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H-measures and semiclassical (Wigner) measures were introduced in earlyn 1990s and since then they have found numerous applications in problems involving $\mathrm{L}^2$ weakly converging sequences. Although they are similar objects, neither of them is a generalisation of the other, the fundamental difference between them being the fact that semiclassical measures have a characteristic length, while H-measures have none. Recently introduced objects, the one-scale H-measures, generalise both of them, thus encompassing properties of both. The main aim of this paper is to fully develop this theory to the $\mathrm{L}^p$ setting, $p\in(1,\infty)$, by constructing one-scale H-distributions, a generalisation of one-scale H-measures and, at the same time, of H-distributions, a generalisation of H-measures to the $\mathrm{L}^p$ setting, without any characteristic length. We also address an alternative approach to $\mathrm{L}^p$ extension of semiclassical measures via the Wigner transform, introducing new type of objects (semiclassical distributions). Furthermore, we derive a localisation principle in a rather general form, suitable for problems with a characteristic length, as well as those without a specific characteristic length, providing some applications., 41 pages, 3 figures

Osteoarthritis (OA) can be treated using either a pharmacological or non-pharmacological approach, or a combination of both. The purpose of the present study was to investigate the efficacy of crystalline glucosamine sulfate (CGS) in patients with knee OA. This open-label prospective study (with a 12-month follow-up) included 111 patients of both genders suffering from knee OA, who attended the Special Hospital for Rheumatic Diseases in Novi Sad, Serbia during the 2011-2013 period. Patients were divided into the experimental (n=52) and the control (n=59) group. While the former was prescribed CGS 1500 mg/day, the latter was treated with nonsteroidal anti-inflammatory drugs (NSAIDs) according to the standard protocol. The efficacy of both treatment modes was assessed using the Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) and Lequesne index, along with the radiological findings which involved knee joint space width (JSW) measurements. One year following the initial assessment, all patients reported pain intensity reduction; however, those in the CGS group experienced significantly lower pain intensity when compared with controls. At the end of the study, no reduction in the progression of joint structure damage (p>0.5) was noted in either group. Thus, while CGS demonstrated symptomatic efficacy, it failed to delay the progression of knee OA., Osteoartritis (OA) može se liječiti farmakološkim ili ne-farmakološkim pristupom, ili kombinacijom oba pristupa. Cilj ove studio bio je istražiti djelotvornost kristaliziranog glukozamin sulfata (CGS) u pacijenata sa OA koljena. Ova open-label prospektivna studija (s 12 mjeseci praćenja) uključivala je 111 pacijenata oba spola koji su patili od OA koljena te koji su liječeni u Specijalnoj bolnici za reumatološke bolesti u Novome Sadi u Srbiji u razdoblju od 2011. do 2013. godine. Pacijenti su podijeljeni u eksperimentalnu (n=52) i kontrolnu (n=59) skupinu. Eksperimentalnoj je skupini propisano 1500 mg/dnevno CGS-a, dok je kontrolna skupina liječena nesteroidnim protuupalni, lijekovima (engl. non-steroidal anti-inflammatory drugs; NSAID) prema standardnom protokolu. Djelotvornost oba modaliteta liječenja analizirana je pomoću Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) i Lequesne indexa, uz radiološke rezultate koji su uključivali mjerenje širine zajedničkog prostora (JSW) koljena. Godinu dana nakon prvog pregleda, svi pacijenti su prijavili smanjenje intenziteta boli; no pacijenti u CGS skupini iskusili su značajno niži intenzitet boli u usporedbi s kontrolnom skupinom. Na kraju studije nije zamijećena progresija oštećenja strukture zgloba (p>0.5) ni u jednoj skupini. Stoga, iako je CGS pokazao djelotvornost u smanjivanju simptoma, nije spriječio progresiju OA koljena.

The theory of homogenisation can be seen as a way to simplify complicated heterogeneous models (i.e.~to average heterogeneous media in order to derive effective properties) or to pass from the micro-scale to the macro-scale (e.g. to derive physical laws on a macro-scale starting from the atomic scale). In the mathematical theory of homogenisation this is achieved by studying differential equations with parameter-dependent coefficients for which we want to pass to the limit (with respect to that parameter) and get the effective equation. In this talk we are interested in nonlocal linear elliptic operators with non-symmetric kernels.

