of distortions in the electronic structure at the nanoscale. Here we consider two-dimensional graphene as an excellent nanoscale carbon material for exploring the interaction between electronic and thermal transport phenomena, by presenting a direct and quantitative measurement of the Peltier component to electronic cooling and heating in graphene. Thanks to an architecture including nanoscale thermometers, we detected Peltier component modulation of up to 15 mK for currents of 20 $\mu$A at room temperature and observed a full reversal between Peltier cooling and heating for electron and hole regimes. This fundamental thermodynamic property is a complementary tool for the study of nanoscale thermoelectric transport in two-dimensional materials.
Graphene is an interesting material for spintronics, showing long spin relaxation lengths even at room temperature. For future spintronic devices it is important to understand the behavior of the spins and the limitations for spin transport in structures where the dimensions are smaller than the spin relaxation length. However, the study of spin injection and transport in graphene nanostructures is highly unexplored. Here we study the spin injection and relaxation in nanostructured graphene with dimensions smaller than the spin relaxation length. For graphene nanoislands, where the edge length to area ratio is much higher than for standard devices, we show that enhanced spin-flip processes at the edges do not seem to play a major role in the spin relaxation. On the other hand, contact induced spin relaxation has a much more dramatic effect for these low dimensional structures. By studying the nonlocal spin transport through a graphene quantum dot we observe that the obtained values for spin relaxation are dominated by the connecting graphene islands and not by the quantum dot itself. Using a simple model we argue that future nonlocal Hanle precession measurements can obtain a more significant value for the spin relaxation time for the quantum dot by using high spin polarization contacts in combination with low tunneling rates.
It has been shown recently that in spin precession experiments, the interaction of spins with localized states can change the response to a magnetic field, leading to a modified, effective spin relaxation time and precession frequency. Here, we show that also the shape of the Hanle curve can change, so that it cannot be fitted with the solutions of the conventional Bloch equation. We present experimental data that shows such an effect arising at low temperatures in epitaxial graphene on silicon carbide with localized states in the carbon buffer layer. We compare the strength of the effect between materials with different growth methods, epitaxial growth by sublimation and by chemical vapor deposition. The presented analysis gives information about the density of localized states and their coupling to the graphene states, which is inaccessible by charge transport measurements and can be applied to any spin transport channel that is coupled to localized states.
Single electron spins in semiconductor quantum dots (QDs) are a versatile platform for quantum information processing, however controlling decoherence remains a considerable challenge. Recently, hole spins have emerged as a promising alternative. Holes in III-V semiconductors have unique properties, such as strong spin-orbit interaction and weak coupling to nuclear spins, and therefore have potential for enhanced spin control and longer coherence times. Weaker hyperfine interaction has already been reported in self-assembled quantum dots using quantum optics techniques. However, challenging fabrication has so far kept the promise of hole-spin-based electronic devices out of reach in conventional III-V heterostructures. Here, we report gate-tuneable hole quantum dots formed in InSb nanowires. Using these devices we demonstrate Pauli spin blockade and electrical control of single hole spins. The devices are fully tuneable between hole and electron QDs, enabling direct comparison between the hyperfine interaction strengths, g-factors and spin blockade anisotropies in the two regimes.
In a multicellular organism different cell types express a gene in different amounts. Samples from which gene expression levels can be measured typically contain a mixture of different cell types, the resulting measurements thus give only averages over the different cell types present. Based on fluctuations in the mixture proportions from sample to sample it is in principle possible to reconstruct the underlying expression levels of each cell type: to deconvolute the sample. We use a statistical mechanics approach to the problem of deconvoluting such partial concentrations from mixed samples, give analytical results for when and how well samples can be unmixed, and suggest an algorithm for sample deconvolution.
Due to the strong spin-orbit interaction in indium antimonide, orbital motion and spin are no longer separated. This enables fast manipulation of qubit states by means of microwave electric fields. We report Rabi oscillation frequencies exceeding 100 MHz for spin-orbit qubits in InSb nanowires. Individual qubits can be selectively addressed due to intrinsic dierences in their g-factors. Based on Ramsey fringe measurements, we extract a coherence time T_2* = 8 +/- 1 ns at a driving frequency of 18.65 GHz. Applying a Hahn echo sequence extends this coherence time to 35 ns.
