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Moving averages, also termed convolution filters, are widely applied in science and engineering at large. As moving averages transform inputs to outputs by convolution, they induce correlation. In effect, moving averages are perhaps the most fundamental and ubiquitous mechanism of transforming uncorrelated inputs to correlated outputs. In this paper we study the correlation structure of general moving averages, unveil the Rényi-entropy meaning of a moving-averageʼs overall correlation, address the maximization of this overall correlation, and apply this overall correlation to the dispersion-measurement and to the classification of regular and anomalous diffusion transport processes.

In this work we generalize the etching model (Mello et al 2001 Phys. Rev. E 63 041113) to d + 1 dimensions. The dynamic exponents of this model are compatible with those of the Kardar–Parisi–Zhang universality class. We investigate the roughness dynamics with surfaces up to d = 6. We show that the data from all substrate lengths and for all dimensions can be collapsed into one common curve. We determine the dynamic exponents as a function of the dimension. Moreover, our results suggest that d = 4 is not an upper critical dimension for the etching model, and that it fulfills the Galilean invariance.

We solve the problem of a chain, modeled as a self-avoiding walk (SAW), grafted to the wall limiting a semi-infinite Bethe lattice of arbitrary coordination number q. In particular, we determine the pressure exerted by the polymer on the wall, as a function of the distance to the grafting point. The pressure, in general, decays exponentially with the distance, at variance with what is found for SAWs and directed walks on regular lattices and gaussian walks. The adsorption transition, which is discontinuous, and its influence on the pressure are also studied.

We provide a renormalization analysis of correlations in a quasi-periodically forced two-level system in a time dependent field with periodic kicks whose amplitude is given by a general class of discontinuous modulation function. For certain intensities of modulation, we give a complete understanding of the autocorrelation function. Furthermore, once the locations of the discontinuities of the modulation function are known, aperiodic orbits lead to correlations on renormalization strange sets which are determined by two specified features of the modulation function of which there are only a finite number of variations.

We find that a symbolic walk (SW)—performed by a walker with memory given by a Bernoulli shift—is able to distinguish between the random or chaotic topology of a given network. We show this result by means of studying the undirected baker network, which is defined by following the Ulam approach for the baker transformation in order to introduce the effect of deterministic chaos into its structure. The chaotic topology is revealed through the central role played by the nodes associated with the positions corresponding to the shortest periodic orbits of the generating map. They are the overwhelmingly most visited nodes in the limit cycles at which the SW asymptotically arrives. Our findings contribute to linking deterministic chaotic dynamics with the properties of networks constructed using the Ulam approach.

A theoretical description of the final stage of Ostwald ripening given by Lifshitz and Slyozov (LS) predicts that after long times the distribution of particles over sizes tends to a universal form. A qualitative behavior of their theory has been confirmed, but experimental particle size distributions are more broad and squat than the LS asymptotic solution. The origin of discrepancies between the theory and experimental data is caused by the relaxation of solutions from the early to late stages of Ostwald ripening. In other words, the initial conditions at the ripening stage lead to the formation of a transition region near the blocking point of the LS theory and completely determine the distribution function. A new theoretical approach of the present analysis based on the Slezov theory (Slezov 1978 Formation of the universal distribution function in the dimension space for new-phase particles in the diffusive decomposition of the supersaturated solid solution J. Phys. Chem. Solids39 367–74; Slezov 2009 Kinetics of First-Order Phase Transitions (Weinheim: Wiley, VCH)) focuses on a relaxation dynamics of analytical solutions from the early stage of Ostwald ripening to its concluding state, which is described by the LS asymptotic regime. An algebraic equation for the boundaries of a transition layer independent of all material parameters is derived. A time-dependent function $\varepsilon (\tau )$ responsible for the evolution of solutions at the ripening stage is found. The distribution function obtained is more broad and flat than the LS asymptotic solution. The particle radius, supersaturation and number density as functions of time are determined. The analytical solutions obtained are in good agreement with experimental data.

