Table of contents

Synapses in real neural circuits can take discrete values including zero (silent or potential) synapses. The computational role of zero synapses in unsupervised feature learning of unlabeled noisy data is still unclear, thus it is important to understand how the sparseness of synaptic activity is shaped during learning and its relationship with receptive field formation. Here, we formulate this kind of sparse feature learning by a statistical mechanics approach. We find that learning decreases the fraction of zero synapses, and when the fraction decreases rapidly around a critical data size, an intrinsically structured receptive field starts to develop. Further increasing the data size refines the receptive field, while a very small fraction of zero synapses remain to act as contour detectors. This phenomenon is discovered not only in learning a handwritten digits dataset, but also in learning retinal neural activity measured in a natural-movie-stimuli experiment.

Instances of negative mobility, where a system responds to a perturbation in a way opposite to naive expectation, have been studied theoretically and experimentally in numerous nonequilibrium systems. In this work we show that absolute negative mobility (ANM), whereby current is produced in a direction opposite to the drive, can occur around equilibrium states. This is demonstrated with a simple one-dimensional lattice model with a driven tracer. We derive analytical predictions in the linear response regime and elucidate the mechanism leading to ANM by studying the high-density limit. We also study numerically a model of hard Brownian disks in a narrow planar channel, for which the lattice model can be viewed as a toy model. We find that the model exhibits negative differential mobility (NDM), but no ANM.

Perceptrons are the building blocks of many theoretical approaches to a wide range of complex systems, ranging from neural networks and deep learning machines, to constraint satisfaction problems, glasses and ecosystems. Despite their applicability and importance, a detailed study of their Langevin dynamics has never been performed yet. Here we derive the mean-field dynamical equations that describe the continuous random perceptron in the thermodynamic limit, in a very general setting with arbitrary noise and friction kernels, not necessarily related by equilibrium relations. We derive the equations in two ways: via a dynamical cavity method, and via a path-integral approach in its supersymmetric formulation. The end point of both approaches is the reduction of the dynamics of the system to an effective stochastic process for a representative dynamical variable. Because the perceptron is formally very close to a system of interacting particles in a high dimensional space, the methods we develop here can be transferred to the study of liquid and glasses in high dimensions. Potentially interesting applications are thus the study of the glass transition in active matter, the study of the dynamics around the jamming transition, and the calculation of rheological properties in driven systems.

We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.

Application of multivariate creative telescoping to a finite triple sum representation of the discrete space-time Green's function for an arbitrary numeric (non-symbolic) lattice point on a 3D simple cubic lattice produces a fast, no-neighbours, seventh-order, eighteenth-degree, discrete-time recurrence scheme.

For arbitrary numeric lattice points outside the diagonal symmetry planes, the seven numeric eighteenth-degree polynomial coefficients of the recurrence scheme are products of polynomials with integer coefficients that are linear in the recurrence index n, and two polynomials of degree four, and five polynomials of degree twelve that are irreducible over the field of integers. Owing to the symmetry of the scalar Green's function upon interchanging any of the lattice point coordinates, the twelfth degree polynomials with integer coefficients may each be expanded in terms of 102 elementary symmetric polynomials in symbolic lattice point coordinates. The recurrence schemes determined by the telescoper for 102 distinct numeric lattice points can be used to form linear systems of equations. These are solved for the coefficients of the elementary symmetric polynomials required to construct the symbolic polynomial coefficients of the generic 3D recurrence scheme.

Given its compact and straightforward 2D counterpart, this 3D recurrence scheme is far more intricate than expected, and is most efficiently presented through tables of coefficients. However, the scheme and the resulting lattice Green's function sequences also exhibit more features. The complexity reduces for lattice points on diagonal symmetry planes, yielding a fast no-neighbours, fifth-order, twelfth-degree, discrete-time recurrence scheme. An illustrative example reveals unexpected phenomena, e.g. a late-time, high-frequency interplay of resonances that appears anomalous but can be fully explained, and the possible occurrence of removable recurrence scheme singularities. These effects are studied in detail in separate papers.

We construct the coherent states and Schrödinger cat states associated with new types of ladder operators for a particular case of a rationally extended harmonic oscillator involving type III Hermite exceptional orthogonal polynomials. In addition to the coherent states of the annihilation operator, c, we form the linearised version, $\tilde{c},$ and obtain its coherent states. We find that while the coherent states defined as eigenvectors of the annihilation operator c display only quantum behaviour, those of the linearised version, $\tilde{c},$ have position probability densities displaying distinct wavepackets oscillating and colliding in the potential. The collisions are certainly quantum, as interference fringes are produced, but the remaining evolution indicates a classical analogue.

We simplify the nonlinear equations of motion of charged particles in an external electromagnetic field that is the sum of a plane travelling wave $F_t^{\mu\nu}(ct - z)$ and a static part $F_s^{\mu\nu}(x, y, z)$: by adopting the light-like coordinate $\xi=ct - z$ instead of time t as an independent variable in the Action, Lagrangian and Hamiltonian, and deriving the new Euler–Lagrange and Hamilton equations accordingly, we make the unknown $z(t)$ disappear from the argument of $F_t^{\mu\nu}$. We first study and solve the single particle equations in a few significant cases of extreme accelerations. In particular, we obtain a rigorous formulation of a Lawson–Woodward-type (no-final-acceleration) theorem and a compact derivation of cyclotron autoresonance, beside new solutions in the presence of uniform $F_s^{\mu\nu}$. We then extend our method to plasmas in hydrodynamic conditions, and apply it to plane problems: the system of (Lorentz–Maxwell  +  continuity) partial differential equations may be partially solved or sometimes even completely reduced to a family of decoupled systems of ordinary ones; this occurs e.g. with the impact of the travelling wave on a vacuum-plasma interface (what may produce the slingshot effect).

