The existence and asymptotic behavior of positive increasing solutions of the cyclic second-order nonlinear difference system O(p(i)(n)|Ox(i)(n)|(alpha)(i)-1 Delta x(i)(n))= q(i)(n)|x(i+1)(n + 1)|(beta)(i)-1x(i+1)(n + 1), i = (-)1, N, are studied, where x(N+1) = x(1), and the sequences p(i) = p(i)(n) and q(i) = q(i)(n) are positive for all n is an element of N, while the constants alpha(i) and beta(i), i = (-)1, N, are positive and satisfy the sublinearity condition alpha(1)alpha(2) alpha(N) > beta(1)beta(2) beta N. We consider two types of positive increasing solutions: those converging to a positive constant and those diverging to infinity, whose associated quasi-differences tend to a positive constant. For both classes of solutions, necessary and sufficient conditions for existence are established using fixed point methods. In addition, under the assumption that the coefficient sequences are regularly varying, we investigate positive increasing solutions for which both the solution components and their quasi-differences tend to infinity. In this case, the corresponding existence conditions are also derived, along with precise asymptotic formulas, based on the theory of discrete regular variation.

The existence and asymptotic behavior of positive decreasing solutions to the cyclic second-order nonlinear difference system Delta(p(i)(n)|Delta x(i)(n)(|alpha)(i-)1 Delta x(i)(n))=q(i)(n)|x(i+1)(n+1)|(beta)(i)-1x(i+1)(n+1), i=1,N-, are studied, where x(N+1)=x(1), p(i)={p(i)(n)} and q(i)={q(i)(n)} are positive real sequences, and the constants alpha(i) and beta(i), i=1,N- are positive and satisfy the sublinear condition alpha(1)alpha(2).....alpha(N)>beta(1)beta(2).....beta(N). Two distinct types of positive decreasing solutions are considered, depending on whether the series Sigma(infinity)(n=1)p(i)(n)(-1/alpha)(i) is divergent or convergent. In the first case, necessary and sufficient conditions for the existence of solutions tending to a positive constant as well as solutions tending to zero, while their associated quasi-differences approach a nonzero limit, are rigorously derived using fixed point techniques. In the second case, the analysis is focused on solutions whose components and quasi-differences both tend to zero. Under the additional assumption that the coefficient sequences are regularly varying, necessary and sufficient conditions for the existence of such solutions are obtained, and their precise asymptotic behavior is determined using the theory of discrete regular variation.

We propose a predator-prey system with Holling type II functional response incorporating both the Allee effect in the growth of the prey population and a nonlinear Michaelis-Menten type harvesting in predator. We provide a detailed mathematical analysis of the proposed model, including, positivity and boundedness of solutions, uniform persistence, existence, and local and global asymptotic stability of equilibria. Detailed bifurcation analysis is carried out and it is observed that the proposed system exhibits very complex dynamics and many local and global bifurcations as transcritical, pitchfork, saddle-node, Hopf, homoclinic, and Bogdanov-Takens have been identified. We observe the bi-stability and tri-stability in the system, so the basins of attraction in all possible cases of the existence of multiple attractors are discussed in detail. The system shows different types of bi-stabilities behavior in the case of strong and weak Allee effect and different types of tri-stability in the case of strong Allee effect. Extensive numerical simulations are performed for supporting evidence of our analytical findings. According to our analysis, the proposed model allows the development of a harvesting policy that can prevent the extinction of predator and prey populations. In the case of weak Allee effect, the maximum threshold for continuous predator harvesting without the extinction risk of both species is obtained. In the case of strong Allee effect, the optimal harvesting threshold has been also determined, but optimal harvesting rate of the predator population can only promote the coexistence of the population whenever the Allee effect is quite low, otherwise, the predator harvesting ceases to have any stabilizing effect.

Vještačka inteligencija danas, htjeli mi to ili ne, dio je našeg društva u mnogim sferama života, tako da ni pravo odnosno pravosuđe nije izuzetak. U doba povećanog i brzog protoka informacija i digitalizacije, najnovija i najinovativnija tehnološka i informativna dostignuća sve se više koriste i u pravosuđu. Upotreba vještačke inteligencije u pravosuđu nije više izbor već neminovnost, koja ima brojne izazove, ukoliko se tim procesima odgovorno ne upravlja. Naime, vještačka inteligencija i algoritmi mašinskog učenja imaju mnoge specifičnosti koje mogu dovesti do diskriminacije i kršenja osnovnih ljudskih prava, zbog čega se toj oblasti mora posvetiti velika i posebna pažnja. Cilj ovog rada je da ukaže na pravce razvoja i izazove vještačke inteligencije u pravosuđu, te daje smjernice kako da se na bazi evropskih standarda u ovoj oblasti doprinese unapređenju pravosuđa u opsegu koji podržava poštovanje osnovnih ljudskih prava, transparentnost, nediskriminaciju, odgovornost i vladavinu prava.