In this talk we study entropy solutions of degenerate (homogeneous) scalar parabolic equations on a bounded domain and prove that, under certain conditions, all such solutions admit the strong trace at the boundary. Here the degeneracy appears as the diffusion matrix is only positive semi-definite, i.e. it can be equal to zero in (variable-dependent) directions. Although the well-posedness for the Cauchy problems (under the strong trace) corresponding to such equations is well established in quite general situations, our result implies that the weak trace suffices for uniqueness. Moreover, this result could be an important step into formulating the initial boundary value problem in the sense of Bardos, LeRoux and Nédélec.

In this talk we study entropy solutions to the degenerate parabolic equation $$ \partial_t u + \operatorname{; ; div}; ; _x f(u) = \operatorname{; ; div}; ; _x\bigl(a(u)\nabla u\bigr) \, , \quad (t, x)\in (0, +\infty)\times \mathbb{; ; R}; ; ^d \ ; , $$ subject to the initial condition $u(0, \cdot)=u_0$. Here the degeneracy appears as the matrix $a(\lambda)$ is only positive semi- definite, i.e.~it can be equal to zero in some directions and the directions can depend on $\lambda$. Equations of this form often occur in modelling flows in porous media and sedimentation- consolidation processes. Since in the special case $a=0$ (which is allowed) the equations becomes a (multidimensional) scalar conservation law, in the talk we start by recalling some known results in this setting (e.g.~non-uniqueness of weak solutions (see Figure \ref{; ; fig:burgers}; ; ), notion of entropy solutions, existence of solutions and traces), and emphasising some open problems (e.g.~heterogeneous case). \begin{; ; figure}; ; [h!] \centering \includegraphics{; ; burgers2.mps}; ; \vskip1cm \includegraphics{; ; burgers1.mps}; ; \caption{; ; Visualisation of the characteristics for Burgers' equation ($d=1$, $f(\lambda)=\lambda^2/2$, $a=0$) for two initial data. On the first one we have a problem of intersecting characteristics, while on the second one the problem is how to uniquely extend the solution to the region not covered by any characteristic.}; ; \label{; ; fig:burgers}; ; \end{; ; figure}; ; The notion of entropy solutions easily generalises for the degenerate parabolic equations, and it is well-established in the literature. Moreover, an equivalent kinetic formulation is also available, which will be used in this talk. Our contributions consist of obtaining the existence of entropy solutions for the starting problem by applying a suitable velocity averaging result, and proving that all such solutions admit the strong trace at $t=0$. The latter results implies that the weak trace suffices for uniqueness and could be an important step into formulating the initial boundary value problem in the sense of Bardos, LeRoux and N\'ed\'elec. This is joint work with Marin Mišur and Darko Mitrović.

In this talk we study solutions to the degenerate parabolic equation $$ \partial_t u + \operatorname{; ; div}; ; _x f(u) = \operatorname{; ; div}; ; _x(a(u)\nabla u) \, , $$ subject to the initial condition $u(0, \cdot)=u_0$. Here the degeneracy appears as the matrix $a(\lambda)$ is only positive semi- definite, i.e.~it can be equal to zero in some directions. Moreover, the directions can depend on $\lambda$. %, which is the main novelty. Equations of this form often occur in modelling flows in porous media and sedimentation- consolidation processes. As a consequence of the degeneracy, solutions could be singular, so one needs to justify the meaning of the initial condition. A standard way is to show that $u_0$ is the strong trace of a solution $u$ at $t=0$. The notion of strong traces proved to be very useful in showing the uniqueness of the solution to scalar conservation laws with discontinuous flux. We prove existence of strong traces for entropy solutions to the equation above. No non-degeneracy conditions on $f$ and $a$ are required. The proof is based on the blow-up techniques and the velocity averaging result for ultra-parabolic equations, applied to the kinetic formulation. In some special situations, we can allow even for heterogeneous fluxes and diffusion matrices, i.e.~$f$ and $a$ dependent of $(t, x)$ as well.

