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V magistrskem delu je predstavljena obravnava verjetnosti v osnovnih, srednje poklicnih in strokovnih šolah ter gimnazijah. Magistrsko delo je razdeljeno na tri dele. V prvem delu so predstavljeni osnovni pojmi in defnicije, ki se obravnavajo v osnovnih, srednje poklicnih in strokovnih šolah ter gimnazijah. V drugem delu so predstavljene vsebine, ki zajemajo področja verjetnosti in so zapisana v učnem načrtu za osnovne, srednje poklicne in strokovne šole ter gimnazije. V tretjem delu magistrskega dela smo pregledali večino osnovnošolskih, srednješolskih in gimnazijskih učbenikov in opisali, katere vsebine iz področja verjetnosti obravnavajo. Pri- kazali smo tudi morebitne nejasnosti in napake, ki se pojavijo v pregledanih učbenikih. Ob koncu smo spoznali, da so v učbenikih uporabljeni različni pristopi pri defniranju poj- mov. Opazili smo, da se največja razlika pojavi med osnovnošolskimi in gimnazijskimi učbeniki. The following master's thesis presents the treatment of probability in primary schools, secondary technical and vocational schools, and grammar schools. The master's thesis is divided into three parts. The frst part introduces the basic concepts and defnitions covered in primary schools, secondary technical and vocational and grammar schools. The second part presents the content dealing with areas of probability, which is determined by the syllabus for primary schools, secondary technical and vocational schools, and grammar schools. The third part of the master's thesis deals with an examination of most primary, secondary and grammar school textbooks, describing what content from the area of probability they are likely to address. We have also pointed to possible ambiguities and errors occurring in the reviewed textbooks. To conclude, we found that different approaches were used in the textbooks to defne the terms and that the biggest difference appears between primary and secondary school textbooks.

Statistika velja za eksaktno vejo znotraj matematike, ki uporablja preverjene in zanesljive metode. Kljub temu dogaja, da znotraj nje prihaja do napak, prirejanj in zlorab, ki so bodisi naključne bodisi namerne narave. Napake v statistiki niso posledica samih matematičnih/statističnih metod te so zanesljive in preverjene, ampak posameznikov, ki delajo napake iz površnosti, pomanjkljivega znanja ali pa manipulirajo vedo samo, zaradi finančnih, gospodarstvenih, političnih in osebnih razlogov z namenom koristi. Magistrsko delo se deli na dva dela, in sicer smo v prvem delu utemeljili osnovne statistične pojme, ki so nam koristili v nadaljevanju. Pregledali smo položaj statistike v družbi, jo preučili z vidika posameznika in se osredotočili na mnenja, ki ga gojijo posamezniki o vedi. Na podlagi primerov iz resničnega življenja smo razložili, zakaj prihaja do negativnega mnenja o vedi sami in se osredotočili na napake in zlorabe vede, jih utemeljili in v drugem delu tudi matematično podkrepili ter podali rešitve/napotke za kvalitetnejšo statistično raziskovanje. Statistics is a precise branch within mathematics, using established and reliable methods. However, despite all efforts, errors or misconceptions both accidental and intentional can happen. Errors in statistics are not a result of faulty mathematical or statistical methods – these are reliable and verified errors are instead a consequence of mistakes made by individuals, as a result of, for example, superficiality, lack of knowledge and manipulation due to economic, personal or political reasons. The master's thesis is divided into two parts: in the first part, we establish the basic statistical concepts that are used in the field of statistics. Furthermore, we examine the position of statistics in society, focusing on the aspects of both individual and social perspectives. Based on real life examples, we explain the origin of generally negative perceptions of statistics. We focus on mistakes/manipulation and abuse of science as well. In the second part, we provide the mathematical background of misused methods and suggest possible solutions for achieving better quality of statistical practice.

