Crystallization of random matrix orbits

Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta=1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices can be extrapolated to general values of $\beta>0$ through associated special functions. We show that $\beta\to\infty$ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta$ self-adjoint matrix with fixed eigenvalues is known as $\beta$-corners process. We show that as $\beta\to\infty$ these eigenvalues crystallize on the irregular lattice of all the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field (dGFF) put on top of this lattice, which provides a new explanation of why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.Comment: 25 pages. v2: misprints corrected, to appear in IMR

Cutting corners

We define a class of subshifts defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. For such a subshift, a locally legal pattern of convex shape is globally legal, and there is a measure that samples uniformly on convex sets. We show by example that these subshifts need not admit a group structure by shift-commuting continuous operations. Our approach to convexity is axiomatic, and only requires an abstract convex geometry that is “midpointed with respect to the shape”. We construct such convex geometries on several groups, in particular strongly polycyclic groups and free groups. We also show some other methods for sampling finite patterns, and show a link to conjectures of Gottshalk and Kaplansky.</p

Two-Loop ϕ4\phi^4-Diagrams from String Theory

Using the {\em cutting and sewing} procedure we show how to get Feynman diagrams, up to two-loop order, of $\Phi^{4}$-theory with an internal SU(N) symmetry group, starting from tachyon amplitudes of the open bosonic string theory. In a properly defined field theory limit, we easily identify the corners of the string moduli space reproducing the correctly normalized field theory amplitudes expressed in the Schwinger parametrization.Comment: 28 pages, 12 figure

Геометрия спирального сверла в различных системах координат

В работе рассмотрена геометрия режущей части спиральных сверл в разных системах координат. Установлен характер изменения этих углов на режущей кромке.In the article geometry of cutting part of spiral drills is considered in the different systems of coordinates. Character of change of these corners is determined on a cutting edge

Demand for On-Farm Permanent Hired Labor in Family Holdings: A Comment

This comment critically discusses the theoretical and empirical treatment of corner solutions in the analysis of labor decisions on farm households. As more and more labor decisions are analyzed jointly, the more ambitious becomes the theoretical justification of empirical applications. "Cutting corners" in theoretical models puts the validity of empirical conclusions in doubt. In such cases relying on intuitive theoretical justification of empirical modeling is preferred.Community/Rural/Urban Development,

An efficient NC tool path planning approach

One major problem encountered in NC machining is that cutting load fluctuates in twisty tool paths and increases momentarily when milling concave corners where stock material concentrates. Much research has been conducted on controlling milling forces but the problem remains unsolved. This paper presents a solution to reduce and stabilise cutting load in NC milling operation by applying computational techniques to isolate the concave corners of a machining region, thereby forming a major region and a number of disjoint sub-regions. Due to the suppression of concave corners, the boundary profile of the major region will become smoother. Contour-parallel offset tool paths can then be used to remove the material in the major region. For the isolated sub-regions, special corner removing tool paths with the use of different cutter sizes will be employed. By using this approach, the adverse effects caused by excessive undulation of milling tool path can be reduced.published_or_final_versio

Micro milling performance assessment of diamond-like carbon coatings on a micro-end mill

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ 2013 Institution of Mechanical Engineers.In micro milling, unpredictable tool life and premature tool failures are the major constraints for its industrial applications, and prolongation of the tool life so as to enhance the tooling performance presents great challenges. Appropriate coating techniques potentially offer a feasible and promising solution. In this study, diamond-like carbon films are deposited on a Ø500 µm diameter tungsten carbide (WC) micro-end mill by the plasma-enhanced chemical vapour deposition process. Coating characterisation has been undertaken and the diamond-like carbon coatings are found in good coverage on the tool except for a slight delaminating on the edge corners. Besides, the surface of the amorphous coatings is much smoother than that of WC. In addition, comprehensive cutting performance of the diamond-like carbon coated tool in dry slot milling of Al 6061-T6 has been compared with those of uncoated tools in both dry and wet conditions. It is observed that the use of diamond-like carbon coatings can reduce the cutting forces, lessen the tool wear, improves the surface roughness and minimise the micro-burr formation as compared to the corresponding performance of an uncoated tool in dry cutting. However, the performance improvement is still unreachable to those resulting from the cutting fluid influence.UK Technology Strategy Board and Kistler UK Ltd

Определение кинематических задних углов при нарезании внутренних резьб головками

Formulas for definition of kinematic back corners on cutting and calibrating parts of combs are received depending on design data of the head and parameters of sharpening of combs. Influence of these factors on back corners is analysed. The recommendations are resulted, allowing to raise {increase} quality of grooves.Получены формулы для определения кинематических задних углов на режущей и калибрующей частях гребенок в зависимости от конструктивных параметров головки и параметров заточки гребенок. Проанализировано влияние этих факторов на задние углы. Приведены рекомендации, позволяющие повысить качество резьб