We use electric dipole spin resonance to measure dynamic nuclear polarization in InAs nanowire quantum dots. The resonance shifts in frequency when the system transitions between metastable high and low current states, indicating the presence of nuclear polarization. We propose that the low and the high current states correspond to different total Zeeman energy gradients between the two quantum dots. In the low current state, dynamic nuclear polarization efficiently compensates the Zeeman gradient due to the $g$-factor mismatch, resulting in a suppressed total Zeeman gradient. We present a theoretical model of electron-nuclear feedback that demonstrates a fixed point in nuclear polarization for nearly equal Zeeman splittings in the two dots and predicts a narrowed hyperfine gradient distribution.
We developed a spin transport model for a diffusive channel with coupled localized states that result in an effective increase of spin precession frequencies and a reduction of spin relaxation times in the system. We apply this model to Hanle spin precession measurements obtained on monolayer epitaxial graphene on SiC(0001) (MLEG). Combined with newly performed measurements on quasi-free-standing monolayer epitaxial graphene on SiC(0001) our analysis shows that the different values for the diffusion coefficient measured in charge and spin transport measurements in MLEG and the high values for the spin relaxation time can be explained by the influence of localized states arising from the buffer layer at the interface between the graphene and the SiC surface.
The large amounts of data from molecular biology and neuroscience have lead to a renewed interest in the inverse Ising problem: how to reconstruct parameters of the Ising model (couplings between spins and external fields) from a number of spin configurations sampled from the Boltzmann measure. To invert the relationship between model parameters and observables (magnetisations and correlations) mean-field approximations are often used, allowing to determine model parameters from data. However, all known mean-field methods fail at low temperatures with the emergence of multiple thermodynamic states. Here we show how clustering spin configurations can approximate these thermodynamic states, and how mean-field methods applied to thermodynamic states allow an efficient reconstruction of Ising models also at low temperatures.
We developed an easy, upscalable process to prepare lateral spin-valve devices on epitaxially grown monolayer graphene on SiC(0001) and perform nonlocal spin transport measurements. We observe the longest spin relaxation times tau_S in monolayer graphene, while the spin diffusion coefficient D_S is strongly reduced compared to typical results on exfoliated graphene. The increase of tau_S is probably related to the changed substrate, while the cause for the small value of D_S remains an open question.
Double quantum dot in the few-electron regime is achieved using local gating in an InSb nanowire. The spectrum of two-electron eigenstates is investigated using electric dipole spin resonance. Singlet-triplet level repulsion caused by spin-orbit interaction is observed. The size and the anisotropy of singlet-triplet repulsion are used to determine the magnitude and the orientation of the spin-orbit effective field in an InSb nanowire double dot. The obtained results are confirmed using spin blockade leakage current anisotropy and transport spectroscopy of individual quantum dots.
We apply the Bethe-Peierls approximation to the problem of the inverse Ising model and show how the linear response relation leads to a simple method to reconstruct couplings and fields of the Ising model. This reconstruction is exact on tree graphs, yet its computational expense is comparable to other mean-field methods. We compare the performance of this method to the independent-pair, naive mean- field, Thouless-Anderson-Palmer approximations, the Sessak-Monasson expansion, and susceptibility propagation in the Cayley tree, SK-model and random graph with fixed connectivity. At low temperatures, Bethe reconstruction outperforms all these methods, while at high temperatures it is comparable to the best method available so far (Sessak-Monasson). The relationship between Bethe reconstruction and other mean- field methods is discussed.
