Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

Über den Satz von Poincaré in der Störungstheorie Briot-Bouquetscher Differentialgleichungen

The well known theorem of H. Poincaré concerning the analytic continuation and representation along a curve of solutions of differential equations depending on a parameter is generalized to solutions of Differential Equations of Briot-Bouquet type depending on a parameter.

---

---

Herrn Professor Dr. N. Hofreiter zum 70. Geburtstag     

Four-wave mixing in semiconductor laser amplifiers: Applications for optical communication systems

Four-wave mixing in semiconductor laser amplifiers enables various applications in photonic switching including wavelength conversion, space and time switching, optical phase conjugation, and demultiplexing. Switching times faster than 1 ps are possible.     

A framework for the automated real-time detection of short tonal sounds from ocean observatories

Acquisition of acoustic data from ocean observatories is expected to play a key role for the long-term monitoring of marine mammals and anthropogenic noise. It typically requires processing of a large volume of acoustic data and it must rely on automated identification of signals. We present an algorithmic framework for the detection of short tonal sounds (e.g. cetacean calls, anthropogenic pings) intended to act as a first stage in a system for the automated real-time detection, classification, and localisation of acoustic sources. The algorithm was validated under a diversity of scenarios expected at ocean observatories. Using simulated signals that emulate a variety of cetacean call-types, perfect identification of signal position was obtained for signal to noise ratios of ?15 to ?5 dB, depending on the signal-type. Separation of real-world data segments with short tonal sounds (mainly cetacean calls) from segments with other sounds or noise resulted in Area Under the ROC Curve values between 0.96 and 0.98. The algorithm can be used to automatically identify cetacean calls and anthropogenic short tonal sounds much faster than in real-time, thereby reducing the burden put on data transmission, storage, or processing by classification and localisation algorithms.     

Stochastic optimization for power system configuration with renewable energy in remote areas

This paper presents a stochastic mixed integer programming model for a comprehensive hybrid power system design problem, including renewable energy generation, storage device, transmission network, and thermal generators, for remote areas. Given the complexity of the model, we developed a Benders’ decomposition algorithm with two additional types of cutting planes: Pareto-optimal cuts generated using a modified Magnanti-Wong method and cuts generated from a maximum feasible subsystem. Computational results show significant improvement in our ability to solve this type of problem in comparison to a state-of-the-art professional solver. This model and the solution algorithm provide an analytical decision support tool for the hybrid power system design problem.     

Associative formal power series in two indeterminates

Investigating the associativity equation for formal power series in two variables we show that the transcendental associative formal power series are of order one or two and that they can be represented by an invertible formal power series in one variable. We also discuss the convergence of associative formal power series.     

Synthesis,Crystal Structure,and Selected Properties of [Au(S2CNH2)2]SCN: A Precursor for Gold Macro-Needles Consisting of Gold Nanoparticles Glued by Graphitic Carbon Nitride

A new preparation route is developed for the synthesis of needle-like crystals of [Au(S₂CNH₂)₂]SCN, which avoids disproportionation of the Au^I salt used as a starting material. In the crystal structure, the two crystallographically independent Au^III centers are in a square-planar environment of two S₂CNH₂ ligands. The Hirshfeld surface analysis reveals the presence of noncovalent intermolecular S⋅⋅⋅S interactions, which are essential for the spatial arrangement of the molecules. Density functional theory (DFT) calculations including dispersion and damping corrections result in a unit cell volume very close to the value determined experimentally. Thermal decomposition in an inert atmosphere generates black needles with lengths of up to 500 μm. X-ray powder diffraction and pair distribution function analyses demonstrate that the needles are composed of nanosized crystals with a volume-weighted average domain size of 20(1) nm. According to results of X-ray photoemission experiments, the black needles are covered by a nitrogen-rich carbon nitride with composition near (CN)₂N. ¹³C solid-state NMR investigations indicate that two different carbon species are present, with signals corresponding well to heptazine units as in melon and triazine units as in poly(triazin imide) type compounds. Scanning transmission electron microscopy tomography evidences that the needles are composed of slightly elongated nanoparticles.     

Investigation of hard magnetic properties in the Fe-Pt system by combinatorial deposition of thin film multilayer libraries

The magnetic properties of annealed Fe-Pt multilayer thin films with a broad composition range were investigated in order to identify the effects of composition and annealing temperature on the achievable coercive field, and to identify its maximum at low processing temperatures. Two types of multilayer systems were deposited as materials libraries to vary the composition from Fe₂₀Pt₈₀ to Fe₇₅Pt₂₅. The first type of multilayer was comprised of alternating opposing wedges, whereas the second type consisted of repeated uniform Fe and Pt layers interspersed periodically with Fe wedge layers. It was found that coercive fields μ₀H_C > 0.7 T can be achieved at an annealing temperature of about 300 °C (60 min) for both types of multilayers as long as the composition is close to 50:50. Higher annealing temperatures are needed for films, which deviate from this composition. Increasing the annealing temperature up to 700 °C leads to increased coercivity values. Multilayers with additional Fe layers showed increased remanence but reduced coercive fields.