Non-trivial Distribution Research Articles

Finding binary star fractions in any distribution

Abstract Candidate visual binary systems are often found by identifying two stars that are closer together than would be expected by chance. However, in regions with non-trivial density distributions, the ‘random’ distances between stars varies because of the background distribution, as well as the presence of binaries. We show that when no binaries are present, the distribution of the ratios of the distances to the nearest and tenth nearest neighbours, d1/d10, is always well approximated by a Gaussian with mean 0.2–0.3 and variance 0.16–0.19 for any underlying density distribution. The introduction of binaries causes some (or all) nearest neighbours to become closer than expected by random chance, introducing a component to the distribution where d1/d10 is much lower than expected. We show how a simple single or double Gaussian fit to the distribution of d1/d10 can be used to find the binary fraction in any underlying density distribution quickly and simply.

A tight $$\sqrt{2}$$-approximation for linear 3-cut

We investigate the approximability of the linear 3-cut problem in directed graphs. The input here is a directed graph $$D=(V,E)$$ with node weights and three specified terminal nodes $$s,r,t\in V$$ , and the goal is to find a minimum weight subset of non-terminal nodes whose removal ensures that s cannot reach r and t, and r cannot reach t. The precise approximability of linear 3-cut has been wide open until now: the best known lower bound under the unique games conjecture (UGC) was 4 / 3, while the best known upper bound was 2 using a trivial algorithm. In this work we completely close this gap: we present a $$\sqrt{2}$$ -approximation algorithm and show that this factor is tight under UGC. Our contributions are twofold: (1) we analyze a natural two-step deterministic rounding scheme through the lens of a single-step randomized rounding scheme with non-trivial distributions, and (2) we construct integrality gap instances that meet the upper bound of $$\sqrt{2}$$ . Our gap instances can be viewed as a weighted graph sequence converging to a “graph limit structure”. We complement our results by showing connections between the linear 3-cut problem and other fundamental cut problems in directed graphs.

Statistical dynamics of early creep stages in disordered materials

When materials are loaded below their short-term strength over extended periods, a slow time-dependent process known as creep deformation takes place. During creep deformation, the structural properties of a material evolve as a function of time. By means of a generic coarse-grained mesoscopic elastoplastic model which envisages deformation as a sequence of stochastically activated discrete events, we study the creep deformation of disordered materials. We find that the structural evolution of the material during creep modifies not only the average material properties but also changes the statistics of those properties. We analyze the emergence of correlations in the strain localization and deformation activity patterns, the variation of the event rate and the evolution of the inter-event time distribution. We find that the event rate follows the Omori law of aftershocks, which is the discrete counterpart of Andrade's transient creep law, and that the exponent of these laws only depends on the microstructural heterogeneity. Finally, we find during the initial stages of transient creep a transition from Poisson distributed inter-event times towards a non-trivial power law distribution.

Self-Destructing Dark Matter

We present Self-Destructing Dark Matter (SDDM), a new class of dark matter models which are detectable in large neutrino detectors. In this class of models, a component of dark matter can transition from a long-lived state to a short-lived one by scattering off of a nucleus or an electron in the Earth. The short-lived state then decays to Standard Model particles, generating a dark matter signal with a visible energy of order the dark matter mass rather than just its recoil. This leads to striking signals in large detectors with high energy thresholds. We present a few examples of models which exhibit self destruction, all inspired by bound state dynamics in the Standard Model. The models under consideration exhibit a rich phenomenology, possibly featuring events with one, two, or even three lepton pairs, each with a fixed invariant mass and a fixed energy, as well as non-trivial directional distributions. This motivates dedicated searches for dark matter in large underground detectors such as Super-K, Borexino, SNO+, and DUNE.

Performance of large-scale polling systems with branching-type and limited service

Motivated by emerging Internet-of-Things (IoT) applications and smart building environments, we analyze the performance of large-scale symmetric polling systems where the number of queues grows large. We consider a scenario in which the total arrival rate is kept fixed and the individual switch-over time and service time distributions remain the same. This asymptotic regime leads to cycles of infinite length and queue lengths with non-trivial distributions. We show that for most traditional service policies the scaled cycle times converge to a deterministic value in the limit, which in turn implies that the queue lengths at the various nodes become asymptotically independent. Using these insights, we find that the behavior of individual queues simplifies to that of a discrete-time bulk service queue in the limit, so that the marginal queue length and waiting-time distributions become considerably easier to analyze. Additionally, we propose a new flexible k-limited service discipline aimed at striking a good balance between short mean queue lengths and predictable cycle times for deadline-critical applications.