In this talk we study solutions to the degenerate parabolic equation $$ \partial_t u + \operatorname{; ; ; div}; ; ; _x f(u) = \operatorname{; ; ; div}; ; ; _x(a(u)\nabla u) \, , $$ subject to the initial condition $u(0, \cdot)=u_0$. Here the degeneracy appears as the matrix $a(\lambda)$ is only positive semi- definite, i.e.~it can be equal to zero in some directions. Moreover, the degeneracy can occur in any direction, which is the main novelty. Equations of this form often occure in modelling flows in porous media and sedimentation- consolidation processes. As a consequence of the degeneracy, solutions could be singular, so one needs to justify the meaning of the initial condition. A standard way is to show that $u_0$ is the strong trace of a solution $u$ at $t=0$. The notion of strong traces proved to be very useful in showing the uniqueness of the solution to scalar conservation laws with discontinuous flux. We prove existence of strong traces for entropy solutions to the equation above under the non- degeneracy conditions. The proof is based on the blow-up techniques, where a variant of microlocal defect functional is used and applied to the kinetic formulation of the equation above. This is joint work with Darko Mitrovi\'c.

The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems via the universal operator extension theory of dual pairs (Grubb, 1968). For a given Friedrichs system the existence of a boundary condition such that the problem is well-posed is shown, as well as a classification of all such boundary conditions.

Teoriju pozitivnih simetričnih sustava ili Friedrichsovih sustava uveo je sam Kurt Otto Friedrichs pedesetih godina prošlog stoljeća. Riječ je o posebnoj klasi sustava parcijalnih diferencijalnih jednadžbi prvog reda koja je ipak dovoljno općenita da obuhvaća većinu (polu)linearnih jednadžbi matematičke fizike, neovisno o njihovom tipu. U novije vrijeme su posebno zanimljivi pri numeričkom rješavanju parcijalnih diferencijalnih jednadžbi, što je motiviralo uvođenje apstraktne formulacije u terminima Hilbertovih prostora u radovima Ern, Guermond i Caplain (2007.), te Antonić i Burazin (2010.). U predavanju ćemo prezentirati nedavne rezultate povezivanja Friedrichsovih operatora s teorijom proširenja operatora, kao i nekoliko primjera od interesa. Predavanje je temeljeno na zajedničkom radu s N. Antonićem i A. Michelnagelom.

The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems via the universal operator extension theory of dual pairs (Grubb, 1968). For a given Friedrichs system the existence of a boundary condition such that the problem is well-posed is shown, as well as a classification of all such boundary conditions. Within this framework we study self-adjoint (or even only closed) realisations of the Hamiltonians of contact interactions.

We prove a velocity averaging result for the L^p-bounded, p>=2, sequence of solutions to degenerate dissipative transport equations. As a consequence, we prove existence of weak solution to degenerate parabolic equations with discontinuous flux. In the proofs a new variant of H-measures (or microlocal defect measures) is introduced and used, which is adapted to equations that change type.

Osnovni cilj ovog istraživanja je utvrditi funkcionalne osobitosti nogometaša pionirskog, kadetskog i juniorskog uzrasta. Istraživanje je provedeno na uzorku od 66 nogometaša koji se natječu u Prvoj hrvatskoj nogometnoj ligi. Ispitanici su podijel jeni u tri dobne skupine: U-15; pioniri (N=22), U-17; kadeti (N=22) i U-19; juniori (N=22). Nogometaši pojedine dobne skupine podijeljeni su na tri subuzorka: obrambeni igrači, vezni igrači i napadači, te po kvaliteti na 1. ekipu i 2. ekipu. Na svim ispitanicima je primijenjena skupina od 3 testa za procjenu morfoloških obilježja, 15 testova za procjenu funkcionalnih sposobnosti, te 6 testova za procjenu ventilacijskih karakteristika. Kako bi se ispunili postavljeni ciljevi i testirale hipoteze korištene su sljedeće metode obrade podataka: deskriptivna statistika, faktorska analiza varijance s Fischer LSD post-hoc testom, kao i faktorska analiza varijance s Tukey-Kramer Unequal N HSD post-hoc testom. Temeljem dobivenih rezultata moguće je konstatirati slijedeće: o Nogometnim treningom moguće je unaprijediti rad srčano-žilnog i dišnog sustava, te tako povećati specifični radni kapacitet, odnosno radnu sposobnost (izdržljivost) sportaša. Dobivene razlike u parametrima za procjenu morfoloških karakteristika, spiroergometrijskih te spirometrijskih parametara između nogometaša različitih uzrasnih kategorija mogu se pored nogometnog treninga, također pripisati i rastu i sazrijevanju; o Razlike unutar svake od navedenih skupina nogometaša po igračkim pozicijama (obrana, vezni red, napad), prvenstveno su uvjetovane pozicijskim specifičnostima; o Od početka igranja nogometa do ulaska u pubertet, djeca imaju uravnotežen rast i razvoj. U pubertetu dolazi do ubrzanog fizičkog rasta i razvoja, stoga su biološki starija djeca fizički superiornija od ostalih vršnjaka i imaju bitno veće šanse da budu odabrani u ekipu. Ključne riječi, The main aim of this research was to establish the functional characteristics of young soccer players: pioneers, cadets and juniors. The study included 66 first-league football players from Croatia. The participants were divided in three age groups: U-15; pioneers (N=22), U-17; cadets (N=22) and U-19; juniors (N=22). Further, every age group was divided in three subsamples: defense players, midfielders and attack players, and by quality in first and second team. All participants were subdued to following tests: 3 tests for anthropometric characteristics assessment, 15 tests for functional characteristics assessment and 6 tests for assessment of ventilatory characteristics. For fulfilling the goals of the study and testing the hypothesis, the following statistical analysis were used: basic statistics, factorial analysis of variance with Fischer LSD post-hoc test and factorial analysis of variance with Tukey-Kramer Unequal N HSD post-hoc test. According to the obtained results it is possible to conclude: o It is possible to improve the cardio-respiratory and pulmonary system by football training and to increase specific work capacity – player’s endurance, respectively. Besides the training, differences in anthropometric, functional and ventilative parameters between age groups can be assigned to growth and maturation. o The differences among every age group in player's positions (defense, midfield, attack) are primarily caused by positions specificities. o By the time children reach puberty they have well-balanced growth and development. Reaching puberty rapid physical growth occurs, so biologically older children are physically superior to their peers and have higher chances to be picked for the team.