This paper presents a study of dynamic behavior and bifurcation analysis of a predator-prey system with the functional response proposed by Cosner et al. (Theor Popul Biol 56:65-75, 1999) and Allee effect in prey population. The functional response used is specific in compare with the conventional functional responses according to its monotonicity for both prey and predator density, and moreover it increases as predator density increase. This function response describes a behavioral mechanism which a group of predators foraging in linear formation, contacts and then hunts gathering around the herd or a school of prey. Mainly, our aim is to demonstrate the impact of strong and weak Allee effect on the system dynamics. Mathematically our analysis primarily focuses on the stability of coexisting equilibrium points and all possible bifurcations that the system may exhibit. Actually, we consider the existence of equilibria and analyze their stability. The possibility of extinction of both populations is also considered, by studying dynamics of the system near the origin. The bifurcation of the system will be analyzed, including the occurrence of saddle-node bifurcation, Hopf and degenerate Hopf bifurcation, and Bogdanov-Takens bifurcation. The theoretical results are verified by numerical simulations. We observe the bi-stability and tri-stability, so that we further discuss the basins of attraction in all possible cases of existence of multiple attractors.

The performance analysis of an energy constrained Internet of Things (IoT) system with unmanned aerial vehicle (UAV) is provided in this paper. In the considered system, a power beacon is used for the energy supply of a sensor node that has no other power sources, while the UAV is used for the collection of sensor data. The outage and capacity performances are analyzed under the assumption of a Nakagami-m fading environment, for the case when the power and information transfer are performed based on the time-switching protocol and the UAV is randomly positioned at a certain height. Based on the provided analysis we derive the exact closed-form expressions for the outage probability, the outage capacity and the ergodic capacity of the power beacon assisted IoT system. The analytical results are confirmed using an independent simulation method. The performed analysis demonstrates the impact of various system and channel parameters on system performances.

One of the main motivations for our research was to find a connection between the Brownian motion of microorganisms within fractal nature, with the idea of developing an appropriate procedure and method to control the microorganism's motion direction and predict the position of the microorganism in time. In this paper, we have followed the results of the very rear microorganism's motion sub-microstructures in the experimental microstructure analysis already observed and published. All of these data have been good basis to describe the motion trajectory by time interval method and fractals. We successfully defined the diagrams in two and three-dimensions and we were able to establish the control of Brownian chaotic motion as a bridge between chaotic disorders to control disorder. This significant study opens a new possibility for future investigation and the new potential of total control of the microorganism motion. These perspectives and findings provide significant data for getting more information from these bio systems. They can also be applied, based on self-similarities and biomimetics, to particle physical systems and matter, generally.

The two-dimensional systems of first order nonlinear differential equations (S1) x? = p(t)y?, y? = q(t)x? and (S2) x? + p(t)y? = 0, y? + q(t)x? = 0 are analyzed using the theory of rapid variation. This approach allows us to prove that all strongly increasing solutions of system (S1) (and, respectively, all strongly decreasing solutions of system (S2) ) are rapidly varying functions under the assumption that p and q are rapidly varying. Also, the asymptotic equivalence relations for these solutions are given.

We discuss sublinear differential equations of the Emden–Fowler type $x''=q(t) x^{\gamma }$ under the assumption that the coefficient q is a rapidly varying function. We show that all of their strongly decreasing and strongly increasing solutions are rapidly varying functions and are in the asymptotic equivalence relation with a precisely defined function determined by the coefficient q.

We study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation (p(t)vertical bar x'vertical bar(alpha) sgn x')' + q(t)vertical bar x vertical bar(alpha) sgn x = 0, where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions integral(infinity)(alpha) ds/p(s)(1/alpha) = infinity or integral(infinity)(alpha) ds/p(s)(1/alpha) < infinity. The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

The electrical conductivity, phase, and space group of property of the NMT substance and grain boundary of samples were approximated from complex compound by Raman spectroscopy, X-ray, SEM, TEM, and differential scanning calorimeter (DSC), samples prepared and analyzed in the laboratory in the frequency ranges from 9.76 GHz. The solid-state conventional method utilized. All the samples were tested by using XRD from 1250°C to 1550°C by increasing 50 degrees for every sample and cooling to the room temperature. The contraction of structure did not happen even at elevated temperature. The increase in Nd + ion polarizability is 0.50 nm compared to La + rare elements material, which is 0.60, this contributes to change of perovskite cubic. This is result of the smaller size of Nd + 0.127 nm cation compared to La + which increase number of dipoles despite the reduction of diploes and expands relative permittivity (ε r ). The single-phase material produced which shows quality factor Q×f = 60000 saturated at 10 GHz and temperature coefficient of (τ f ) = -77 MK -1 .

