In this note we nd the exact solution for the minimal recurrence S n = min bn=2c k=1 faS n?k + bS k g; where a and b are positive integers and a b. We prove that the solution is the same as that of the recurrence relation S n = aS dn=2e + bS bn=2c. In other words, S n = S 1 +(a+b?1)S 1 P n?1 i=1 a z(i) b blg nc?z(i) , where z(i) is the number of zeros in the binary representation of i. The proo...

We consider an uncertain variant of the knapsack problem in which the weight of the items is not exactly known in advance, but belongs to a given interval, and an upper bound is imposed on the number of items whose weight differs from the expected one. For this problem, we provide a dynamic programming algorithm and present techniques aimed at reducing its space and time complexities. Finally, ...

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology. We ...

Abstract The susceptible-infectious-recovered (SIR) model describes the evolution of three species of individuals which are subject to an infection and recovery mechanism. A susceptible S can become infectious with an infection rate β by an infectious Itype provided that both are in contact. The Itype may recover with a rate γ and from then on stay immune. Due to the coupling between the differ...

Using a simple transfer matrix approach we have derived long series expansions for the perimeter generating functions of both three-choice polygons and punctured staircase polygons. In both cases we find that all the known terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We report on an analysis of the properties of the differential equ...

In this work we show, that for each permuted submodular MinSum problem (Energy Minimization Task) the corresponding submodular MinSum problem can be found in polynomial time. It follows, that permuted submodular MinSum problems are exactly solvable by transforming them into corresponding submodular tasks followed by applying standart approaches (e.g. using MinCutMaxFlow based techniques).

In this paper the first exact interior solution to Einstein’s field equations for a static and non-uniform braneworld star with local and non-local bulk terms is presented. It is shown that the bulk Weyl scalar U(r) is always negative inside the stellar distribution, in consequence it reduces both the effective density and the effective pressure. It is found that the anisotropy generated by bul...

In this paper we propose exact solution methods for a bilevel uncapacitated lot-sizing problem with backlogs. This is an extension of the classical uncapacitated lot-sizing problem with backlogs, in which two autonomous and self-interested decision makers constitute a twoechelon supply chain. The leader buys items from the follower in order to meet external demand at lowest cost. The follower a...

The exact solution is presented for the "susceptibility," χ (the number of sites covered by the maximally extended eigenfunction), for the zero-energy solutions of a hopping model on a randomly dilute Cayley tree. If p is the concentration, then χ~(p*−p)−1 with p*~pce1/ξ1, where pc is the critical percolation concentration and ξ1 the one-dimensional localization length. This result is argued to...

Fifteen years ago Chvatal conjectured that if 9 is a family of k subsets of an nset, 191 > (;I;), d is an arbitrary integer with d< k 1 and (d+ 1) k n,(k). in a more general framework. Another problem which is solved asymp...