Several proofs have been published of the modZ gluing formula for the η–invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the η–invariant is left obscure in the literature. In this article we present a gluing formula for the η–invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderón pr...

Motivated by the propagation of thin bacterial films around planar obstacles, this paper considers the dynamics of travelling wave solutions to the Fisher-KPP equation [Formula: see text] in a planar strip [Formula: see text], [Formula: see text]. We examine the propagation of fronts in the presence of a mixed boundary condition (also referred to as a 'partially absorbing' or 'reactive' boundar...

We prove an adiabatic limit formula for the eta invariant of a manifold with boundary. The eta invariant is defined using the Atiyah-PatodiSinger boundary condition and the underlying manifold is fibered over a manifold with boundary. Our result extends the work of Bismut-Cheeger to manifolds with boundary.

We derive a decomposition formula for the spectral flow of a 1-parameter family of self-adjoint Dirac operators on an odd-dimensional manifold M split along a hypersurface Σ (M = X ∪Σ Y ). No transversality or stretching hypotheses are assumed and the boundary conditions can be chosen arbitrarily. The formula takes the form SF (D) = SF (D|X , BX) + SF (D|Y , BY ) + μ(BY , BX) + S where BX and B...

In this paper, we consider the existence of least energy sign-changing solutions for a class of Kirchhoff-type problem [Formula: see text]where [Formula: see text] is a bounded domain in [Formula: see text], [Formula: see text], with a smooth boundary [Formula: see text], [Formula: see text] and [Formula: see text]. By using variational approach and some subtle analytical skills, the existence ...

Relations are found between grain boundary character distribution, misorientation distribution and two-point orientation distribution. Some of these relations are analogies for polycrystals of the Debye-Anderson-Brumberger formula and Berryman formula established for two-phase composites.

In the paper, we derive a formula for computing the determinant of a Schrödinger operator on a compact metric graph. This formula becomes very explicit in the case of the Laplacian with the Neumann boundary conditions.

In MSSM models with various boundary conditions for the soft breaking terms ([Formula: see text]) and for a Higgs mass of 126 GeV, there is a (minimal) electroweak fine-tuning [Formula: see text] to [Formula: see text] for the constrained MSSM and [Formula: see text] for non-universal gaugino masses. These values, often regarded as unacceptably large, may indicate a problem of supersymmetry (SU...

In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman–Kac formula for the Robin problem which employs the standard reflecting Brownian motion (SRBM) and its boundary local time arising from the Skorokhod problem. By simulating SRBM paths through Brownian motio...