Our method can be seen as an application of the light-front approach. Since Fourier analysis plays no role in our general framework, the method can be applied to all kinds of travelling waves, ranging from almost monochromatic to so-called 'impulses', which contain few, one or even no complete cycles.

The renewed Green's function approach to calculating the angular Fock coefficients, $\psi_{k, p}(\alpha, \theta)$ is presented. The final formulas are simplified and specified to be applicable for analytical, as well as numerical calculations. The Green's function formulas with the hyperspherical angles $\theta=0, \pi$ (arbitrary α) or $\alpha=0, \pi$ (arbitrary θ) are indicated as corresponding to the angular Fock coefficients possessing physical meaning. The most interesting case of $\theta=0$ corresponding to a collinear arrangement of the particles is studied in detail. It is emphasized that this case represents the generalization of the specific cases of the electron–nucleus ($\alpha=0$) and electron–electron ($\alpha=\pi/2$) coalescences. It is shown that the Green's function method for $\theta=0$ enables us to calculate any component/subcomponent of the angular Fock coefficient in the form of a single series representation with arbitrary angle θ. Those cases where the Green's function approach cannot be applied, are thoroughly studied, and the corresponding solutions are found.

We explore the existence of a class of generalised Laplace maps for third-order partial differential operators of the form

∂₁∂₂ ∂₃ + a₁∂₂∂₃ + a₂∂₁∂₃ + a₃∂₁∂₂ + a₁₂∂₃ + a₂₃∂₁ + a₁₃∂₂ + a₁₂₃

and related first order $3\times 3$ systems and show that they require the satisfaction of constraints on the invariants for such operators.

A mixed quantum state is represented by a Hermitian positive semi-definite operator ρ with the unit trace. The positivity requirement is responsible for a highly nontrivial geometry of the set of quantum states. A known way to satisfy this requirement automatically is to use the map $\newcommand{\tr}{{\rm tr}\,} \rho=\tau^2/\tr\tau^2$, where τ can be an arbitrary nonzero Hermitian operator. We elaborate the parametrization of the set of quantum states induced by the parametrization of the linear space of Hermitian operators by virtue of this map. In particular, we derive an equation for the boundary of the set. Further, we discuss how this parametrization can be applied to a set of quantum states constrained by some symmetry. We consider several examples of the squaring parametrisation of sets of qubits and qutrits constrained by various symmetries.

Bohmian trajectories are considered for a particle that is free (i.e. the potential energy is zero), except for a half-line barrier. On the barrier, both Dirichlet and Neumann boundary conditions are considered. The half-line barrier yields one of the simplest cases of diffraction. Using the exact time-dependent propagator found by Schulman, the trajectories are computed numerically for different initial Gaussian wave packets. In particular, it is found that different boundary conditions may lead to qualitatively different sets of trajectories. In the Dirichlet case, the particles tend to be more strongly repelled. The case of an incoming plane wave is also considered. The corresponding Bohmian trajectories are compared with the trajectories of an oil drop hopping on the surface of a vibrating bath.

Two very similar proposals have been made recently for witnessing nonclassical features of gravity, by Bose et al and by Marletto and Vedral. However, while these proposals are asserted to be very general, they are in fact based on a very strong claim: that quantum systems cannot become entangled via a classical intermediary. We point out that the support provided for this claim is only applicable to a very limited class of quantum-classical interaction models, corresponding to Koopman-type dynamics. We show that the claim is also valid for mean-field models, but that it is contradicted by explicit counterexamples based on the configuration-ensemble model. Thus, neither proposal provides a definitive test of nonclassical gravity.

Wave-particle duality demonstrates a competition relation between wave and particle behavior for a particle going through an interferometer. This duality can be formulated as an inequality, which upper bounds the sum of interference visibility and path information. However, if the particle is entangled with a quantum memory, then the bound may decrease. Here, we find the duality relation between coherence and path information for a particle going through a multipath interferometer in the presence of a quantum memory, offering an upper bound on the duality relation which is directly connected with the amount of entanglement between the particle and the quantum memory.

In this paper, using the techniques of Gauge/gravity duality we explore the hydrodynamic regime of z  =  3 Lifshitz fixed points in 1  +  1 dimensions. The speed of sound in the non relativistic plasma turns out to be $\sqrt{3}$, which clearly violates the conjectured upper bound. We identify this as a natural consequence of the Lorentz symmetry breakdown at z  =  3 Lifshitz fixed point. In our analysis, we compute bulk viscosity to entropy ratio for quantum field theory dual to Lifshitz gravity in 2  +  1 dimensions and it turns out that for this particular holographic model this ratio is above the conjectured lower bound.

In an article in 2010, Kim et al introduced the definition of Rényi-α entanglement for bipartite quantum states and established an analytic formula of Rényi-α entanglement for arbitrary two-qubit states with $\alpha\geqslant 1$. They also derived a monogamy of entanglement in multi-qubit systems in terms of Rényi-α entanglement for $\alpha\geqslant 2$ Kim et al (2010 J. Phys. A: Math. Theor. 43 445305). We find the proofs of theorems 2 and 3 contain some errors and we also present an improved derivation to overcome this flaw. The alternative derivation shows that the main conclusions remain valid despite the invalidity of the proofs.