Hobby–Rice theorem

Hobby–Rice theorem In mathematics, and in particular the necklace splitting problem, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by Charles R. Hobby and John R. Rice;[1] a simplified proof was given in 1976 durch eine. Pinkus.[2] The theorem Given an integer n, define a partition of the interval [0,1] as a sequence of numbers which divide the interval to {displaystyle n+1} subintervals: {displaystyle 0=z_{0}Define a signed partition as a partition in which each subinterval {Anzeigestil i} has an associated sign {displaystyle delta _{ich}} : {displaystyle delta _{1},Punktec ,Delta _{k+1}in left{+1,-1Rechts}} The Hobby-Rice theorem says that for every n continuously integrable functions: {Anzeigestil g_{1},Punktec ,g_{n}colon [0,1]longrightarrow mathbb {R} } there exists a signed partition of [0,1] so dass: {Anzeigestil Summe _{i=1}^{n+1}Delta _{ich}!int _{z_{i-1}}^{z_{ich}}g_{j}(z),dz=0{Text{ zum }}1leq jleq n.} (mit anderen Worten: for each of the n functions, its integral over the positive subintervals equals its integral over the negative subintervals). Application to fair division The theorem was used by Noga Alon in the context of necklace splitting[3] in 1987. Suppose the interval [0,1] is a cake. There are n partners and each of the n functions is a value-density function of one partner. We want to divide the cake into two parts such that all partners agree that the parts have the same value. This fair-division challenge is sometimes referred to as the consensus-halving problem.[4] The Hobby-Rice theorem implies that this can be done with n cuts. References ^ Hobby, C. R.; Rice, J. R. (1965). "A moment problem in L1 approximation". Verfahren der American Mathematical Society. Amerikanische Mathematische Gesellschaft. 16 (4): 665–670. doi:10.2307/2033900. JSTOR 2033900. ^ Pinkus, Allan (1976). "A simple proof of the Hobby-Rice theorem". Verfahren der American Mathematical Society. Amerikanische Mathematische Gesellschaft. 60 (1): 82–84. doi:10.2307/2041117. JSTOR 2041117. ^ Alon, Noga (1987). "Splitting Necklaces". Fortschritte in der Mathematik. 63 (3): 247–253. doi:10.1016/0001-8708(87)90055-7. ^ F.W. Simmons and F.E. Sie sind (2003). "Consensus-halving via theorems of Borsuk-Ulam and Tucker" (Pdf). Mathematische Sozialwissenschaften. 45: 15–25. doi:10.1016/S0165-4896(02)00087-2. Dieser Artikel zur mathematischen Analyse ist ein Stummel. Sie können Wikipedia helfen, indem Sie es erweitern. Kategorien: Theorems in measure theoryFair divisionCombinatorics on wordsTheorems in analysisMathematical analysis stubs