Hard Edge Research Articles

ОСОБЕННОСТИ ОБРАБОТКИ ДЕТАЛЕЙ НЕФТЕГАЗОВОЙ ОТРАСЛИ, ВЫПОЛНЕННЫХ ИЗ ВОЛЬФРАМОВЫХ СПЛАВОВ

The purpose of the work carried out is to increase the effectiveness of hard alloy machining. Within the limits of the paper there is solved a matter of electro-erosion treatment and turning TC 3 alloy parts. Basic investigation methods are: - experiment and comparison, - computer simulation The work resulted in recommendations for the choice of cutters and cutting modes for hard alloy edge machining and also electrode material at electro-erosion treatment. One of the ways to increase productivity and operation quality equipment consists in the application of new materials and alloys including tungsten alloys. Within the limits of this work there are considered the peculiarities in machining TC3 alloy parts for oil-gas industry by the example of the “sleeve” part. For machining inner and outer surfaces there was used a turning operation, and for complex grooves – electro-erosion treatment as the only possible choice for processing because of a small size of the part. The peculiarities hard alloy machining through turning and basic reasons of hard alloy hard-to-workability are described. There are shown approaches to the choice of cutters, and the dependence of surface roughness and tool durability upon cutting mode choice is shown. During turning plate material must have higher strength than alloy under machining. One of the best choices for hard alloy machining is cubic boron nitride. For processing it is necessary to use tool holders with the increased rigidity. There are presented principles of electrode designing for hard alloy electro-erosion treatment and ways for process productivity increase. The application of copper-tungsten pseudo-alloys and m1 grade copper is compared from the point of view of economic purposefulness in their use. In spite of copper-tungsten alloy high durability which gives advantages in processing rate and economy in manufacturing electrodes its use is economically inexpedient and it is more rational to use copper which is cheaper.

Universality of local spectral statistics of products of random matrices

We derive exact analytical expressions for correlation functions of singular values of the product of M Ginibre matrices of size N in the double scaling limit M,N→∞. The singular value statistics is described by a determinantal point process with a kernel that interpolates between Gaussian unitary ensemble statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit N→∞, the interpolation parameter is given by the limiting quotient a=N/M. One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk, and at the soft edge for any a. We find that in addition to the standard scaling regimes, there is a transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant positions, with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.

Operator level hard-to-soft transition for β-ensembles

The soft and hard edge scaling limits of β-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators [12, 15]. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit [3, 12, 14]. We prove that this limit can be realized on the level of the corresponding random operators. More precisely, the random operators can be coupled in a way so that the scaled versions of the hard edge operators converge to the soft edge operator a.s. in the norm resolvent sense.

On boundary confinements for the Coulomb gas

We introduce a family of boundary confinements for Coulomb gas ensembles, and study them in the two-dimensional determinantal case of random normal matrices. The family interpolates between the free boundary and hard edge cases, which have been well studied in various random matrix theories. The confinement can also be relaxed beyond the free boundary to produce ensembles with more fuzzy boundaries, i.e., where the particles are more and more likely to be found outside of the boundary. The resulting ensembles are investigated with respect to scaling limits and distribution of the maximum modulus. In particular, we prove existence of a new point field - a limit of scaling limits to the ultraweak point when the droplet ceases to be well defined.

Hard-edge asymptotics of the Jacobi growth process

Nous introduisons une famille a deux parametres $(\alpha,\beta>-1)$ de systemes de particules en interaction, avec des noyaux de correlation determinantaux qui s’expriment en termes des polynomes de Jacobi $\{P^{(\alpha,\beta)}_{k}\}_{k\geq 0}$. Cette famille comprend les mesures de Plancherel pour les groupes orthogonal et symplectique de dimension finie, qui avaient ete decouvertes auparavant. La construction utilise certains polynomes orthogonaux multivaries de type BC qui generalisent les caracteres de ces derniers groupes. Les asymptotiques locales pres du bord dur, ou l’on attend un comportement caracteristique, font intervenir le noyau de Jacobi discret multi-temps dependant de $(\alpha,\beta)$, et le noyau de Pearcey du bord dur multi-temps dependant de $(\alpha,\beta)$. Le noyau de Pearcey du bord dur etait apparu precedemment dans les asymptotiques au bord dur de processus de carres de Bessel s’evitant mutuellement.

