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In this paper we present estimation formulas for the expectations of power means of large data and associate them with means of probability distribution and means of random sample. The proposed method follows from the asymptotic expansion of power means which is applicable for sufficiently large data and it is especially useful when value of such expectation is hard to obtain. We will show the accuracy of these approximations for random samples which have uniform and normal distribution and analyse their behavior for large sample volume.

Kao uvod u glavni dio ovog rada izveden je asimptotski razvoj integralne sredine poligama funkcije. Dana je rekurzivna relacija koja određuje koeficijente u tom razvoju i dokazana su neka njihova svojstva. Taj je pristup potom generaliziran na integralne sredine funkcija koje imaju asimptotski razvoj u red potencija. U tu svrhu dobiven je algoritam za rješavanje jednadžbi oblika \(B(A(x)) = C(x)\), gdje su asimptotski razvoji funkcija B i C poznati, a zatim su rezultati ilustrirani na primjerima nekih bitnih integralnih sredina. Proučavanje asimptotskih razvoja funkcija u red potencija pokazalo se korisnim i kod usporedbe sredina. Tako su izvedeni nužni uvjeti za usporedbu parametarskih sredina. Rad je podijeljen u nekoliko cjelina. U uvodnom poglavlju definirani su asimptotski razvoji i opisana su osnovna svojstva vezana uz operacije s asimptotskim razvojima. Drugo poglavlje vezano je uz gama funkciju. Korištenjem poznatih rezultata i zapisa pomoću kvocijenta gama funkcija dobiven je asimptotski razvoj Wallisovog niza \(W_n\) na temelju čega su poboljšane neke postojeće nejednakosti vezane za ovaj niz. Na sličan način su dobiveni koeficijenti u asimptotskom razvoju Wallisove potencije i opisana su neka njihova svojstva. Izveden je asimptotski razvoj Wallisove funkcije preko poligama funkcija. Zatim su dobiveni i analizirani koeficijenti u razvoju integralne sredine poligama funkcije što dovodi do promatranja integralnih sredina općenito. U trećem poglavlju opisan je algoritam za računanje koeficijenata u asimptotskom razvoju integralne sredine funkcije koja ima asimptotski razvoj te su dobiveni rezultati objašnjeni na nekim konkretnim primjerima. Slične tehnike korištene su u četvrtom poglavlju kod analize klasičnih te parametarskih sredina. Poznavajući koeficijente u asimptotskim razvojima izvedeni su nužni uvjeti za usporedbu tih sredina. Također su promatrane konveksne kombinacije dviju sredina, posebno kada su uključene Seiffertove ili Neuman-Sándorova sredina. Dokazane su neke nejednakosti dok su druge ostavljene u obliku slutnji. As an introduction to the main part of this work, the asymptotic expansion of the integral mean of polygamma function was derived. The recursive relation which determines coefficients in this expansion was given. This approach is then generalized to the integral mean of the functions which possess asymptotic power series expansion. To this end, algorithm for solving equations of the form \(B(A(x)) = C(x)\) is obtained, where asymptotic expansions of functions B and C are known, and the results are illustrated by the examples of some relevant integral means. Studying asymptotic expansions turned out to be useful when comparing means. Thus, the necessary conditions for the comparison of parametric means were derived. This thesis is divided in several parts. In the introduction, asymptotic expansions were defined and some basic properties about manipulations with asymptotic series were described. Second chapter deals with gamma and related functions. Using known results and expressions through quotient of gamma functions, asymptotic expansion of Wallis sequence \(W_n\) was derived, which provided improvements of some existing inequalities related to this sequence. In the same manner, coefficients in asymptotic expansion of the Wallis power function were found and some properties were described. Asymptotic expansion of the Wallis function through polygamma function was derived. Then the coefficients in asymptotic expansion of integral mean of polygamma function were found which motivated the studying of integral means in general. In the third and main chapter the algorithm for calculating coefficients in asymptotic expansion of integral mean of a function possessing asymptotic expansion was described and obtained results were explained on some specific examples. Similar techniques were used in the fourth chapter in the analysis of classical and parameter means. By knowing coefficients in asymptotic expansions, necessary conditions for comparison of those means were derived. Convex combinations of two means which include Seiffert or Neuman-Sándor mean were also observed. Some new inequalities were proved while others were stated in form of conjectures.

