We have deduced an energy function for a Hindmarsh-Rose model neuron and we have used it to evaluate the energy consumption of the neuron during its signaling activity. We investigate the balance of energy in the synchronization of two bidirectional linearly coupled neurons at different values of the coupling strength. We show that when two neurons are coupled there is a specific cost associated to the cooperative behavior. We find that the energy consumption of the neurons is incoherent until very near the threshold of identical synchronization, which suggests that cooperative behaviors without complete synchrony could be energetically more advantageous than those with complete synchrony.

We study a competitive reaction-diffusion system with initially separated components. In this system, two similar species on one side of the system compete to react with the species on the other side. The competition is due to significant differences in the microscopic reaction constants and the initial densities of the two competing species. In the short-time limit, each of the competitive reactions is considered as perturbation with respect to the diffusion, the latter is essential for the effective mixing of the reactants. We identify the small parameters required for the perturbation analysis of the competitive scheme. The resulting perturbative expressions provide the rich spatiotemporal reaction front patterns, which were experimentally observed for ${\mathrm{Cr}}^{3+}+\text{Xylenol}\text{Orange}\left(\mathrm{XO}\right)\to \text{products}$, where the aggregated and nonaggregated forms of ${\mathrm{Cr}}^{3+}$ in aqueous solution compete to react with the XO.

There is a recent thesis in the literature that an important organization of a physical system precedes a catastrophic event. In this context, one can search for signatures that imply the transition from a normal state to a main catastrophic event (e.g., earthquake). Experimental techniques are thus useful in corroborating theories from observed data. For example, recent results indicate that preseismic electromagnetic time series contain information characteristic of an ensuing earthquake event. Hereby, we attempt to demonstrate that an easily computable complexity measure, such as $T$-complexity or approximate entropy, gives evidence of state changes leading to the point of global instability. The appearance of a precatastrophic state is characterized by significant lower complexity in terms of $T$-complexity and approximate entropy. The present study confirms the conclusions of previous works based on an independent linear fractal spectral analysis. This convergence between nonlinear and linear analysis provides a more reliable detection concerning the emergence of the last phase of the earthquake preparation process. More precisely, we claim that our results suggest an important principle: significant complexity decrease and accession of persistency in electromagnetic (EM) time series can be confirmed at the tail of the preseismic EM emission, which could be used as diagnostic tools for the Earth's impending crust failure. Direct laboratory and field experimental data as well as theoretical arguments support the conclusions of the present analysis.

We present a nonfeedback method to tame or enhance crisis-induced intermittency in dynamical systems. By adding a small harmonic perturbation to a parameter of the system, the intermittent behavior can be suppressed or enhanced depending on the value of the phase difference between the main driving and the perturbation. The validity of the method is shown both in the model and in an experiment with a ${\mathrm{CO}}_2$ laser. An analysis of this scheme applied to the quadratic map near crisis illustrates the role of phase control in nonlinear dynamical systems.

The gyrotron traveling-wave amplifier employing the distributed-loss scheme is capable of very high gain and effective in suppressing the global absolute instabilities. This study systematically characterizes the local absolute instabilities and their transitional behavior. The local absolute instabilities are analyzed using a model that incorporates the penetration of the field from the copper section into the lossy section. The axial modes were characterized from the perspective of beam-wave interaction and were found to share many characteristics with the global modes. The transition from global modes to local modes as the distributed loss increases was demonstrated. The electron transit angle in the copper section, which determines the feedback criterion, governs the survivability of an oscillation. In addition, the oscillation thresholds predicted using this model are more accurate than those obtained using a simplified model.

We study analytically and numerically the self-focusing arrest of femtosecond laser pulses in air at different pressures in the presence of an external focus lens. Analytical estimations as well as results of simulations show that the intensity at which the self-focusing arrest occurs is almost independent of the gas pressure. However, a dependence on the temperature is found. The Raman effect is taken into account, and an estimation of the intensity inside of filaments at high altitudes is given.