Edge Scaling Limit of the Spectral Radius for Random Normal Matrix Ensembles at Hard Edge

We investigate a random normal matrix model with eigenvalues forced to be in the droplet, the support of the equilibrium measure associated with an external field. For radially symmetric external fields, we show that the fluctuations of the spectral radius around a hard edge tend to follow an exponential distribution as the number of eigenvalues tends to infinity. As a corollary, we obtain the order statistics of the moduli of eigenvalues.

Periodic Table of the Ordinary and Supersymmetric Sachdev-Ye-Kitaev Models.

We develop a unified minimal scheme to classify quantum chaos in the Sachdev-Ye-Kitaev (SYK) and supersymmetric (SUSY) SYK models and also work out the structure of the energy levels in one periodic table. The SYK with even q-body or SUSY SYK with odd q-body interaction, with N even or odd number of sites, are put on an equal footing in the minimal Hilbert space; N (mod 8), q (mod 4) double Bott periodicity, and a reflection condition are identified. Exact diagonalizations (EDs) are performed to study both the bulk energy level statistics and hard-edge behaviors. Excellent agreements between the ED results and the symmetry classifications are demonstrated. Our compact and systematic methods can be transformed to map out more complicated periodic tables of SYK models with more degrees of freedom, tensor models, or symmetry protected topological phases.

Art Centers and Peripheral Art [A Lecture at the University of Hamburg, October 15, 1982

Change in the history of art has many causes, but one often overlooked by art historical institutions is the complex, unequal set of relationships that subsist between art centers and peripheries. These take many forms, from powerful penetration of peripheral art by the subjects, styles and modes of the relevant center, through accommodation to this penetration to various degrees and kinds of resistance to it. Mapping these relationships should be a major task for art historians, especially those committed to tracing the reception of works of art and the dissemination of ideas about art. This lecture, delivered by Nicos Hadjinicolaou in 1982, outlines a “political art geography” approach to these challenges, and demonstrates it by exploring four settings: the commissioning of paintings commemorating key battles during the Greek War of Independence; the changes in Diego Rivera's style on his return to Mexico from Paris in the 1920s; the impact on certain Mexican artists in the 1960s of “hard edge” painting from the United States; and the differences between Socialist Realism in Moscow and in the Soviet Republics of Asia during the mid-twentieth century. The lecture is here translated into English for the first time and is introduced by Terry Smith, who relates it to its author's long-term art historical quest, as previously pursued in his book Art History and Class Struggle (1973).

An Introduction to Nicos Hadjinicolaou's “Art Centers and Peripheral Art” (1982)

Change in the history of art has many causes, but one often overlooked by art historical institutions is the complex, unequal set of relationships that subsist between art centers and peripheries. These take many forms, from powerful penetration of peripheral art by the subjects, styles and modes of the relevant center, through accommodation to this penetration to various degrees and kinds of resistance to it. Mapping these relationships should be a major task for art historians, especially those committed to tracing the reception of works of art and the dissemination of ideas about art. This lecture, delivered by Nicos Hadjinicolaou in 1982, outlines a “political art geography” approach to these challenges, and demonstrates it by exploring four settings: the commissioning of paintings commemorating key battles during the Greek War of Independence; the changes in Diego Rivera's style on his return to Mexico from Paris in the 1920s; the impact on certain Mexican artists in the 1960s of “hard edge” painting from the United States; and the differences between Socialist Realism in Moscow and in the Soviet Republics of Asia during the mid-twentieth century. The lecture is here translated into English for the first time and is introduced by Terry Smith, who relates it to its author's long-term art historical quest, as previously pursued in his book Art History and Class Struggle (1973).