Self-steering partially coherent vector beams.

We introduce a class of self-steering partially coherent vector optical beams with the aid of a generalized complex Gaussian representation. We show that such partially coherent vector beams have mobile guiding centers of their intensity and polarization state distributions on the beam free space propagation that could be employed to generate far-field polarization arrays. Further, we introduce theoretically and realize experimentally a class of vector beams with inhomogeneous statistical and nontrivial far-field angular distributions, which we term cylindrically correlated partially coherent (CCPC) vector beams. We find that such novel beams possess, in general, cylindrically polarized, far-field patterns of an adjustable degree of polarization. The steering control of the intensity and polarization of the self-steering CCPC vector beam is also demonstrated in experiment. Our findings can find important applications, such as trapping of neutral microparticles and excitation of novel surface waves.

Stochastic particle production in a de Sitter background

We explore non-adiabatic particle production in a de Sitter universe for a scalar spectator field, by allowing the effective mass m2(t) of this field and the cosmic time interval between non-adiabatic events to vary stochastically. Two main scenarios are considered depending on the (non-stochastic) mass M of the spectator field: the conformal case with M2=2H2, and the case of a massless field. We make use of the transfer matrix formalism to parametrize the evolution of the system in terms of the “occupation number”, and two phases associated with the transfer matrix; these are used to construct the evolution of the spectator field. Assuming short-time interactions approximated by Dirac-delta functions, we numerically track the change of these parameters and the field in all regimes: sub- and super-horizon with weak and strong scattering. In all cases a log-normally distributed field amplitude is observed, and the logarithm of the field amplitude approximately satisfies the properties of a Wiener process outside the horizon. We derive a Fokker-Planck equation for the evolution of the transfer matrix parameters, which allows us to calculate analytically non-trivial distributions and moments in the weak-scattering limit.

Distinct magnetotransport and orbital fingerprints of chiral bobbers

While chiral magnetic skyrmions have been attracting significant attention in the past years, recently, a new type of a chiral particle emerging in thin films $-$ a chiral bobber $-$ has been theoretically predicted and experimentally observed. Here, based on theoretical arguments, we provide a clear pathway to utilizing chiral bobbers for the purposes of future spintronics by uncovering that these novel chiral states possess inherent transport fingerprints that allow for their unambiguous electrical detection in systems comprising several types of chiral states. We reveal that unique transport and orbital characteristics of bobbers root in the non-trivial magnetization distribution in the vicinity of the Bloch points, and demonstrate that tuning the details of the Bloch point topology can be used to drastically alter the emergent response properties of chiral bobbers to external fields, which bears great potential for engineering chiral dynamics and cognitive computing.

Glassy phase of the frustrated Bethe lattice models

As a short range frustrated spin model, we study the ferromagnetic spin models on the Bethe lattice with next nearest neighbor (n.n.n.) interactions with negative sign. The iteration equations which are similar to belief propagation are derived and studied by population dynamics. At low temperatures, we find the ferromagnetic phase for zero or small n.n.n. interactions. As the strength of them increases, ferromagnetic phase transition temperature becomes zero and non-trivial distributions of the local fields arise, implying that there are glassy phases in these models.

Repulsively coupled Kuramoto-Sakaguchi phase oscillators ensemble subject to common noise.

We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, the formation of stable multicluster states has been observed in this regime. However, we argue here that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of clusters is impossible. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. By monitoring the induced change in certain integrals of motion, we quantify these errors. We support these observations by showing, on the basis of the analysis of the corresponding Fokker-Planck equation, that two-cluster states are non-attractive. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function as well as additional amplitude dynamics, multiclusters can occur naturally.

