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We show that general integral triangle inequality does not hold for shifted q-integrals. Furthermore, we obtain a triangle inequality for shifted qintegrals. We also give an estimate for q-integrable product and use it to refine some recently obtained Ostrowski inequalities for quantum calculus.

Trapezoid inequality estimates the difference of the integral mean of a function on the finite interval [a, b] and the arithmetic mean of its values at the endpoints a and b. Quantum calculus is the calculus based on finite diference principle or without the concept of limits. Euler-Jackson q-difference operator and q-integral operator are discretization of ordinary derivatives and integrals and they can be generalized to its shifted versions on arbitrary domain [a, b]. In this paper we disprove a trapezoid inequality for shifted quantum integral operator appearing in the literature by giving two counterexamples. We point out some differences between the definite q-integral and Riemann integral to explain why the mistake is made and obtain corrected results. We also prove the sharpness of our new bounds in estimating the value of the quantum integral mean. Further we derive generalized sharp trapezoid inequality in which we point out the case with tightest bounds.

Ovaj priručnik je izrađen u sklopu projekta „Primjena Hrvatskog kvalifikacijskog okvira za sveučilišne studijske programe u području elektrotehnike HKO-ELE“. Cilj ovog projekta je na temelju Zakona o Hrvatskom kvalifikacijskom okviru (HKO) te metodologije propisane Pravilnikom o Registru HKO-a, definirati standarde kvalifikacija za sveučilišne studijske programe u području elektrotehnike, uključujući informacijsku i komunikacijsku tehnologiju. U sklopu projekta izradit će se standardi zanimanja i kvalifikacija, a u skladu s definiranim standardima, unaprijedit će se 14 postojećih studijskih programa koji se trenutačno izvode na visokoškolskim partnerskim ustanovama. Projekt je sufinancirala Europska unija iz Europskog socijalnog fonda prema Pozivu „Provedba HKO-a na razini visokog obrazovanja“.

We present new identities and derive related weighted Ostrowski-type inequalities by using Lidstone interpolation

We present a weighted Montgomery identity for fractional integral of a function f with respect to another function g and use it to obtain weighted Ostrowski type inequalities for fractional integrals involving functions whose first derivatives belong to Lp spaces. These inequalities are generally sharp in case p > 1 and best possible in case p = 1. Applications for the Hadamard fractional integrals are given.

New generalizations of Steffensen's inequality are obtained by means of weighted Montgomery identity with n different weight functions. Instead for a nondecreasing (1-convex) function our generalization hold for a n-convex function. Further, functionals associated to these new generalizations are observed and used to generate n exponentially and exponentially convex functions as well as to obtain new Stolarsky type means related to these functionals.

The main objective of this book is to expose some recent research in the theory of weighted integral and discrete inequalities of Ostrowski type, with emphasis on its applications such as generalizations of classical numerical quadrature rules. General identities from which the related inequalities are deduced are various kinds of different generalizations of Pečarić weighted Montgomery identity and Euler integral formula.

Error estimates of the approximations in complex domain for several integral transforms (Laplace transform, Mellin transform, Laplace-Carson transform) are given for functions f which vanish beyond a finite domain and whose derivative belongs to Lp spaces. New inequalities concerning these integral transforms of f, integral mean of f, exponential mean and logarithmic mean of the endpoints of the domain of f are presented and used to obtain associated numerical rules and error bounds of their remainders.

We use recently obtained generalization of weighted Montgomery identity to obtain new sharp weighted generalization of Ostrowski type inequality for functions whose derivatives belong to $L_p$ spaces. We use these results to obtain error bounds for the weighted two-point quadrature formulae of open and semi-open type. In particular, we apply all the results with some well-known weight functions. As special cases, the quadrature formulae of Radau and Gauss type and related inequalities are obtained.

In this paper the Mellin transform in complex domain is considered for functions f which vanish beyond a finite domain [a, b]⊂[0, ∞> and such that f′∈L_p[a, b]. New inequalities involving the Mellin transform of f, integral mean of f, exponential mean and logarithmic mean of the endpoints of the domain of f are presented.

In this paper error estimates of approximations in complex domain for the Laplace transform are given for functions which vanish beyond a finite domain and whose derivatives are from Lp spaces. New inequalities concerning the Laplace transform, as well as estimates of the difference between the two Laplace transforms are presented and used to obtain two associated numerical rules and error bounds of their remainders.