We report three-dimensional parallel Lagrangian particle simulations using the lattice Boltzmann method, conducted at a low Reynolds number. Using modified Lees-Edwards boundary conditions and directly calculated viscous dissipation, we show that it is possible to recover excellent agreement with the Einstein viscosity formula in the low concentration limit and to predict viscosity corrections for larger concentrations.

A numerical method to determine the history of particle paths is presented and its application for mixing quantification is illustrated. When more than one source exists in a flow field, the current technique can reveal the particle’s identity found in any time and place in the field, by backward tracking its origin. Since the particle position at a preceding time is not known, the velocity vector is implicit. To resolve this uncertainty, a three-stage iterative procedure is developed and implemented. The current particle velocity is multiplied by the time increment and the product is used to estimate the previous time increment particle position. Two velocity matrices are generated on a mesh around the estimated position. The first matrix is the Eulerian velocity field interpolated on the mesh. The second matrix contains velocity vectors that point to the current particle position. A correlation matrix is calculated from the two velocity matrices in order to resolve the actual particle position in the previous time increment. Determination of the time increments' size is performed by checking the maximum of the correlation matrix. The new algorithm was validated using a numerical solution of the confined twin-jet flow at low Reynolds number. This flow performs a Hopf bifurcation at a Reynolds number of about 30, therefore chaotic trajectories might exist in the flow. First, the convergence of the backward particle path to the streamlines in steady flow was demonstrated. Convergence of the particle paths for various time increments was achieved also at the unsteady two-dimensional confined twin-jet flow. When the flow has more than one source, the proposed tracking method can be applied to generate complete and ordered particle source images in any desired position and time in the flow field. Maps of particle sources are used to visualize the flow patterns and the stretched interface between the two fluid sources. Those maps are demonstrated as a powerful tool for mixing quantification and can be implemented also for pollution source detection and many other fluid dynamics applications.

Using exact computations we study the classical hard-core monomer-dimer models on $m\times n$ plane lattice strips with free boundaries. For an arbitrary number $v$ of monomers (or vacancies), we found a logarithmic correction term in the finite-size correction of the free energy per lattice site. The coefficient of the logarithmic correction term depends on the number of monomers present $\left(v\right)$ and the parity of the width $n$ of the lattice strip: the coefficient equals to $v$ when $n$ is odd, and $v∕2$ when $n$ is even. The results are generalizations of the previous results for a single monomer in an otherwise fully packed lattice of dimers. We also study the finite-size correction in the low dimer density limit, where the number of dimers $d$ is fixed. In this case the coefficient of the logarithmic correction term equals to $d$, for both odd and even $n$.

We analyze the simplest problem of electrochemical relaxation in more than one dimension—the response of an uncharged, ideally polarizable metallic sphere (or cylinder) in a symmetric, binary electrolyte to a uniform electric field. In order to go beyond the circuit approximation for thin double layers, our analysis is based on the Poisson-Nernst-Planck (PNP) equations of dilute solution theory. Unlike most previous studies, however, we focus on the nonlinear regime, where the applied voltage across the conductor is larger than the thermal voltage. In such strong electric fields, the classical model predicts that the double layer adsorbs enough ions to produce bulk concentration gradients and surface conduction. Our analysis begins with a general derivation of surface conservation laws in the thin double-layer limit, which provide effective boundary conditions on the quasineutral bulk. We solve the resulting nonlinear partial differential equations numerically for strong fields and also perform a time-dependent asymptotic analysis for weaker fields, where bulk diffusion and surface conduction arise as first-order corrections. We also derive various dimensionless parameters comparing surface to bulk transport processes, which generalize the Bikerman-Dukhin number. Our results have basic relevance for double-layer charging dynamics and nonlinear electrokinetics in the ubiquitous PNP approximation.

Differential optical reflectivity (DOR) was used to study the temperature dependence of the short helical pitch in freestanding films of a liquid crystal compound. The experimentally measured DOR signal was fitted using Berreman’s $4\times 4$ matrix method to get the pitch value in the smectic-$C_{\alpha }^{*}$ $\left(\mathrm{Sm}{C}_{\alpha }^{*}\right)$ phase. The results show continuous evolution of the pitch between the smectic-$C^{*}$ and $\mathrm{Sm}{C}_{\alpha }^{*}$ phases. In $\mathrm{Sm}{C}_{\alpha }^{*}$, the pitch decreases as temperature increases and is found to level off at $16\pm 1$ smectic layers at the $\mathrm{Sm}{C}_{\alpha }^{*}$ to smectic-$A^{*}$ transition.