With the aim of determining their psychological coping skills, 85 futsal players from the Croatian UniSport (Academic) First League of Futsal have been measured using the ACSI-28 questionnaire. Low levels of reliability have been determined on few questionnaire scales, but all given data has been analyzed because this kind of study on a population of student futsal players hadn’t been conducted in Croatia before. Average results from scales of psychological coping skills of futsal players vary between 3.06 on ‘freedom from worry’ scale up to medium high 3.81 on confidence/motivation scale. Groups of players of different chronological age and various futsal playing experience differ in psychological skill significance: a group of the youngest players has the lowest expression of ‘freedom from worry’ characteristic (F=3.97 ; p=0.02), while a group of the oldest players has a significantly lower expression of ‘coping with adversity’ characteristic (F=3.45 ; p=0.04) ; a group of players with a playing experience only at the lowest competitive rank has a significantly lower expression of concentration characteristics (F=3.38 ; p=0.04) and ‘peaking under pressure’ (F=3.14 ; p=0.05) than the others ; a group of players with the longest playing experience in competitive futsal has a significantly higher expression of ‘goal- setting/mental preparation’ characteristic (F=4.34 ; p=0.02) then other players. Given results indicate the need for conducting further research of psychological characteristics of futsal players.

Microlocal defect functionals (H-measures, H- distributions, semiclassical measures etc.) are objects which determine, in some sense, the lack of strong compactness for weakly convergent Lp sequences. H-measures are suitable to treat problems where all partial derivatives are of the same order [4]. Recently, parabolic H- measures are introduced in order to treat 1:2 ratio between orders of partial derivatives [1], and also fractional H-measures which treat arbitrary ratios [2, 3]. We generalise Second commutation lemmas introduced in [1] and [4] to fractional Hmeasures, from which we are able to derive the propagation principle for the following fourth order partial differential equation: iu_t + (a(x)u_xx)_xx = f. References [1] N. Antoni´c, M. Lazar, Parabolic H-measures, Journal of Functional Analysis, 265 (2013) 1190– 1239. [2] M. Erceg, I. Ivec, On generalisation of H-measures, accepted for publication in Filomat, 18 pp. [3] D. Mitrovi´c, I. Ivec, A generalization of H-measures and application on purely fractional scalar conservation laws, Comm. Pure Appl. Analysis, 10 (2011) (6) 1617– 1627. [4] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proceedings of the Royal Society of Edinburgh, 115A (1990) 193–230.

Application of a variant of the Schwartz kernel theorem for anisotropic distributions to the Peetre's result on differential operators.

