On Hamiltonian balanced dynamics and the slowest invariant manifold

The concept of a slowest invariant manifold is investigated for the five-component model of Lorenz under conservative dynamics. It is shown that Lorenz's model is a two-degree-of-freedom canonical Hamiltonian system, consisting of a nonlinear vorticity-triad oscillator coupled to a linear gravity wave oscillator, whose solutions consist of regular and chaotic orbits. When either the Rossby number or the rotational Froude number is small, there is a formal separation of timescales, and one can speak of fast and slow motion. In the same regime, the coupling is weak, and the Kolmogorov–Arnold-Moser theorem is shown to apply. The chaotic orbits are inherently unbalanced and are confined to regions sandwiched between invariant tori consisting of quasi-periodic regular orbits. The regular orbits generally contain free fast motion, but a slowest invariant manifold may be geometrically defined as the set of all slow cores of invariant tori (defined by zero fast action) that are smoothly related to such cores in the uncoupled system. This slowest invariant manifold is not global; in fact, its structure is fractal; but it is of nearly full measure in the limit of weak coupling. It is also nonlinearly stable. As the coupling increases, the slowest invariant manifold shrinks until it disappears altogether. The results clarify previous definitions of a slowest invariant manifold and highlight the ambiguity in the definition of “slowness.” An asymptotic procedure, analogous to standard initialization techniques, is found to yield nonzero free fast motion even when the core solutions contain none. A hierarchy of Hamiltonian balanced models preserving the symmetries in the original low-order model is formulated; these models are compared with classic balanced models, asymptotically initialized solutions of the full system and the slowest invariant manifold defined by the core solutions. The analysis suggests that for sufficiently small Rossby or rotational Froude numbers, a stable slowest invariant manifold can be defined for this system, which has zero free gravity wave activity, but it cannot be defined everywhere. The implications of the results for more complex systems are discussed.

Effect of disease-selective predation on prey infected by contact and external sources

We propose and analyze a simple mathematical model for susceptible prey (S)–infected prey (I)–predator (P) interaction, where the susceptible prey population (S) is infected directly from external sources as well as through contact with infected class (I) and the predator completely avoids consuming the infected prey. The model is analyzed to obtain different thresholds of the key parameters under which the system exhibits stability around the biologically feasible equilibria. Through numerical simulations we display the effects of external infection and the infection through contact on the system dynamics in the absence as well as in the presence of the predator. We compare the system dynamics when infection occurs only through contact, with that when it occurs through contact and external sources. Our analysis demonstrates that under a disease-selective predation, stability and oscillations of the system is determined by two key parameters: the external infection rate and the force of infection through contact. Due to the introduction of external infection, the predator and the prey population show limit-cycle oscillations over a range parametric values. We suggest that while predicting the dynamics of such an eco-epidemiological system, the modes of infection and the infection rates might be carefully investigated.

Studying the contact point and interface moving in a sinusoidal tube with lattice Boltzmann method

The multicomponent nonideal gas lattice Boltzmann model by Shan and Chen (S-C) is used to study the immiscible displacement in a sinusoidal tube. The movement of interface and the contact point (contact line in three-dimension) is studied. Due to the roughness of the boundary, the contact point shows "stick-slip" mechanics. The "stick-slip" effect decreases as the speed of the interface increases. For fluids that are nonwetting, the interface is almost perpendicular to the boundaries at most time, although its shapes at different position of the tube are rather different. When the tube becomes narrow, the interface turns a complex curves rather than remains simple menisci. The velocity is found to vary considerably between the neighbor nodes close to the contact point, consistent with the experimental observation that the velocity is multi-values on the contact line. Finally, the effect of three boundary conditions is discussed. The average speed is found different for different boundary conditions. The simple bounce-back rule makes the contact point move fastest. Both the simple bounce-back and the no-slip bounce-back rules are more sensitive to the roughness of the boundary in comparison with the half-way bounce-back rule. The simulation results suggest that the S-C model may be a promising tool in simulating the displacement behaviour of two immiscible fluids in complex geometry.

