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...

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 (...

This paper discusses cake-cutting protocols when the cake is a heterogeneous good that is represented by an interval in the real line. We propose a new desirable property, the meta-envy-freeness of cake-cutting, which has not been formally considered before. Though envy-freeness was considered to be one of the most important desirable properties, envy-freeness does not prevent envy about role a...

The current generation of CASE tools mainly provides rigid and unnexible support for editing, storage, veriication and transformation of speciications. Information engineers have to adapt their way of working to CASE tools instead of vice versa. To enable a more exi-ble, method independent, support of information engineers, the concept of a CASE shell has emerged. Crucial for CASE shells is the...

We study classic cake-cutting problems, but in discrete models rather than using infiniteprecision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and movingknife), we roughly partition the various fair-allocation problems into 3 classes: “easy” (constant number of rounds of logar...

We consider the classic cake-divison problem when the cake is a heterogeneous good represented by an interval in the real line. We provide a mechanism to implement, in an anonymous way, an envy-free and e¢cient allocation when agents have private information on their preferences. The mechanism is a multistep sequential game form in which each agent at each step receives a morsel of the cake tha...

isatin, n-methyl isatin and n-benzylisatin were smoothly monobrominated in the 5-position by treatment with pyridinium bromochromate (pbc) in acetic acid medium. in the similar conditions indigo underwent bromination and oxidation to give 5-bromoisatin in high yield.