Loss of energy concentration in nonlinear evolution beam equations

Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation $$u_{tt} + u_{xxxx} + f(u)= g(x, t)$$ in bounded space-time intervals. We give a new definition of instability for these particular solutions, based on the loss of energy concentration on their prevailing mode. We distinguish between two different forms of energy transfer, one physiological (unavoidable and depending on the nonlinearity) and one due to the insurgence of instability. We then prove a theoretical result allowing to reduce the study of this kind of infinite-dimensional stability to that of a finite-dimensional approximation. With this background, we study the occurrence of instability for three different kinds of nonlinearities $f$ and for some forcing terms $g$, highlighting some of their structural properties and performing some numerical simulations

Thresholds for hanger slackening and cable shortening in the Melan equation for suspension bridges

The Melan equation for suspension bridges is derived by assuming small displacements of the deck and inextensible hangers. We determine the thresholds for the validity of the Melan equation when the hangers slacken, thereby violating the inextensibility assumption. To this end, we preliminarily study the possible shortening of the cables: it turns out that there is a striking difference between even and odd vibrating modes since the former never shorten the cables. These problems are studied both on beams and plates

One or Two Statement Approach for the Income Statement of Czech Companies?

Purpose of the article: The article examines the IAS 1 revised that allows the company to choose between a reclassification of items by nature or by function (destination), it also provides two alternative presentations of the income statement: a single statement of comprehensive income or two statements, the first containing the components of net income, and the second that, starting from net income, shows the other components of expanded income. Methodology/methods: First we outline the principal theories of the statement of comprehensive income, secondly we concentrate the research to obtain the empirical data. The research is developed in the following steps: selection of the companies surveyed; selection of the documents to be analysed; election of the empirical aspects to be studied. Scientific aim: The aim of the research is to define which format has been adopted by companies listed on the Czech Republic Stock Exchange for the new Income Statement to be prepared according to the International Accounting Standards – IAS 1 revised in 2007. In particular, we examine the Consolidated Financial Statements 2010 published online. Findings: This article illustrates the two choices that have the companies to present items of net income, items of other comprehensive income and total comprehensive income: they can create one continuous statement of comprehensive income or two separate consecutive statements. Conclusions: The results show that the Czech financial statements have not a firms’ propensity to separate the section of the income statement in two statements rather than to integrate it into a single one, while the reclassification of the income statements privileges the reclassification by nature

THE DRIVERS OF PRODUCTIVITY

This paper will try to demonstrate that productivity is the basis of all productive systems, which are viewed as transformers of utility and value, since the search for maximum productive efficiency is necessary to reduce production costs and thus produce value. After presenting a coherent frame of reference, we shall examine the drivers of productivity and then move on to discuss the consequences of the continual growth in productivity and the non-economic aspects linked to the gradual improvement in productivit

Resonance tongues for the Hill equation with Duffing coefficients and instabilities in a nonlinear beam equation

We consider a class of Hill equations where the periodic coefficient is the squared solution of some Duffing equation plus a constant. We study the stability of the trivial solution of this Hill equation and we show that a criterion due to Burdina (V.I. Burdina, Boundedness of solutions of a system of differential equations) is very helpful for this analysis. In some cases, we are also able to determine exact solutions in terms of Jacobi elliptic functions. Overall, we obtain a fairly complete picture of the stability and instability regions. These results are then used to study the stability of nonlinear modes in some beam equations.Comment: Communications in contemporary mathematics. Print ISSN: 0219-1997. Online ISSN: 1793-668

Higher order linear parabolic equations

We first highlight the main differences between second order and higher order linear parabolic equations. Then we survey existing results for the latter, in particular by analyzing the behavior of the convolution kernels. We illustrate the updated state of art and we suggest several open problems.Comment: Dedicated to Patrizia Pucci. To appear in the the Contemporary Mathematics series of the AM