Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic

It was shown in a series of recent publications that the eigenvalues of $n\times n$ Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of $n+1$. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol $g(x)=(2\sin(x/2))^4$, which does not satisfy the simple-loop conditions, and derived asymptotic expansions of a more complicated form. We here use these results to show that the eigenvalues of the pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic expansion. This also delivers a counter-example to a conjecture by Ekstr\"{o}m, Garoni, and Serra-Capizzano and reveals that the simple-loop condition is essential for the existence of the regular asymptotic expansion.Comment: 28 pages, 7 figure

Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order $n$, where one edge has weight $\alpha\in\mathbb{C}$, with $\operatorname{Re}(\alpha)<0$, and all the others have weights $1$. This paper is a sequel of a previous one where we considered $\operatorname{Re}(\alpha) \in[0,1]$ (Eigenvalues of laplacian matrices of the cycles with one weighted edge, Linear Algebra Appl. 653, 2022, 86--115). We prove that for $\operatorname{Re}(\alpha)<0$ and $n>\operatorname{Re}(\alpha-1)/\operatorname{Re}(\alpha)$, one eigenvalue is negative while the others belong to $[0,4]$ and are distributed as the function $x\mapsto 4\sin^2(x/2)$. Additionally, we prove that as $n$ tends to $\infty$, the outlier eigenvalue converges exponentially to $4\operatorname{Re}(\alpha)^2/(2\operatorname{Re}(\alpha)-1)$. We give exact formulas for the half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as $n$ tends to $\infty$, both for the eigenvalues belonging to $[0,4]$ and the outlier. We also compute the eigenvectors and their norms.Comment: 28 pages, 8 figure

Environmental risks for children oral health in Low Danube region

The study was aimed to assess the environmental risks for children oral health in Low Danube region.Material and methods. The study was conducted in 2011-2021. Information on the state of drinking water, the level of environmental safety of food, qualitative and quantitative composition of food was obtained from the reports of territorial institutions of the sanitary-epidemiological service of Odessa region. Actual nutrition was assessed using standard questionnaires. The assessment of the level of environmental and hygienic safety was based on the recommendations of the EPA. Statistical processing was performed by methods of analysis of variance and correlation using specialized software Statistica 10.0.Results. Excessive carbohydrate intake and excessive nitrate load are the main environmental risk factors for dental pathology in children living in the Danube region. Qualitative composition of diets plays a bigger role than mineral composition of drinking water.Conclusion. 1. Drinking water of suboptimal mineral composition was consumed by 41.7% of people, high content of refined carbohydrates was inherent in the diets of 63.0% of surveyed children.2. Children’s diet is characterized with subdeficiency of B vitamins, excessive consumption of refined carbohydrates and a significant nitrate load

Low-frequency interaction between horizontal and overturning gyres in the ocean

Author Posting. © American Geophysical Union, 2008. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Geophysical Research Letters 35 (2008): L18614, doi:10.1029/2008GL035206.Low-frequency variability of the horizontal circulation in an idealized, eddy-permitting, numerical model drives the dominant mode of low-frequency variability in the meridional overturning circulation. This coupling takes place through the influence of lateral advection in the cyclonic high-latitude boundary current on the mixed layer depth along the boundary. The mean and low-frequency variability of the meridional overturning circulation are well predicted by a diagnostic estimate that assumes the downwelling is controlled by the thermal wind shear within the mixed layer along the boundary, which is in turn determined by a simple balance between lateral advection and surface cooling. The more general result is the demonstration that the mean and low frequency variability of the meridional overturning streamfunction are controlled by the baroclinic pressure gradient within the mixed layer along the boundary, which may be influenced by numerous factors such as low-frequency variability in lateral advection, wind stress, surface buoyancy fluxes, or ice melt and freshwater runoff.This work was supported by NSF grants OCE-0423975 and OCE-0726339

On a possibility of inelasticity partial coefficient K sub gamma determination in pi C and pi Pb interactions at 10 to the 14th power eV (experiment PAMIR 1)

The investigation of hadron-nuclear interactions in Pamir experiment is carried out by means of X-ray emulsion chambers of two types: carbon (C) and lead (Pb). While comparing the results from the chambers of both types it was found a discrepancy in n sub h and E sub h(1)R values. The observed discrepancy in C and Pb chambers is connected with the difference in values of effective coefficients of energy transfer to the soft component K sub eff for C and Pb chambers