V magistrskem delu je predstavljena obravnava statistike v osnovni šoli, srednji poklicni in strokovni šoli ter v gimnaziji. Magistrsko delo je razdeljeno na tri dele. V prvem delu so predstavljeni osnovni pojmi in definicije statistike, ki se obravnavajo v osnovni šoli, srednji poklicni in strokovni šoli in v gimnaziji. V drugem delu je predstavljena vsa vsebina, ki zajema področje statistike in je zapisana v osnovnošolskem, srednje poklicnem in strokovnem ter v gimnazijskem učnem načrtu. Prav tako so predstavljeni cilji po posameznih razredih in letnikih izobraževanja. Nazadnje smo pregledali večino osnovnošolskih, srednje poklicnih in strokovnih ter gimnazijskih učbenikov ter opisali katero snov statistike obravnavajo in česa ne. Prikazali smo tudi različne pristope ter nejasnosti in napake pri opisovanju in definiranju pojmov. Spoznali smo, da imajo učbeniki zelo različen pristop pri definiranju pojmov. Največja razlika se pojavi med osnovnošolskimi in gimnazijskimi učbeniki. Največ napak in nejasnosti smo zasledili pri vpeljavi kvartilov. Sicer so pa v večini učbeniki napisani korektno in matematično pravilno. The master's thesis presents the discussion and teaching of statistics in primary and secondary education. It is split into three parts. The first part deals with basic concepts and definitions of statistics, that are taught in primary and secondary schools. The second part consists of content regarding statistics, that can be found in the curriculum of primary and secondary schools, as well as the end goals of every grade. Lastly, we take a look at the most of primary and secondary school textbooks, and describe what statistics concepts they do and don't include. We also show some approaches, obscurities and mistakes when defining concepts. We recognized, that textbooks use a lot of varying approaches when describing concepts. The biggest difference appears when comparing primary and secondary school textbooks. We saw the most mistakes and obscurities happen when introducing the subject of quartiles. Otherwise, most of the textbooks are written concretely and mathematically correct.

Kadar v vsakdanjem ˇzivljenju govorimo o povsem logiˇcnih sklepih, dostikrat uporabljamo teorijo homogenega Poissonovega procesa, ki ni niˇc drugega kot ime za teorijo ˇstetja pojavov, z doloˇcenimi lastnostmi, ki so najveˇckrat povsem oˇcitne in samoumevne za vsakega posameznika. Po drugi strani pa je za dokazovanje teh oˇcitnih lastnosti, sklepov, potrebne zelo veliko matematike, natanˇcneje teorije verjetnosti. Podobno velja za sestavljen Poissonov model, le da si ga je teˇzje predstavljati in poslediˇcno teˇzje sklepati. Sestavljen Poissonov model govori o gibanju neke vrednosti, katero linearno zvezno poveˇcujemo in hkrati diskretno zmanjˇsujemo v nekih nakljuˇcnih ˇcasih za nakljuˇcne vrednosti. V prvem delu se predstavi homogen Poissonov proces. Zaˇcne se z izrekom, ki pove, kdaj ˇstejemo dogodke, ki so porazdeljeni Poissonovo. Prvi del se nadaljuje z definiranjem lastnosti in konˇca z nazornim primerom. V drugem delu magistrskega dela se najprej navedejo predpostavke sestavljenega Poissonovega modela, ˇcemur sledi definicija. Za predstavitev uporabe sestavljenega Poissonovega modela, sta definirani tudi zelo pomembni porazdelitveni funkciji sluˇcajnih spremenljivk ”verjetnosti in ˇcasa propada”. Delo se nadaljuje z zelo pomembno formulo, s katero se raˇcuna verjetnost propada in konˇca s primeri, katerih verjetnost propada je moˇc izraˇcunati analitiˇcno. When we talk about logics we often think of theory named Homogenious Poisson process. Homogenious Poisson process is nothing more then a name of counting theory with some properties, which are at most obvious and granted for most human beings. On the other hand prooving theese obvious properties is one needs plenty of mathematical knowleadge especially probability theory. The same stands for compound Poisson model just it is harder to imagine it and conclude from it. Compound Poisson model is about a value that is continiously linearly rising and at the same time discretly falling at random times and for random values. In the first part homogenious Poisson process is defined. It starts with the theorem that indicates wheater events are Poisson distributed. It goes on defining properties and ends with ilustrating example. In the second part first the assumptions of compound Poisson model are made which are followed by a definition of a compound Poisson model. For the need of illustrating the useage of compound Poisson model two very important distributions, probability of ruin and time to ruin, are defined. Second part goes on with a very important Pollaczeck-Khinchine formula wich is used for calculating probability of ruin and ends with examples where probability of ruin can be calculated analitically.

