We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex k-plane. This Riemann-Hilbert problem has explicit (x, t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of t...

Turkmenistan’s economic performance during the first decade after independence isinteresting because it is an extreme case, regularly ranking last among all the former centrallyplanned economies by transition indicators measuring speed of reform or degree of economicliberalization. This paper analyses the evolution of the economic system and Turkmenistan’seconomic development si...

In the book Proofs that Really Count [1], the authors use combinatorial arguments to prove many identities involving Fibonacci numbers, Lucas numbers, and their generalizations. Among these, they derive 91 of the 118 identities mentioned in Vajda’s book [2], leaving 27 identities unaccounted. Eight of these identities, presented later in this paper, have such a similar appearance, the authors r...

Condition (1.3) says that the pressure p vanishes outside the domain and condition (1.4) says that the boundary moves with the velocity V of the fluid particles at the boundary. Given a domain D0 ⊂ R, that is homeomorphic to the unit ball, and initial data v0, satisfying the constraint (1.2), we want to find a set D = ∪ 0≤t≤T {t} × Dt, Dt ⊂ R and a vector field v solving (1.1)-(1.4) with initia...

We propose a mechanism for solving the horizon and entropy problems of standard cosmology which does not make use of cosmological inflation. Crucial ingredients of our scenario are brane gases, extra dimensions, and a confining potential due to string gas effects which becomes dominant at string-scale brane separations. The initial conditions are taken to be a statistically homogeneous and isot...

We study the spherical collapse model in dark energy cosmologies, in which dark energy is modelled as a minimally coupled scalar field. We first follow the standard assumption that dark energy does not cluster on the scales of interest. Investigating four different popular potentials in detail, we show that the predictions of the spherical collapse model depend on the potential used. We also in...

as y′′ + t t2−16y ′ + 9 (t2−16)(t−3)y = 0. So p(t) = t t2−16 and q(t) = 9 (t2−16)(t−3) . Hence p(t) is continuous if t ∈ (−∞,−4) ∪ (4,∞) and q(t) is continuous if t ∈ (−∞,−4) ∪ (−4, 3) ∪ (3,∞). Therefore p(t) and q(t) are continuous if t ∈ (−∞,−4)∪(−4, 3)∪(3,∞). The initial conditions are y(2) = 1 and y′(2) = 2. We have 2 ∈ (−4, 3). Thus the solution exists on the interval (−4, 3). (b) The init...

The time-integrated current of the TASEP has non-Gaussian fluctuations of order t. The recently discovered connection to random matrices and the Painlevé II RiemannHilbert problem provides a technique through which we obtain the probability distribution of the current fluctuations, in particular their dependence on initial conditions, and the stationary two-point function. Some open problems ar...

mν . In particular we explore the impact of a cosmological parameters, the importance of priors, the uncertainties due to biasing in large scale structure and the dependence on choice of initial conditions. We find that the mass constraints are independent of the choice of parameterization as well as the inclusion of spatial curvature. Yet the difference between an upperbound of 2.2 eV, assumin...

Abstract: The anthropic fine-tuning of the cosmos refers to the claim that the laws of nature, the constants of physics, and the initial conditions of the universe must be set to an enormous degree of precision for life to exist. Three major responses have been offered to this fine-tuning: the multiverse explanation; theism; and the claim that it is just a brute fact that requires no further ex...