A Test for the Stability of Networks

A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p) and o(p) denote the even and the odd part of p [25].Rowinska-Schwarzweller Agnieszka - Chair of Display Technology University of Stuttgart Allmandring 3b, 70596 Stuttgart, GermanySchwarzweller Christoph - Institute of Computer Science University of Gdansk Wita Stwosza 57, 80-952 Gdansk, PolandGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265-269, 2001.Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.Christoph Schwarzweller. Introduction to rational functions. Formalized Mathematics, 20 (2):181-191, 2012. doi:10.2478/v10037-012-0021-1.Christoph Schwarzweller and Agnieszka Rowinska-Schwarzweller. Schur’s theorem on the stability of networks. Formalized Mathematics, 14(4):135-142, 2006. doi:10.2478/v10037-006-0017-9.Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Rolf Unbehauen. Netzwerk- und Filtersynthese: Grundlagen und Anwendungen. Oldenbourg-Verlag, fourth edition, 1993.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

A multifactor approach to modelling the impact of wind energy on electricity spot prices

We introduce a three-factor model of electricity spot prices, consisting of a determinis- tic seasonality and trend function as well as short- and long-term stochastic components, and derive a formula for futures prices. The long-term component is modelled as a L ́evy process with increments belonging to the class of generalised hyperbolic distributions. We de- scribe the short-term factor by L ́evy semistationary processes: we start from a CARMA(2,1), i.e. a continous-time ARMA model, and generalise it by adding a short-memory stochastic volatility. We further modify the model by including the information about the wind energy production as an exogenous variable. We fit our models to German and Austrian data in- cluding spot and futures prices as well as the wind energy production and total load data. Empirical studies reveal that taking into account the impact of the wind energy generation on the prices improves the goodness of fit

Some Algebraic Properties of Polynomial Rings

In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/ is isomorphic to the field of polynomials with degree smaller than the one of p.Schwarzweller Christoph - Institute of Computer Science University of Gdansk, PolandKorniłowicz Artur - Institute of Informatics University of Białystok, PolandRowinska-Schwarzweller Agnieszka - Sopot PolandGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1.H. Heuser. Lehrbuch der Analysis. B.G. Teubner Stuttgart, 1990.Steven H. Weintraub. Galois Theory. Springer Verlag, 2 edition, 2009

Solution structure and metal ion binding sites of the human CPEB3 ribozyme's P4 domain

Three ribozymes are known to occur in humans, the CPEB3 ribozyme, the CoTC ribozyme, and the hammerhead ribozyme. Here, we present the NMR solution structure of a well-conserved motif within the CPEB3 ribozyme, the P4 domain. In addition, we discuss the binding sites and impact of Mg2+ and [Co(NH3)6]3+, a spectroscopic probe for [Mg(H2O)6]2+, on the structure. The well-defined P4 region is a hairpin closed with a UGGU tetraloop that shows a distinct electrostatic surface potential and a characteristic, strongly curved backbone trajectory. The P4 hairpin contains two specific Mg2+ binding sites: one outer-sphere binding site close to the proposed CPEB3 ribozyme active site with potential relevance for maintaining a compact fold of the ribozyme core, and one inner-sphere binding site, probably stabilizing the tetraloop structure. The structure of the tetraloop resembles an RNase III recognition structure, as previously described for an AGUU tetraloop. The detailed knowledge of the P4 domain and its metal ion binding preferences thus brings us closer to understanding the importance of Mg2+ binding for the CPEB3 ribozyme's fold and function in the cell

unexpected impact of the number of glutamine residues on metal complex stability

The emerging question which this study aims to answer is: what impact do glutamines have on the stability of metal–peptide complexes? We focused our attention on the N-terminal domain of Hpn and Hpn-like proteins from Helicobacter pylori. Cu2+ and Ni2+ complexes of the model peptides MAHHE-NH2, MAHHEEQ-NH2, MAHHEQQ-NH2 and MAHHEQQHQA-NH2 were studied by means of different thermodynamic and spectroscopic techniques, as well as through molecular modelling computation. Experimental results, in very good agreement with theoretical findings, lead to the not obvious conclusion that the stability of metal complexes distinctly increases with the number of glutamine residues present in the peptide, although glutamine side-chains do not directly take part in coordination. This peculiar finding allows one to look at polyglutamine sequences, not only the ones present in some bacterial chaperones but also those involved in several neurodegenerative diseases, from a new perspective

Technical performance and diagnostic utility of the new Elecsys (R) neuron-specific enolase enzyme immunoassay

This international multicenter study was designed to evaluate the technical performance of the new double-monoclonal, single-step Elecsys neuron-specific enolase (NSE) enzyme immunoassay (EIA) and to assess its utility as a sensitive and specific test for the diagnosis of small-cell lung cancer (SCLC). Intra and interassay coefficients of variation, determined in five control or serum specimens in six laboratories, ranged from 0.7 to 5.3 (interlaboratory median: 1.3%) and from 1.3 to 8.5 (interlaboratory median: 3.4%), respectively. Laboratory-to-laboratory comparability was excellent with respect to recovery and interassay coefficients of variation. The test was linear between 0.0 and 320 ng/ml (highest measured concentration). There was a significant correlation between NSE concentrations measured using the Elecsys NSE and the established Cobas Core NSE EIA II in all subjects (n=723) and in patients with lung cancer (n=333). However, NSE concentrations were systematically lower (approximately 9%) with the Elecsys NSE than with the comparison test. Based on a specificity of 95% in comparison with the group suffering from benign lung diseases (n=183), the cutoff value for the discrimination between malignant and benign conditions was set at 21.6 ng/ml. NSE was raised in 73.4% of SCLC patients (n=188) and was significantly higher (p<0.01) in extensive (87.8%) as opposed to limited disease (56.7%). NSE was also elevated in 16.0% of the cases with non-small cell lung cancer (NSCLC, n=374). It is concluded that the Elecsys NSE EIA is a reliable and accurate diagnostic procedure for the measurement of NSE in serum samples. The special merits of this new assay are the wide measuring range (according to manufacturers declaration up to 370 ng/ml) and a short incubation time of 18 min

Effects of COVID-19 in the financial statements of selected companies listed in Warsaw stock exchange

Purpose: The objective of this article is to evaluate the quality of the presented content concerning the impact of the COVID-19 pandemic on continuing activities of selected companies listed in Warsaw Stock Exchange. Design/Methodology/Approach: In the first part we applied critical review of subject literature and the monographic method. In the second part, the content analysis method was used, and the basic source material were the financial statements - annual for 2019 and interim for 2020. Findings: Research demonstrates that disclosures of events following the balance sheet date did not allow users of financial statements to assess the risk associated with the impact of COVID-19. Some companies completely ignored the risk related to the pandemic in their financial statements, and in some cases the scope of disclosures is symbolic. Practical Implications: Indicating the gap between the expected and actual scope of disclosures regarding the consequences of the pandemic in financial statements, which are the primary source of information about a listed company that are further authenticated by an auditor. Originality/value: Demonstration of the insufficient level of disclosures regarding the impact of the pandemic on the financial condition and performance of the entities audited, including a lack of numerical analyses of its impact on the entity's ability to continue operations, and the valuation of its assets and liabilities.peer-reviewe