Improvement and generalization via integral power means of an recently published integral inequality similar to Qi's inequality are given.

In this talk error estimates of the approximations in complex domain for the Mellin transform M(f) are given for real functions f which vanish beyond a finite domain [a, b]⊂ and such that f′∈Lp[a, b]. The estimate of the difference between E(zln a, zln b)(L(a, b))⁻¹∫_{;a};^{;b};f(t)dt and M(f)(z) is given, where E(z, w) is exponential mean of complex numbers z and w, and L(a, b) is logarithmic mean of real numbers a and b. New inequalities concerning the estimate of the difference between two Mellin transforms are presented. These are used to obtain two associated numerical quadrature rules and error bounds for their reminders.

The weighted Montgomery identity states that $$f\left(x\right)=\int_a^b w\left(t\right)f\left(t\right)dt+\int_a^b P_w\left(x, t\right)f'\left(t\right)dt, $$ where $f:\left[a, b\right]\to \mathbf{; ; R}; ; $ is a differentiable function such that $f'$ is integrable on $\left[a, b\right]$, $w:\left[a, b\right]\to\left[0, \infty\right\rangle$ is some normalized weighted function and $P_w\left(x, t\right)$ is the weighted Peano kernel. We construct an extension of this identity for Riemann-Stieltjes integral and obtain some new inequalities for functions of bounded variation and for functions whose first derivatives belong to $L_p$ spaces. We use these results to obtain bounds of the remainder $E\left(f\right)$ of the weighted two-point quadrature formula of semiclosed type $$\int_a^b w\left(t\right)f\left(t\right)dt=A_1 f\left(a\right)+A_2f\left(x_2\right)+E\left(f\right), $$ where $A_1+A_2=1.$ In particular, we apply all the results with some well-known weighted functions. As special cases, the weighted quadrature formulae of Radau type and related inequalities are obtained.

Some new inequalities concerning approximations of the Laplace transform of functions in Lp spaces are presented. Also, new estimates of the difference between the two Laplace transforms are given. These results are used to obtain bounds of associated numerical quadrature formula.

We present some new inequalities concerning approximations of the Fourier transform and estimates of the difference between the two Fourier transforms of functions in Lp spaces. These are used to obtain bounds of associated numerical quadrature formula.

Some of the recent results of various authors concerning weighted generalizations of integral and discrete Euler identity are presented. In most cases these identities are used to obtain weighted generalizations of the Ostrowski-type inequalities.

Some new generalizations of the integral and discrete weighted Euler identity are presented as well as related generalizations of the integral and discrete Ostrowski type inequalities.

In this paper we establish some new Ostrowski type integral inequalities, by using the Montgomery identity and Taylor's formula.

Some new extensions of the weighted Montgomery identity are given, and used to generalize the Ostrowski inequality and the estimates of the difference of two integral means for alpha-L-Holder type functions

In this paper we established some Čebišev-Gruss type integral inequalities, by using the extension of the Montgomery identity.

Few different versions of Landau type inequalities are given, each using the Ostrowski inequality or Ostrowski type inequalities.

Four different versions of Landau type inequalities are given. First one is obtained using extension of the Montgomery identity via Taylor's formula, then another for functions with Holder continuous derivatives, then one using the Euler identity and the one obtained via the Fink identity.

Few different versions of Gruss type inequalities are given, using Euler identity, Fink identity and Anastassiou identity

Discrete analogue of the weighted Montgomery identity (i.e. Euler identity) for finite seqences of vectors in normed linear space is given as well as discrete analogue of Ostrowski type inequalities and estimations of difference of two arithmetic means

We present discrete analogue of the weighted Montgomery identity as well as it's generalizations for finite and infinite sequences. Further we use theese identities to obtain some discrete Ostrowski type inequalities and the estimates of the difference between two arithmetic means.

U ovom radu dana su težinska poopćenja poznate Eulerove integralne formule, kao i njene diskretne inačice. Ove formule su korištene za poboljšanje nekih poznatih nejednakosti, kao i za dobivanje novih nejednakosti tipa Ostrowskog, Eulerovih trapeznih nejednakosti, Eulerovih nejednakosti o središnjoj točki, ocjena razlika integralnih sredina i to u kontinuiranom i u diskretnom slučaju.

New inequalities concerning the Fourier transform are given. Estimates of the difference between two Fourier transforms and even bounds to the associated numerical quadrature formulae are provided as well.

