A simple construction of complex equiangular lines

A set of vectors of equal norm in $\mathbb{C}^d$ represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in $\mathbb{C}^d$ for every $d \geq 2$. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.Comment: 11 pages; minor revisions and comments added in section 1 describing a link to previously known results; correction to Theorem 1 and updates to reference

Algebro-geometric approach in the theory of integrable hydrodynamic type systems

The algebro-geometric approach for integrability of semi-Hamiltonian hydrodynamic type systems is presented. This method is significantly simplified for so-called symmetric hydrodynamic type systems. Plenty interesting and physically motivated examples are investigated

Equivalence problem for the orthogonal webs on the sphere

We solve the equivalence problem for the orthogonally separable webs on the three-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved the corresponding canonical forms problem. The solution to the equivalence problem together with the results by Olevsky forms a complete solution to the problem of orthogonal separation of variables to the Hamilton-Jacobi equation defined on the three-sphere via orthogonal separation of variables. It is based on invariant properties of the characteristic Killing two-tensors in addition to properties of the corresponding algebraic curvature tensor and the associated Ricci tensor. The result is illustrated by a non-trivial application to a natural Hamiltonian defined on the three-sphere.Comment: 32 page

Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions

We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless Lax pairs and an infinity of hydrodynamic reductions.Comment: 34 page

Strict selection alone of patients undergoing liver transplantation for hilar cholangiocarcinoma is associated with improved survival

Liver transplantation for hilar cholangiocarcinoma (hCCA) has regained attention since the Mayo Clinic reported their favorable results with the use of a neo-adjuvant chemoradiation protocol. However, debate remains whether the success of the protocol should be attributed to the neo-adjuvant therapy or to the strict selection criteria that are being applied. The aim of this study was to investigate the value of patient selection alone on the outcome of liver transplantation for hCCA. In this retrospective study, patients that were transplanted for hCCA between1990 and 2010 in Europe were identified using the European Liver Transplant Registry (ELTR). Twenty-one centers reported 173 patients (69%) of a total of 249 patients in the ELTR. Twenty-six patients were wrongly coded, resulting in a study group of 147 patients. We identified 28 patients (19%) who met the strict selection criteria of the Mayo Clinic protocol, but had not undergone neo-adjuvant chemoradiation therapy. Five-year survival in this subgroup was 59%, which is comparable to patients with pretreatment pathological confirmed hCCA that were transplanted after completion of the chemoradiation protocol at the Mayo Clinic. In conclusion, although the results should be cautiously interpreted, this study suggests that with strict selection alone, improved survival after transplantation can be achieved, approaching the Mayo Clinic experience

The Nontriviality of Trivial General Covariance: How Electrons Restrict 'Time' Coordinates, Spinors (Almost) Fit into Tensor Calculus, and 7/16 of a Tetrad Is Surplus Structure

It is a commonplace that any theory can be written in any coordinates via tensor calculus. But it is claimed that spinors as such cannot be represented in coordinates in a curved space-time. What general covariance means for theories with fermions is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov (OP) constructed spinors in coordinates in 1965, helping to spawn nonlinear group representations. Locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient. The tetrad formalism is de-Ockhamized, with 6 extra fields and 6 compensating gauge symmetries. OP spinors, as developed nonperturbatively by Bilyalov, admit any coordinates at a point, but "time" must be listed first: the product of the metric components and the matrix diag(-1,1,1,1) must have no negative eigenvalues to yield a real symmetric square root function of the metric. Thus the admissible coordinates depend on the types and values of the fields. Apart from coordinate order and spinorial two-valuedness, OP spinors form, with the metric, a nonlinear geometric object, with Lie and covariant derivatives. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in GR. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of a matrix with 16 components, one can use weighted OP spinors coupled to the 9-component symmetric unimodular square root of the conformal metric density. The surprising mildness of the restrictions on coordinates for the Schwarzschild solution is exhibited. (edited)Comment: Forthcoming in \textit{Studies in History and Philosophy of Modern Physics

Gauging the spacetime metric -- looking back and forth a century later

H. Weyl's proposal of 1918 for generalizing Riemannian geometry by local scale gauge (later called {\em Weyl geometry}) was motivated by mathematical, philosophical and physical considerations. It was the starting point of his unified field theory of electromagnetism and gravity. After getting disillusioned with this research program and after the rise of a convincing alternative for the gauge idea by translating it to the phase of wave functions and spinor fields in quantum mechanics, Weyl no longer considered the original scale gauge as physically relevant. About the middle of the last century the question of conformal and/or local scale gauge transformation were reconsidered by different authors in high energy physics (Bopp, Wess, et al.) and, independently, in gravitation theory (Jordan, Fierz, Brans, Dicke). In this context Weyl geometry attracted new interest among different groups of physicists (Omote/Utiyama/Kugo, Dirac/Canuto/Maeder, Ehlers/Pirani/Schild and others), often by hypothesizing a new scalar field linked to gravity and/or high energy physics. Although not crowned by immediate success, this ``retake'' of Weyl geometrical methods lives on and has been extended a century after Weyl's first proposal of his basic geometrical structure. It finds new interest in present day studies of elementary particle physics, cosmology, and philosophy of physics.Comment: 56 pages, contribution to Workshop Hundred Years of Gauge Theory Bad Honnef, July 30 - August 3, 201