On the Validations of the Asymptotic Matching Conjectures

In this paper we review the asymptotic matching conjectures for r-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in d-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of r-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density p, for any p∈[0,1], for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for certain infinite r-regular bipartite graphs.     

Phase coherence of the electrons in δ-doped gaAs

The phase coherence length of the electrons is determined by magnetoconductivity for a wide range of sheet concentrations of Si-donors and free electrons. For diffusive carrier motion a comparison of the experimental and theoretical results for weak fields shows that the weak localisation is limited by inelastic electron-electron-scattering as expected for disturbed systems. For higher magnetic fields the role of mutual electron-electron interaction is also significant.     

On an inverse problem for nonnegative and eventually nonnegative matrices

Let σ= (λ₁,···λ_n)⊂C. We discuss conditions for which σ is the spectrum of a nonnegative or eventually nonnegative matrix. This brings us to study rational functions with nonnegative Maclaurin coefficients. A conjecture for special sets σ is stated and some evidence in support of this conjecture is given. Dedicated to my teacher Professor Menachem Schiffer on the occasion of his sixty-fifth birthday.     

A generalization of the Motzkin-Taussky theorem

In this paper we generalize the Motzkin-Taussky theorem to matrices with polynomial entries.     

Inverse eigenvalue problems

The classical inverse additive and multiplicative inverse eigenvalue problems for matrices are studied. Using general results on the solvability of polynomial systems it is shown that in the complex case these problems are always solvable by a finite number of solutions. In case of real symmetric matrices the inverse problems are reformulated to have a real solution. An algorithm is given to obtain this solution.     

Die Reaktion7Li (d, α)5He im Energiegebiet von 0,6 bis 2,0 MeV

The excitation function of the reaction⁷Li(d, α)⁵He has been measured between 0.6 and 2.0 MeV in steps of 50 keV. The two resonances at 0.75 and 1.00 MeV corresponding to the levels at 17.28 and 17.48 of the compound nucleus⁹Be were resolved. Angular distribution measurements were performed at 14 different energies mainly near the 1.00 MeV resonance. From the analysis of the data the 17.28 MeV level seems to be a positive parity state (3/2⁺ → 7/2⁺) whilst a negative parity (7/2^? or 9/2^?) is indicated for the 17.48 MeV level. This is contrary to the order of parities as given in the literature.     

Small Ball Estimates for Quasi-Norms

This note contains two types of small ball estimates for random vectors in finite-dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain Littlewood–Offord type estimates for quasi-norms. This generalizes results which were previously obtained in Friedland and Sodin (C R Math Acad Sci Paris 345(9):513–518, 2007), and Rudelson and Vershynin (Commun Pure Appl Math 62(12):1707–1739, 2009).     

On pointwise and analytic similarity of matrices

LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈ _p B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈_a B(ε) ifB(ε)=T(ε)A(ε)T ⁻¹(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈ _a B(ε) provided thatA(ε)≈ _p B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the origin and was first stated by Wasow. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024