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By Bouchitte G., Buttazzo G., De Pascale L.

We exhibit that the matter of discovering the simplest mass distribution, either in conductivity and elasticity situations, will be approximated by way of recommendations of a p-Laplace equation, as p→+S. This turns out to supply a variety criterion whilst the optimum options are nonunique.

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A p-Laplacian Approximation for Some Mass Optimization Problems

We exhibit that the matter of discovering the simplest mass distribution, either in conductivity and elasticity circumstances, should be approximated through ideas of a p-Laplace equation, as p→+S. This turns out to supply a variety criterion whilst the optimum ideas are nonunique.

The Diophantine Frobenius Problem

In the course of the early a part of the final century, Ferdinand Georg Frobenius (1849-1917) raised he following challenge, often called the Frobenius challenge (FP): given rather major optimistic integers a1,. .. ,an, locate the biggest average quantity (called the Frobenius quantity and denoted via g(a1,. .. ,an) that's not representable as a nonnegative integer mixture of a1,.

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With the above theorem, the complete characterization of all solutions and non-solutions to nj=1 xj aj = L is obtained from the function E(L) and thus g(a1 , . . , an ) = max{E(L)|L = 1, . . , a1 − 1} − a1 . 11 Let a1 = 5, a2 = 7 and a7 = 9. We compute E(j) = min{5x1 + 7x2 + 9x3 |7x2 + 9x3 ≡ j mod 5, x1 , x2 , x3 ≥ 0} for each j = 1, . . , 4. We have that E(1) = 16 (with x1 = 0, x2 = x3 = 1), E(2) = 7 (with x1 = x3 = 0, x2 = 1), E(3) = 18 (with x1 = x2 = 0, x3 = 2) and E(4) = 9 (with x1 = 0 = x2 = 0, x3 = 1).

E. in counterclockwise sense) from the present one is on, we leave it on if it was already on, otherwise we leave it off. The process halts as soon as any a1 consecutive lights are on. Let s(lai ) be the number of times light lai is visited during the procedure and let lr be the last visited off light just before ending the process. Then, g(a1 , . . , an ) = r + (s(lr ) − 1)an . 6). 13 Let a1 = 5, a2 = 6 and a3 = 7. In Fig. 9 the procedure is represented where the arrow marks the encountered light during the sweeping and the full (resp.

An in positive integers. Then, equality g(a1 , . . , an ) = dg( ad1 , . . , an−1 d , an ) + (d − 1)an holds if and only if n G(a1 , . . , , an = dG d d G(a1 , . . , an ) = dG n−1 holds. Notice that G(a1 , . . , an ) = i=1 ai xi with xi > 0 (this follows n−1 ai xi + an xn from the fact that we can write an + G(a1 , . . , an ) = i=1 n−1 with xi > 0 and thus G(a1 , . . , an ) = i=1 ai xi + an (xn − 1) that contradicts the definition of G unless xn = 1). Let ai = dai , i = 1, . . , n − 1.

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