Ovaj rad bavi se svojstvima sparivanja u bipartitnim grafovima i njihovim primjenama. Uvodno se definira bipartitan graf, daje algoritam za provjeru bipartitnosti grafa i navode se osnovni oblici sparivanja. U glavnom dijelu se pomnije obrađuju sparivanja u bipartitnim grafovima i pokazuju se neke primjene. Potom se analiziraju Mađarski i Hopcroft-Karpov algoritam kao učinkoviti algoritmi za pronalazak savršenih sparivanja i sparivanja maksimalne kardinalnosti u bipartitnom grafu. Na kraju se povezuju maskimalna sparivanja s problemom parkiranja i s molekulskim strukturama vinilnih polimera. This thesis focuses on the properties of matchings in bipartite graphs and their applications. The introduction defines a bipartite graph, provides an algorithm for checking whether a given graph is bipartite, and lists the basic types of matchings. The main part explains matchings in bipartite graphs in more detail and shows some of the applications. Then, the Hungarian and Hopcroft-Karp algorithms are analyzed as efficient algorithms for finding perfect matchings and maximum cardinality matchings in a bipartite graph. Finally, maximal matchings are associated with the parking problem and with the molecular structures of vinyl polymers.

Benfordov, Zipfov i Paretov zakon opisuju razdiobu velikih skupova podataka – Benfordov u kontekstu numeričkih vrijednosti bez definiranih maksimuma i minimuma, poput cijena dionica na tržištu ili broja stanovnika po gradovima, a Zipfov u okviru lingvističkih analiza i procjene odnosa učestalosti riječi, dok Paretov utvrđuje numerički model odnosa utjecaja ulaznih faktora nekog procesa na krajnji rezultat. U radu su prikazani i grafovi raspodjele stvarnih podataka preuzetih s web stranica. Benford's, Zipf's, and Pareto's laws all describe the distribution of large data sets — Benford's in the context of numerical values without defined maximums and minimums, such as market share prices or population per country, Zipf's in linguistic analysis and word frequency estimation and Pareto's establishes a numeric model which shows how input values influence final results of a certain process. The paper also presents charts of the distribution of actual data taken from statistical web pages.

U ovom radu prezentirani su osnovni algoritmi pretraživanja grafa u širinu i dubinu, te kako se njihova struktura može iskoristiti za napredniji algoritam A* te algoritme testiranja 2-obojivosti grafa i topološkog sortiranja koji se koriste za mnoge učestale probleme u teoriji grafova. Definirani su osnovni pojmovi teorije grafova potrebni za razumijevanje danih algoritama. Dodatno, pokazana je implementacija algoritama u Javi te je objašnjeno kako koristiti isti programski kod za više algoritama. This paper presents basic graph search algorithms Breadth-First Search and Depth- First Search and how their structure can be used for more advanced algorithms such as A*, bipartiteness testing and topological search algorithm which are used for many graph theory problems. Basic graph theory definitions are given in order to understand the given algorithms. Additionally, the implementation of the algorithms is shown in Java and it is explained how to reuse programming code for multiple algorithms.

U moderno vrijeme, primjene teorije grafova su sveprisutne. Pojam grafa, osnovna klasifikacija grafova te reprezentacija grafa u memoriji, nužni su razumijevanje složenijih postupaka obilaska grafa i pronalaska najkraćeg puta, pronalaska povezanih komponenti i minimalnog razapinjućeg stabla. Pretraživanjem u širinu ili BFS algoritmom sustavno se pretražuju vrhovi jedne razine grafa prije prelaska na sljedeću razinu. Suprotno, pretraživanjem u dubinu ili DFS algoritmom obilaze se vrhovi najdublje razine, prije nego se backtrackingom algoritam vraća na pliće razine. U slučaju težinskog grafa, Dijkstrinim algoritmom pronalazi se put s najkraćom cijenom. Pretraživanje u širinu specijalan je slučaj Dijkstrinog algoritma, u kojem su težine bridova konstantne. Osnovna povezanost komponenti definirana nad neusmjerenim grafom, u slučaju usmjerenog grafa dijeli se na slabu i jaku povezanost. Svi oblici povezanosti rješivi su algoritmima BFS i DFS. Neovisan o njima, problem pronalaska minimalnog razapinjućeg stabla rješava se Kruskalovim algoritmom. Efikasno rješenje ostvaruje se specifičnom podatkovnom strukturom za upravljanje nepreklapajućim podskupovima inicijalnog skupa, tzv. strukturom Union-find. In modern times, applications of graph theory are omnipresent. Concept of graph, basic graph classification and the representation of the graph in the memory are necessary to understand the more complex procedures of traversing the graph and finding the shortest path, finding connected components and the minimum spanning tree. Breadth-first search or BFS systematically scans vertices of one level before going to the next level. On the contrary, depth-first search or DFS visits the vertices at the deepest level before the algorithm, using backtracking technique, returns to shallower levels. In the case of a weighted graph, Dijkstra's algorithm finds the path with the lowest cost. Breadth-first search is a special case of Dijkstra's algorithm, in which edge weights are constant. Basic definition of connectivity defined for an undirected graph, in case of a directed graph is divided into weak and strong connectivity. All forms of connectivity are solved by BFS or DFS. Regardless of them, the problem of finding the minimum spanning tree is solved by Kruskal's algorithm. An efficient solution is achieved through usage of the specific data structure for managing non-overlapping subsets of the initial set, so-called Union-find.