The present article completes the mathematical description initiated in the paper by Dhont and Zhilinskií (2013 The action of the orthogonal group on planar vectors: invariants, covariants and syzygies J. Phys. A: Math. Theor.46 455202) of the algebraic structures that emerge from the symmetry-adapted polynomials in the $({{x}_{i}},{{y}_{i}})$ coordinates of n planar vectors under the action of the SO(2) group. The set of $\left( m \right)$-covariant polynomials contains all the polynomials that transform according to the weight $m\in \mathbb{Z}$ of SO(2) and is a free module for $|m|\leqslant n-1$ but a non-free module for $|m|\geqslant n$. The sum of the rational functions of the Molien function for $\left( m \right)$-covariants describes the decomposition of the ring of invariants or the module of $\left( m \right)$-covariants as a direct sum of submodules. A method for extracting the generating function for $\left( m \right)$-covariants from the comprehensive generating function for all polynomials is introduced. The approach allows the direct construction of the integrity basis for the module of $\left( m \right)$-covariants decomposed as a direct sum of submodules and gives insight into the expressions for the Molien functions found in our earlier paper. In particular, a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed, where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed.

For a certain infinite family $\mathcal{F}$ of knots or links, we study the growth power ratios of their stick number, lattice stick number, minimum lattice length and minimum ropelength compared with their minimum crossing number c(K) for every $K\in \mathcal{F}$. It is known that the stick number and lattice stick number grow between the $\frac{1}{2}$ and linear power of the crossing number, and minimum lattice length and minimum ropelength grow with at least the $\frac{3}{4}$ power of crossing number (which is called the four-thirds power law). Furthermore, the minimal lattice length and minimum ropelength grow at most as O $(c(K){{[{\rm ln} (c(K))]}^{5}})$, but it is unknown whether any family exhibits superlinear growth. For any real number r between $\frac{1}{2}$ and 1, we give an infinite family of non-splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number. Furthermore for any real number r between $\frac{3}{4}$ and 1, we give another infinite family of non-splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number.

We construct the coherent states in the sense of Gilmore and Perelomov for the fermionic Fock space. Our treatment is from the outset adapted to the infinite-dimensional case. The fermionic Fock space becomes in this way a reproducing kernel Hilbert space of continuous holomorphic functions.

We present an inverse scattering transform (IST) approach for the (differentiated) Ostrovsky–Vakhnenko equation

$$\begin{eqnarray*}&&{{u}_{txx}}-3{{u}_{x}}+3{{u}_{x}}{{u}_{xx}}+u{{u}_{xxx}}=0.\end{eqnarray*}$$

We present a recursive method to generate the expansion of the lattice Green function of the d-dimensional face-centred cubic (fcc) lattice. We produce a long series for d = 7. Then we show (and recall) that, in order to obtain the linear differential equation annihilating such a long power series, the most economic way amounts to producing the non-minimal order differential equations. We use the method to obtain the minimal order linear differential equation of the lattice Green function of the seven-dimensional fcc lattice. We give some properties of this irreducible order-eleven differential equation. We show that the differential Galois group of the corresponding operator is included in $SO(11,\;\mathbb{C})$. This order-eleven operator is non-trivially homomorphic to its adjoint, and we give a 'decomposition' of this order-eleven operator in terms of four order-one self-adjoint operators and one order-seven self-adjoint operator. Furthermore, using the Landau conditions on the integral, we forward the regular singularities of the differential equation of the d-dimensional lattice and show that they are all rational numbers. We evaluate the return probability in random walks in the seven-dimensional fcc lattice. We show that the return probability in the d-dimensional fcc lattice decreases as d⁻² as the dimension d goes to infinity.

