Homogenized description of defect modes in periodic structures with localized defects

A spatially localized initial condition for an energy-conserving wave equation with periodic coefficients disperses (spatially spreads) and decays in amplitude as time advances. This dispersion is associated with the continuous spectrum of the underlying differential operator and the absence of discrete eigenvalues. The introduction of spatially localized perturbations in a periodic medium leads to defect modes, states in which energy remains trapped and spatially localized. In this paper we study weak, localized perturbations of one-dimensional periodic Schr\"odinger operators. Such perturbations give rise to such defect modes, and are associated with the emergence of discrete eigenvalues from the continuous spectrum. Since these isolated eigenvalues are located near a spectral band edge, there is strong scale-separation between the medium period and the localization length of the defect mode. Bound states therefore have a multi-scale structure: a "carrier Bloch wave" times a "wave envelope", which is governed by a homogenized Schr\"odinger operator with associated effective mass, depending on the spectral band edge which is the site of the bifurcation. Our analysis is based on a reformulation of the eigenvalue problem in Bloch quasi-momentum space, using the Gelfand-Bloch transform and a Lyapunov-Schmidt reduction to a closed equation for the near-band-edge frequency components of the bound state. A rescaling of the latter equation yields the homogenized effective equation for the wave envelope, and approximations to bifurcating eigenvalues and eigenfunctions.Comment: The title differs from version 1. To appear in Communications in Mathematical Science

Contribution to the development of the method of forecasting the frequency and duration of "failure" of technical systems

An active approach to the market means the full adaptation of the production system to the demands of the consumers, and the high profitability. To realize the above, production system should create the conditions for the simulation of the behavior of the system in real conditions. In the opposite the behavior of the system will be highly uncertain because of the disturbance in the system. The simulation of the behavior of the system in real conditions includes the existence of reliable information, obtained based on previous research. A series of such information also includes the information on the production cycle of a series of products. The components of the production cycle are also different disturbances. This paper analyses the failures of the technical system. This fact points to the significance of creating the conditions for forecasting the frequency and duration of the "failure". This paper, based on research of two formatisers, presents the methodology of carrying out the forecasting in real conditions

Simulation of primary log cutting plan (PPRT) in sawmill wood processing

A specification of sawn timber is realized based on the previously designed log cutting plan, primary and secondary. The plan of primary log cutting (PPRT) has its verbal and mathematical interpretations. The theoretical aspect of PPRT design reported in the literature has great disadvantages. As it is based on the description and empirical combinations, such PPRT could hardly be evaluated as optimal. In addition to the above weakness, there is another one, i.e., the fact that the problem of PPRT design is reduced to a mono-criterion problem, i.e. the minimisation of total waste. If the problem of PPRT design is approached systematically, it is clear that the minimisation of total waste is only one out of a series of goals that a PPRT should fulfil. Nevertheless, the procedure presented in the literature is deeply rooted in practice. This paper, based on the on simulation principle, presents a modern procedure of PPRT design which consists of a number of phases. The first phase includes computer support in the design procedure of log cutting plans and also, a mono-criterion decision making in the selection of log cutting plans which ensure the minimal total waste per PPRT. The second phase includes the definition of another two criteria which are decisive for the optimal quality of PPRT. They are the log prices and the profit. Altogether 84 (eighty-four) cutting plans were designed and 4 (four) PPRTs. The solution of the problem, i.e. the design of an optimal PPRT in the conditions of three criteria, is based on multicriteria decision making using the method of Analytical Hierarchy Processes - AHP. The calculation is computed by the software Program Criterion Decision Plus - Version 3.0 Student version, InfoHarvest.Inc. In the study simulation of PPRT design, the optimal plan was PPRT3. The presented three criteria are not the final number. The inclusion of additional relevant criteria will depend on the inventiveness of the manager

Forecasting of the processing time as the base of simulation of the production system behavior in real conditions

The absence of precise information on the magnitudes that determine the behavior of the production system generates the disturbances of the system. The consequence is the low efficacy of the system and the high costs. Therefore, it is necessary to create the base for the prediction of individual magnitudes and thus enable the simulation of the production system behavior in real conditions. The information on time norms has a special significance. It is the base of planning the terms and of defining a part of direct costs. Modern approach in the identification of standard times should be established on new foundations. It should appreciate the specificities of the present moment, as well as the future tendencies in wood processing. They are the production system dynamistic, conditioned predominantly by discontinuous production, as well as by the necessity of cooperation of the production systems. In this study, the approach to the identification of standard times is original, supporting the modern tendencies in wood processing and it has an applicative character

Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes

Let $Q(x)$ denote a periodic function on the real line. The Schr\"odinger operator, $H_Q=-\partial_x^2+Q(x)$, has $L^2(\mathbb{R})-$ spectrum equal to the union of closed real intervals separated by open spectral gaps. In this article we study the bifurcation of discrete eigenvalues (point spectrum) into the spectral gaps for the operator $H_{Q+q_\epsilon}$, where $q_\epsilon$ is spatially localized and highly oscillatory in the sense that its Fourier transform, $\widehat{q}_\epsilon$ is concentrated at high frequencies. Our assumptions imply that $q_\epsilon$ may be pointwise large but $q_\epsilon$ is small in an average sense. For the special case where $q_\epsilon(x)=q(x,x/\epsilon)$ with $q(x,y)$ smooth, real-valued, localized in $x$, and periodic or almost periodic in $y$, the bifurcating eigenvalues are at a distance of order $\epsilon^4$ from the lower edge of the spectral gap. We obtain the leading order asymptotics of the bifurcating eigenvalues and eigenfunctions. Underlying this bifurcation is an effective Hamiltonian associated with the lower edge of the $(b_*)^{\rm th}$ spectral band: $H^\epsilon_{\rm eff}=-\partial_x A_{b_*,\rm eff}\partial_x - \epsilon^2 B_{b_*,\rm eff} \times \delta(x)$ where $\delta(x)$ is the Dirac distribution, and effective-medium parameters $A_{b_*,\rm eff},B_{b_*,\rm eff}>0$ are explicit and independent of $\epsilon$. The potentials we consider are a natural model for wave propagation in a medium with localized, high-contrast and rapid fluctuations in material parameters about a background periodic medium.Comment: To appear in SIAM Journal on Mathematical Analysi

On Responsibility in University Teaching: Aspects Contributing to the Enhancement of Teaching Quality Provided at Universities

This paper will highlight the intrinsic link between university educators’ general responsibilities and the three domains of responsibilities. The text initially presents perspectives regarding responsibility and the act of teaching. It is then followed by an analysis of the concept of responsibility in teaching, which is examined across three distinct domains: the domain of encouraging students’ curiosity, the domain of keeping abreast of the prospective career requirements for students, and the domain of language usage. In brief, it is crucial to enhance teachers' sense of responsibility toward their students, as it can ultimately result in enhanced academic achievement and personal development for students

Elektrohemijsko generisanje katalizatora za Majklovu adiciju dikarbonilnih jedinjenja i cijanida na akriloilferocen

Michael addition of diethyl malonate, ethyl acetoacetate, acetylacetone, and cyanide anion to acryloylferrocene promoted by a catalyst in situ generated from a sacrificial zirconium anode is described. Most of the obtained compounds were identified by comparison of their spectral and physical data with those published elsewhere, whereas the only newly synthesized compound - diethyl 2,2-bis(3-ferrocenyl-3-oxopropyl)malonate - was completely characterized by spectral (IR, 1H- and 13C-NMR), physical and crystallographic (single-crystal X-ray) data.Opisana je Majklova adicija dietil-malonata, etil-acetoacetata, acetilacetona i cijanidnog anjona na akriloilferocen pomoću katalizatora generisanog in situ sa rastvorne anode od cirkonijuma. Većina dobijenih jedinjenja je identifikovana poređenjem njihovih fizičkih i spektroskopskih podataka sa literaturnim, a jedino novo jedinjenje je potpuno opisano spektroskopskim (IR, 1H- i 13C-NMR) i fizičkim podacima, kao i podacima iz kristalografske analize X-zracima

Electrochemical Synthesis of Keto and Enol Forms of Tetraketones of the Type 3,4-Diaroyl-2,5-hexanedione by Oxidative Dimerization of Aroylacetones

Tetraketones of the type 3,4-diaroyl-2,5-hexanedione: Ar = CsH;-, p-CHs-CsH4-, 3,4-(CHsh-CsHs-, p-CHaO-CsH4-, p- Cl-CsH4-, p-Br-C6H4- and 2-C4H3S- were prepared in good yield by oxidative anodic dimerization of aroylacetones. Electrochemical oxidation was carried out at constant current (4-5 A) in cylindrical cell without diaphragm with Pt-gauze anode and Ni-cathode. Depending on the method used (A or B) and depending on the properties of the electrolysed aroylacetones, dimer-products were isolated as stable keto or enol tautomers. Electrolysis in 80°/o aqueous methanol (Method A) gave keto or enol tautomers, while electrolysis in absolute methanol (Method B) yielded only enol tautomers

Electrochemical Synthesis of Keto and Enol Forms of Tetraketones of the Type 3,4-Diaroyl-2,5-hexanedione by Oxidative Dimerization of Aroylacetones

Tetraketones of the type 3,4-diaroyl-2,5-hexanedione: Ar = CsH;-, p-CHs-CsH4-, 3,4-(CHsh-CsHs-, p-CHaO-CsH4-, p- Cl-CsH4-, p-Br-C6H4- and 2-C4H3S- were prepared in good yield by oxidative anodic dimerization of aroylacetones. Electrochemical oxidation was carried out at constant current (4-5 A) in cylindrical cell without diaphragm with Pt-gauze anode and Ni-cathode. Depending on the method used (A or B) and depending on the properties of the electrolysed aroylacetones, dimer-products were isolated as stable keto or enol tautomers. Electrolysis in 80°/o aqueous methanol (Method A) gave keto or enol tautomers, while electrolysis in absolute methanol (Method B) yielded only enol tautomers