The recent observation of fractional quantum Hall effect in high mobility suspended graphene devices introduced a new direction in graphene physics, the field of electron-electron interaction dynamics. However, the technique used currently for the fabrication of such high mobility devices has several drawbacks. The most important is that the contact materials available for electronic devices are limited to only a few metals (Au, Pd, Pt, Cr and Nb) since only those are not attacked by the reactive acid (BHF) etching fabrication step. Here we show a new technique which leads to mechanically stable suspended high mobility graphene devices which is compatible with almost any type of contact material. The graphene devices prepared on a polydimethylglutarimide based organic resist show mobilities as high as 600.000 cm^2/Vs at an electron carrier density n = 5.0 10^9 cm^-2 at 77K. This technique paves the way towards complex suspended graphene based spintronic, superconducting and other types of devices.
This paper addresses the statistical significance of structures in random data: Given a set of vectors and a measure of mutual similarity, how likely does a subset of these vectors form a cluster with enhanced similarity among its elements? The computation of this cluster p-value for randomly distributed vectors is mapped onto a well-defined problem of statistical mechanics. We solve this problem analytically, establishing a connection between the physics of quenched disorder and multiple testing statistics in clustering and related problems. In an application to gene expression data, we find a remarkable link between the statistical significance of a cluster and the functional relationships between its genes.
Alexander S. Zyazin, Johan W.G. van den Berg, Edgar A. Osorio, Herre S.J. van der Zant, Nikolaos P. Konstantinidis, Martin Leijnse, Maarten R. Wegewijs, Falk May, Walter Hofstetter, Chiara Danieli, Andrea Cornia We have measured quantum transport through an individual Fe$_4$ single-molecule magnet embedded in a three-terminal device geometry. The characteristic zero-field splittings of adjacent charge states and their magnetic field evolution are observed in inelastic tunneling spectroscopy. We demonstrate that the molecule retains its magnetic properties, and moreover, that the magnetic anisotropy is significantly enhanced by reversible electron addition / subtraction controlled with the gate voltage. Single-molecule magnetism can thus be electrically controlled.
Gene expression is a stochastic process governed by the presence of specific transcription factors. Here we study the dynamics of gene expression in the presence of feedback, where a gene regulates its own expression. The nonlinear coupling between input and output of gene expression can generate a dynamics different from simple scenarios such as the Poisson process. This is exemplified by our findings for the time intervals over which genes are transcriptionally active and inactive. We apply our results to the lac system in E. coli, where parametric inference on experimental data results in a broad distribution of gene activity intervals.
In order to express specific genes at the right time, the transcription of genes is regulated by the presence and absence of transcription factor molecules. With transcription factor concentrations undergoing constant changes, gene transcription takes place out of equilibrium. In this paper we discuss a simple mapping between dynamic models of gene expression and stochastic systems driven out of equilibrium. Using this mapping, results of nonequilibrium statistical mechanics such as the Jarzynski equality and the fluctuation theorem are demonstrated for gene expression dynamics. Applications of this approach include the determination of regulatory interactions between genes from experimental gene expression data.
Complex interactions between genes or proteins contribute a substantial part to phenotypic evolution. Here we develop an evolutionarily grounded method for the cross-species analysis of interaction networks by \em alignment, which maps bona fide functional relationships between genes in different organisms. Network alignment is based on a scoring function measuring mutual similarities between networks taking into account their interaction patterns as well as sequence similarities between their nodes. High-scoring alignments and optimal alignment parameters are inferred by a systematic Bayesian analysis. We apply this method to analyze the evolution of co-expression networks between human and mouse. We find evidence for significant conservation of gene expression clusters and give network-based predictions of gene function. We discuss examples where cross-species functional relationships between genes do not concur with sequence similarity.
In this article, we review some of our approaches to granular dynamics, now well known to consist of both fast and slow relaxational processes. In the first case, grains typically compete with each other, while in the second, they cooperate. A typical result of \it cooperation is the formation of stable bridges, signatures of spatiotemporal inhomogeneities; we review their geometrical characteristics and compare theoretical results with those of independent simulations. \it Cooperative excitations due to local density fluctuations are also responsible for relaxation at the angle of repose; the \it competition between these fluctuations and external driving forces, can, on the other hand, result in a (rare) collapse of the sandpile to the horizontal. Both these features are present in a theory reviewed here. An arena where the effects of cooperation versus competition are felt most keenly is granular compaction; we review here a random graph model, where three-spin interactions are used to model compaction under tapping. The compaction curve shows distinct regions where 'fast' and 'slow' dynamics apply, separated by what we have called the \it single-particle relaxation threshold. In the final section of this paper, we explore the effect of shape -- jagged vs. regular -- on the compaction of packings near their jamming limit. One of our major results is an entropic landscape that, while microscopically rough, manifests \it Edwards' flatness at a macroscopic level. Another major result is that of surface intermittency under low-intensity shaking.
