Uopšteni stohastički procesi sa primenama u rešavanju jednačina

In this dissertation stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. Such processes are called Colombeau stochastic processes.The notion of point values of Colombeau stochastic processes in compactly supported generalized points is established. The Colombeau algebra of compactly supported generalized constants is endowed with the topology generated by sharp open balls. The measurability of the corresponding random variables with values in the Colombeau algebra of compactly supported generalized constants is shown. The generalized correlation function and the generalized characteristic function of Colombeau stochastic processes are introduced and their properties are investigated. It is shown that the characteristic function of classical stochastic processes can be embedded into the space of generalized characteristic functions. Examples of generalized characteristic function related to gaussian Colombeau stochastic processes are given. The structural representation of the generalized correlation function which is supported on the diagonal is given. Colombeau stochastic processes with independent values are introduced. Strictly stationary and weakly stationary Colombeau stochastic processes are studied. Colombeau stochastic processes with stationary increments are characterized via their stationarity of the gradient of the process.Gaussian stationary solutions are analyzed for linear stochastic partial differential equations with generalized constant coefficients in the framework of Colombeau stochastic processes.U disertaciji se stohastički procesi posmatraju u okviru Kolomboove algebre uopštenih funkcija. Takve procese nazivamo Kolomboovi stohastički procesi. Pojam vrednosti Kolomboovog stohastičkog procesa u tačkama sa kompaktnim nosačem je uveden. Dokazana je merljivost odgovarajuće slučajne promenljive sa vrednostima u Kolomboovoj algebri uopštenih konstanti sa kompaktnim nosačem,  snabdevenom topologijom generisanom oštrim otvorenim loptama. Uopštena korelacijska funkcija i uopštena karakteristična funkcija Kolomboovog stohastičkog procesa su definisane i njihove osobine su izučavane. Pokazano je da  se karakteristična funkcija klasičnog stohastičkog procesa može potopiti u prostor uopštenih karakterističnih funkcija. Dati su primeri uopštenih karakterističnih funkcija  gausovskih Kolomboovih stohastičkih procesa. Data je strukturna reprezentacija uopštene korelacijske funkcije sa nosačem na dijagonali. Kolomboovi stohastički procesi sa nezavisnim vrednostima su predstavljeni. Izučavani su strogo stacionarni i  slabo stacionarni Kolomboovi stohastički procesi. Kolomboovi stohastički procesi sa stacionarnim priraštajima su okarakterisani preko stacionarnosti gradijenta procesa. Gausovska stacionarna rešenja za linearnu stohastičku parcijalnu diferencijalnu jednačinu sa uopštenim konstantnim koeficijentima su analizirana u okvirima Kolomboovih stohastičkih procesa

