On the Rate of Convergence in the Central Limit Theorem for Linear Statistics of Gaussian, Laguerre, and Jacobi Ensembles

Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an estimate for the characteristic functions of the linear statistics; this estimate is uniform over the growing interval. The proof of the lemma relies on the Riemann--Hilbert approach.Comment: 45 pages, 5 figures. Final version. Cleared up exposition, added new section "Outline of proof and discussion", fixed minor typo

Functional Limit Theorems for Constrained Mittag-Leffler Ensemble in Hard Edge Scaling

We consider the hard-edge scaling of the Mittag-Leffler ensemble confined to a fixed disk inside the droplet. Our primary emphasis is on fluctuations of rotationally-invariant additive statistics that depend on the radius and thus give rise to radius-dependent stochastic processes. For the statistics originating from bounded measurable functions, we establish a central limit theorem in the appropriate functional space. By assuming further regularity, we are able to extend the result to a vector functional limit theorem that additionally includes the first hitting "time" of the radius-dependent statistic. The proof of the first theorem involves an approximation by exponential random variables alongside a coupling technique. The proof of the second result rests heavily on Skorohod's almost sure representation theorem and builds upon a result of Galen Shorack (1973).Comment: 26 page

Planar Orthogonal Polynomials As Type I Multiple Orthogonal Polynomials

A recent result of S.-Y. Lee and M. Yang states that the planar orthogonal polynomials orthogonal with respect to a modified Gaussian measure are also multiple orthogonal polynomials of type II on a contour in the complex plane. We show that the same polynomials are also type I orthogonal polynomials on a contour, provided the exponents in the weight are integer. From this orthogonality, we derive several equivalent Riemann-Hilbert problems. The proof is based on the fundamental identity of Lee and Yang, which we establish using a new technique.Comment: 20 pages, 1 figur

Antenna Array Pattern Synthesis via Coordinate Descent Method

This paper presents an array pattern synthesis algorithm for arbitrary arrays based on coordinate descent method (CDM). With this algorithm, the complex element weights are found to minimize a weighted L2 norm of the difference between desired and achieved pattern. Compared with traditional optimization techniques, CDM is easy to implement and efficient to reach the optimum solutions. Main advantage is the flexibility. CDM is suitable for linear and planar array with arbitrary array elements on arbitrary positions. With this method, we can configure arbitrary beam pattern, which gives it the ability to solve variety of beam forming problem, e.g. focused beam, shaped beam, nulls at arbitrary direction and with arbitrary beam width. CDM is applicable for phase-only and amplitude-only arrays as well, and furthermore, it is a suitable method to treat the problem of array with element failures

Geochemistry of beryl varieties: comparative analysis and visualization of analytical data by principal component analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE)

A study of the trace element composition of beryl varieties (469 SIMS analyses) was carried out. Red beryls are distinguished by a higher content of Ni, Sc, Mn, Fe, Ti, Cs, Rb, K, and B and lower content of Na and water. Pink beryls are characterized by a higher content of Cs, Rb, Na, Li, Cl, and water with lower content of Mg and Fe. Green beryls are defined by the increased content of Cr, V, Mg, Na, and water with reduced Cs. A feature of yellow beryls is the reduced content of Mg, Cs, Rb, K, Na, Li, and Cl. Beryls of various shades of blue and dark blue (aquamarines) are characterized by higher Fe content and lower Cs and Rb content. For white beryls, increased content of Na and Li has been established. Principal Component Analysis (PCA) for the CLR-transformed dataset showed that the first component separates green beryls from other varieties. The second component divides pink and red beryls. The stochastic neighborhood embedding method with t-distribution (t-SNE) with CLR-transformed data demonstrated the contrasting compositions of green beryls relative to other varieties. Red and pink beryls form the most compact clusters

The robustness of interdependent clustered networks

It was recently found that cascading failures can cause the abrupt breakdown of a system of interdependent networks. Using the percolation method developed for single clustered networks by Newman [Phys. Rev. Lett. {\bf 103}, 058701 (2009)], we develop an analytical method for studying how clustering within the networks of a system of interdependent networks affects the system's robustness. We find that clustering significantly increases the vulnerability of the system, which is represented by the increased value of the percolation threshold $p_c$ in interdependent networks.Comment: 6 pages, 6 figure