The complexity of tropical matrix factorization

The tropical arithmetic operations on R are defined by a⊕b=min{a,b}$a\oplus b=\min \{a,b\}$ and a⊗b=a+b$a\otimes b=a+b$. Let A be a tropical matrix and k   a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist tropical matrices B∈R^m×k$B\in {\mathbf{R}}^{m\times k}$ and C∈R^k×n$C\in {\mathbf{R}}^{k\times n}$ satisfying B⊗C=A$B\otimes C=A$. We show that the TMF problem is NP-hard for every k≥7$k\ge 7$ fixed in advance, thus resolving a problem proposed by Barvinok in 1993.     

Inverse bifurcation problems for diffusive Holling–Tanner population model

We consider the equation which is called Holling–Tanner population model where is a bifurcation parameter and are unknown constants. In this paper, we determine the unknown constants from the asymptotic behavior of the bifurcation curve , where .     

Sodium-23 MAS NMR study on nuclear quadrupole interaction in NA(1-x)Ag(x)NO2.

Ag-impurity effects on the first- and second-order quadrupole interaction (QI) at ²³Na site in an isomorphic mixed system, Na_1−xAg_xNO₂ (x=0, 0.0084, 0.026, 0.079, 0.094, 0.16), have been investigated by employing ²³Na (I=3/2) magic angle spinning nuclear magnetic resonance (MAS NMR) technique. The central transition (CT) and satellite transition (ST) are simultaneously observed with this system. From the spectral analysis, the quadrupole parameter and its distribution width are obtained as a function of Ag concentration. From the intensity loss of CT MAS centerband and of the envelope function of ST MAS sidebands due to impurities, the range of their influence on the second- and first-order QI is estimated. The estimated ranges contain the second and first neighbouring Na sites from the resonating ²³Na nucleus for the first- and second-order QI, respectively.     

Categories of partial frames

This article discusses the basic categorical algebra for categories of partial frames. Categories of partial frames are labelled by subset selectors that indicate which joins exist. Constructions for limits, colimits, and free functors connecting various categories of partial frames are given. Examples of partial frame categories are given. Subset selectors which preserve surjections are virtually the same as rules which select all subsets smaller than a given cardinal. Received December 1, 2004; accepted in final form March 28, 2005.     

Determination of highly polar compounds in water samples around former ammunition plants

Summary A method was developed for the determination of nitrobenzoic acids and nitrophenols as well as of diaminoaromatics and was applied to the analysis of water samples from the former ammunition plant at Elsnig (Saxony, Germany). The procedure is based on a preseparation into a neutral, an acidic and a basic fraction by multi-step extraction at different pH values followed by HPLC analysis with UV and electrochemical detectors, coupled in series. Applying optimized enrichment conditions, all investigated compounds were extracted from spiked distilled water with recoveries >80% and variation coefficients <7%. Similar results were obtained with spiked ground water samples. After enrichment, all compounds can be analysed by HPLC with UV detection at concentrations below 100 ngL⁻¹. The electrochemical detector (ELCD) allowed a selective and sensitive detection of the nitrophenols and especially of the diaminoaromatics and, therefore, provides, some advantages in the analysis of real samples. Presented at the 21^st ISC held in Stuttgart, Germany, 15^th–20^th September, 1996     

Higher relative class number formulae

Let E be a totally complex abelian number field with maximal real subfield F, and let denote the non-trivial character of . Similar to the classical case n=1 the value of the Artin L-function at for odd is given by a relative class number formula of the form Here is a higher Q-index, which is equal to 1 or 2 and is a higher relative class number. Here for any number field L the higher class number is the order of the finite group closely related to the order of the higher K-theory group of the ring of integers in L. Received: 4 June 1999 / Revised version: 27 September 2001 / Published online: 26 April 2002     

Numerical Properties of Shifted Tridiagonal LU Factorizations

In this paper, we consider shifted tridiagonal matrices. We prove that the standard algorithm to compute the LU factorization in this situation is mixed forward-backward stable and, therefore, componentwise forward stable. Moreover, we give a formula to compute the corresponding condition number in O(n) flops. This research has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain through grants BFM2003-06335-C03-02 and MTM2006-06671 as well as by the Postdoctoral Fellowship EX2004-0658 provided by Ministerio de Educación y Ciencia of Spain.     

Backward errors for eigenvalue and singular value decompositions

Summary. We present bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two-norm and in the Frobenius norm. Through different orthogonal decompositions of the computed eigenvectors we can define different symmetric backward errors for the eigenvalue decomposition. When the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small. Consequently it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured. Analogous results hold for the singular vectors. We indicate the effect of our error bounds on implementations for eigenvector and singular vector computation. In a more general context we prove that the distance of an appropriately scaled matrix to its orthogonal QR factor is not much larger than its distance to the closest orthogonal matrix. Received July 19, 1993     

Local field enhancements on gold and silver nanostructures for aperture near field spectroscopy

Optical properties of silver and gold semi-continuous films near the percolation threshold are investigated by scanning near field optical microscopy (SNOM) operating in the collection/transmission mode. The electromagnetic enhanced fields, due to plasmon-polariton localizations, are observed and the intensity distribution is shown to present high intensity variations that increase with the wavelength. The local fields are also demonstrated to be polarization dependent. The intensity distribution is investigated as function of the incident wavelength both for gold and silver films. The calculated probability distribution function (PDF) exhibits an exponential decay for large enhancement factors and is proved to be wavelength dependent.