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Even though Kemeny’s constant was first discovered in Markov chains and expressed by Kemeny in terms of mean first passage times on a graph, it can also be expressed using the pseudo-inverse of the Laplacian matrix representing the graph, which facilitates the calculation of a sharp upper bound of Kemeny’s constant. We show that for certain classes of graphs, a previously found bound is tight, which generalises previous results for bipartite and (generalised) windmill graphs. Moreover, we show numerically that for real-world networks, this bound can be used to find good numerical approximations for Kemeny’s constant. For certain graphs consisting of up to 100 K nodes, we find a speedup of a factor 30, depending on the accuracy of the approximation that can be achieved. For networks consisting of over 500 K nodes, the approximation can be used to estimate values for the Kemeny constant, where exact calculation is no longer feasible within reasonable computation time.

In this paper, we focus on option pricing models based on time-fractional diffusion with generalized Hilfer-Prabhakar derivative. It is demonstrated how the option is priced for fractional cases of European vanilla option pricing models. Series representations of the pricing formulas and the risk-neutral parameter under the time-fractional diffusion are also derived.

Fish populations subject to heavy exploitation are expected to evolve over time smaller average body sizes. We introduce Stackelberg evolutionary game theory to show how fisheries management should be adjusted to mitigate the potential negative effects of such evolutionary changes. We present the game of a fisheries manager versus a fish population, where the former adjusts the harvesting rate and the net size to maximize profit, while the latter responds by evolving the size at maturation to maximize the fitness. We analyze three strategies: i) ecologically enlightened (leading to a Nash equilibrium in game-theoretic terms); ii) evolutionarily enlightened (leading to a Stackelberg equilibrium) and iii) domestication (leading to team optimum) and the corresponding outcomes for both the fisheries manager and the fish. Domestication results in the largest size for the fish and the highest profit for the manager. With the Nash approach the manager tends to adopt a high harvesting rate and a small net size that eventually leads to smaller fish. With the Stackelberg approach the manager selects a bigger net size and scales back the harvesting rate, which lead to a bigger fish size and a higher profit. Overall, our results encourage managers to take the fish evolutionary dynamics into account. Moreover, we advocate for the use of Stackelberg evolutionary game theory as a tool for providing insights into the eco-evolutionary consequences of exploiting evolving resources.

The fractional Schrödinger equation has recently received substantial attention. We generalize the fractional Schrödinger equation to the Hilfer time derivative and the Caputo space derivative, and solve this equation for an infinite potential by using the Adomian decomposition method. The infinite domain solution of the space-time fractional Schrödinger equation in the case of Riesz space fractional derivative is obtained in terms of the Fox H-functions. We interpret our results for the fractional Schrödinger equation by introducing a complex effective potential in the standard Schrödinger equation, which can be used to describe quantum transport in quantum dots.

The traditional secondary frequency control of power systems restores nominal frequency by steering Area Control Errors (ACEs) to zero. Existing methods are a form of integral control with the characteristic that large control gain coefficients introduce an overshoot and small ones result in a slow convergence to a steady state. In order to deal with the large frequency deviation problem, which is the main concern of the power system integrated with a large number of renewable energy, a faster convergence is critical. In this paper, we propose a secondary frequency control method named Power-Imbalance Allocation Control (PIAC) to restore the nominal frequency with a minimized control cost,in which a coordinator estimates the power imbalance and dispatches the control inputs to the controllers after solving an economic power dispatch problem. The power imbalance estimation converges exponentially in PIAC, both overshoots and large frequency deviations are avoided. In addition, when PIAC is implemented in a multi-area controlled network, the controllers of an area are independent of the disturbance of the neighbor areas, which allows an asynchronous control in the multi-area network. A Lyapunov stability analysis shows that PIAC is locally asymptotically stable and simulation results illustrates that it effectively eliminates the drawback of the traditional integral control based methods.

