We derive general properties of the finite-size scaling of probability density functions and show that when the apparent exponent τ̃ of a probability density is less than 1, the associated finite-size scaling ansatz has a scaling exponent τ equal to 1, provided that the fraction of events in the universal scaling part of the probability density function is non-vanishing in the thermodynamic limi...

Exactly solvable models of planar polygons, weighted by perimeter and area, have deepened our understanding of the critical behaviour of polygon models in recent years. Based on these results, we derive a conjecture for the exact form of the critical scaling function for planar self-avoiding polygons. The validity of this conjecture was recently tested numerically using exact enumeration data f...

We consider large-scale Markov decision processes (MDPs) with parameter uncertainty, under the robust MDP paradigm. Previous studies showed that robust MDPs, based on a minimax approach to handling uncertainty, can be solved using dynamic programming for small to medium sized problems. However, due to the “curse of dimensionality”, MDPs that model real-life problems are typically prohibitively ...

We describe a procedure which aims at determining, for all the values of the coupling constant, all the generalised scaling functions arising in the high spin limit for long operators in the sl(2) sector of N = 4 SYM. All the relevant information is encoded in infinite linear systems, each easily connected with the previous ones. Therefore, the solution may be written in a recursive form and th...

Important brain functions need to be conserved throughout organisms of extremely varying sizes. Here we study the scaling properties of an essential component of computation in the brain: the single neuron. We compare morphology and signal propagation of a uniquely identifiable interneuron, the HS cell, in the blowfly (Calliphora) with its exact counterpart in the fruit fly (Drosophila) which i...

We study hyperbolic maps depending on a parameter ε. Each of them has an invariant Cantor set. As ε tends to zero, the map approaches the boundary of hyperbolicity. We analyze the asymptotics of scaling function of the invariant Cantor set as ε goes to zero. We show that there is a limiting scaling function of the limiting map and this scaling function has dense jump discontinuities because the...

Finding a universal description of the algorithm optimization is one of the key challenges in personalized recommendation. In this article, for the first time, we introduce a scaling-based algorithm (SCL) independent of recommendation list length based on a hybrid algorithm of heat conduction and mass diffusion, by finding out the scaling function for the tunable parameter and object average de...

A method for determining the generalised scaling function(s) arising in the high spin behaviour of long operator anomalous dimensions in the planar sl(2) sector of N = 4 SYM is proposed. The all-order perturbative expansion around the strong coupling is detailed for the prototypical third and fourth scaling functions, showing the emergence of the O(6) Non-Linear Sigma Model mass-gap from differ...

abstract introduction: scaling and root planing is one of the most commonly used procedures during periodontal treatments. removal of calculus using conventional hand instruments is incomplete and rather time consuming. to find more efficient and less difficult instrumentation method, investigators have proposed lasers as an alternative or adjunct to hand instruments for scaling and root planin...