On Minimal Valid Inequalities for Mixed Integer Conic Programs

We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically defined class of K- sublinear inequalities. We establish a close connection between K-sublinear inequalities and the support functions of sets with a particular structure. This connection results in practical ways of showing that a given inequality is K-sublinear and K-minimal. Our framework generalizes some of the results from the mixed integer linear case. It is well known that the minimal inequalities for mixed integer linear programs are generated by sublinear (positively homogeneous, subadditive and convex) functions that are also piecewise linear. This result is easily recovered by our analysis. Whenever possible we highlight the connections to the existing literature. However, our study unveils that such a cut generating function view treating the data associated with each individual variable independently is not possible in the case of general cones other than nonnegative orthant, even when the cone involved is the Lorentz cone

Disjunctive cuts for cross-sections of the second-order cone

Abstract In this paper we study general two-term disjunctions on affine cross-sections of the secondorder cone. Under some mild assumptions, we derive a closed-form expression for a convex inequality that is valid for such a disjunctive set, and we show that this inequality is sufficient to characterize the closed convex hull of all two-term disjunctions on ellipsoids and paraboloids and a wide class of two-term disjunctions-including split disjunctions-on hyperboloids. Our approach relies on the work of Kılınç-Karzan and Yıldız which considers general two-term disjunctions on the second-order cone

Polynomial cubic differentials and convex polygons in the projective plane

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie-Loftin parameterization of convex RP^2 structures on a compact surface by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report. v2: Corrections in section 5 and related new material in appendix

On the multi-orbital band structure and itinerant magnetism of iron-based superconductors

This paper explains the multi-orbital band structures and itinerant magnetism of the iron-pnictide and chalcogenides. We first describe the generic band structure of an isolated FeAs layer. Use of its Abelian glide-mirror group allows us to reduce the primitive cell to one FeAs unit. From density-functional theory, we generate the set of eight Fe $d$ and As $p$ localized Wannier functions for LaOFeAs and their tight-binding (TB) Hamiltonian, $h(k)$. We discuss the topology of the bands, i.e. allowed and avoided crossings, the origin of the d6 pseudogap, as well as the role of the As $p$ orbitals and the elongation of the FeAs$_{4}$ tetrahedron. We then couple the layers, mainly via interlayer hopping between As $p_{z}$ orbitals, and give the formalism for simple and body-centered tetragonal stackings. This allows us to explain the material-specific 3D band structures. Due to the high symmetry, several level inversions take place as functions of $k_{z}$ or pressure, resulting in linear band dispersions (Dirac cones). The underlying symmetry elements are, however, easily broken, so that the Dirac points are not protected, nor pinned to the Fermi level. From the paramagnetic TB Hamiltonian, we form the band structures for spin spirals with wavevector $q$ by coupling $h(k)$ and $h (k+q)$. The band structure for stripe order is studied as a function of the exchange potential, $\Delta$, using Stoner theory. Gapping of the Fermi surface (FS) for small $\Delta$ requires matching of FS dimensions (nesting) and $d$-orbital characters. The origin of the propeller-shaped FS is explained. Finally, we express the magnetic energy as the sum over band-structure energies, which enables us to understand to what extent the magnetic energies might be described by a Heisenberg Hamiltonian, and the interplay between the magnetic moment and the elongation of the FeAs4 tetrahedron

Output-Adaptive Tetrahedral Cut-Cell Validation for Sonic Boom Prediction

A cut-cell approach to Computational Fluid Dynamics (CFD) that utilizes the median dual of a tetrahedral background grid is described. The discrete adjoint is also calculated, which permits adaptation based on improving the calculation of a specified output (off-body pressure signature) in supersonic inviscid flow. These predicted signatures are compared to wind tunnel measurements on and off the configuration centerline 10 body lengths below the model to validate the method for sonic boom prediction. Accurate mid-field sonic boom pressure signatures are calculated with the Euler equations without the use of hybrid grid or signature propagation methods. Highly-refined, shock-aligned anisotropic grids were produced by this method from coarse isotropic grids created without prior knowledge of shock locations. A heuristic reconstruction limiter provided stable flow and adjoint solution schemes while producing similar signatures to Barth-Jespersen and Venkatakrishnan limiters. The use of cut-cells with an output-based adaptive scheme completely automated this accurate prediction capability after a triangular mesh is generated for the cut surface. This automation drastically reduces the manual intervention required by existing methods

3-D Oropharyngeal Airway Analysis of Different Antero-Posterior and Vertical Craniofacial Skeletal Patterns in Children and Adolescents

