We introduce partition equilibrium and study its existence in resource selection games (RSG). In partition equilibrium the agents are partitioned into coalitions, and only deviations by the prescribed coalitions are considered. This is in difference to the classical concept of strong equilibrium according to which any subset of the agents may deviate. In resource selection games, each agent sel...

Polarisabilities of atoms within molecules can be calculated from molecular electron densities using the density partitioning method known as Iterated Stockholder Atoms (ISA). Non-local polarisation effects are removed by optimising ISA functional for perturbed density, with additional constraint that atomic charges do not change. The resulting polarisabilities chemically reasonable, tend to be...

Let P1, . . . ,Pn be properties of graphs. A (P1, . . . ,Pn)-partition of a graph G is a partition of the vertex set V (G) into subsets V1, . . . , Vn such that the subgraph G[Vi] induced by Vi has property Pi; i = 1, . . . , n. A graph G is said to be uniquely (P1, . . . ,Pn)-partitionable if G has exactly one (P1, . . . ,Pn)-partition. A property P is called hereditary if every subgraph of ev...

3. A lower bound To obtain a lower bound we count only those sequences of refinements which include the partition 1 .2.3 . . . d . r of n into d or d + 1 parts, d of which are of different size, where 0 < r = n-$d(d + 1) < d, so that J(2n) > d > J(2n)-3. Moreover we only count sequences in which we split off 1 from each of the d 1 parts of different size greater than 1. These d 1 steps can be m...

This review emphasizes the importance of classification surface reservoirs represented by building and indoor materials. Models developed in past 45 years to study mainly non-reactive thermodynamic partitioning air pollutants surfaces as well their validation through analytical measurements are discussed. Analytical techniques aiming at characterization have been summarized. Applications releva...

ABSTRACT: Crop residues decomposition are controlled by chemical tissue components. This study evaluated changes on plant components, separated the Van Soest partitioning method, during cover crop decomposition. The soluble fraction was first to be released from residues, followed cellulose and hemicellulose. Lignin residue component that suffered least degradation, for certain types, lignin de...

We call a sum a1 + a2 + · · ·+ ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ · · · ≤ ak and n = a1 + a2 + · · ·+ ak. For i = 1, 2, . . . , k, we call ai the i-th part of the sum a1 + a2 + · · ·+ ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1 + a2 + · · · + ak and b1 + b2 + · · ·+ bk strongly inter...

We give a simple combinatorial proof of three identities of Warnaar. The proofs exploit involutions due to Franklin and Schur.

Theorems in the theory of partitions are closely related to basic hypergeometric series. Some identities arising in basic hypergeometric series can be interpreted in the theory of partitions using Fpartitions. In this paper, Ramanujan’s 1ψ1 summation and the q-Gauss summation are established combinatorially.