In reverse chronological order:

• with R. Hardt, Size minimization and approximating problems, Prépublication d'Orsay 2001-64.

  We consider Plateau type variational problems related to the size minimization of rectifiable currents. We realize the limit of a size minimizing sequence as a stationary varifold and a minimal set. Other examples of functionals to be minimized include the integral over the underlying carrying set of a power q of the multiplicity function, with 0 < q ≤ 1. Because minimizing sequences may have unbounded mass we make use of a more general object called a rectifiable scan for describing the limit. This concept is motivated by the possibility of recovering a flat chain from a sufficiently large collection of its slices. In case the given boundary is smooth and compact, the limiting scan has finite mass and corresponds to a rectifiable current.

  
  

• Nearly flat almost monotone measures are big pieces of Lipschitz graphs, to appear in J. Geom. Anal. 12(1), 2002. 29-61.

  A concentrated (ξ,m) almost monotone measure in Rn is a Radon measure φ satisfying the two following conditions: (1) Θm(φ,x) ≥ 1 for every x ∈ spt(φ) and (2) for every x ∈ Rn the ratio exp[ξ(r)]r-mφ(B(x,r)) is increasing as a function of r > 0. Here ξ is an increasing function such that ξ(0+)=0. We prove that there is a relatively open dense set Reg(ξ) ⊂ spt(φ) such that at each x ∈ Reg(φ) the support of φ has the following regularity property: given ε > 0 and λ > 0, there is an m dimensional subspace W of Rn and a λ Lipschitz function from x + W to the orthogonal of W so that 100-ε % of spt(φ) ∩ B(x,r) coincides with the graph of f, at some scale r > 0, depending on x, ε and λ.

  
  

• with P. Huovinen, Points of ε differentiability of Lipschitz functions Rⁿ → R^n-1 , to appear in Bull. London Math. Soc.

  This paper proves that for every Lipschitz function f : Rn → Rn-1 there exists at least one point of ε differentiability of f which is the union of all n-1 dimensional affine subspaces of the form q0 + span{q1,...,qn-1} where qj, j=0,...,n-1, are points in Rn with rational coordinates.

  
  

• with Z. Buczolich and W.F. Pfeffer, Charges, BV functions and multipliers for generalized Riemann integrals, Indiana Univ. Math. J. 48(1), 1999. 1471-1511.

  For an invariant generalized Riemann integral in Rm, we obtain the following results. (1) A function is a multiplier for the space of locally integrable functions if and only if it is locally bounded and locally BV. (2) The dual of the space of all functions integrable in a bounded BV set A is linearly isomorphic to the space of all bounded BV functions vanishing oustide A, and each element of the dual has the usual integral representation. (3) On Lipschitz domains an integration by parts formula holds for any continuous function that is pointwise Lipschitz everywhere except on a set of \sigma finite n-1 dimensional Hausdorff measure.

  
  

• with Z. Buczolich and W.F. Pfeffer, Multipliers for generalized Riemann integrals, C.R. Math. Acad. Sci. Soc. R. Can. 21(4), 1999. 139-145.

• Some geometric problems from the calculus of variation, Mémoire de l'Académie royale de Belgique, 1998.

• with D. Smets, On explicit solutions for the problem of Mumford and Shah, Comm. Contemp. Math. 1(2), 1999. 201-212.

  We look for explicit image segmentations in the framework of the variational model proposed by Mumford and Shah. We first treat the symmetric case when the "screen" is a disk D and the image is a concentric disk C ⊂ D. We prove the optimal segmentation is either the given disk D or the solution of the associated Neumann problem, depending on both the difference of intensity between the background and the disk, and the distance separating Bdry D and Bdry C. Both segmentations are optimal in some critical cases which we characterize. Our main result is a first step towards a generalization of this behaviour. In case D and C are convex, we prove the following for an optimal segmentation(u,K) such that K ⊂ C : K tends to Bdry C (in the Hausdorff distance) when the difference of intensity between D and C goes to infinity.

  
  

• Multipliers for one-dimensional non absolutely convergent integrals, Atti Sem. Mat. Fis. Univ. Modena, XLVII, 1999. 327-335.

  We provide a unifying method proving BV functions to be multipliers for several 1 dimensional non absolutely convergent integration theories.

  
  

• with V. Beiu and S. Draghici, A constructive approach to calculating lower entropy bounds, Neural Processing Letters, 9, 1999. 1-12.

  This paper presents a constructive approach to estimating the size of a neural network necessary to solve a given classification problem. The results are derived using an information entropy approach in the context of limited precision integer weights. Such weights are particularly suited for hardware implementations since the area they occupy is limited, and the computations performed with them can be efficiently implemented in hardware. The considerations presented use an information entropy perspective and calculate lower bounds on the number of bits needed in order to solve a given classification problem. These bounds are obtained by approximating the classification hypervolumes with the volumes of several regular (highly symmetric) n dimensional bodies. The bounds given here allow the user to choose the appropriate size of a neural network such that: (i) the given classification problem can be solved, and (ii) the network architecture is not oversized. All considerations presented take into account the restrictive case of limited precision integer weights, and therefore can be directly applied when designing VLSI implementations of neural networks.

  
  

• On SBV dual, Indiana Univ. Math. J. 47(1), 1998. 99-121.

  This paper addresses the question of describing the dual of the space of functions having bounded variation in Rn. In particular we are interested in how a linear continuous functional acts on the jump part of the distributional derivative of a BV function. It is shown that representing this action as the flux of an Hn-1 measurable vectorfield through the jump set is independent of ZFC. Analogous undecidable results about various spaces of 1 dimensional currents are stated. Finally we apply our results to obtain an improvement of Whitney's integral representation of flat cochains.

  
  

• Topologies for the space of BV integrable functions in Rⁿ, J. Funct. Anal., 144(1), 1997. 190-231.

  We introduce two topologies on the space of BV integrable functions in Rn. Among the studied properties are barelledness and integral representation of linear continuous functionals. In one case we characterize the dual space. We show the relations with the study of multipliers. Finally we also introduce and study a topology on the space of bounded BV functions in Rn.

  
  

• A concept of generalized absolute continuity for the F integral, Real Anal. Exch., 22(1), 1996-97. 350-361.

  We define a concept of generalized absolute continuity for additive functions of figures. This makes it possible to give a descriptive definition of the F integral introduced by W.F. Pfeffer. Finally we discuss a possible extension to additive functions of sets of bounded variation.