A short proof is given of the theorem by Wik, who, in 1965, showed that the nowhere-dense strong Ditkin subsets of$T$, the circle group, are finite.

This paper introduces a new central trapdoor for multivariate quadratic (MQ) public-key cryptosystems that allows for encryption, in contrast to time-tested MQ primitives such as Unbalanced Oil and Vinegar or Hidden Field Equations which only allow for signatures. Our construction is a mixed-field scheme that exploits the commutativity of the extension field to dramatically reduce the complexity of the extension field polynomial implicitly present in the public key. However, this reduction can only be performed by the user who knows concise descriptions of two simple polynomials, which constitute the private key. After applying this transformation, the plaintext can be recovered by solving a linear system. We use the minus and projection modifiers to inoculate our scheme against known attacks. A straightforward C++ implementation confirms the efficient operation of the public key algorithms.

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrdinger operator in L^2(\mathbb{R}^2) with an attractive interaction supported on a closed curve , formally given by (<i>x</i>); we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread in L^2(\mathbb{R}^2), homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems reduce to purely geometric questions about inequalities for mean values of chords of . We prove an isoperimetric theorem for <i>p</i>-means of chords of curves when <i>p</i> 2, which implies in particular that the global extrema for the physical problems are always attained when is a circle. The letter concludes with a discussion of the <i>p</i>-means of chords when <i>p</i> &gt; 2.

We consider an electron with an anomalous magnetic moment g&gt; 2 confined to a plane and interacting with a non-zero magnetic field B perpendicular to the plane. We show that if B has compact support and the magnetic flux in natural units is, the corresponding Pauli Hamiltonian has at least bound states, without making any assumptions about the field profile. Furthermore, in the zero-flux case there is a pair of bound states with opposite spin orientations. Using a Birman-Schwinger technique, we extend the last claim to a weak rotationally symmetric field with, thus correcting a recent result. Finally, we show that under mild regularity assumptions existence of the bound states can also be proved for non-symmetric fields with tails.

We investigate a periodic quantum graph in the form of a square lattice with a general self-adjoint coupling at the vertices. We analyse the spectrum's high-energy behaviour. Depending on the coupling type, bands and gaps have different asymptotics. Bands may be flat even if the edges are coupled, and non-flat band widths may behave like\mathcal O (n^ j),\, j= 1, 0,-1,-2,-3, as the band index n. The gaps may be of asymptotically constant width or linearly growing with the latter case being generic.