V magistrskem delu so predstavljene osnove diskriminantne analize. Magistrsko delo je razdeljeno v štiri dele. V prvem delu so predstavljeni osnovni pojmi statistike, potrebni za obrazložitev diskriminantne analize. V drugem delu sta obrazložena postopka analize variance (ANOVA) in multivariatne analize variance (MANOVA). Tretji del je namenjen obrazložitvi diskriminantne analize. Podrobno sta obrazloženi diskriminantna analiza za dve skupini kot tudi diskriminantna analiza za več skupin. V zadnjem delu smo na podatkih, pridobljenih iz aplikacije mOIDom, opravili diskriminantno analizo. The master thesis presents basics of discriminant analysis. It is divided into four parts. The first part presents basic notions of statistics that we need to explain the discriminant analysis. The second part describes the process of analysis of variance (ANOVA) and multivariate analysis of variance (MANOVA). The third part presents discriminant analysis where discriminant analysis for two or more groups is explained. In the fourth part, the discriminant analysis based on data from mOIDom application is made.

V magistrskem delu so na algebri zgornje trikotnih matrik obravnavani in karakterizirani avtomorfizmi, jordanski izomorfizmi in Liejevi avtomorfizmi. V delu dokažemo,da je vsak avtomorfizem na algebri zgornje trikotnih matrik Tn(K), kjer je K komutativen kolobar z enoto, notranji. Vsak jordanski izomorfizem ki slika iz algebre Tn(K) v poljubno algebro A, je bodisi izomorfizem bodisi antiizomorfizem natanko tedaj, ko je kolobar K povezan. Vsak Liejev avtomorfizem na algebri Tn(F), kjer je F polje, se lahko zapiše kot vsota avtomorfizma in linearne preslikave, ki slika v center algebre Tn(F) in uniči komutatorje ali pa kot vsota negativnega antiavtomorfizma in linearne preslikave, ki slika v center algebre Tn(F) in uniči komutatorje. In the master`s thesis, automorphisms, Jordan isomorphisms and Lie automorphisms of the upper triangular matrix algebra are discussed and characterized. We prove that every automorphism on the upper triangular matrix algebra Tn(K), where K is a commutative ring with unity, is an inner automorphism. Each Jordan isomorphism, which maps from algebra Tn(K) into an algebra A, is either an isomorphism or an antiisomorphism precisely when the ring K is connected. Each Lie automorphism on algebra Tn(F), where F is a field, can be written as a sum of an automorphism and linear mapping which maps into the centre of algebra Tn(F) and vanishes on all commutators of algebra Tn(F) or a sum of an negative antiautomorphism and linear mapping which maps into the centre of algebra Tn(F) and vanishes on all commutators of algebra Tn(F).

V uvodu predstavimo posamezna poglavja diplomskega dela. V drugem poglavju podamo deÖnicije in opiöemo vrste matrik ter operacije, ki jih opravljamo na matrikah. DeÖniramo rang matrike, transponiranje in Gauss-Jordanovo eliminacijsko metodo, ki jo bomo potrebovali pri izraµcunu Moore-Penroseovega inverza. V tretjem poglavju predstavimo Moore-Penroseov inverz poljubne matrike in zapiöemo njegove lastnosti. Zadnje poglavje je namenjeno izraµcunu Moore-Penroseovega inverza. Predstavimo tri algoritme, s katerimi lahko izraµcunamo Moore-Penroseov inverz. Opiöemo njihovo raµcunsko zahtevnost in jih predstavimo na primeru. In the introduction each chapter of the graduation thesis is presented. In the second chapter we give deÖnitions and descriptions of the types of matrices and operations that are performed with them. We deÖne the rank of a matrix, the transpose of a matrix, and the Gauss-Jordan elimination method, which will be needed in the calculation of the Moore-Penrose inverse. In the third chapter we present the Moore-Penrose inverse of a matrix and its characteristics. The last chapter deals with the calculation of the Moore-Penrose inverse. We present three algorithms, which can be used to calculate the Moore-Penrose inverse. We describe their computational complexity and present them through examples.

