Fourier-Feynman transform, convolution and first variation

We examine several interesting relationships and expressions involving Fourier-Feynman transform, convolution product and first variation for functionals in the Fresnel class F(B) of an abstract Wiener space B. We also prove a translation theorem and Parseval's identity for the analytic Feynman integral. This revised version was published online in June 2006 with corrections to the Cover Date.     

Shape Optimization and Fictitious Domain Approach for Solving Free Boundary Problems of Bernoulli Type

This contribution deals with an efficient method for the numerical realization of the exterior and interior Bernoulli free boundary problems. It is based on a shape optimization approach. The state problems are solved by a fictitious domain solver using boundary Lagrange multipliers.     

Detecting Mass Points of Representing Measures

We discuss a possibility of deciding whether measures representing a moment sequence or realizing orthogonality of polynomials have atoms. This is done on the real line and in several variables.     

ON THE EQUATION □φ = |vφ|2 IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in n spacedimensions□φ = F( φ),φ(0, x) = f(x), tφ(0, x) = g(x),The minimal value of s is determined such that the above Cauchy problem is locally well-posed in Hs. It turns out that for the general equation s must satisfyThis is due to Ponce and Sideris (when n = 3) and Tataru (when n ≥ 5). The purpose of thispaper is to supplement with a proof in the case n = 2, 4.     

Dirac index classes and the noncommutative spectral flow

We present a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various index-theoretic situations, extending to the noncommutative context results of Booss–Wojciechowski, Melrose–Piazza and Dai–Zhang. In particular, we prove a variational formula, in K_*(C_r^*(Γ)), for the index classes associated to 1-parameter family of Dirac operators on a Γ-covering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K_*(C_r^*(Γ)), for the index class defined by a Dirac-type operator associated to a closed manifold M and a map r:M→BΓ when we assume that M is the union along a hypersurface F of two manifolds with boundary M=M_{+ F} M₋. Finally, we prove a defect formula for the signature-index classes of two cut-and-paste equivalent pairs (M₁,r₁:M₁→BΓ) and (M₂,r₂:M₂→BΓ), where

M₁=M_{+ (F,φ1)} M₋, M₂=M_{+ (F,φ2)} M₋

and φ_jDiff(F). The formula involves the noncommutative spectral flow of a suitable 1-parameter family of twisted signature operators on F. We give applications to the problem of cut-and-paste invariance of Novikov's higher signatures on closed oriented manifolds.     

Discrete Sturm–Liouville problems, Jacobi matrices and Lagrange interpolation series

The close relationship between discrete Sturm–Liouville problems belonging to the so-called limit-circle case, the indeterminate Hamburger moment problem and the search of self-adjoint extensions of the associated semi-infinite Jacobi matrix is well known. In this paper, all these important topics are also related with associated sampling expansions involving analytic Lagrange-type interpolation series.     

The non-approximability of bicriteria network design problems

Let G be an undirected graph with two edge costs (c-cost and d-cost). We want to minimize the diameter of a spanning subgraph S (under d-cost) subject to the constraint that the total cost of the edges in S (with respect to c) does not exceed a given budget. We prove that this problem is non-approximable, even in some special cases. Similar results are proved if the stretch factor or the root stretch factor is considered instead of the diameter.     

Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques

We develop two implementable algorithms, the first for the solution of finite and the second for the solution of semi-infinite min-max-min problems. A smoothing technique (together with discretization for the semi-infinite case) is used to construct a sequence of approximating finite min-max problems, which are solved with increasing precision. The smoothing and discretization approximations are initially coarse, but are made progressively finer as the number of iterations is increased. This reduces the potential ill-conditioning due to high smoothing precision parameter values and computational cost due to high levels of discretization. The behavior of the algorithms is illustrated with three semi-infinite numerical examples.     

Use of the Minimum-Norm Search Direction in a Nonmonotone Version of the Gauss-Newton Method

In this work, a new stabilization scheme for the Gauss-Newton method is defined, where the minimum norm solution of the linear least-squares problem is normally taken as search direction and the standard Gauss-Newton equation is suitably modified only at a subsequence of the iterates. Moreover, the stepsize is computed by means of a nonmonotone line search technique. The global convergence of the proposed algorithm model is proved under standard assumptions and the superlinear rate of convergence is ensured for the zero-residual case. A specific implementation algorithm is described, where the use of the pure Gauss-Newton iteration is conditioned to the progress made in the minimization process by controlling the stepsize. The results of a computational experimentation performed on a set of standard test problems are reported.     

On Vector Quasi-Equilibrium Problems with Set-Valued Maps

In this paper, we introduce a new class of vector quasi-equilibrium problems with set-valued maps. Almost all the vector equilibrium models of the Blum-Oettli type in the literature are special cases of our new class of equilibrium problems under consideration. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the -diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for such vector equilibrium problems.