Narain transform for spectral deformations of random matrix models

We start from applying the general idea of spectral projection (suggested by Olshanski and Borodin and advocated by Tao) to the complex Wishart model. Combining the ideas of spectral projection with the insights from quantum mechanics, we derive in an effortless way all spectral properties of the complex Wishart model: first, the Marchenko-Pastur distribution interpreted as a Bohr-Sommerfeld quantization condition for the hydrogen atom; second, hard (Bessel), soft (Airy) and bulk (sine) microscopic kernels from properly rescaled radial Schrödinger equation for the hydrogen atom. Then, generalizing the ideas based on Schrödinger equation to the case when Hamiltonian is non-Hermitian, we propose an analogous construction for spectral projections of universal kernels for bi-orthogonal ensembles. In particular, we demonstrate that the Narain transform is a natural extension of the Hankel transform for the products of Wishart matrices, yielding an explicit form of the universal kernel at the hard edge. We also show how the change of variables of the rescaled kernel allows us to make the link to the universal kernel of the Muttalib-Borodin ensemble. The proposed construction offers a simple alternative to standard methods of derivation of microscopic kernels. Finally, we speculate, that a suitable extension of the Bochner theorem for Sturm-Liouville operators may provide an additional insight into the classification of microscopic universality classes in random matrix theory.

Modeling the fringe field of permanent magnet multipoles using numerical simulations.

Permanent magnet multipoles (PMMs) are widely used in accelerators to either focus particle beams or confine plasma in ion sources. The real magnetic field created by PMMs is calculated by magnetic field simulation software and then used in particle tracking codes by means of a three dimensional magnetic field map. A common alternative is to use the so-called "hard-edge" model, which gives an approximation of the magnetic field inside the PMM assuming a null fringe field. This work proposes an investigation of the PMM fringe field properties. An analytical model of the PMM magnetic field is developed using the Fourier multipole expansion. A general axial potential function with a unique parameter λ, able to reproduce the actual PMM magnetic field (including its two fringe fields) with an explicit dependence on the PMM length, is proposed. An analytical first order model including the axial fringe field is derived. This simple model complies with the Maxwell equations [curl(B) = 0 and div(B) = 0] and can replace advantageously the "hard-edge" model when fast analytical calculations are required. The higher order analytical multiple expansion model quality is assessed by means of χ2 estimators. The general dependence of the potential function parameter λ is given as a function of the PMM geometry for quadrupole, hexapole, and multipole, allowing one to use the developed model in simulation programs where the multipole geometry is an input parameter.

Extended transfer matrix through aligned solenoidal fields using the traditional formalism

The traditional transfer matrix used in particle accelerators has been modified to easily take into account a non-zero angular canonical momentum, without the hard edge approximation of the field. To this end, we reformulate the transfer matrix based on an existing method, but by using both the explicit real transformation matrix in the Larmor frame of the 4D transverse phase space, and the real 4D transfer matrix of a solenoidal field in this frame and in the laboratory frame. This reformulation allows to clearly identify differences with the traditional matrix when performing analytic inter-comparisons. The implementation is also easier to perform. A comparison with computer simulation further illustrates its usefulness.

Scaling limits of random normal matrix processes at singular boundary points

We introduce a method for taking microscopic limits of normal matrix ensembles and apply it to study the behaviour near certain types of singular points on the boundary of the droplet. Our investigation includes ensembles without restrictions near the boundary, as well as hard edge ensembles, where the eigenvalues are confined to the droplet. We establish in both cases existence of new types of determinantal point fields, which differ from those which can appear at a regular boundary point, or in the bulk.

A power law study of the edge influence on the perceived filling-in brightness magnitude

BackgroundEdge plays a special role in spatial perception and as well as in determining the brightness of a surface within borders. The aim of our study was to measure threshold brightness in different levels of edges thickness.MethodsSteven’s power law for circles modulating in luminance was estimated for 30 subjects (mean age 24 years, SD 3.3, 13 female). Stimuli were presented on the iMac display using the 11-bit graphic board and consisted of two circles of 3° of visual angle, separated by 10°. We tested 7 levels of Michelson contrast: 7, 8, 10, 15, 26, 50, and 100. Three edges filtering were tested (0.3, 0.8, and 1.5° of smoothing). The subjects’ task was to judge the brightness of the edge filtered circle compared with the circle of the hard edge which was considered the modulus and received an arbitrary level of 50, representing the amount of brightness perception. In each trial, the same contrast level was presented in both circles. Five judgments were performed for each contrast level in edge filtering.ResultsWe found an increase in the power law exponent as the increase of the edge filtering (for sigma of 0.3 = 0.43, sigma of 0.8 = 0.73, and sigma 1.5 = 0.97). All power function fitting had high correlation coefficients (r2 = .94, r2 = .95, r2 = .97, respectively to sigma 0.3, 0.8, and 1.5) passing to the model’s adhesion criteria.ConclusionsThere was a progressive distortion on the figure brightness perception as increasing the edge filtering suggesting the control of edges on the polarity of the overall brightness. Also, perceived brightness was increasingly veridical with increased filtering, approaching 1:1 correspondence at 1.5 sigmas.

