Crystal approach to affine Schubert calculus

We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-$A$ affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a $k$-Schur function, consequently proving that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb C^n$ enumerate the highest weight elements under these operators. Included in this class are the Schubert structure constants in the (quantum) product of a Schubert polynomial with a Schur function $s_\lambda$ for all $|\lambda^\vee|< n$. Another by-product gives a highest weight formulation for various fusion coefficients of the Verlinde algebra and for the Schubert decomposition of certain positroid classes.Comment: 42 pages; version to appear in IMR

Effective charge-spin models for quantum dots

It is shown that at low densities, quantum dots with few electrons may be mapped onto effective charge-spin models for the low-energy eigenstates. This is justified by defining a lattice model based on a many-electron pocket-state basis in which electrons are localised near their classical ground-state positions. The equivalence to a single-band Hubbard model is then established leading to a charge-spin ($t-J-V$) model which for most geometries reduces to a spin (Heisenberg) model. The method is refined to include processes which involve cyclic rotations of a ``ring'' of neighboring electrons. This is achieved by introducing intermediate lattice points and the importance of ring processes relative to pair-exchange processes is investigated using high-order degenerate perturbation theory and the WKB approximation. The energy spectra are computed from the effective models for specific cases and compared with exact results and other approximation methods.Comment: RevTex, 24 pages, 7 figures submitted as compressed and PostScript file

Cluster adjacency beyond MHV

We explore further the notion of cluster adjacency, focussing on non-MHV amplitudes. We extend the notion of adjacency to the BCFW decomposition of tree-level amplitudes. Adjacency controls the appearance of poles, both physical and spurious, in individual BCFW terms. We then discuss how this notion of adjacency is connected to the adjacency already observed at the level of symbols of scattering amplitudes which controls the appearance of branch cut singularities. Poles and symbols become intertwined by cluster adjacency and we discuss the relation of this property to the $\bar{Q}$-equation which imposes constraints on the derivatives of the transcendental functions appearing in loop amplitudes.Comment: 51 pages, 25 figures, 4 table

Using a Conformal Water Bolus to Adjust Heating Patterns of Microwave Waveguide Applicators

Background: Hyperthermia, i.e., raising tissue temperature to 40-45°C for 60 min, has been demonstrated to increase the effectiveness of radiation and chemotherapy for cancer. Although multi-element conformal heat applicators are under development to provide more adjustable heating of contoured anatomy, to date the most often used applicator to heat superficial disease is the simple microwave waveguide. With only a single power input, the operator must be resourceful to adjust heat treatment to accommodate variable size and shape tumors spreading across contoured anatomy. Methods: We used multiphysics simulation software that couples electromagnetic, thermal and fluid dynamics physics to simulate heating patterns in superficial tumors from commercially available microwave waveguide applicators. Temperature distributions were calculated inside homogenous muscle and layered skin-fat-muscle-tumor-bone tissue loads for a typical range of applicator coupling configurations and size of waterbolus. Variable thickness waterbolus was simulated as necessary to accommodate contoured anatomy. Physical models of several treatment configurations were constructed for comparison of simulation results with experimental specific absorption rate (SAR) measurements in homogenous muscle phantom. Results: Accuracy of the simulation model was confirmed with experimental SAR measurements of three unique applicator setups. Simulations demonstrated the ability to generate a wide range of power deposition patterns with commercially available waveguide antennas by controllably varying size and thickness of the waterbolus layer. Conclusion: Heating characteristics of 915 MHz waveguide antennas can be varied over a wide range by controlled adjustment of microwave power, coupling configuration, and waterbolus lateral size and thickness. The uniformity of thermal dose delivered to superficial tumors can be improved by cyclic switching of waterbolus thickness during treatment to proactively shift heat peaks and nulls around under the aperture, thereby reducing patient pain while increasing minimum thermal dose by end of treatment. © (2017) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE)

kk-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website: http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates and corrections since v1. This material is based upon work supported by the National Science Foundation under Grant No. DMS-065264

Experimental and Numerical Evaluation of Residual Displacement and Ductility in Ratcheting and Shakedown of an Aluminium Beam

Safety assessment of structures can be obtained employing limit design to overcome uncertainties concerning actual response due to inelastic constitutive behavior and more generally to non-linear structural response and loads' random variability. The limit analysis is used for evaluating the safety of the structures, starting directly from load level without any knowledge of the load history. In the paper, the lower bound calculation is proposed where a new strain-based approach is used that allowed describing the residual stress and displacement in terms of permanent strain. The strategy uses the permanent strain as effective parameters of the procedure so that it is possible to assess the ductility requirements for the complete load program developed till collapse or shakedown. The procedure is compared to experimental results obtained on aluminum beams in shakedown

Experimental and Numerical Evaluation of Residual Displacement and Ductility in Ratcheting and Shakedown of an Aluminum Beam

Safety assessment of structures can be obtained employing limit design to overcome uncertainties concerning actual response due to inelastic constitutive behavior and more generally to non-linear structural response and loads’ random variability. The limit analysis is used for evaluating the safety of the structures, starting directly from load level without any knowledge of the load history. In the paper, the lower bound calculation is proposed where a new strain-based approach is used that allowed describing the residual stress and displacement in terms of permanent strain. The strategy uses the permanent strain as effective parameters of the procedure so that it is possible to assess the ductility requirements for the complete load program developed till collapse or shakedown. The procedure is compared to experimental results obtained on aluminum beams in shakedown