EphB2 regulates contact-dependent and independent signalling to control platelet function

The Eph kinases, EphA4 and EphB1 and their ligand, ephrinB1 have been previously reported to be present in platelets where they contribute to thrombus stability. While thrombus formation allows for Eph-ephrin engagement and bidirectional signalling, the importance specifically of Eph kinase or ephrin signalling in regulating platelet function remained unidentified. In the present study, a genetic approach was used in mice to establish the contribution of signalling orchestrated by the cytoplasmic domain of EphB2 (a newly discovered Eph kinase in platelets) in platelet activation and thrombus formation. We conclude that EphB2 signalling is involved in the regulation of thrombus formation and clot retraction. Furthermore, the cytoplasmic tail of this Eph kinase regulates initial platelet activation in a contact-independent manner in the absence of Eph-ephrin ligation between platelets. Together these data demonstrate that EphB2 signalling not only modulates platelet function within a thrombus but is also involved in the regulation of the function of isolated platelets in a contact-independent manner.

Oxytocin increases eye contact during a real-time, naturalistic social interaction in males with and without autism

Autism spectrum conditions (autism) affect ~1% of the population and are characterized by deficits in social communication. Oxytocin has been widely reported to affect social-communicative function and its neural underpinnings. Here we report the first evidence that intranasal oxytocin administration improves a core problem that individuals with autism have in using eye contact appropriately in real-world social settings. A randomized double-blind, placebo-controlled, within-subjects design is used to examine how intranasal administration of 24 IU of oxytocin affects gaze behavior for 32 adult males with autism and 34 controls in a real-time interaction with a researcher. This interactive paradigm bypasses many of the limitations encountered with conventional static or computer-based stimuli. Eye movements are recorded using eye tracking, providing an objective measurement of looking patterns. The measure is shown to be sensitive to the reduced eye contact commonly reported in autism, with the autism group spending less time looking to the eye region of the face than controls. Oxytocin administration selectively enhanced gaze to the eyes in both the autism and control groups (transformed mean eye-fixation difference per second=0.082; 95% CI:0.025–0.14, P=0.006). Within the autism group, oxytocin has the most effect on fixation duration in individuals with impaired levels of eye contact at baseline (Cohen’s d=0.86). These findings demonstrate that the potential benefits of oxytocin in autism extend to a real-time interaction, providing evidence of a therapeutic effect in a key aspect of social communication.

Sparse density estimation on the multinomial manifold

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion for the finite mixture model. Since the constraint on the mixing coefficients of the finite mixture model is on the multinomial manifold, we use the well-known Riemannian trust-region (RTR) algorithm for solving this problem. The first- and second-order Riemannian geometry of the multinomial manifold are derived and utilized in the RTR algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with an accuracy competitive with those of existing kernel density estimators.

Low rank representation on Riemannian manifold of symmetric positive definite matrices

Sparse coding aims to find a more compact representation based on a set of dictionary atoms. A well-known technique looking at 2D sparsity is the low rank representation (LRR). However, in many computer vision applications, data often originate from a manifold, which is equipped with some Riemannian geometry. In this case, the existing LRR becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to applications. In this paper, we generalize the LRR over the Euclidean space to the LRR model over a specific Rimannian manifold—the manifold of symmetric positive matrices (SPD). Experiments on several computer vision datasets showcase its noise robustness and superior performance on classification and segmentation compared with state-of-the-art approaches.

Spectral analysis and the Aharonov-Bohm\~ effect on certain almost-Riemannian manifold

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

Heterogeneous tensor decomposition for clustering via manifold optimization

Tensor clustering is an important tool that exploits intrinsically rich structures in real-world multiarray or Tensor datasets. Often in dealing with those datasets, standard practice is to use subspace clustering that is based on vectorizing multiarray data. However, vectorization of tensorial data does not exploit complete structure information. In this paper, we propose a subspace clustering algorithm without adopting any vectorization process. Our approach is based on a novel heterogeneous Tucker decomposition model taking into account cluster membership information. We propose a new clustering algorithm that alternates between different modes of the proposed heterogeneous tensor model. All but the last mode have closed-form updates. Updating the last mode reduces to optimizing over the multinomial manifold for which we investigate second order Riemannian geometry and propose a trust-region algorithm. Numerical experiments show that our proposed algorithm compete effectively with state-of-the-art clustering algorithms that are based on tensor factorization.

Sparse density estimation on multinomial manifold combining local component analysis

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion combining local component analysis for the finite mixture model. We start with a Parzen window estimator which has the Gaussian kernels with a common covariance matrix, the local component analysis is initially applied to find the covariance matrix using expectation maximization algorithm. Since the constraint on the mixing coefficients of a finite mixture model is on the multinomial manifold, we then use the well-known Riemannian trust-region algorithm to find the set of sparse mixing coefficients. The first and second order Riemannian geometry of the multinomial manifold are utilized in the Riemannian trust-region algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with competitive accuracy to existing kernel density estimators.