Fourier spectral methods for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator

Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime

We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its ``purification''. As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein-Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its ``purification'' induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein-Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III_1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.Comment: 42 pages, LaTeX. The Def. 3.3 was incomplete and this has been corrected. Several misprints have been removed. All results and proofs remain unchange

The influence of 6 weeks of maximal eccentric plantarflexor training on muscle-tendon mechanics

Resistance training can influence muscle-tendon properties including strength, flexibility, stretch tolerance and muscle-tendon stiffness; however the specific influence of eccentric-only training is unknown. Therefore, the aims of the present study were to examine the effects of a 6-week maximal eccentric resistance training programme on isometric plantarflexor moment (MVC), dorsiflexion range of motion (ROM), stretch tolerance (peak passive moment), muscle and tendon stiffness and running economy. Thirteen recreationally active men (age = 20.0 ± 0.9 yr, mass = 75.9 ± 8.5 kg, height = 1.8 ± 0.1 m) volunteered for the study after giving written informed consent; ethical approval was granted from the University of Northampton. Training was performed twice weekly for six weeks and consisted of 5 sets of 12 repetitions of 3-s maximal eccentric contractions at 10°•s-1 from 20° plantarflexion to 10° dorsiflexion. Maximal isometric plantarflexor moment, dorsiflexion ROM, stretch tolerance, and muscle, tendon and muscle-tendon unit (MTU) stiffness were measured using isokinetic dynamometry, real-time ultrasound and 3D motion analyses before and after the training. Running economy (VO2) was determined at a running speed equating to 70%VO2max using online gas analysis. Repeated measures t-tests were used to determine significant differences between pre- and post-training data, significance accepted at p0.05). Analysis of ultrasound data revealed a significant decrease in muscle stiffness (20.6%; p0.05). While the training-induced increase in plantarflexor strength was expected, the substantial increases in ROM, stretch tolerance and tendon stiffness, and the reduction in passive muscle stiffness, were important and novel findings. Interestingly, when measured during passive stretch, MTU stiffness remained unchanged while tendon stiffness increased and muscle stiffness decreased. These disparate findings have clear implications for testing methodologies, and indicate that imaging techniques must be utilised in order to examine the effects of interventions on specific tissues. As the training clearly enhanced the capacity of the muscle to tolerate both tissue loading and deformation, which are commonly associated with muscle strain injury, these data have clear implications for both muscular performance and injury risk

Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization

Structural heterogeneity constitutes one of the main substrates influencing impulse propagation in living tissues. In cardiac muscle, improved understanding on its role is key to advancing our interpretation of cell-to-cell coupling, and how tissue structure modulates electrical propagation and arrhythmogenesis in the intact and diseased heart. We propose fractional diffusion models as a novel mathematical description of structurally heterogeneous excitable media, as a mean of representing the modulation of the total electric field by the secondary electrical sources associated with tissue inhomogeneities. Our results, validated against in-vivo human recordings and experimental data of different animal species, indicate that structural heterogeneity underlies many relevant characteristics of cardiac propagation, including the shortening of action potential duration along the activation pathway, and the progressive modulation by premature beats of spatial patterns of dispersion of repolarization. The proposed approach may also have important implications in other research fields involving excitable complex media

The influence of acute variable resistance loading on subsequent free-weight maximal squat performance

Elastic bands attached to a loaded barbell during a squat exercise create a variable resistance (VR), thus changing the mechanical loading and stress placed through the musculoskeletal system. Preconditioning the neuromuscular system using near-maximal or maximal voluntary contractions (MVC) can induce a phenomenon known as post-activation potentiation (PAP) to enhance performance to ‘supramaximal’ levels. However, the potentiating effects of VR on subsequent free-weight resistance (FWR) squat performance have not been examined. Thus, the aim of the present study was to examine the influence of VR exercise using elastic bands on subsequent FWR squat performance. Sixteen recreationally active men (age = 26.0 ± 7.8 yr, height = 1.7 ± 0.2 m, mass 82.6 ± 12.7 kg) experienced in squatting (>3yr) volunteered for the study after giving written informed consent; ethical approval was granted from the University of Northampton. Subjects’ 1-RM were determined then on two subsequent days either a 3-RM FWR (control) or a 3-RM VR (experimental) squat exercise was performed at 85% 1-RM (35% of the load generated from band tension in the VR condition). Five minutes later, motion analysis recorded knee joint kinematics during a subsequent FWR 1-RM squat, with vastus medialis, vastus lateralis, rectus femoris and semitendinosus electromyograms (EMG) simultaneously recorded. Paired t-tests were used to determine significance, accepted at p0.05) or EMG amplitude (5.9%; p>0.05) occurred. No subjects increased 1-RM in the FWR condition, however 13 of 16 (81%) increased 1-RM by ~10% following VR. Preconditioning the neuromuscular system using VR significantly increased 1-RM without changes in knee extensor muscle activity or knee flexion angle, however eccentric and concentric velocities were reduced. Thus, VR can potentiate the neuromuscular system to enhance subsequent maximal lifting performance. The lack of change in EMG suggests that changes in muscle activity were small or non-existent, which may be explained by force-velocity effects (slower movement = larger forces). Alternatively a greater activation of hip musculature (not measured in the present study) may allow a greater total lower limb force to be developed. Regardless, as 1-RM increased greater lower-limb loading occurred, thus VR potentiated the neuromuscular system and could enhance training stimuli

Limits of sensing temporal concentration changes by single cells

Berg and Purcell [Biophys. J. 20, 193 (1977)] calculated how the accuracy of concentration sensing by single-celled organisms is limited by noise from the small number of counted molecules. Here we generalize their results to the sensing of concentration ramps, which is often the biologically relevant situation (e.g. during bacterial chemotaxis). We calculate lower bounds on the uncertainty of ramp sensing by three measurement devices: a single receptor, an absorbing sphere, and a monitoring sphere. We contrast two strategies, simple linear regression of the input signal versus maximum likelihood estimation, and show that the latter can be twice as accurate as the former. Finally, we consider biological implementations of these two strategies, and identify possible signatures that maximum likelihood estimation is implemented by real biological systems.Comment: 11 pages, 2 figure

Orientation and solvatochromism of dyes in liquid crystals.

The orientation and solvatochromism of some dye molecules in a liquid crystal have been investigated. Interactions with the host and the structure of the dye molecule affect the macroscopic alignment of dichroic dye molecules in a liquid crystal: It was observed that some dye molecules show a large bathochromic shift of their absorption maxima in the liquid crystal host relative to the situation in isotropic solvents. It is suggested that this is due to the occurrence of a much weaker reaction field in the anisotropic, rigid host. These dye molecules show little or no apparent order in the anisotropic host despite the observation of a reduction in the electro optic switching time when the dye is present. The highest degree of macroscopic alignment was observed for a merocyanine compound, which showed the smallest solvatochromic shift in the liquid crystal host. These results are discussed in terms of the steric, dipolar and hydrogen bond interactions between the guest and the host

Cellular Ability to Sense Spatial Gradients in the Presence of Multiple Competitive Ligands

Many eukaryotic and prokaryotic cells can exhibit remarkable sensing ability under small gradient of chemical compound. In this study, we approach this phenomenon by considering the contribution of multiple ligands to the chemical kinetics within Michaelis-Menten model. This work was inspired by the recent theoretical findings from Bo Hu et al. [Phys. Rev. Lett. 105, 048104 (2010)], our treatment with practical binding energies and chemical potential provides the results which are consistent with experimental observations.Comment: 5 pages, 4 figure