A study of finite gap solutions to the nonlinear Schrödinger equation

The vector nonlinear Schrödinger equation is an envelope equation which models the propagation of ultra-short light pulses and continuous-wave beams along optical fibres. Previous work has focused almost entirely on soliton solutions to the equation using a Lax representation originally developed by Manakov. We prove recursion formulae for the family of higher-order nonlinear Schrödinger equations, along with its associated Lax hierarchy, before investigating finite gap solutions using an algebrogeometric approach which introduces Baker-Akhiezer functions defined upon the Riemann surface of the relevant spectral curve. We extend this approach to account for solutions of arbitrary genus and compare it with an alternative method describing solutions of genus two. The scalar nonlinear Schrödinger and Heisenberg ferromagnet equations were shown to be equivalent following work by Lakshmanan; we generalise this idea by introducing the Heisenberg ferromagnet hierarchy and show it is entirely gauge equivalent to the scalar nonlinear Schrödinger hierarchy in the attractive case. We also investigate the polarisation state evolution of general solutions to the vector nonlinear Schrödinger equation and study possible degenerations to the Heisenberg ferromagnet equation

An Energy Bound in the Affine Group

We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set P for which no line contains a positive proportion of points from P, there may be at most one line, meeting the set of lines defined by P in at most a constant multiple of |P| points.Comment: 16 pages, 1 figur

Importance of dynamics in small scale mechanical testing: Fast constant strain rate and ballistic testing

Recent advances in electronics have enabled nanomechanical measurements with very low noise levels at fast time constants and high data acquisition rates. Nanomechanical testers with sub-nanometer noise levels at a displacement time constant of 20µs are currently available which open the doors for a wide range of ultra-fast nanomechanical testing. It not only enables accurate measurement of materials’ response during a dynamic event such as strain burst (pop-in) during micropillar or indentation testing, but also enables fast indentation tests. However, fast testing requires a precise understanding of the instrument’s dynamic response along with the time constants of the measurement signals. One of the novel experiments that can be performed using a system with fast time constants is the constant high strain rate indentation test. This enables generating four dimensional (4D) mechanical property maps, wherein the hardness and modulus are measured as a function of depth and the spatial location. Results from such high speed 4D mapping of a silica grating on silicon substrate with each indentation taking less than 3.5 seconds will be presented as shown in the figure below. This is a very powerful tool for mapping the mechanical properties of integrated circuits including sub-surface features. Another area of small scale testing that greatly benefits from current advancements in nanomechanical testing instrumentation is the rate dependence of strength. One of the major limitations of the current methods to measure the strain rate sensitivity by indentation over any significant range of strain rates is the lack of access to high strain rates. With an accurate dynamic model and an instrument with fast time constants, step load tests can be performed which enable access to indentation strain rates (ḣ / h) approaching ballistic levels (\u3e 1000 s-1). Results from high strain rate testing on aluminum and tin will be presented and compared to the results from Split Hopkinson bar experiments for fcc metals [1

Propagation of Pike in Multi-Purpose Lake Management

Methods for rearing, harvesting, and management use of pike (Esox lucius L) in Minnesota are described, especially as related to multiple use of shallow water areas. Principal emphasis is placed on the method of removing fingerling and yearling pike in winter from shallow lakes. Such lakes often raise wild rice and also are utilized by waterfowl. The pike, spawned naturally in these lakes, are harvested by pumping aerated water through fish traps when oxygen levels in the lake become low. Pumping is most effective when oxygen levels in the lake fall below 2 parts per million but is higher in the trap area

A direct comparison of high temperature nanoindentation and tensile creep measurements for aluminum

Scanning electron microscope (SEM) in situ high temperature nanomechanical testing has received a lot of interest in recent times. Performing nanomechanical tests in non-ambient conditions presents several challenges. To meet those challenges a thorough understanding of the instruments’ performance both from static as well as dynamic point of view is required. In this regard, data from dynamic nanoindentation testing at temperatures up to 550 C on commercial purity aluminum in the SEM will be presented. The activation energy for creep was found to be 140 KJ/mol/K, matching the value determined with high temperature tensile creep experiments extremely well. The stress exponent for creep at the highest temperatures is determined to be approximately six. This result compares well to the value determined with tensile creep experiments [1]. In addition, the work of Su et.al. [2] can be used for predicting the pre-exponential term and the pile-up/sink-in factor that is appropriate for the stress exponents measured. Thus the indentation results and the tensile results can be directly compared. This comparison is show in the figure bellow. Over at least 15 orders of magnitude, the results are very close. At the lowest temperatures the stresses are under estimated. This is most likely due to the relatively small strains available with the indentation results when compared to the very large strain torsion experiments that were performed to accurately measure steady state creep stresses at low temperatures [1]. New testing and analysis strategies for improving the reliability of the results from high temperature testing will be presented. These results demonstrate that existing models can be used to facilitate accurate comparisons of high temperature tensile creep tests with indentation tests done with a Berkovich indenter over more than ten orders of magnitude in strain rate

Exploring accurate structure, composition and mechanical properties of η carbides in high tungsten iron-based alloy: High-throughput mapping and DFT calculations

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Measurement of hardness and elastic modulus by depth sensing indentation: Further advances in understanding and refinements in methodology

Depth sensing indentation technique has been widely used to measure small scale mechanical properties over the years. Starting from the seminal work of Oliver & Pharr [1], there have been many improvements / modifications to the test methodology and also significant advances in measurement electronics / testing instrumentation. These advancements provide opportunities to not only develop novel testing capabilities but also further improve the precision and accuracy of the most common measurement parameters – hardness and elastic modulus. In this regard, this work presents a comprehensive study on the various steps involved in a typical depth sensing indentation test, viz., surface approach, surface detection, load-time history including superimposing an oscillatory force on broad band load, unloading and drift rate measurement. The effect of each of these steps on the accuracy and precision of the hardness and elastic modulus measurement will be discussed with specific focus on frequency specific testing techniques such as continuous stiffness measurement. The effect of the instrument’s measurement time constants and dynamic parameters such as mass, spring constant and damping coefficient during different steps of an indentation test and thereby on the hardness and elastic modulus will be presented. A simple model is developed to simulate a depth sensing indentation test that incorporates the material and instrumentation parameters to help visualize the overall process and provide new insights for pushing the limits of the currently available instrumentation for improved precision and accuracy. This involves performing tests beyond the traditional boundaries of parameter space such as increased oscillation amplitude, strain rate, oscillation frequency, etc. For instance, if the indentation strain rate gets high compared to the oscillation frequency, inaccuracies can occur. This work presents the critical experimental parameters and the associated first order corrections for the potential errors. The model predictions and corrections are validated on different classes of materials. Finally, guidelines for measuring hardness and elastic modulus using a depth sensing indentation test with significantly improved precision and accuracy within the limitations of the currently available instrumentation will be discussed. [1] Oliver & Pharr, Journal of Materials Research, 7(6),1992