Analysis of math textbooks by SPUR

Udžbenički komplet je osnovni didaktički materijal u nastavi matematike i veoma je važno kakve zadatke sadrži. U radu smo analizirali udžbeničke komplete matematike za 4. razred osnovne škole kako bismo utvrdili zastupljenost zadataka primenjujući SPUR pristup. SPUR je multidimenzionalni pristup učenju matematike koji podstiče razvoj učenika kroz četiri dimenzije: veštine (Skills) podrazumevaju poznavanje procedura za rešavanje zadatka; osobine (Properties) se odnose na poznavanje matematičkih principa i činjenica; upotreba (Uses) podrazumeva primenu znanja u svakodnevnom životu, dok se predstavljanje (Representations) odnosi na upotrebu vizuelnih prikaza. Uzorak čine udžbenički kompleti matematike za 4. razred osnovne škole – tri izdavačke kuće – koji se najviše koriste na teritoriji Školske uprave Sombor. Instrument predstavlja ček lista sastavljena za potrebe ovog istraživanja, a metoda je analiza sadržaja. Zadaci su svrstani u jednu dimenziju ili u više dimenzija SPUR-a. Rezultati pokazuju da su zadaci iz svake od navedenih dimenzija podjednako zastupljeni u svim udžbeničkim kompletima. Najzastupljeniji su zadaci koji se odnose na dimenziju veštine (više od 65% ukupnog broja zadataka), zatim osobine (više od 18%), upotrebu (više od 9%) i najmanje zadaci koji se odnose na predstavljanje (manje od 4%). Udžbenički kompleti najviše podstiču razvoj veština, a najmanje sposobnost grafičkog predstavljanja postupaka i rešavanja zadataka. Nedovoljno su zastupljeni zadaci koji podstiču primenu matematičkih znanja u svakodnevnom životu. Ujednačen broj zadataka za svaku dimenziju bi omogućio učenicima lakše razumevanje i primenu matematičkih znanja kada se suoče sa realnim problemima. U budućim istraživanjimabi mogla biti ispitana povezanost zastupljenosti zadataka u udžbeničkim kompletima primenom SPUR pristupa i postignuća učenika na testu u kome bi bile ravnomerno zastupljene posmatrane dimenzije.A textbook is basic didactic material in teaching mathematics and it is very important what kind of tasks it contains. In this study we analysed mathematics textbooks for the 4th grade of primary school in order to determine representation of tasks according to SPUR. SPUR is a multi-dimensional approach to learning mathematics that encourages development of students through four dimensions. The “Skills” include knowledge of procedures for solving a task, “Properties” include mathematical principles and facts, “Uses” imply the application of knowledge in everyday life and “Representations” involve the use of visual representations. The sample consists of mathematics textbook sets for the 4th grade of primary school by three publishing houses that are most frequently used in the territory of the School Administration of Sombor. The instrument is a checklist compiled for the purpose of this research, and the method is content analysis. Every task is sorted in one or more SPUR dimensions. Results show that tasks from each of the given dimensions are equally represented in all textbooks. The most common tasks are those related to the dimension of Skills (more than 65% of the total number of tasks), then Properties (more than 18%), Uses (more than 9%) and the least common are the tasks related to Representations (less than 4%). Textbook sets predominantly stimulate development of skills, and most infrequently the ability to graphically present procedures and solve tasks. In textbook sets, there is insufficient number of tasks that encourage development of the ability to apply mathematical knowledge in everyday life. A uniform number of tasks in textbook sets for each dimension would enable students to better understand and apply mathematical knowledge in face of real problems. Further researches should analyse relationship between the number of tasks in textbook sets by SPUR and students` achievements in the test consisted of equal number of tasks for every dimension.Zbornik rezimea / 24. Međunarodna naučna konferencija "Pedagoška istraživanja i školska praksaBook of abstracts / 24th International Scientific Conference "Educational Research and School Practice

Stochastic parabolic equations with singular potentials

In this work we consider a class of stochastic parabolic equations with singular space depending potential, random driving force and random initial condition. For the analysis of these equations we combine the chaos expansion method from the white noise analysis and the concept of very weak solutions. For given stochastic parabolic equation we introduce the notion of a stochastic very weak solution, prove the existence and uniqueness of the very weak solution to corresponding stochastic initial value problem and show its independence of a regularization on given singular potential. In addition, the consistency of a stochastic very weak solution with a stochastic weak solution is shown.Comment: 29 page

On a Wick-type stochastic parabolic equations with random potentials

Stochastic parabolic equations with random potentials, where the Wick product is used to give sense to the product of generalized stochastic processes, are considered. The existence and uniqueness of solutions are proved via the chaos expansion method from white noise analysis and explicit estimates on the coefficients in the chaos expansion representation of the solutions are provided

New Equivalents of Kurepa’s Hypothesis for Left Factorial

Kurepa’s hypothesis for the left factorial has been an unsolved problem for more than 50 years. In this paper, we have proposed new equivalents for Kurepa’s hypothesis for the left factorial. The connection between the left factorial and the continued fractions is given. The new equivalent based on the properties of the integer part of real numbers is proven. Moreover, a new equivalent based on the properties of two well-known sequences is given. A new representation of the left factorial is listed. Since derangement numbers are closely related to Kurepa’s hypothesis, we made some notes about the derangement numbers and defined a new sequence of natural numbers based on the derangement numbers. In this paper, we indicate a possible direction for further research through solving quadratic equations