We extend the classic cake-cutting problem to a situation in which the ”cake” is divided among families. Each piece of cake is owned and used simultaneously by all members of the family. A typical example of such a cake is land. We examine three ways to assess the fairness of such a division, based on the classic no-envy criterion: (a) Average envy-freeness means that for each family, the avera...

Properties of discrete cake-cutting procedures that use a minimal number of cuts (n – 1 if there are n players) are analyzed. None is always envy-free or efficient, but divideand-conquer (D&C) minimizes the maximum number of players that any single player may envy. It works by asking n ≥ 2 players successively to place marks on a cake that divide it into equal or approximately equal halves, the...

We develop the rst explicit procedures for exact envy-free chore division for 4 or more players. We also give a simple 3-person procedure. Our 4-person moving-knife scheme and n-person discrete algorithm are accomplished by exploiting a notion of \irrevocable advantage" for chores, and are motivated by similar solutions for cake-cutting by Brams, Taylor, and Zwicker. We discuss the diierences b...

It is well-known that the central limit theorem holds for partial sums of a stationary sequence (Xi) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if Var(Xi) 6= 0. We show that this happens only in the case when Xi − EXi = Yi − Yi−1 for an (m − 1)-dependent stationary sequence (Yi) with finite variance (a result implicit in earli...

The problem of fairly dividing a cake (as a metaphor for a heterogeneous divisible good) has been the subject of much interest since the 1940’s, and is of importance in multiagent resource allocation. Two fairness criteria are usually considered: proportionality, in the sense that each of the n agents receives at least 1/n of the cake; and the stronger property of envy-freeness, namely that eac...

The relationship between phosphorus (P) uptake and growth rate of chrysanthemum plants grown hydroponically for cut flowers was investigated. In Exp. 1, fresh weight increase of chrysanthemum cv. Seiun plants at flower budding phase administered 0 to 40 ppm in culture solution for 16 days was evaluated. 1. P uptake by the chrysanthemum plants was nearly proportionate to P concentration in the c...

Different PCR protocols have been established for detection of European fruit trees phytoplasmas; however the majority of the procedures for extracting phytoplasma DNA are complex, time consuming, and expensive, with a risk of contamination or loss of target DNA. In present study, a crude extract preparation method previously used to detect other plant pathogens was adapted to samples from appl...

Phosphorus (P) is an essential macronutrient for plants and considered as one of the limiting nutrients to plants. The majority P in soil found form insoluble phosphate complexes, making unavailable In present study, we report two highly efficient solubilizing bacteria isolated from closed flower Hedychium coronarium L., a medicinal plant commonly known Gulbakawali. Based on morphological, bioc...

There is a heterogeneous resource that contains both good parts and bad parts, for example, a cake with some parts burnt, a land-estate with some parts heavily taxed, or a chore with some parts fun to do. The resource has to be divided fairly among n agents, each of whom has a personal value-density function on the resource. The value-density functions can accept any real value positive or nega...

If Ill' ... , ~tn are non-atomic probability measures on the same measurable space (S, ff"), then there is an ff"-measurable partition {AJ7= 1 of S so that IlJAi)~(n-l+m)-l for all i=I, ... ,n, where m=l l i 01 lli l l is the total mass of the largest measure dominated by each of the Il/s; moreover, this bound is attained for all n ~ 1 and all m in [0,1]. This result is an analog of the bound (...