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 di A. 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 {stile di visualizzazione n+1} subintervals: {displaystyle 0=z_{0}Define a signed partition as a partition in which each subinterval {stile di visualizzazione i} has an associated sign {displaystyle delta _{io}} : {displaystyle delta _{1},punti ,delta _{k+1}in left{+1,-1Giusto}} The Hobby-Rice theorem says that for every n continuously integrable functions: {stile di visualizzazione g_{1},punti ,g_{n}colon [0,1]longrightarrow mathbb {R} } there exists a signed partition of [0,1] tale che: {somma dello stile di visualizzazione _{io=1}^{n+1}delta _{io}!int _{z_{i-1}}^{z_{io}}g_{j}(z),dz=0{testo{ per }}1leq jleq n.} (in altre parole: 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". Atti dell'American Mathematical Society. Società matematica americana. 16 (4): 665–670. doi:10.2307/2033900. JSTOR 2033900. ^ Pinkus, Allan (1976). "A simple proof of the Hobby-Rice theorem". Atti dell'American Mathematical Society. Società matematica americana. 60 (1): 82–84. doi:10.2307/2041117. JSTOR 2041117. ^ Alon, Noga (1987). "Splitting Necklaces". Progressi in matematica. 63 (3): 247–253. doi:10.1016/0001-8708(87)90055-7. ^ F.W. Simmons and F.E. Sono (2003). "Consensus-halving via theorems of Borsuk-Ulam and Tucker" (PDF). Scienze Matematiche Sociali. 45: 15–25. doi:10.1016/S0165-4896(02)00087-2. Questo articolo relativo all'analisi matematica è solo un abbozzo. Puoi aiutare Wikipedia espandendolo. Categorie: Theorems in measure theoryFair divisionCombinatorics on wordsTheorems in analysisMathematical analysis stubs