Interaction networks are of central importance in post-genomic molecular biology, with increasing amounts of data becoming available by high-throughput methods. Examples are gene regulatory networks or protein interaction maps. The main challenge in the analysis of these data is to read off biological functions from the topology of the network. Topological motifs, i.e., patterns occurring repeatedly at different positions in the network have recently been identified as basic modules of molecular information processing. In this paper, we discuss motifs derived from families of mutually similar but not necessarily identical patterns. We establish a statistical model for the occurrence of such motifs, from which we derive a scoring function for their statistical significance. Based on this scoring function, we develop a search algorithm for topological motifs called graph alignment, a procedure with some analogies to sequence alignment. The algorithm is applied to the gene regulation network of E. coli.
The regulation of a gene depends on the binding of transcription factors to specific sites located in the regulatory region of the gene. The generation of these binding sites and of cooperativity between them are essential building blocks in the evolution of complex regulatory networks. We study a theoretical model for the sequence evolution of binding sites by point mutations. The approach is based on biophysical models for the binding of transcription factors to DNA. Hence we derive empirically grounded fitness landscapes, which enter a population genetics model including mutations, genetic drift, and selection. We show that the selection for factor binding generically leads to specific correlations between nucleotide frequencies at different positions of a binding site. We demonstrate the possibility of rapid adaptive evolution generating a new binding site for a given transcription factor by point mutations. The evolutionary time required is estimated in terms of the neutral (background) mutation rate, the selection coefficient, and the effective population size. The efficiency of binding site formation is seen to depend on two joint conditions: the binding site motif must be short enough and the promoter region must be long enough. These constraints on promoter architecture are indeed seen in eukaryotic systems. Furthermore, we analyse the adaptive evolution of genetic switches and of signal integration through binding cooperativity between different sites. Experimental tests of this picture involving the statistics of polymorphisms and phylogenies of sites are discussed.
The structure of molecular networks derives from dynamical processes on evolutionary time scales. For protein interaction networks, global statistical features of their structure can now be inferred consistently from several large-throughput datasets. Understanding the underlying evolutionary dynamics is crucial for discerning random parts of the network from biologically important properties shaped by natural selection. We present a detailed statistical analysis of the protein interactions in Saccharomyces cerevisiae based on several large-throughput datasets. Protein pairs resulting from gene duplications are used as tracers into the evolutionary past of the network. From this analysis, we infer rate estimates for two key evolutionary processes shaping the network: (i) gene duplications and (ii) gain and loss of interactions through mutations in existing proteins, which are referred to as link dynamics. Importantly, the link dynamics is asymmetric, i.e., the evolutionary steps are mutations in just one of the binding parters. The link turnover is shown to be much faster than gene duplications. According to this model, the link dynamics is the dominant evolutionary force shaping the statistical structure of the network, while the slower gene duplication dynamics mainly affects its size. Specifically, the model predicts (i) a broad distribution of the connectivities (i.e., the number of binding partners of a protein) and (ii) correlations between the connectivities of interacting proteins.
We develop a statistical theory of networks. A network is a set of vertices and links given by its adjacency matrix $\c$, and the relevant statistical ensembles are defined in terms of a partition function $Z=\sum_{\c} \exp {[}-\beta \H(\c) {]}$. The simplest cases are uncorrelated random networks such as the well-known Erdös-Rény graphs. Here we study more general interactions $\H(\c)$ which lead to \em correlations, for example, between the connectivities of adjacent vertices. In particular, such correlations occur in \em optimized networks described by partition functions in the limit $\beta \to \infty$. They are argued to be a crucial signature of evolutionary design in biological networks.
