arXiv subjects

Michal Gnacik

Publications and source records attributed to Michal Gnacik.

9 recordsLinked to original sources

arXiv2021

Public efforts to reduce disease transmission implied from a spatial game

One approach to understand people's efforts to reduce disease transmission, is to consider the effect of behaviour on case rates. In this paper we present a spatial infection-reducing game model of public behaviour, formally equivalent to a Hopfield neural network coupled to SIRS disease dynamics. Behavioural game parameters can be precisely calibrated to geographical time series of Covid-19 active case numbers, giving an implied spatial history of behaviour. This is used to investigate the effects of government intervention, quantify behaviour area by area, and measure the effect of wealth on behaviour. We also demonstrate how a delay in people's perception of risk levels can induce behavioural instability, and oscillations in infection rates.

physics.soc-ph↗

arXiv2020

Approximate modularity: Kalton's constant is not smaller than 3

Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), 803--816] proved that there exists a universal constant $K\leqslant 44.5$ such that for every set algebra $\mathcal{F}$ and every 1-additive function $f\colon \mathcal{F}\to \mathbb R$ there exists a finitely-additive signed measure $μ$ defined on $\mathcal{F}$ such that $|f(A)-μ(A)|\leqslant K$ for any $A\in \mathcal{F}$. The only known lower bound for the optimal value of $K$ was found by Pawlik [Colloq. Math., 54 (1987), 163--164], who proved that this constant is not smaller than $1.5$; we improve this bound to $3$ already on a non-negative 1-additive function.

math.FA↗

arXiv2020

Size and shape of tracked Brownian bridges

We investigate the typical sizes and shapes of sets of points obtained by irregularly tracking two-dimensional Brownian bridges. The tracking process consists of observing the path location at the arrival times of a non-homogeneous Poisson process on a finite time interval. The time varying intensity of this observation process is the tracking strategy. By analysing the gyration tensor of tracked points we prove two theorems which relate the tracking strategy to the average gyration radius, and to the asphericity -- a measure of how non-spherical the point set is. The act of tracking may be interpreted either as a process of observation, or as process of depositing time decaying "evidence" such as scent, environmental disturbance, or disease particles. We present examples of different strategies, and explore by simulation the effects of varying the total number of tracking points.

math.PR↗

arXiv2020

Inverse problems for symmetric doubly stochastic matrices whose Suleĭmanova spectra are bounded below by 1/2

A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is still open in its full generality. It is proved that whenever $λ_2, \ldots, λ_n$ are non-positive real numbers with $1 + λ_2 + \ldots + λ_n \geqslant 1/2$, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely $(1, λ_2, \ldots, λ_n)$. We point out that this criterion is incomparable to the classical sufficient conditions due to Perfect-Mirsky, Soules, and their modern refinements due to Nader et al. We also provide some examples and applications of our results.

math.SP↗

arXiv2019

Quasifree stochastic cocycles and quantum random walks

The theory of quasifree quantum stochastic calculus for infinite-dimensional noise is developed within the framework of Hudson-Parthasarathy quantum stochastic calculus. The question of uniqueness for the covariance amplitude with respect to which a given unitary quantum stochastic cocycle is quasifree is addressed, and related to the minimality of the corresponding stochastic dilation. The theory is applied to the identification of a wide class of quantum random walks whose limit processes are driven by quasifree noises.

math-ph↗

arXiv2018

Statistical Physics of Language Maps in the USA

Spatial linguistic surveys often reveal well defined geographical zones where certain linguistic forms are dominant over their alternatives. It has been suggested that these patterns may be understood by analogy with coarsening in models of two dimensional physical systems. Here we investigate this connection by comparing data from the Cambridge Online Survey of World Englishes to the behaviour of a generalised zero temperature Potts model with long range interactions. The relative displacements of linguistically similar population centres reveals enhanced east-west affinity. Cluster analysis reveals three distinct linguistic zones. We find that when the interaction kernel is made anisotropic by stretching along the east-west axis, the model can reproduce the three linguistic zones for all interaction parameters tested. The model results are consistent with a view held by some linguists that, in the USA, language use is, or has been, exchanged or transmitted to a greater extent along the east-west axis than the north-south.

physics.soc-ph↗

arXiv2018

The shape of a memorised random walk

We view random walks as the paths of foraging animals, perhaps searching for food or avoiding predators while forming a mental map of their surroundings. The formation of such maps requires them to memorise the locations they have visited. We model memory using a kernel, proportional to the number of locations recalled as a function of the time since they were first observed. We give exact analytic expressions relating the elongation of the memorised walk to the structure of the memory kernel, and confirm these by simulation. We find that more slowly decaying memories lead to less elongated mental maps.

math.PR↗

arXiv2018

Strong convergence of quantum random walks via semigroup decomposition

We give a simple and direct treatment of the strong convergence of quantum random walks to quantum stochastic operator cocycles, via the semigroup decomposition of such cocycles. Our approach also delivers convergence of the pointwise product of quantum random walks to the quantum stochastic Trotter product of the respective limit cocycles, thereby revealing the algebraic structure of the limiting procedure. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.

math.PR↗

arXiv2017

Infrequent social interaction can accelerate the spread of a persuasive idea

We study the spread of a persuasive new idea through a population of continuous-time random walkers in one dimension. The idea spreads via social gatherings involving groups of nearby walkers who act according to a biased "majority rule": After each gathering, the group takes on the new idea if more than a critical fraction $\frac{1-\varepsilon}{2} < \frac{1}{2}$ of them already hold it; otherwise they all reject it. The boundary of a domain where the new idea has taken hold expands as a traveling wave in the density of new idea holders. Our walkers move by Lévy motion, and we compute the wave velocity analytically as a function of the frequency of social gatherings and the exponent of the jump distribution. When this distribution is sufficiently heavy tailed, then, counter to intuition, the idea can propagate faster if social gatherings are held less frequently. When jumps are truncated, a critical gathering frequency can emerge which maximizes propagation velocity. We explore our model by simulation, confirming our analytical results.

nlin.AO↗