U ovom radu opisan je i analiziran problem prebrojavanja svih razmještaja. Problem je podijeljen u 12 slučajeva. Korištene su pretpostavke o različitosti ili identičnosti lopta (objekta) i kutija. Također su korištene pretpostavke o tome jesu li prazne kutije dozvoljene ili nisu. Objašnjeni i uvedeni su termini poput particija broja, particija skupa, Stirlingov broj druge vrste, itd. koji su korišteni u rješavanju različitih slučajeva ovog problema. This paper describes and analyzes the problem of counting all placements. The problem is divided into 12 cases. The assumptions on the difference or identity of the ball (object) and the box are used. The assumptions on whether empty boxes are allowed or not are also used. New terms such as partition numbers, partitions, Stirling's number of other types, etc. are explained and introduced which are used to solve different cases of this problem.

U ovom radu obrađena je Laplaceova transformacija, važan matematički alat za rješavanje diferencijalnih i integralnih jednadžbi. Dan je pregled svojstava, primjera, glavnih formula i teorema Laplaceove transformacije i inverznog postupka te najvažnijih Laplaceovih transformata. Prikazana je važnost transformacije u elektrotehnici kroz rješavanje diferencijalnih jednadžbi koje se tipično pojavljuju u problemima struke, kao što su opisivanje sustava upravljanja i električnih mreža. This paper deals with Laplace transform, an important mathematical tool for solving differential and integral equations. The overview of the properties, examples, main formulas and theorems of Laplace’s transformation and its inverse method is given. The importance of L. transformation in electrical engineering is demonstrated by solving differential equations typically occurring in the profession's problems, such as those describing control systems and electrical networks.

Možemo reći da je to zapravo znanstvena disciplina koja proučava racionalno ponašanje međuovisnih igrača, odnosno načine na koji strateške interakcije između racionalnih igrača rezultiraju ishodima ovisno o prioritetima i sklonostima (tj. korisnosti) igrača. Formalno nastaje početkom 18. stoljeća, a najveći uzlet doživljava tijekom 20. stoljeća. Neki od najvažnijih teoretičara igara bili su John Nash i John van Neumann. Postoji više vrsta igara koje se proučavaju i o tome koju vrstu igru proučavamo postoje različite strategije koje možemo primijeniti da bismo došli do nekog ishoda. Neke od vrsta igra su kooperativne i nekooperativne, simetrične i asimetrične, sekvencijske i simultane, igre s nultom i promjenjivom sumom... Postoje i različite strategije koje možemo primjenjivati, a glavne kategorije strategija su mješovita, čista i dominantna. Izbor strategije ovisi o vrsti igrača, njihovoj sklonosti riziku i različitim drugim faktorima. Na kraju smo prošli kroz razne primjene teorije igara u komunikacijskim mrežama i uvidjeli da se već koristi u npr. WLAN-ovima i za kontrolu pristupa, a njezine će primjene u budućnosti biti samo još veće. We can say that this is actually a scientific discipline that examines the rational behavior of interdependent players, that is, the ways in which strategic interactions between rational players result in outcome based on the priorities and preferences (the usefulness) of the player. It formally originated at the beginning of the 18th century, but it took off in the 20th century. Some of the most important theoreticians of the game theory were John Nash and John van Neumann. There are several types of games being studied and several types of strategies we can apply to achieve a goal. Some of the game types are cooperative and noncooperative, symmetrical and asymmetrical, sequential and simultaneous, games with a zero sum and variable sum… There are different strategies we can apply, and the main categories of strategies are mixed, pure, and dominant. The choice of strategy depends on the type of player, their likelihood of risk and various other factors. At the end of this paper, I went through the various applications of game theory in communication networks and showed that it is already in use in, for example, WLANs and access control, and its applications will be even greater in the future.

