Acta Mathematica Hungarica最新文献

We study Markov population processes on large graphs, with the local state transition rates of a single vertex being a linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. An explicit error bound is given and it is (frac{1}{sqrt{N}}) plus the largest discrepancy of the graph. For Erdős–Rényi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process.

Suppose K is a knot in a 3-manifold Y, and that Y admits a pair of distinct contact structures. Assume that K has Legendrian representatives in each of these contact structures, such that the corresponding Thurston-Bennequin framings are equivalent. This paper provides a method to prove that the contact structures resulting from Legendrian surgery along these two representatives remain distinct. Applying this method to the situation where the starting manifold is (-Sigma(2,3,6m+1)) and the knot is a singular fiber, together with convex surface theory we can classify the tight contact structures on certain families of Seifert fiber spaces.

The mixed area of a Reuleaux polygon and its symmetric with respect to the origin is expressed in terms of the mixed area of two explicit polygons. This gives a geometric explanation of a classical proof due to Chakerian. Mixed areas and volumes are also used to reformulate the minimization of the volume under constant width constraint as isoperimetric problems. In the two dimensional case, the equivalent formulation is solved, providing another proof of the Blaschke–Lebesgue theorem. In the three dimensional case the proposed relaxed formulation involves the mean width, the area and inclusion constraints.

Let ((a(n) : n in mathbb{N})) denote a sequence of nonnegative integers. Let (0.a(1)a(2) ldots ) denote the real number obtained by concatenating the digit expansions, in a fixed base, of consecutive entries of ((a(n) : n in mathbb{N})). Research on digit expansions of this form has mainly to do with the normality of (0.a(1)a(2) ldots ) for a given base. Famously, the Copeland-Erdős constant (0.2357111317 ldots {}), for the case whereby (a(n)) equals the (n^{text{th}}) prime number (p_{n}), is normal in base 10. However, it seems that the “inverse” construction given by concatenating the decimal digits of ((pi(n) : n in mathbb{N})), where (pi) denotes the prime-counting function, has not previously been considered. Exploring the distribution of sequences of digits in this new constant (0.0122 ldots 9101011 ldots ) would be comparatively difficult, since the number of times a fixed (m in mathbb{N} ) appears in ((pi(n) : n in mathbb{N})) is equal to the prime gap (g_{m} = p_{m+1} - p_{m}), with the behaviour of prime gaps notoriously elusive. Using a combinatorial method due to Szüsz and Volkmann, we prove that Cramér’s conjecture on prime gaps implies the normality of (0.a(1)a(2) ldots ) in a given base (g geq 2), for (a(n) = pi(n)).