We prove uniqueness of the weak solution of the Euler equations for compactly supported, single signed and bounded initial vorticity in simply connected planar domains with corners forming angles larger than π/2. In this type of domain, the velocity is not log-lipschitz and does not belong to W 1,p for all p, which is the standard regularity for the Yudovich’s arguments. Thanks to the explicit ...

In this paper we study the controllability problem for a system that models the vibrations of a controlled tree-shaped network of vibrating elastic strings. The control acts through one of the exterior nodes of the network. With the help of the d’Alembert representation formula for the solutions of the 1-d wave equation we find certain linear relations between the traces of the solutions at the...

We show that on every Spin(7)-manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an exlicit formula for the Riemannian covariant derivative of the fundamental 4form in terms of its exterior di...

The representation of a function by its circular Radon transform (CRT) and various related problems arise in many areas of mathematics, physics and imaging science. There has been a substantial spike of interest towards these problems in the last decade mainly due to the connection between the CRT and mathematical models of several emerging medical imaging modalities. This paper contains some n...

In this paper after recalling some essential tools concerning the theory of differential forms in the Cartan, Hodge and Clifford bundles over a Riemannian or Riemann-Cartan space or a Lorentzian or Riemann-Cartan spacetime we solve with details several exercises involving different grades of difficult. One of the problems is to show that a recent formula given in [10] for the exterior covariant...

Let 0 ∈ D+ be a connected domain with analytic boundary on the complex plane C. Then according to the Riemann theorem there exists a function w(z) = 1 r z + ∑∞ j=0 pjz −j , mapping biholomorphically D− = C \D+ to the exterior of the unit disk {w ∈ C : |w| > 1}. From Wiegmann’s and Zabrodin’s rezults it follows that this function is described by the formula logw = log z − ∂t0( 1 2∂t0 + ∑ k>1 z−k...

Analytical solutions are constructed for an assembly of any finite number of bubbles in steady motion in a Hele-Shaw channel. The solutions are given in the form of a conformal mapping from a bounded multiply connected circular domain to the flow region exterior to the bubbles. The mapping is written as the sum of two analytic functions-corresponding to the complex potentials in the laboratory ...

The Hochschild and cyclic homology groups are computed for the algebra of ‘cusp’ pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and evaluated in terms of extensions of the trace functionals on the two natural ideals, corresponding to the two filtrations by interior order and vanishing degree at t...

Using memoization and various other optimization techniques, the number of dissections of the n × n square into n polyominoes of size n is computed for n ≤ 8. On this task our method outperforms Donald Knuth’s Algorithm X with Dancing Links. The number of jigsaw sudoku puzzle solutions is computed for n ≤ 7. For every jigsaw sudoku puzzle polyomino cover with n ≤ 6 the size of its smallest crit...

We show that on every Spin(7)-manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Riemannian covariant derivative of the fundamental 4-form in terms of its exterior ...