V magistrskem delu je obravnavan problem določitve cp-ranga dane popolnoma pozitivne matrike. Uvodoma so opisane osnovne lastnosti pozitivno semidefinitnih matrik in predstavljeni so konveksni stožci evklidskega prostora V. V osrednjem delu se osredotočimo na popolnoma pozitivne matrike. Matrika A je popolnoma pozitivna, če jo lahko zapišemo kot A=BB^{T} za neko nenegativno matriko B. Dokažemo osnovne lastnosti popolnoma pozitivnih matrik ter definiramo diagonalno dominantne in primerjalne matrike. Delo zaključimo z obravnavo problema določitve cp-ranga popolnoma pozitivne matrike. Obravnavamo primer za matrike manjše velikosti ter določimo zgornjo mejo za cp-rang matrike danega ranga in matrike dane velikosti. In the master thesis the problem of determining the cp-rank of a given completely positive matrix is discussed. In the introduction the basic properties of positive semidefinite matrices are described and convex cones in euclidean space V are presented. In the main part we focus on completely positive matrices. Matrix A is completely positive if it can be decomposed as A=BB^{T}, where B is a nonnegative matrix. We prove the basic properties of totally positive matrices and define diagonally dominant and comparative matrix. The thesis is concluded with a discussion of a problem of determining the cp-rank of a completely positive matrix. We consider a case of a matrix of a smaller size and set an upper bound for cp-rank matrix of a given rank and a matrix of a given order.

Diplomsko delo se osredotoča na preučevanje Kroneckerjevih grafov. Najprej je predstavljena motivacija za vpeljavo in študij Kroneckerjevih grafov. V nadaljevanju je definiran Kroneckerjev ali tenzorski produkt matrik ter Kroneckerjev produkt grafov in njune osnovne lastnosti. V naslednjih poglavjih se pozornost nameni lastnostim Kroneckerjevih in stohastičnih Kroneckerjevih grafov. Predstavljen je porazdelitveni zakon stopnje posameznih vozlišč teh grafov. Dokazana sta zgostitveni potenčni zakon med številom vozlišč in številom povezav ter ohranjanje efektivnega premera glede na začetni graf. Pri stohastičnih Kroneckerjevih grafih so podani potrebni in zadostni pogoji za povezanost ter obstoj velike povezane komponente tega grafa. Dokazano je tudi, če je graf povezan, je premer v tem grafu konstanten. Na koncu so prikazani primeri praktične uporabe teorije, predstavljene skozi vso diplomsko nalogo. This graduation thesis focuses on the study of Kronecker graphs. First the motivation for introduction and investigation of Kronecker graphs is presented. Next are the definitions of the Kronecker or tensor product of matrices and the Kronecker product of graphs, introduced together with their basic properties. In the following chapters the focus is oriented to the study of the properties of Kronecker and stochastic Kronecker graphs. One of the important properties is the behaviour of the degree distribution. This result is folowed by the proof of the densification power law between the number of edges and the number of nodes and the proof of the conservation of the size of the effective diameter regarding the initiator graph. Next, necessary and sufficient conditions are proven for the connectivity and the existance of a giant component in the stochastic Kronecker graphs. From this follows that: under the parameters that the graph is connected, it also has a constant diameter. For conclusion examples for practical use of the theory presented throughout the thesis are given.