A general elliptic N × N matrix Lax scheme is presented, leading to two classes of elliptic lattice systems, one which we interpret as the higher-rank analogue of the Landau–Lifschitz equations, while the other class we characterize as the higher-rank analogue of the lattice Krichever–Novikov equation (or Adlerʼs lattice). We present the general scheme, but focus mainly on the latter type of models. In the case N = 2 we obtain a novel Lax representation of Adlerʼs elliptic lattice equation in its so-called 3-leg form. The case of rank N = 3 is analyzed using Cayleyʼs hyperdeterminant of format $2\times 2\times 2$, yielding a multi-component system of coupled 3-leg quad-equations.

The dynamics is investigated of a free particle on a sphere (rigid rotor or rotator) that is initially in a coherent state. The instability of coherent states with respect to the free evolution leads to nontrivial temporal development of averages of observables representing the position of a particle on a sphere that can be interpreted as quantum beats of the rotor. The beats are related to occuring quantum coherent state wave packet revivals on a sphere.

A zero-thickness limit for two-terminal and three-terminal devices from the quantum electronics domain is analysed. The study is focused on heterostructures composed of a single barrier with, adjacent, one or two prewells. The point interactions obtained in this limit are shown to be described by a family of 'resonant' diagonal matrices that connect the two-sided boundary conditions at the device origin, which are a subclass of the whole four-parameter family of point interactions. Transmission through such a device is absent almost everywhere, except at a few points, whose number and position can be controlled by a gate voltage applied externally to the barrier subsystem. It is remarkable that the existence of resonances in the zero-thickness limit occurs only if a squeezing sequence is constructed in a '$\delta ^{\prime}$-like' way. In this case, the $\delta ^{\prime}$-limit describes adequately the resonant behaviour of a barrier–well heterostructure with realistic parameters. Simple analytical expressions obtained for resonance sets are supported by direct numerical calculations of the transmission being in agreement with the results of experiments on semiconductor devices. The zero-thickness $\delta ^{\prime}$-like limiting procedure can be used in the design of nanodevices or contacting quantum wires.

Here we consider one- and two-dimensional nonlinear Schrödinger equations with double well potential and a Stark-type perturbation term. In the semiclassical limit we give an explicit solution to these equations for times of the order of the unperturbed beating period, up to an exponentially small remainder term. In particular, it turns out that the solution has a periodic behavior and the period is explicitly computed.

In previous papers about searches on star graphs several patterns have been made apparent; the speed up only occurs when graphs are 'tuned' so that their time step operators have degenerate eigenvalues, and only certain initial states are effective. More than that, the searches are never faster than $O(\sqrt{N})$ time. In this paper the problem is defined rigorously, the causes for all of these patterns are identified, sufficient and necessary conditions for quadratic-speed searches for any connected subgraph are demonstrated, the tolerance of these conditions is investigated, and it is shown that (unfortunately) we can do no better than $O(\sqrt{N})$ time. Along the way, a useful formalism is established that may be useful in future work involving highly symmetric graphs.

We compute the spectrum of radiation emitted by a generic quantum system interacting with an external classic noise. Our motivation is a wish to understand this phenomenon within the framework of collapse models. However, the computation is general and applies to practically any situation in which a quantum system interacts with a noise. The computation is carried out at a perturbative level. This poses problems as regards the correct way of performing the analysis, as repeatedly discussed in the literature. We will also clarify this issue.

We analyze the metric properties of conditioned quantum state spaces $\mathcal{M}_{\eta }^{(n\times m)}.$ These spaces are the convex sets of $nm\times nm$ density matrices that, when partially traced over m degrees of freedom, respectively yield the given n × n density matrix η. For the case n = 2, the volume of $\mathcal{M}_{\eta }^{(2\times m)}$ equipped with the Hilbert–Schmidt measure can be conjectured to be a simple polynomial of the radius of η in the Bloch-ball. Remarkably, for $m=2,3$ we find numerically that the probability $p_{{\rm sep}}^{(2\times m)}(\eta )$ to find a separable state in $\mathcal{M}_{\eta }^{(2\times m)}$ is independent of η (except for η pure). For $m\gt 3$, the same holds for $p_{{\rm PosPart}}^{(2\times m)}(\eta )$, the probability to find a state with a positive partial transpose in $\mathcal{M}_{\eta }^{(2\times m)}$. These results are proven analytically for the case of the family of 4 × 4 X-states, and thoroughly numerically investigated for the general case. The important implications of these findings for the clarification of open problems in quantum theory are pointed out and discussed.

