On Ɛ-uniform convergence of exponentially fitted methods

A class of methods constructed to numerically approximate solution of two-point singularly perturbed boundary value problems of the form $varepsilon u\u27\u27 + b u\u27 + c u = f$ use exponentials to mimic exponential behavior of the solution in the boundary layer(s). We refer to them as exponentially fitted methods. Such methods are usually exact on polynomials of certain degree and some exponential functions. Shortly, they are exact on exponential sums. It is often possible that consistency of the method follows from the convergence of interpolating function standing behind the method. Because of that, we consider interpolation error for exponential sums. A main result of the paper is an error bound for interpolation by exponential sum to the solution of singularly perturbed problem that does not depend on perturbation parameter $varepsilon$ when $varepsilon$ is small with the respect to mesh width. Numerical experiment implies that the use of dense mesh in the boundary layer for small meshwidth results with $varepsilon$-uniform convergence

Least squares fitting the three-parameter inverse Weibull density

The inverse Weibull model was developed by Erto [10]. In practice, the unknown parameters of the appropriate inverse Weibull density are not known and must be estimated from a random sample. Estimation of its parameters has been approached in the literature by various techniques, because a standard maximum likelihood estimate does not exist. To estimate the unknown parameters of the three-parameter inverse Weibull density we will use a combination of nonparametric and parametric methods. The idea consists of using two steps: in the first step we calculate an initial nonparametric density estimate which needs to be as good as possible, and in the second step we apply the nonlinear least squares method to estimate the unknown parameters. As a main result, a theorem on the existence of the least squares estimate is obtained, as well as its generalization in the $l_p$ norm ($1leq p<infty$). Some simulations are given to show that our approach is satisfactory if the initial density is of good enough quality

On Ɛ-uniform convergence of exponentially fitted methods

A class of methods constructed to numerically approximate solution of two-point singularly perturbed boundary value problems of the form $varepsilon u\u27\u27 + b u\u27 + c u = f$ use exponentials to mimic exponential behavior of the solution in the boundary layer(s). We refer to them as exponentially fitted methods. Such methods are usually exact on polynomials of certain degree and some exponential functions. Shortly, they are exact on exponential sums. It is often possible that consistency of the method follows from the convergence of interpolating function standing behind the method. Because of that, we consider interpolation error for exponential sums. A main result of the paper is an error bound for interpolation by exponential sum to the solution of singularly perturbed problem that does not depend on perturbation parameter $varepsilon$ when $varepsilon$ is small with the respect to mesh width. Numerical experiment implies that the use of dense mesh in the boundary layer for small meshwidth results with $varepsilon$-uniform convergence

Mathematical modeling of the spread of the epidemic

Prikazan je izvod standardnog SIR modela. Ovo je osnovni epidemiološki model i njegovom modifikacijom izgrađen je veliki dio epidemioloških modela.We explain derivation of the standard SIR model. This is basic epidemiological model. Many epidemiological models are derived as a modification of this standard model

Large Chondrosarcoma of the Lumbar Spine – A Rare yet Important Cause of Lower Back Pain

We report a case of a large chondrosarcoma of an L4 vertebral body causing iliac vein thrombosis. The slow-growing tumor eluded definitive diagnosis early in its development since the main symptom it caused was only lower back pain. Five years after onset of the disease, the patient presented with fever, tenderness and swelling in the leg, the tumor was diagnosed and found to be exerting a mass effect causing further pain and compressing the left common iliac vein. Due to inoperability of the tumor, a multidisciplinary surgical approach was used to resect the majority of the tumor as a palliative measure and rid the patient of her symptoms. Due to the chemoresistance and relative radioresistance of these tumors, prompt full surgical resection before the tumor invades vital structures remains the mainstay of successful treatment of chondrosarcoma of the spine

Metoda evaluacije kvalitete akustike sobe temeljenoj na energetskim relacijama zvuka