The subject of this paper is linear combinations of reciprocal values of classical bivariate means and their behavior for translated values of the arguments. Based on the known asymptotic expansions some optimal parameters were obtained.

Da bi se dobio odgovor na problem predviđanja budućeg uspjeha mladih sportaša, isti su nasumično podijeljeni u dvije skupine - kontrolnu i radnu. Radna se skupina koristila za razvoj i optimizaciju novog višedimenzionalnog modela koji je kombinirao SVD i ROC analizu. Nakon optimizacije, model je testiran na kontrolnoj skupini. Korištenjem SVD analize nad antropometričkim podacima i podacima fizičke spreme, objektivno su određeni kriteriji za razlikovanje budućeg uspjeha u karijeri sa senzitivnošću većom od 80%. Zbog toga zaključujemo da bi predložena analiza mogla biti korisna prilikom raspoznavanja talenta igrača. To solve the problem of young players’ future career attainment prediction, players were blindly and randomly divided into two datasets - explanatory and validation. The exploratory dataset was used to develop and optimize a novel higher-dimensional model, which combined SVD with ROC analysis. Once optimized, the model was tested using the validation dataset. Through the use of SVD analysis it was possible to objectively identify criteria to distinguish future career attainment with a sensitivity over 80% using anthropometric and fitness data alone. As such, this suggests that proposed analysis may be a useful analysis tool for research and practice within talent identification.

U ovom radu opisani su osnovni principi Monte Carlo metode, dana je teorijska podloga vjerojatnosti, te opisan i implementiran program temeljen na Monte Carlo metodi koji može odigrati igru potapanja brodova na jednakoj ili boljoj razini u odnosu na čovjeka. Program je implementiran u programskom jeziku Python, uz biblioteke Numpy i Matplotlib. Uspored̄ena je Monte Carlo metoda s determinističkom kroz igru potapanja brodova i grafički su prikazani rezultati koje postiže Monte Carlo metoda. This paper describes basic principles of the Monte Carlo method. The paper also includes the theoretical base of probability, as well as the description and implementa- tion of a program created on the basis of Monte Carlo method. The program can play the Battle Shop game at the same level as a human being, or, potentially, at an even higher level. The program is developed using Python programming language, inclu- ding Numpy and Malplotib libraries. Finally, Monte Carlo method is compared with the deterministic method via “Battleship” game, and the achieved results of the Monte Carlo method are graphically presented.

U ovom radu pružana je alternativa klasičnom pristupu rangiranja sportskih timova koji uključuje iscrpna natjecanja timova svaki sa svakim, generalno jednak broj puta. Alternativa se pokazala vremenski učinkovitijom, te uz korisnikovu intuiciju potencijalno i jednaka klasičnom svatko sa svakim načinu rangiranja. Korišten je problem kvadratnog pridodjeljivanja uz posebne heurističke metode umjetne inteligencije radi bržeg pronalaska optimaliteta koji nije garantiran. Postupak se razlikuje od skupa podataka do skupa podataka, i ovisan je o korisnički unesenim parametrima, no pruža veću fleksibilnost, poštenost u vidu pristranosti, te brzinu nauštrb težine uporabe nad klasičnim pristupom. This work provides an alternative way of ranking sports teams to the classic round-robin extensive team by team competitive rankings. This alternative has shown to be more time efficient, coupled with the user's intuition potentially an equal way of ranking sports teams to the classic round-robin rankings. The work utilises the quadratic assignment problem coupled with specialised heuristic methods of artificial intelligence in order to speed up the process of finding optimality which is not guaranteed. The alternative approach provides different results depending on the provided dataset and the user's input of parameters, but it provides a greater degree of flexibility, fairness in the case of bias, and speed at the expense of ease of use over the classic approach.