U ovom radu obrađeni su Catalanovi, Motzkinovi i Narayanini brojevi, te Dyckovi putevi. Pokazali smo da su Catalanovi brojevi niz brojeva. Catalanove brojeve smo koristili kao uvod u Dyckove puteve, te Dyckov jezik. Pokazali smo kako je broj Dyckovih puteva duljine n jednak Catalanovom broju Cn. Uveli smo i Naranayine brojeve koji određuju broj Dyckovih puteva duljine n s k vrhova. Zadnji pojam bili su Motzkinovi brojevi koji određuju broj istoimenih puteva. Uz to, napravljen je i računalni program koji računa i iscrtava Motzkinove i Dyckove puteve. This paper is about Catalan, Motzkin and Narayana numbers, as well as Dyck paths. We have shown that Catalan numbers are a sequence. Catalan numbers have been used as an introduction to Dyck paths and the Dyck language. We have shown that the number of Dyck paths for a certain n is equal to the Catalan number Cn, and that the number of Dyck paths for a certain n with k peaks is equal to the Narayana number N(n, k). The last term we introduced was Motzkin numbers, which count the number of paths with the same name. In addition to that, a computer program has been written that counts and draws Motzkin and Dyck paths.

We consider the elements a of a unital C*- algebra A which are Roberts orthogonal to the unit e of A. We present a characterization of this orthogonality in terms of the Davis- Wielandt shell of a and show that, for certain classes of elements of A, the Roberts orthogonality of a and e is equivalent to the symmetry of the numerical range of a with respect to the origin. When A is the C*-algebra of all 2x2 complex matrices, we give a complete description of the form of 2x2 matrices which are Roberts orthogonal.

In this talk we give improvements of several Jensen-type inequalities, giving the necessary and sufficient conditions for them to hold in case of the real Stieltjes measure d \lambda, not necessarily positive. As an application, we derive corresponding mean-value theorems, investigate the exponential and logarithmic convexity of the differences between the left-hand and the right-hand side of the obtained inequalities, also giving several examples of the families of functions for which those results can be applied.

The classical Jensen inequality is expressed by internally dividing points and so are non- commutative Jensen inequalities. Considering that it is expressed by externally dividing points, we shall discuss complementary to Choi-Davis-Jensen type inequalities from this point of view. \\ For example, looking at the external version of the Choi-Davis-Jensen inequality given in [1], we obtain complementary to this inequality under condition that f is a real valued concave function. Moreover, we observe improved and generalized inequalities by using results without concavity [2]. \\ References: \\ [1] J. I. Fujii, J. Pečarić and Y. Seo, The Jensen inequality in an external formula, J. Math. Inequal., 4 (2012), 473--480. \\ [2] J. Mićić, H.R. Moradi and S. Furuichi, Some complementary inequalities to Jensen's operator inequality, J. Inequal. Appl., 2018:25 (2018), 1--20.

A convex combination is the basic notion related to the discrete form of Jensen's inequality. We consider infinite convex combinations and Jensen's inequality. By applying the theory of series, the notion of a convex combination (containing finite number of terms) can be extended to the series (containing infinite number of terms). We present an infinite convex combination on the bounded closed interval of real numbers.

The main purpose of this work is to give the uniform treatment of the Jensen type inequalities. Using different Green functions, we give new conditions for Jensen's inequality to hold in case of the real Stieltjes measure d \lambda, which is not necessarily positive. Using the same method we also give adequate results for the converses of the Jensen inequality and as a consequence we get the results concerning the Hermite-Hadamard inequality. We also give some further interesting results and applications.

In this work, the Green's function of order two is used together with Fink's approach in Ostrowski's inequality to represent the difference between the sides of the Sherman's inequality. Čebyšev, Grüss and Ostrowski-type inequalities are used to obtain several bounds of the presented Sherman-type inequality.