A semi-analytical solution for the transient temperature field generated by a volumetric heat source developed for the simulation of friction stir welding

The accurate prediction of transient temperature fields, induced in alloy systems during advanced manufacturing processes, is critical. These fields influence the magnitude and distribution of residual stresses, the evolution of material microstructures, and material properties such as fracture toughness. Such predictions in the vicinity of a concentrated heat source require precise knowledge of the associated heat flux as a function of position and time. If the applied thermal load is time-dependent this can have a significant effect on the resulting temperature fields and microstructures. In this work a novel three-dimensional heat source is proposed to represent the friction stir welding process along with the semi-analytical solution for the temperature field. The volumetric heat source model has a nontrivial spatial distribution constructed from physical arguments and may account for complex mass transfer, and the associated thermal effects, without explicitly solving the flow equations. A method for incorporating a time-dependent heating scenario into analytical solutions generated by this heat source is also presented. Predicted temperatures are compared with those measured experimentally for two cases reported in the literature and good agreement is observed. Example solutions for various time-dependent heat inputs are also presented.

Capital Requirements in a Quantitative Model of Banking Industry Dynamics

We develop a model of banking industry dynamics to study the quantitative impact of capital requirements on bank risk taking, commercial bank failure, and market structure. We propose a market structure where big, dominant banks interact with small, competitive fringe banks. The paper extends our previous work by letting banks accumulate securities like treasury bills and to undertake short-term borrowing when there are cash flow shortfalls. A nontrivial size distribution of banks arises out of endogenous entry and exit, as well as banks' buffer stocks of securities. We find that a 50% rise in capital requirements leads to a 45% reduction in exit rates of small banks and a more concentrated industry. Aggregate loan supply falls by 8.71% and interest rates rise by 50 basis points. The lower exit rate causes the tax/output rate necessary to fund deposit insurance to drop in half. Higher interest rates, on the other hand, result in a higher default frequency as well as a lower level of intermediated output. We also use the model to study the effect of lowering the rate different sized banks are charged when borrowing on their lending behavior studied by Kashyap and Stein (2001).

Controlled Ordering of Topological Charges in an Exciton-Polariton Chain.

We demonstrate, experimentally and theoretically, controlled loading of an exciton-polariton vortex chain into a 1D array of trapping potentials. Switching between two types of vortex chains, with topological charges of the same or alternating signs, is achieved by appropriately shaping an off-resonant pump beam that drives the system to the regime of bosonic condensation. In analogy to spin chains, these vortex sequences realize either a "ferromagnetic" or an "antiferromagnetic" order, whereby the role of spin is played by the orbital angular momentum. The ferromagnetic ordering of vortices is associated with the formation of a persistent chiral current. Our results pave the way for the controlled creation of nontrivial distributions of orbital angular momentum and topological order in a periodic exciton-polariton system.

Statistical signatures of the spatial imprints of road network growth

We present a statistical characterization of the morphological features emerging from the complex processes governing the growth of the road network, particularly in a mostly self-organized urban setting. Apart from the usual fractal analysis, the roads are quantified by their lengths and straightnesses, while the segmented blocks are characterized by their areas, perimeters and circularities. When applied to the Metro Manila conurbation, one of the megacities in Asia with the fastest growing populations, we observe dense space-filling and nontrivial statistical distributions of roads and blocks that can be attributed to the geographical constraints of the metropolis. The emergence of the scale-free regimes is explained using a simple rule-based model patterned after the assumed dynamical interplay between the local and global factors involved in individual street formation. By viewing road network growth from a quantitative complex systems perspective, we can gain insights into the underlying rules operating at the local scales that give rise to the global spatial patterns.

Eigenfunction distribution for the Rosenzweig-Porter model

The statistical distribution of eigenfunctions for the Rosenzweig-Porter model is derived for the region where eigenfunctions have fractal behaviour. The result is based on simple physical ideas and leads to transparent explicit formulas which agree very well with numerical calculations. It constitutes a rare case where a non-trivial eigenfunction distribution is obtained in a closed form.

Phase structuring of the complex degree of coherence.

