Lecture notes on ridge regression

The linear regression model cannot be fitted to high-dimensional data, as the high-dimensionality brings about empirical non-identifiability. Penalized regression overcomes this non-identifiability by augmentation of the loss function by a penalty (i.e. a function of regression coefficients). The ridge penalty is the sum of squared regression coefficients, giving rise to ridge regression. Here many aspect of ridge regression are reviewed e.g. moments, mean squared error, its equivalence to constrained estimation, and its relation to Bayesian regression. Finally, its behaviour and use are illustrated in simulation and on omics data. Subsequently, ridge regression is generalized to allow for a more general penalty. The ridge penalization framework is then translated to logistic regression and its properties are shown to carry over. To contrast ridge penalized estimation, the final chapter introduces its lasso counterpart

Bose-Glass Phases in Disordered Quantum Magnets

In disordered spin systems with antiferromagnetic Heisenberg exchange, transitions into and out of a magnetic-field-induced ordered phase pass through a unique regime. Using quantum Monte Carlo simulations to study the zero-temperature behavior, these intermediate regions are determined to be a Bose-Glass phase. The localization of field-induced triplons causes a finite compressibility and hence glassiness in the disordered phase.Comment: 4 pages, 4 figure

Progress in the development of an 88-mm bore 10 Tn3Sn dipole magnet

A 10 T, 2-layer cos(&thetas;)-dipole model magnet with an 88 mm clear bore utilizing an advanced powder-in-tube Nb3Sn conductor is being developed for the LHC. A dedicated conductor development program has resulted in a well performing Rutherford cable containing strands that uniquely exhibit both an overall current density of 600 A/mm2 @ 11 T and filaments with a diameter of 20 ¿m. The resistance between crossing strands amounts to 30-70 ¿¿ by insertion of a stainless steel core. After being exposed to a transverse pressure of 200 MPa identical cables show negligible permanent degradation of the critical current. The mechanical support structure is further optimized in order to reduce the peak stress in the mid-plane to below 130 MPa at full excitation and to control the pre-stress build-up during system assembly. Prior to the manufacturing of the final coils a dummy 2-layer pole is wound, heat-treated at 675°C and vacuum resin impregnated. This paper presents the current status of the magnet development program and highlights in particular the successful conductor developmen

Wang-Landau sampling for quantum systems: algorithms to overcome tunneling problems and calculate the free energy

We present a generalization of the classical Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)] to quantum systems. The algorithm proceeds by stochastically evaluating the coefficients of a high temperature series expansion or a finite temperature perturbation expansion to arbitrary order. Similar to their classical counterpart, the algorithms are efficient at thermal and quantum phase transitions, greatly reducing the tunneling problem at first order phase transitions, and allow the direct calculation of the free energy and entropy.Comment: Added a plot showing the efficiency at first order phase transition

High yield fusion in a Staged Z-pinch

We simulate fusion in a Z-pinch; where the load is a xenon-plasma liner imploding onto a deuterium-tritium plasma target and the driver is a 2 MJ, 17 MA, 95 ns risetime pulser. The implosion system is modeled using the dynamic, 2-1/2 D, radiation-MHD code, MACH2. During implosion a shock forms in the Xe liner, transporting current and energy radially inward. After collision with the DT, a secondary shock forms pre-heating the DT to several hundred eV. Adiabatic compression leads subsequently to a fusion burn, as the target is surrounded by a flux-compressed, intense, azimuthal-magnetic field. The intense-magnetic field confines fusion $\alpha$-particles, providing an additional source of ion heating that leads to target ignition. The target remains stable up to the time of ignition. Predictions are for a neutron yield of $3.0\times 10^{19}$ and a thermonuclear energy of 84 MJ, that is, 42 times greater than the initial, capacitor-stored energy

Field-Induced Magnetic Order in Quantum Spin Liquids

We study magnetic field-induced three-dimensional ordering transitions in low-dimensional quantum spin liquids, such as weakly coupled, antiferromagnetic spin-1/2 Heisenberg dimers and ladders. Using stochastic series expansion quantum Monte Carlo simulations, thermodynamic response functions are obtained down to ultra-low temperatures. We extract the critical scaling exponents which dictate the power-law dependence of the transition temperature on the applied magnetic field. These are compared with recent experiments on candidate materials and with predictions for the Bose-Einstein condensation of magnons obtained in mean-field theory.Comment: RevTex, 4 pages with 5 figure

Supersolid Order from Disorder: Hard-Core Bosons on the Triangular Lattice

We study the interplay of Mott localization, geometric frustration, and superfluidity for hard-core bosons with nearest-neighbor repulsion on the triangular lattice. For this model at half-filling, we demonstrate that superfluidity survives for arbitrarily large repulsion, and that diagonal solid order emerges in the strongly correlated regime from an order-by-disorder mechanism. This is thus an unusual example of a stable supersolid phase of hard-core lattice bosons at a commensurate filling.Comment: 4 pages, 2 figures; finite-size scaling discussion adde