A simple microscopic mean-field model for a flat nematic liquid crystal (NLC) sample in a contact to the solid substrate surface is offered. An interaction between NLC molecules is simulated by the well known McMillan model potential, and an orienting action of the solid substrate surface on NLC molecules is modeled by a short-range external field which acts directly only on molecules within the first molecular layer of the nematic sample adjacent to the substrate surface. For an undistorted NLC sample, the model allows the calculation of local order parameter profiles for different values of strength of this external orienting field and temperature of the sample. These profiles are used in a description of a director field distortion caused by a certain external action and in the calculation of the anchoring energy coefficient $W$ used in a macroscopic description of the anchoring at the nematic-solid substrate interface. Dependence of this coefficient on the strength of the short-range orienting field is obtained, and an unequivocal relation between the magnitude of $W$ and the orientational order parameter profile near the substrate surface is established. The temperature dependence of the coefficient $W$ calculated from the offered microscopic model is in good agreement with the experimental data on NLC MBBA.

Chemotaxis is ubiquitously performed by bacteria. They sense and move toward a region with a higher concentration of an attractive chemical by changing the rate of tumbling for random walk. We numerically studied several models with internal adaptive dynamics to examine the validity of the condition for chemotaxis proposed by Oosawa and Nakaoka [J. Theor. Biol. 66, 747 (1977)], which states that the time scale of tumbling frequency must be smaller than that of adaptation and greater than that of sensing. Suitable renormalization of the time scales showed that the condition holds for a variety of environments and for both short- and long-term behavior.

We develop a deterministic mathematical model to describe the way in which polymers bind to DNA by considering the dynamics of the gap distribution that forms when polymers bind to a DNA plasmid. In so doing, we generalize existing theory to account for overlaps and binding cooperativity whereby the polymer binding rate depends on the size of the overlap. The proposed mean-field models are then solved using a combination of numerical and asymptotic methods. We find that overlaps lead to higher coverage and hence higher charge neutralizations, results which are more in line with recent experimental observations. Our work has applications to gene therapy where polymers are used to neutralize the negative charges of the DNA phosphate backbone, allowing condensation prior to delivery into the nucleus of an abnormal cell.

We perform a comparative study of applicability of the multifractal detrended fluctuation analysis (MFDFA) and the wavelet transform modulus maxima (WTMM) method in proper detecting of monofractal and multifractal character of data. We quantify the performance of both methods by using different sorts of artificial signals generated according to a few well-known exactly soluble mathematical models: monofractal fractional Brownian motion, bifractal Lévy flights, and different sorts of multifractal binomial cascades. Our results show that in the majority of situations in which one does not know a priori the fractal properties of a process, choosing MFDFA should be recommended. In particular, WTMM gives biased outcomes for the fractional Brownian motion with different values of Hurst exponent, indicating spurious multifractality. In some cases WTMM can also give different results if one applies different wavelets. We do not exclude using WTMM in real data analysis, but it occurs that while one may apply MFDFA in a more automatic fashion, WTMM must be applied with care. In the second part of our work, we perform an analogous analysis on empirical data coming from the American and from the German stock market. For this data both methods detect rich multifractality in terms of broad $f\left(\alpha \right)$, but MFDFA suggests that this multifractality is poorer than in the case of WTMM.

We consider numerically the $\mathsf{L}$-, $\mathsf{T}$-, and $\mathsf{X}$-shaped elastic waveguides with the Dirichlet boundary conditions for in-plane deformations (displacements) which obey the vectorial Navier-Cauchy equation. In the $\mathsf{X}$-shaped waveguide we show the existence of a doubly degenerate bound state with frequency below the first symmetrical cutoff frequency, which belongs to the two-dimensional irreducible representation $E$ of symmetry group $C_{4v}$. Moreover the next bound state is below the next antisymmetric cutoff frequency. This bound state belongs to the irreducible representation $A_2$. The $\mathsf{T}$-shaped waveguide has only one bound state while the $\mathsf{L}$-shaped one has no bound states.

