The Riemannian metric on the manifold of positive de nite matrices is de ned by a kernel function in the form K D(H;K) = P i;j ( i; j) 1TrPiHPjK when P i iPi is the spectral decomposition of the foot point D and the Hermitian matrices H;K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7! G(D) is ...

Recently we introduced a family of U N invariant random matrix ensembles which is characterized by a parameter describing logarithmic soft-confinement potentials V H ln H 1+ 0 . We showed that we can study eigenvalue correlations of these “ ensembles” based on the numerical construction of the corresponding orthogonal polynomials with respect to the weight function exp − ln x 1+ . In this work,...

Conditional expectiles are becoming an increasingly important tool in finance as well as in other areas of applications. We analyse a support vector machine type approach for estimating conditional expectiles and establish learning rates that are minimax optimal modulo a logarithmic factor if Gaussian RBF kernels are used and the desired expectile is smooth in a Besov sense. As a special case, ...

One particle inclusive cross sections in the target fragmentation region are considered and an explicit calculation is performed in (3) 6 model eld theory. The collinear divergences can be correctly absorbed into a parton density and a fragmentation function but the renormalized cross section gets a large logarithmic correction as expected in a two scale regime. We nd that the coeecient of such...

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C 1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semicircles , is derived. Comments on higher coefficients are made. A separable hemis...

We study asymptotic properties of the Green metric associated with transient random walks on countable groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. The proof relies on integral representations of both quantities with the extended Martin kernel. In the case of finitely generated groups, where this result is known (Benjami...

Supervised machine learning is the classification of new data based on already classified training examples. In this work, we show that the support vector machine, an optimized binary classifier, can be implemented on a quantum computer, with complexity logarithmic in the size of the vectors and the number of training examples. In cases where classical sampling algorithms require polynomial tim...

For surfaces of revolution B in R, we investigate the limit distribution of minimum energy point masses on B that interact according to the logarithmic potential log(1/r), where r is the Euclidean distance between points. We show that such limit distributions are supported only on the “out-most” portion of the surface (e.g., for a torus, only on that portion of the surface with positive curvatu...

We calculate the quark–anti-quark contribution to the next-to-leading logarithmic corrections to the BFKL kernel, retaining the dependence on the momenta of the produced particles. This allows us to study the details of the NLL corrections. We demonstrate that the standard calculation of the NLL corrections to the scattering of two off-shell gluons includes contributions from energies far above...