Sleep apnea disorder has recently emerged as a significant public health issue. While the prevalence of obesity is on the rise among children, it is one of the main risk factors associated with apnea. Upper airway dimensions and morphology seem to be major components of obstructive sleep apnea (OSA) and can be affected by different craniofacial patterns. The purpose of this retrospective, cross-sectional pilot study is to correlate gender, Body Mass Index, risk for OSA, neck circumference, and 3-D oropharyngeal airway dimensions in children and adolescents with different antero-posterior (AP) and vertical craniofacial skeletal patterns. A total of 86 pre-orthodontic treatment records in the age group of 8-16 years were analyzed. 3-D volumetric skeletal tracing and oropharyngeal airway measurements were completed for each scan. Each subject was classified into AP Classes I, II, and III groups; vertical Normodivergent, Hypodivergent, and Hyperdivergent groups; and combined AP-vertical subgroups. Oropharyngeal airway measurements included the total oropharyngeal airway volume, minimum cross-section area, depth, width, and perimeter. Mean, standard deviation, and Pearson\u27s correlation coefficient were performed to evaluate the relationships among variables. There were one or more correlations, but not all, between gender, Body Mass Index, risk for OSA, neck circumference, and 3-D oropharyngeal airway dimensions in children and adolescents among the AP groups, vertical groups, and nine craniofacial subgroups (P \u3c 0.05 and P \u3c 0.01). This investigation aimed to determine whether patients with certain skeletal deficiencies are predisposed to upper airway obstruction. Early identification and management of airway problems in children and adolescents may prevent or minimize the sequelae and adverse dental implications of obstructive sleep apnea. Our small, young groups of sample were mainly in the healthy weight category with normal size neck circumference. Therefore, this limited our overall findings. Currently, sleep disorders are not well researched and understood. Long-term goal of our study is to further investigate this study in larger sample size taken into considerations predisposing factors (i.e. abnormal neural regulation and intrinsic muscle weakness) and pathologic conditions (allergies, polyps, and tumors). The physiology of the airway, influenced by these confounding factors, has an essential role in determining whether patients with certain skeletal deficiencies are predisposed to upper airway obstruction. Sleep apnea is a complex phenomenon that warrants further research regarding the physiology and anatomy of the airway and craniofacial structures

Novel Particle Model for the Prediction of Stability and Episodic Collapse of Coastal Cliffs and Levees

This thesis investigates the WCSPH model by considering fluid entry and exit, and integrates the WCSPH method into a new, novel, particle-based Bluff Morphology Model (BMM). Using the BMM, this thesis investigates the stability, collapse and equilibrium position of soft coastal bluffs (cliffs). Fluid and floating object interaction using a novel adaptation of the WCSPH method is investigated by incorporating a floating object model. In particular, this thesis examines the water impact, hydrodynamic forces, fluid motions, and movement of objects in the conventional case studies of object entry and exit from still water. A two-dimensional wedge drop analysis was examined, and the hydrodynamic forces show acceptable agreement with published experimental and numerical results. Simulations for water entry and exit of a buoyant and neutral density cylinder compares well with the previous experimental, numerical and empirical studies. These results provide a good foundation to evaluate the accuracy and stability of WCSPH for modelling complex flows, and therefore offers a platform for the use of WCSPH in a Bluff Morphology Model. The BMM combines a multiple wedge displacement method with an adapted Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. At first the wedge method is applied to compute the stability of the bluff. Once the critical failure mechanism of the bluff slope has been identified, if the Factor of Safety for the mechanism is less than 1, the adapted WCSPH method is used to predict the failure movement and residual shape of the slope. The model is validated against benchmark test cases of bluff stability for purely frictional, purely cohesive, and mixed strength bluff materials including 2D static water tables. The model predictions give a good correlation with the expected values, with medium resolution models producing errors of typically less than 2.0%. In addition, the prediction of lateral movement of a surveyed cliff and the dynamic collapse of a vertical bluff are computed, and compare well with published literature. This model is further extended to then investigate the effect of two dimensional seepage on the stability and collapse of soil slopes and levees. To incorporate the seepage in the model, Darcy’s Law is applied to the interactions among neighbouring soil particles and ghost particles are introduced along the enclosed soil boundary to ensure that no fluid crosses the boundary. The contribution of partially saturated soils and matric suction, as well as the change in hydraulic conductivity due to seepage, are predicted well by this model. The predicted time evolution of slope stability and seepage induced collapse are in reasonable agreement with the experimental results for homogeneous frictional sand and multiple layered cohesive soils. Rapid drawdown over a sand soil is also investigated, and the location and time of the levee collapse occurrence are captured well. A toe erosion model is incorporated within the numerical model, and the location and quantity of erosion caused by lateral seepage is well predicted. The interplay of erosion, seepage and slope instability is examined