Synchronization is essential for proper functioning of the power grid. We investigate the synchronous state and its stability for a network with a cyclic topology and with the evolution of the states satisfying the swing equations. We calculate the number of stable equilibria and investigate both the linear and nonlinear stability of the synchronous state. The linear stability analysis shows that the stability of the state, determined by the smallest nonzero eigenvalue, is inversely proportional to the size of the network. The nonlinear stability, which we calculated by comparing the potential energy of the type-1 saddles with that of the stable synchronous state, depends on the network size (\\(N\\)) in a more complicated fashion. In particular we find that when the generators and consumers are evenly distributed in an alternating way, the energy barrier, preventing loss of synchronization approaches a constant value. For a heterogeneous distribution of generators and consumers, the energy barrier will decrease with \\(N\\). The more heterogeneous the distribution is, the stronger the energy barrier depends on \\(N\\). Finally, we found that by comparing situations with equal line loads in cyclic and tree networks, tree networks exhibit reduced stability. This difference disappears in the limit of \\(N\\to\\infty\\). This finding corroborates previous results reported in the literature and suggests that cyclic (sub)networks may be applied to enhance power transfer while maintaining stable synchronous operation.

Force-driven translocation of a macromolecule through a nanopore is investigated by taking into account the monomer-pore friction as well as the \"crowding\" of monomers on the {\\it trans} - side of the membrane which counterbalance the driving force acting in the pore. The set of governing differential-algebraic equations for the translocation dynamics is derived and solved numerically. The analysis of this solution shows that the crowding of monomers on the trans side hardly affects the dynamics, but the monomer-pore friction can substantially slow down the translocation process. Moreover, the translocation exponent \\(\\alpha\\) in the translocation time - vs. - chain length scaling law, \\(\\tau \\propto N^{\\alpha}\\), becomes smaller when monomer-pore friction coefficient increases. This is most noticeable for relatively strong forces. Our findings may explain the variety of \\(\\alpha\\) values which were found in experiments and computer simulations.

We present an analytical study of a toy model for shear banding, without normal stresses, which uses a piecewise linear approximation to the flow curve (shear stress as a function of shear rate). This model exhibits multiple stationary states, one of which is linearly stable against general two-dimensional perturbations. This is in contrast to analogous results for the Johnson-Segalman model, which includes normal stresses, and which has been reported to be linearly unstable for general two-dimensional perturbations. This strongly suggests that the linear instabilities found in the Johnson-Segalman can be attributed to normal stress effects.

We examine the behavior of a colloidal particle immersed in a viscoelastic bath undergoing stochastic resetting at a rate \\(r\\). Microscopic probes suspended in viscoelastic environment do not follow the classical theory of Brownian motion. This is primarily because the memory from successive collisions between the medium particles and the probes does not necessarily decay instantly as opposed to the classical Langevin equation. To treat such a system one needs to incorporate the memory effects to the Langevin equation. The resulting equation formulated by Kubo, known as the Generalized Langevin equation (GLE), has been instrumental to describe the transport of particles in inhomogeneous or viscoelastic environments. The purpose of this work, henceforth, is to study the behavior of such a colloidal particle governed by the GLE under resetting dynamics. To this end, we extend the renewal formalism to compute the general expression for the position variance and the correlation function of the resetting particle driven by the environmental memory. These generic results are then illustrated for the prototypical example of the Jeffreys viscoelastic fluid model. In particular, we identify various timescales and intermittent plateaus in the transient phase before the system relaxes to the steady state; and further discuss the effect of resetting pertaining to these behaviors. Our results are supported by numerical simulations showing an excellent agreement.

We suggest a governing equation which describes the process of polymer chain translocation through a narrow pore and reconciles the seemingly contradictory features of such dynamics: (i) a Gaussian probability distribution of the translocated number of polymer segments at time \\(t\\) after the process has begun, and (ii) a sub-diffusive increase of the distribution variance \\(\\Delta (t)\\) with elapsed time, \\(\\Delta(t) \\propto t^{\\alpha}\\). The latter quantity measures the mean-squared number \\(s\\) of polymer segments which have passed through the pore, \\(\\Delta(t) = <[s(t)-s(t=0)]^2>\\), and is known to grow with an anomalous diffusion exponent \\(\\alpha < 1\\). Our main assumption - a Gaussian distribution of the translocation velocity \\(v(t)\\) - and some important theoretical results, derived recently, are shown to be supported by extensive Brownian dynamics simulation which we performed in \\(3D\\). We also numerically confirm the predictions made in ref.\\cite{Kantor_3}, that the exponent \\(\\alpha\\) changes from \\(0.91\\) to \\(0.55\\), to \\(0.91\\), for short, intermediate and long time regimes, respectively.