The dielectric properties of Neodymium zinc titanium oxide (NZT) and Neodymium magnesium titanium oxide (NMT) were investigated. The single-phase ceramic was synthesized at various temperatures below 1650[Formula: see text]C. The result shows that the value of temperature of resonant frequency [Formula: see text] for NMT is higher than NZT. Our findings also indicate that the rare earth materials produce high property dielectric materials, despite the fact some elements produce lower negative value of temperature of resonant frequency [Formula: see text]. By doping a compound such as CaTiO 3 which has a very positive temperature of resonant frequency ([Formula: see text] ppm/[Formula: see text]C) and a very high relative permittivity [Formula: see text], it is possible to tune NZT and MNT to achieve an excellent dielectric material. This work is under consideration. The results of this scientific research could be very important for modern advance applications in microelectronic miniaturization.

The applications of BaTiO3-ceramics are very important and constantly increasing nowadays. In that sense, we analyzed some phenomena related to inter granular effects. We used experimental data based on Murata powders and processing technology. Our original contribution to Heywang-Jonker-Daniels inter-granular capacity model is based on thermodynamic fractal analysis applied on phase transition in ceramic structures. In this case, PTCR effect has a diffuse first-order phase transition character in a modified Landau theory-fractal approach. Its basic properties are considered. This is an original contribution as a bridge between theoretical aspects of BaTiO3-ceramics and experimental results.

Basic reproduction number for deterministic SEIPHAR model and its stochastic counterpart for the spread of SARS-CoV-2 virus are analyzed and compared. For deterministic version of the model, conditions for stability of the disease-free equilibrium are derived and, in addition, conditions for existence of bifurcation state related to endemic equilibrium are established. For stochastic model, conditions for extinction and persistence in mean of the disease are derived. Complete sensitivity analysis of thresholds between the extinction and mean-persistence are performed for both the deterministic and the stochastic version of the model. Influence of variation in parameter values is illustrated for epidemics in Wuhan in early 2020.

The half‐linear q‐difference equation where , , is analyzed in the framework of q‐regular variation. Necessary and sufficient conditions for the existence of q‐regularly varying solutions are given, under the assumption that p is a q‐regularly varying function and with no sign restriction on r. It is examined in the case when r is eventually negative, whether all positive solutions are q‐regularly varying. Using generalized regularly varying sequences, these results are applied to the half‐linear difference equation case.

The main goal of our research is to find the connection between micro particles and microorganisms motion in the Nature, considered as Brownian’s Motion within the fractal’s nature. For ceramics and generally material science it is important to clarify the particles motion and other phenomena, especially for grains and pores. Our idea is to establish control over the relation order–disorder on particle motion and their collision effects by Brownian motion phenomena in the frame of fractal nature matter. We performed some experiments and got interesting results based on microorganism motion initiated by different outer energetic impulses. This is practically the idea of biomimetic correlation between particles and microorganisms Worlds, what is very original and leads towards biunivocal different phenomena’s understanding. Another idea is to establish some controlling effects for electro ceramic particle motion in chemical-materials sciences consolidation by some phenomena in the nature. These important research directions open new frontiers with very specific reflections for future of microelectronics materials.

This paper investigates the existence of positive strongly decaying solutions of second-order nonlinear q-difference equation with q-regularly varying coefficients a and b. By means of a fixed point theorem in partially ordered Banach space, necessary and sufficient conditions for the existence of solutions satisfying and as are established. Moreover, with the help of q-regular varying theory, precise asymptotic behaviour of such solutions is described.

Under the assumptions that p and q are regularly varying functions satisfying conditions ??a t/p(t)1/? dt < ? and ??a (t/p(t))1/? dt = ? existence and asymptotic form of regularly varying intermediate solutions are studied for a fourth-order quasilinear differential equation (p(t)jx??(t)|?-1 x??(t))?? + q(t)|x(t)|?-1 x(t) = 0, ? > ? > 0. It is shown that under certain integral conditions there exist two types of intermediate solutions which according to their asymptotic behavior is to be divided into six mutual distinctive classes, while asymptotic behavior of each member of any of these classes is governed by a unique explicit law.

Positive decreasing solutions of the nonlinear difference equation ?(pn|?xn|?-1?xn)=qn|xn+1|?-1xn+1, n ? 1, ? > ? > 0, are studied under the assumption that p; q are regularly varying sequences. Necessary and sufficient conditions are established for the existence of regularly varying strongly decreasing solutions and it is shown that the asymptotic behavior of all such solutions is governed by a unique formula.

The existence and the precise asymptotic behavior for t -> infinity of all increasing solutions of a class of second order nonlinear equations is proved. For that the use of Karamata class of regularly varying functions is essential.