Glavna tema magistrske naloge so popolnoma pozitivne matrike, ki so posebni primer pozitivno semidefinitnih matrik. Vsaka realna pozitivno semidefinitna matrika A se lahko zapiše kot A=BB^T, kjer je B realna matrika. V primeru, da je B nenegativna matrika, je matrika A popolnoma pozitivna. Na začetku predstavimo osnovne pojme in definicije realnih matrik, s poudarkom na pozitivno semidefinitnih matrikah. Podamo nekaj primerov in dokažemo osnovne lastnosti teh matrik. V nadaljevanju obravnavamo popolnoma pozitivne matrike. Definiramo Hadamardov in Kroneckerjev produkt ter dokažemo, da sta oba produkta popolnoma pozitivnih matrik popolnoma pozitivni matriki. Spoznamo eno izmed metod, s katero pokažemo, da je dvojno nenegativna matrika popolnoma pozitivna. Definiramo pojem konveksni stožec ter dokažemo, da je množica popolnoma pozitivnih matrik zaprt konveksni stožec. Na algebraični in geometrijski način dokažemo, da so t.i. majhne matrike popolnoma pozitivna. Nazadnje obravnavamo diagonalno dominantne matrike ter dokažemo, da so nenegativne simetrične diagonalno dominantne matrike popolnoma pozitivne. Prav tako definiramo primerjalno matriko in dokažemo, da je matrika A popolnoma pozitivna, če je simetrična nenegativna matrika ter je njena primerjalna matrika pozitivno semidefinitna. The main topic of the master thesis are completely positive matrices, which are the special case of a positive semidefinite matrix. Every real positive semidefinite matrix A can be written in the form A=BB^T, where B is a real matrix. In the case of a nonnegative matrix B the matrix A is completely positive. The first chapter includes some basic terms and definitions of specific real matrices with an emphasis on positive semidefinite matrices. We present some examples and prove the basic properties of these matrices. In the next chapter we consider completely positive matrices. We define Hadamard and Kronecker product and show that both of these products of completely positive matrices are completely positive. We introduce one method which enables us to verify whether the doubly nonnegative matrix is completely positive. We define the concept of a convex cone and show that the set of all completely positive matrices is a closed convex cone. With algebraic and geometric approach we show that small matrices are completely positive. At the end of the thesis we treat diagonally dominant matrices and show that nonnegative symmetric diagonally dominant matrices are completely positive. We also define a comparison matrix and show that the matrix A is completely positive if it is a symmetric nonnegative matrix and if its comparison matrix is positive semidefinite.

V magistrskem delu so predstavljene osnove teorije ekstremnih vrednosti. Na začetku dela so povzeti osnovni pojmi iz verjetnosti in statistike, ki so potrebni za razumevanje snovi. Osrednji del magistrskega dela je namenjen opisu glavnih dveh pristopov, ki se uporabljata v teoriji ekstremnih vrednosti. Kot prvi pristop je to model maksimumov skupin podatkov in kot drugi je to model preseganja mejnih vrednosti. Pri vsakem pristopu je zapisana teoretična izpeljava in podan šše praktičen zgled, za lažje razumevanje. Na zgledih so tako obravnavani nekateri ekstremni dogodki v naravi, kot so količina dnevnih padavin, letne maksimalne morske gladine ... In this master thesis the basics of the theory of extreme values are presented. At the beginning of the thesis we summarize some basic concepts of probability and statistics. The central part of the master thesis is devoted to the description of the two main approaches, which are used in the theory of extreme values. The fi…rst approach is called block maxima model and the second one is called peak over threshold model. For each approach we give a theoretical derivation and for a better understanding also a practical example. Through examples we consider some extreme events in nature such as daily rainfall levels, annual maximum sea-levels...

V prvem poglavju zapišemo uvod magistrskega dela. V drugem poglavju so opisani osnovni pojmi iz teorije normiranih prostorov, linearnih preslikav in matrik. V glavnem delu formuliramo Toeplitz-Hausdorffov izrek, ki pravi, da je numerični zaklad konveksna množica. Zapišemo tudi izrek o spektralni inkluziji, ki pove, da spekter operatorja leži v numeričnem zakladu. Dokažemo lastnosti numeričnega zaklada povezanih s sebiadjungiranimi in normalnimi operatorji. Nato definiramo numerični radij, podamo njegov primer in osnovne rezultate. Posebej obravnavamo numerični zaklad operatorjev (matrik) na končno dimenzionalnih vektorskih prostorih in določimo množice, ki vsebujejo numerični zaklad. Zatem so izpeljane ocene numeričnih radijev 0-1 matrik. Na koncu zapišemo posplošitve numeričnega zaklada. In the first chapter is written an introduction of master thesis. The second chapter describes basic concepts from the theory of normed spaces, linear mapping and matrices. In main part is formulated Toeplitz-Hausdorff theorem, which says that the numerical range is convex set. We written also theorem about spectral inclusion, which says that the spectrum of operator lies in the numerical range. We prove properties of numerical range connected with selfadjoint and normal operators. Then is defined numerical radius, his example and basic results. Specially we treat numerical range operators on finite dimensions vector spaces and we determine sets containing numerical range. After that are derived numerical radius estimates of 0-1 matrices. At the end are written generalization of numerical range.