Measuring procedure of achieving room acoustic quality parameters with impulse response is usually used as the basis for acoustical measuring PC based software. The objective parameters: clarity (C), definition (D) and ratio between reflected and direct energy (R) are defined with reflected, direct and total energy of sound. The relations are set in order to enable estimation of other parameters based on measurement of only one energy parameter. Based on measurements in two architectural identical, but according to acoustic characteristics two different rooms, and additional analysis and calculations connected with number of people in a hall, objective parameters are evaluated according to earlier adopted optimal conditions involving certain deviations from the values.Mjerni postupak postizanja parametara kvalitete zvuka prostorije impulsnim odzivom obično se koristi kao osnova za računalni software za akustično mjerenje. Jasnoća (C), definicija (D) i omjer između reflektirane i izravne energije (R) kao objektivni parametri definirani su reflektiranom, izravnom i ukupnom energijom zvuka. Odnosi su postavljeni tako da bi se omogućila procjena ostalih parametara na temelju mjerenja samo jednog parametra energije. Na temelju mjerenja u dvije arhitektonski identične, ali prema akustičnim svojstvima dvije različite prostorije te dodatnoj analizi i izračunima povezanima s brojem ljudi u dvorani, objektivni parametri se ocjenjuju prema ranije usvojenim optimalnim uvjetima koji uključuju određena odstupanja od vrijednosti

Oral Health Awareness in Croatian and Italian Urban Adolescents

Purpose of this study was to investigate and compare differences in oral health awareness between Croatian and Italian urban adolescents. The sample consisted of primary school last grade students aged between 13 and 15 years, 300 children from Zagreb (Croatia) and 298 children from Bari (Italy). Oral health awareness was evaluated using a self-administered standardized questionnaire. Self-perception of oral health proved to be different between the two groups (p<0.001). The Croatians reported that their oral health was »excellent« or »very-good« more often than the Italians (68.6% vs. 50.2%). The reasons given for visiting a dentist were different (p<0.001). The Italians cleaned their teeth more often than the Croatians (»two or more times a day«, 83.1% vs. 72.2%, p<0.003). Wooden toothpicks were preferred by the Croatians (p<0.001), while floss was preferred by the Italians (p=0.03). The awareness regarding the use of fluoridated toothpaste was higher in the Italian group (95.6% vs. 72.5%, p<0.001). The Croatians were consuming sweetened foods more often than the Italians (p<0.001). Croatian adolescents reported more indicators of a lower level of oral health awareness than the Italians, while on the contrary Croatians had higher esteem of their oral health. Defining national preventive strategies is essential for improving adolescents’ attitudes toward oral health in both countries, particularly in Croatia

High-order exponentially fitted difference schemes for singularly perturbed two-point boundary value problems. ETNA - Electronic Transactions on Numerical Analysis

We introduce a family of exponentially fitted difference schemes of arbitrary orderas numerical approximations to the solution of a singularly perturbed two-point boundary valueproblem: \\varepsilon y” + b y\' + c y = f.The difference schemes are derived from interpolation formulae for exponential sums.The so-defined $k$-point differentiation formulae are exact for functions that are alinear combination of $1,x,\\ldots,x^{k-2},\\exp{(-\\rho x)}$.The parameter $\\rho$ is chosen from the asymptotic behavior of the solution in the boundary layer.This approach allows a construction of the method with arbitrary order of consistency.Using an estimate for the interpolation error, we prove consistency of all the schemes fromthe family.The truncation error is bounded by $C h^{k-2}$, where $C$ is a constant independent of $\\varepsilon$and $h$.Therefore, the order of consistency for the $k$-point scheme is $k-2$ ($k \\geq 3$) in case ofa small perturbation parameter $\\varepsilon$.There is no general proof of stability for the proposed schemes.Each scheme has to be considered separately.In the paper, stability, and therefore convergence, is proved for three-point schemesin the case when $c<0$ and $b \\neq 0$