Slices of maps and Lebesgue area

The Additivity of the Lebesgue Area

A triple of continuous functions T: x(ut u ), i = l, 2, 3, defined on a closed square Q [ O g w x ^ l , 0 ^ w 2 ^ l ] represents a surface © (1.6, 1.17, 1.21). If r is any closed rectangle in Q then we may speak of the triple Tr consisting of the above triple T with its range of definition restricted to r. This triple generates a surface @(r). If r\ and r2 have no interior points in common, and...

Maps on positive operators preserving Lebesgue decompositions

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every ...

The Isoperimetric Inequality and the Lebesgue Definition of Surface Area

The literature of these classical isoperimetric inequalities is very extensive (comprehensive presentations may be found in Blaschke [l] and Bonnesen [l](1)). In most instances, only convex curves and convex surfaces are considered, or else it is assumed that the curves and surfaces involved are sufficiently regular to permit the use of the classical formulas for the quantities a(C), 1(C), A(S)...

Topological Methods in the Theory of Lebesgue Area

Conformal maps and non-reversibility of elliptic area-preserving maps

It has been long observed that area-preserving maps and reversible maps share similar results. This was certainly known to G.D. Birkhoff [5] who showed that these two types of maps have periodic orbits near a general elliptic fixed point. The KAM theory, developed by Kolmogorov-ArnoldMoser for Hamiltonian systems [9], [1] and area preserving maps [15], has also been extended a great deal to rev...