Time-delayed feedback is applied to the motions associated with the nonlinear periodic regime generated due to current-driven ion acoustic instability; this is a typical instability in a laboratory plasma, and the dynamical behavior is experimentally investigated using delayed feedback. A time-delayed autosynchronization method is applied. When delayed feedback is applied to the nonlinear periodic orbit, the periodic state changes to various motions depending on the control parameters, namely, the arbitrary time delay and the proportionality constant. Lyapunov exponents are calculated in order to examine the dynamical behavior.

Theoretical investigation of a short electron beam (extended bunch) interaction with a backward wave propagating in a slow wave structure demonstrates the possibility of producing ultrashort superradiance pulses with a peak power which exceeds the power of the driving beam (conversion factor $K>$1). It is shown that a nonuniform slow wave structure with optimized profile is beneficial in order to increase the conversion factor. The results of theoretical analysis are confirmed by the experiments. At $X$ band using the SINUS-150 accelerator ($4\mathrm{ns}$, $330\mathrm{kV}$, $2.6\mathrm{kA}$) 0.6–$0.8\mathrm{ns}$ superradiance pulses with a peak power of $1.2\mathrm{GW}$ and a conversion factor of 1.5 were obtained. Similar experiments at $\mathrm{Ka}$-band based on the RADAN-303 accelerator ($1\mathrm{ns}$, $290\mathrm{kV}$, $2.5\mathrm{kA}$) demonstrated production of the superradiance (SR) pulse with duration $200\mathrm{ps}$ and peak power about $1\mathrm{GW}$ (conversion factor of 1.4).

Optical vortices are shown to be generated in the near-field through interference between a propagating wave and the amplified evanescent field in a slab of lossy left-handed material. While small loss adversely impacts the sub-wavelength performance in the lens application, the vortex character shown relies on some degree of imperfection. These vortices can be controlled by means of gain/loss and the incident field.

In dimension $d⩾3$, the directed polymer in a random medium undergoes a phase transition between a free phase at high temperature and a low-temperature disorder-dominated phase. For the latter phase, Fisher and Huse have proposed a droplet theory based on the scaling of the free-energy fluctuations $\Delta F\left(l\right)\sim l^{\theta }$ at scale $l$. On the other hand, in related growth models belonging to the Kardar-Parisi-Zhang universality class, Forrest and Tang have found that the height-height correlation function is logarithmic at the transition. For the directed polymer model at criticality, this translates into logarithmic free-energy fluctuations $\Delta F_{T_c}\left(l\right)\sim {\left(\ln l\right)}^{\sigma }$ with $\sigma =1∕2$. In this paper, we propose a droplet scaling analysis exactly at criticality based on this logarithmic scaling. Our main conclusion is that the typical correlation length $\xi \left(T\right)$ of the low-temperature phase diverges as $\ln \xi \left(T\right)\sim {[-\ln (T_c-T)]}^{1∕\sigma }\sim {[-\ln (T_c-T)]}^2$, instead of the usual power law $\xi \left(T\right)\sim {(T_c-T)}^{-\nu }$. Furthermore, the logarithmic dependence of $\Delta F_{T_c}\left(l\right)$ leads to the conclusion that the critical temperature $T_c$ actually coincides with the explicit upper bound $T_2$ derived by Derrida and co-workers, where $T_2$ corresponds to the temperature below which the ratio $\overline{Z_L^2}∕{\left(\overline{Z_L}\right)}^2$ diverges exponentially in $L$. Finally, since the Fisher-Huse droplet theory was initially introduced for the spin-glass phase, we briefly mention the similarities with and differences from the directed polymer model. If one speculates that the free energy of droplet excitations for spin glasses is also logarithmic at $T_c$, one obtains a logarithmic decay for the mean square correlation function at criticality, $\overline{C^2\left(r\right)}\sim 1∕{\left(\ln r\right)}^{\sigma }$, instead of the usual power law $1∕r^{d-2+\eta }$.