Finite‐size corrections at the hard edge for the Laguerre β ensemble

AbstractA fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre β ensemble, characterized by the Dyson parameter β, and the Laguerre weight , in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable . Previous work has established the corresponding functional form of various statistical quantities—for example, the distribution of the smallest eigenvalue, provided that . We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling , the rate of convergence to the limiting distribution is , which is optimal. In the case , general the explicit functional form of the distribution of the smallest eigenvalue at this order can be computed, as it can for and general . An iterative scheme is presented to numerically approximate the functional form for general .

The generating function for the Bessel point process and a system of coupled Painlevé V equations

We study the joint probability generating function for [Formula: see text] occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. This generalizes a result of Tracy and Widom [C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Commun. Math. Phys. 161(2) (1994) 289–309], which corresponds to the case [Formula: see text]. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large [Formula: see text] asymptotics for [Formula: see text] Hankel determinants with a Laguerre weight possessing several jump discontinuities near the hard edge.

Professional Competencies – the soft skills to give Systems Engineers a hard edge

AbstractThe publication of the new INCOSE Systems Engineering Competency Framework is notable for many reasons, not least the inclusion of Professional Competencies which is the focus of this paper.This paper reviews the general importance of the professional competencies (also known as “soft skills”), and the specific relevance of the individual competencies that have been included in the INCOSE framework. This provides a review and discussion of the importance of the Professional Competencies to enable those that do Systems Engineering to be successful. Systems Engineering is both done by people and integrates people: to do that the soft skills or Professional Competencies are essential.

Noninteracting fermions in a trap and random matrix theory

We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions , the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension d = 1 and at zero temperature T = 0, these universality classes are in one-to-one correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the ‘Airy kernel’) and hard edge (described by the ‘Bessel kernel’) universality classes. We further discuss extensions of these results to higher dimensions and to finite temperature. Finally, we discuss correlations in the phase space, i.e. in the space of positions and momenta, characterized by the so called Wigner function.

Fox H-kernel and θ-deformation of the Cauchy Two-Matrix Model and Bures Ensemble

Abstract A $\theta $-deformation of the Laguerre weighted Cauchy two-matrix model, and the Bures ensemble, is introduced. Such a deformation is familiar from the Muttalib–Borodin ensemble. The $\theta $-deformed Cauchy–Laguerre two-matrix model is a two-component determinantal point process. It is shown that the correlation kernel, and its hard edge scaled limit, can be written in terms of particular Fox H-functions, generalising the Meijer G-function class known from the study of the case $\theta = 1$. In the $\theta =1$ case, it is shown that the Laguerre–Bures ensemble is related to the Laguerre–Cauchy two-matrix model, notwithstanding the Bures ensemble corresponding to a Pfaffian point process. This carries over to the $\theta $-deformed case, allowing explicit expressions involving Fox H-functions for the correlation kernel, and its hard edge scaling limit, to also be obtained for the $\theta $-deformed Laguerre–Bures ensemble. The hard edge scaling limit is in the Raney class $(\theta /2+1,1/2)$.

Improvement to beam quality of optically pumped terahertz gas lasers with hole-coupling resonators

The hole-coupling method, which is commonly used in the optically pumped THz gas laser, will reduce the beam quality due to the hard-edge diffraction. It is shown that by optimizing the parameters of the resonator a diffraction-limited output beam can be obtained. A method is proposed for the selection of a higher order waveguide mode, whose first zero point along the radial direction coincides with the edge of the output hole coupler and the effect of hard edge diffraction is depressed. Measurements of beam quality are taken on an experimental prototype whose optical setup is designed by this method. A quasi-Gaussian beam output is obtained, and the beam quality M2 factor is about 1.15