We develop a framework which unifies seemingly different extension (or 'joinability') problems for bipartite quantum states and channels. This includes known extension problems such as optimal quantum cloning and quantum marginal problems as special instances. Central to our generalization is a variant of the Jamiołkowski isomorphism between bipartite states and linear transformations, which we term the homocorrelation map: in contrast to the better-known Choi isomorphism which emphasizes the preservation of the positivity constraint, use of the Jamiołkowski isomorphism allows one to characterize the preservation of the statistical correlations of bipartite states and quantum channels. The resulting homocorrelation map thus acquires a natural operational interpretation. We define and analyze state-joining, channel-joining, and local-positive-joining problems in three-party settings with collective $U\otimes U\otimes U$ symmetry, obtaining exact analytical characterizations in low dimensions. We find that bipartite quantum states are limited in the degree to which their measurement outcomes may agree, whereas quantum channels are limited in the degree to which their measurement outcomes may disagree. Loosely speaking, quantum mechanics enforces an upper bound on the strength of positive correlation across two subsystems at a single time, as well as on the strength of negative correlation between the state of a single system across two instants of time. We argue that these general statistical bounds inform the quantum joinability limitations, and show that they are in fact sufficient for the three-party $U\otimes U\otimes U$-invariant setting.

We show that a suitable choice for the potential term in the two-dimensional baby Skyrme model yields solitons that have a short-range repulsion and a long-range attraction. The solitons are therefore aloof, in the sense that static multi-soliton bound states have constituents that preserve their individual identities and are sufficiently far apart that tail interactions yield small binding energies. The static multi-soliton solutions are found to have a cluster structure that is reproduced by a simple binary species particle model. In the standard three-dimensional Skyrme model of nuclei, solitons are too tightly bound and are often too symmetric, due to symmetry enhancement as solitons coalesce to form bound states. The aloof baby Skyrmion results endorse a way to resolve these issues and provides motivation for a detailed study of the related three-dimensional version of the Skyrme model.

We study the relation between the frame-like and metric-like formulation of higher-spin gauge theories in three space–time dimensions. We concentrate on the theory that is described by an $SL(3)\times SL(3)$ Chern–Simons theory in the frame-like formulation. The metric-like theory is obtained by eliminating the generalized spin connection by its equation of motion, and by expressing everything in terms of the metric and a spin-3 Fronsdal field. We give an exact map between fields and gauge parameters in both formulations. To work out the gauge transformations explicitly in terms of metric-like variables, we have to make a perturbative expansion in the spin-3 field. We describe an algorithm for how to do this systematically, and we work out the gauge transformations to cubic order in the spin-3 field. We use these results to determine the gauge algebra to this order, and explain why the commutator of two spin-3 transformations only closes on-shell.

In recent years, open systems with balanced loss and gain that are invariant under the combined parity and time-reversal ($\mathcal{P}\mathcal{T}$) operations have been studied via asymmetries of their solutions. They represent systems as diverse as coupled optical waveguides and electrical or mechanical oscillators. We numerically investigate the asymmetries of incompressible viscous flow in two and three dimensions with 'balanced' inflow-outflow ($\mathcal{P}\mathcal{T}$-symmetric) configurations. By introducing configuration-dependent classes of asymmetry functions in velocity, kinetic energy density, and vorticity fields, we find that the flow asymmetries exhibit power-law scaling with a single exponent in the laminar regime with the Reynolds number ranging over four decades. We show that such single-exponent scaling is expected for small Reynolds numbers, although its robustness at large values of Reynolds numbers is unexpected. Our results imply that $\mathcal{P}\mathcal{T}$-symmetric inflow-outflow configurations provide a hitherto unexplored avenue to tune flow properties.