For cake cutting, cuts are parallel to an axis or segment and yield rectangular pieces. As such, cutting a cake is viewed as dividing a line segment. For pie cutting, cuts are radial from the center of a disc to the circumference and yield sectors or wedge-shaped pieces. As such, cutting a pie is viewed as dividing a circle. This difference in topology is apparent in the minimal number of cuts ...

Suppose fix,... ,fin are nonatomic probability measures on the same measurable space (S, S). Then there exists a measurable partition isi}"=i of 5 such that Pi(Si) > (n + 1 M)'1 for a11 i l,...,n, where M is the total mass of V?=i ßi (tne smallest measure majorizing each m). This inequality is the best possible for the functional M, and sharpens and quantifies a well-known cake-cutting theorem ...

We study the envy-free cake-cutting problem for d+1 players with d cuts, for both the oracle function model and the polynomial time function model. For the former, we derive a θ(( ǫ )) time matching bound for the query complexity of d + 1 player cake cutting with Lipschitz utilities for any d > 1. When the utility functions are given by a polynomial time algorithm, we prove the problem to be PP...

The cake-cutting problem refers to the issue of dividing a cake into pieces and distributing them to players who have different value measures related to the cake, and who feel that their portions should be “fair.” The fairness criterion specifies that in situations where n is the number of players, each player should receive his/her portion with at least 1/n of the cake value in his/her measur...

Propagation in l inear planar wa vegui des is a wel l-known pro blem , whereas pro pagati on in nonl inear waveguides wa s considered onl y in parti cular cases. Tha t is the caseof the nonl inear waveguide wi th a l inear Ùlm on a nonl inear substra te [1], then the l inear Ùlm surro unded by the nonl inear substra te and cover [2{ 7], the nonl inear Ùlm surro unded by l inear isotro pi c m ed...

In the classic cake-cutting problem (Steinhaus, 1948), a heterogeneous resource has to be divided among n agents with different valuations in a proportional way — giving each agent a piece with a value of at least 1/n of the total. In many applications, such as dividing a land-estate or a time-interval, it is also important that the pieces are connected. We propose two additional requirements: ...

PPAD refers to a class of computational problems for which solutions are guaranteed to exist due to a specific combinatorial principle. The most wellknown such problem is that of computing a Nash equilibrium of a game. Other examples include the search for market equilibria, and envy-free allocations in the context of cake-cutting. A problem is said to be complete for PPAD if it belongs to PPAD...