Kružno raskrižje je prometna građevina, gdje je kretanje vozila određeno središnjim kružnim otokom i kružnim kolnikom te privozima s razdjelnim otocima i s prometnim znakovima. Prva pojava raskrižja s kružnim tokom kao oblikovno strukturni elementi javili su se početkom XX. stoljeća, mnogo prije dominacije motornog prometa. Procjenjuje se da je danas u svijetu izgrađeno preko četrdeset tisuća kružnih raskrižja. Taj prirast možemo osjetiti i u Hrvatskoj, gdje kružna raskrižja poprimaju sve veću važnost. S obzirom na veliko povećanje prometa u gradovima kružna raskrižja su idealna alternativna rješenja pri projektiranju i planiranju raskrižja zbog mnogih svojih prednosti. Ukoliko su pravilno primjenjena i oblikovana znatno doprinose povećanju propusne moći i razine usluge raskrižja, kao i povećanju stupnja prometne sigurnosti zbog malog broja konfliktnih točaka. Prilikom eksploatacije smanjuju ukupno vrijeme putovanja, vrijeme čekanja vozila na raskrižju, potrošnju goriva, te smanjuju štetne utjecaje na okoliš putem smanjenja ispušnih plinova.

Fundamental motor skills (FMS) are defined as a standard motor activity that forms the basis for more advanced and more specific motor activities. Mastering FMS allows good preconditions for the successful realization of specific motor skills. Although FMS is acquired through the development process, it has been shown that not only free play stimulates their development, but also quality instruction is necessary. There are different classifications of FMS and the prevailing opinion is that FMS can be divided into two categories: basic locomotor skills and manipulative skills. Measuring instruments for assessing FMS can be divided into tests that are focused on the quantification of performance and the quality of performance. For the purpose of this research, the qualitative test "Test of Gross Motor Development - 2" was used. Some research suggests that there are significant differences in FMS between genders ; however, it is not yet clear to what extent children's activities affect the above mentioned differences. Therefore, the aim of this study is to determine gender differences in manipulative and locomotor FMS as well as differences between basic locomotor and manipulative skills in boys and girls attending athletic sports school.

The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond, and Caplain (2007), and by Antonić and Burazin (2010), while more recently a purely operator-theoretic description of abstract Friedrichs systems is developed. In this talk we shall present this abstract theory of Friedrichs systems and apply it on several boundary value problems of interest. At the end we study the question whether for any differential operator of the Friedrichs type there exist boundary conditions such that the problem is well-posed. This talk is based on joint works with Nenad Antonić, Krešimir Burazin, Ivana Crnjac, and Alessandro Michelangeli.

Some new applications of H-measures and H-distributions suited for problems that change type.

The main objective of this study was to determine the existence of Relative Age Effect (RAE) across top level European soccer leagues. Birthdates from 5507 professional soccer players were collected. The collected data showed unequal distributions of birthdates with the over-representation of players born in first few months of the year and an under-representation of players born late in the year. There were significant differences between expected and observed frequencies for month and quarter distribution of birthdates. RAE remains problem for the large number of players born later in the selection year. Result maintain high on the importance list even in youth academies which ultimately favors players born in the first quarter or in the first half of the year. Consequently, there is potential for “talent loss” because of the extreme dropout of players born in the last quarter or in the second half of the year.

Application of the Schwartz kernel theorem for anisotropic distributions to the Peetre characterisation of partial differential operators.

The main goal of the study was to determine the differences in anthropological status of young soccer players of different chronological, biological and training age. The subject sample included 47 young male soccer players whose mean chronological age was 14.79±1.17 years, mean biological age 14.67±1.20 and mean training age 7.09±1.47 years. Three morphological measurements were taken: body height, sitting height and body mass. Body mass index (BMI) and Peak height velocity (PHV) were calculated subsequently. By applying the K-means taxonomic analysis, two clusters were defined with categorical variables chronological age, indicator of biological age and training age, and by applying the univariate analysis of variance (ANOVA), significant differences were found in all the observed variables of anthropological status. All the applied variables differentiated the formed clusters significantly. Further analysis determined descriptive parameters of anthropometric characteristics, motor abilities and the results of functional test which was represented by three variables. The obtained results indicate that chronologically and biologically more mature soccer player, who have been included in the training process longer, have greater body and sitting height, they are heavier and have better results in all the measured motor abilities, as well as the functional test applied.