Simple conditions on the magnitude and phase of the complex degree of coherence of stationary, scalar, one-dimensional (1D) Schell-like sources are derived for the design of novel optical beam-like fields. Several examples of new sources obtained with the help of signum, odd monomials, and sine phase functions are given. A geometrical representation, termed coherence curve, for the complex coherence state of a 1D, Schell-like source with non-trivial magnitude and phase distributions by means of a parametric curve confined to the unit circle in a complex plane is also introduced.

Probability distributions for the stress tensor in conformal field theories

The vacuum state—or any other state of finite energy—is not an eigenstate of any smeared (averaged) local quantum field. The outcomes (spectral values) of repeated measurements of that averaged local quantum field are therefore distributed according to a non-trivial probability distribution. In this paper, we study probability distributions for the smeared stress tensor in two-dimensional conformal quantum field theory. We first provide a new general method for this task based on the famous conformal welding problem in complex analysis. Secondly, we extend the known moment generating function method of Fewster, Ford and Roman. Our analysis provides new explicit probability distributions for the smeared stress tensor in the vacuum for various infinite classes of smearing functions. All of these turn out to be given in the end by a shifted Gamma distribution, pointing, perhaps, at a distinguished role of this distribution in the problem at hand.

Retrieving the Size of Deep-Subwavelength Objects via Tunable Optical Spin-Orbit Coupling

We propose a scheme to retrieve the size parameters of a nanoparticle on a glass substrate at a scale much smaller than the wavelength. This is achieved by illuminating the particle using two plane waves to create rich and nontrivial local polarization distributions, and observing the far-field scattering pattern into the substrate. By using this illumination to control the induced complex dipole moment, the exponential decay of power radiated into the supercritical region, as well as directional scattering due to spin-orbit coupling can be exploited to retrieve the particle's shape, size, and position directly from the far-field scattering with high sensitivity and without the need for a complicated and time-consuming optimization algorithm. Our method brings about a far-field superresolution nanometrology scheme based on the interaction of vectorial light with nanoparticles.

Skew semi-invariant submanifolds of generalized quasi-Sasakian manifolds

In the present paper, we study a new class of submanifolds of a generalized Quasi-Sasakian manifold, called skew semi-invariant submanifold. We obtain integrability conditions of the distributions on a skew semi-invariant submanifold and also find the condition for a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold to be mixed totally geodesic. Also it is shown that a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold will be anti-invariant if and only if $A_{\xi}=0$; and the submanifold will be skew semi-invariant submanifold if $\nabla w=0$. The equivalence relations for the skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold are given. Furthermore, we have proved that a skew semi-invariant $\xi^\perp$-submanifold of a normal almost contact metric manifold and a generalized Quasi-Sasakian manifold with non-trivial invariant distribution is $CR$-manifold. An example of dimension 5 is given to show that a skew semi-invariant $\xi^\perp$ submanifold is a $CR$-structure on the manifold.

Transient Spiral Arms from Far Out-of-equilibrium Gravitational Evolution

Abstract We describe how a simple class of out-of-equilibrium, rotating, and asymmetrical mass distributions evolve under their self-gravity to produce a quasi-planar spiral structure surrounding a virialized core, qualitatively resembling a spiral galaxy. The spiral structure is transient, but can survive tens of dynamical times, and further reproduces qualitatively noted features of spiral galaxies such as the predominance of trailing two-armed spirals and large pitch angles. As our models are highly idealized, a detailed comparison with observations is not appropriate, but generic features of the velocity distributions can be identified to be the potential observational signatures of such a mechanism. Indeed, the mechanism leads generically to a characteristic transition from predominantly rotational motion, in a region outside the core, to radial ballistic motion in the outermost parts. Such radial motions are excluded in our Galaxy up to 15 kpc, but could be detected at larger scales in the future by GAIA. We explore the apparent motions seen by external observers of the velocity distributions of our toy galaxies, and find that it is difficult to distinguish them from those of a rotating disk with sub-dominant radial motions at levels typically inferred from observations. These simple models illustrate the possibility that the observed apparent motions of spiral galaxies might be explained by non-trivial non-stationary mass and velocity distributions without invoking a dark matter halo or modification of Newtonian gravity. In this scenario the observed phenomenological relation between the centripetal and gravitational acceleration of the visible baryonic mass could have a simple explanation.