The Edwards hypothesis of ergodicity of blocked configurations for gently tapped granular materials is tested for abstract models of spin systems on random graphs and spin chains with kinetic constraints. The tapping dynamics is modeled by considering two distinct mechanisms of energy injection: thermal and random tapping. We find that ergodicity depends upon the tapping procedure (i.e. the way the blocked configurations are dynamically accessed): for thermal tapping ergodicity is a good approximation, while it fails to describe the asymptotic stationary state reached by the random tapping dynamics.
We discuss two athermal types of dynamics suitable for spin-models designed to model repeated tapping of a granular assembly. These dynamics are applied to a range of models characterised by a 3-spin Hamiltonian aiming to capture the geometric frustration in packings of granular matter.
One-flip stable configurations of an Ising-model on a random graph with fluctuating connectivity are examined. In order to perform the quenched average of the number of stable configurations we introduce a global order-parameter function with two arguments. The analytical results are compared with numerical simulations.
We discuss the use of a ferromagnetic spin model on a random graph to model granular compaction. A multi-spin interaction is used to capture the competition between local and global satisfaction of constraints characteristic for geometric frustration. We define an athermal dynamics designed to model repeated taps of a given strength. Amplitude cycling and the effect of permanently constraining a subset of the spins at a given amplitude is discussed. Finally we check the validity of Edwards' hypothesis for the athermal tapping dynamics.
Traders in a market typically have widely different, private information on the return of an asset. The equilibrium price of the asset may reflect this information more accurately if the number of traders is large enough compared to the number of the states of the world that determine the return of the asset. We study the transition from markets where prices do not reflect the information accurately into markets where it does. In competitive markets, this transition takes place suddenly, at a critical value of the ratio between number of states and number of traders. The Nash equilibrium market behaves quite differently from a competitive market even in the limit of large economies.
The dynamics of spins on a random graph with ferromagnetic three-spin interactions is used to model the compaction of granular matter under a series of taps. Taps are modelled as the random flipping of a small fraction of the spins followed by a quench at zero temperature. We find that the density approached during a logarithmically slow compaction - the random-close-packing density - corresponds to a dynamical phase transition. We discuss the the role of cascades of successive spin-flips in this model and link them with density-noise power fluctuations observed in recent experiments.
Using methods from the statistical mechanics of disordered systems we analyze the properties of bimatrix games with random payoffs in the limit where the number of pure strategies of each player tends to infinity. We analytically calculate quantities such as the number of equilibrium points, the expected payoff, and the fraction of strategies played with non-zero probability as a function of the correlation between the payoff matrices of both players and compare the results with numerical simulations.
We use techniques from the statistical mechanics of disordered systems to analyse the properties of Nash equilibria of bimatrix games with large random payoff matrices. By means of an annealed bound, we calculate their number and analyse the properties of typical Nash equilibria, which are exponentially dominant in number. We find that a randomly chosen equilibrium realizes almost always equal payoffs to either player. This value and the fraction of strategies played at an equilibrium point are calculated as a function of the correlation between the two payoff matrices. The picture is complemented by the calculation of the properties of Nash equilibria in pure strategies.
Matrix games constitute a fundamental problem of game theory and describe a situation of two players with completely conflicting interests. We show how methods from statistical mechanics can be used to investigate the statistical properties of optimal mixed strategies of large matrix games with random payoff matrices and derive analytical expressions for the value of the game and the distribution of strategy strengths. In particular the fraction of pure strategies not contributing to the optimal mixed strategy of a player is calculated. Both independently distributed as well as correlated elements of the payoff matrix are considered and the results compared with numerical simulations.
We calculate the multifractal spectrum of the partition of the coupling space of a perceptron induced by random input-output pairs with non-zero mean. From the results we infer the influence of the input and output bias respectively on both the storage and generalization properties of the network. It turns out that the value of the input bias is irrelevant as long as it is different from zero. The generalization problem with output bias is new and shows an interesting two-level scenario. To compare our analytical results with simulations we introduce a simple and efficient algorithm to implement Gibbs learning.
Braneshop, and
the US National Science Foundation.
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