Manifold Theory Research Articles

Dynamics of a class of nonlinear pest–natural enemy discrete model

The inappropriate use of insecticides may lead to disastrous pest outbreaks. Aiming at avoiding the outbreak of pests by optimizing the control strategies, we propose a novel mathematical model based on the idea of pulse in this study, namely, a discrete‐time model of pest–natural enemies, which considers the implementation of spraying insecticides within the time interval between two generations of pests. We first investigate the existence and stability of fixed points, and then using the central manifold theory, we proved that the system can exist period‐doubling bifurcation. The main theoretical results are verified by numerical simulations. The experiments show that if the insecticidal time is within a certain range, the lower insecticidal rate stimulates the growth of pests and natural enemies, while the larger insecticidal rate can inhibit the growth of pests and natural enemies. In addition, the effects of pest growth rate, timing of pesticide spraying, and pesticide intensity on the system are comprehensively discussed. The main research provides good reference significance to choose an appropriate time to prevent pest outbreaks.

Self-adjoint local boundary problems on compact surfaces. II. Family index

The paper presents a first step towards a family index theorem for classical self-adjoint boundary value problems.We address here the simplest non-trivial case of manifolds with boundary, namely the case of two-dimensional manifolds. The first result of the paper is an index theorem for families of first order self-adjoint elliptic differential operators with local boundary conditions, parametrized by points of a compact topological space X . We compute the K^1(X) -valued index in terms of the topological data over the boundary. The second result is the universality of the index: we show that the index is a universal additive homotopy invariant for such families if the vanishing on families of invertible operators is assumed.

A Malware Propagation Model Considering Conformity Psychology in Social Networks

At present, malware is still a major security threat to computer networks. However, only a fraction of users with some security consciousness take security measures to protect computers on their own initiative, and others who know the current situation through social networks usually follow suit. This phenomenon is referred to as conformity psychology. It is obvious that more users will take countermeasures to prevent computers from being infected if the malware spreads to a certain extent. This paper proposes a deterministic nonlinear SEIQR propagation model to investigate the impact of conformity psychology on malware propagation. Both the local and global stabilities of malware-free equilibrium are proven while the existence and local stability of endemic equilibrium is proven by using the central manifold theory. Additionally, some numerical examples and simulation experiments based on two network datasets are performed to verify the theoretical analysis results. Finally, the sensitivity analysis of system parameters is carried out.

Machine Learning Models Can Accurately Detect Crystalloid Contamination of Basic Metabolic Panels

Abstract A common cause of erroneous laboratory results is preanalytical contamination of samples with intravenous crystalloids. Rules-based approaches, such as delta checks or feasibility cut-offs, can detect these errors, but rely on prior laboratory results and lack sensitivity. In this study, we aimed to improve the detection of crystalloid contamination using two machine learning workflows for the detection of contamination by the four most common crystalloids; normal saline (NS), lactated Ringer’s (LR), and their dextrose-containing counterparts (D5NS and D5LR). First, we employed a semi-supervised approach using manifold approximation and contrastive learning. To do this, we aggregated five years' worth of basic metabolic panel results (n = 67 million), then simulated contaminated samples by mixing random subsets of samples in silico with the crystalloid solutions at different ratios. The dataset was partitioned into a 70:30 split between training and testing sets with five-fold cross-validation. An autoencoder was trained to create an embedding of the data by minimizing the triplet loss. This embedding was then projected onto a two-dimensional manifold. When the four crystalloid solutions and the independent test set were mapped to the manifold, decision boundaries were drawn based on the threshold that maximized the F1 statistic, and labels were assigned. This method detected the 10% in silico mixtures with an area under the receiver operating characteristic curve (AUC) of 0.97, 0.94, 0.99, and 0.99 for NS, LR, D5NS, and D5LR respectively. Next, we evaluated a fully-supervised approach by training and validating Random Forest and XGBoost models using the same training and testing sets as above. Both models performed well, with XGBoost slightly outperforming Random Forest. The XGBoost model demonstrated AUCs of 0.992, 0.990, 0.998, and 0.996 for the NS, LR, D5NS, and D5LR mixtures. This high-level performance was maintained even as the proportion of mixed samples in the dataset was decreased from 50% to 1% to more accurately reflect real-world contamination rates. For NS and LR, the variables with the highest relative importance were calcium and chloride. For their D5 counterparts, glucose and calcium were most important. Creatinine and blood urea nitrogen were the least important variables in both approaches. Overall, the fully-supervised learning approach outperformed that of the contrastive embeddings for the specific task of identifying contamination from pre-defined crystalloids. One advantage of the semi-supervised approach is that it could be adapted to identify outliers that are not pre-defined. However, considering the relative ease of translating tree-based models into practice, we conclude that our XGBoost model offers the most promising solution and plan to prospectively validate this model for real-time detection of sample contamination.

Spline estimation of functional principal components via manifold conjugate gradient algorithm

Functional principal component analysis has become the most important dimension reduction technique in functional data analysis. Based on B-spline approximation, functional principal components (FPCs) can be efficiently estimated by the expectation-maximization (EM) and the geometric restricted maximum likelihood (REML) algorithms under the strong assumption of Gaussianity on the principal component scores and observational errors. When computing the solution, the EM algorithm does not exploit the underlying geometric manifold structure, while the performance of REML is known to be unstable. In this article, we propose a conjugate gradient algorithm over the product manifold to estimate FPCs. This algorithm exploits the manifold geometry structure of the overall parameter space, thus improving its search efficiency and estimation accuracy. In addition, a distribution-free interpretation of the loss function is provided from the viewpoint of matrix Bregman divergence, which explains why the proposed method works well under general distribution settings. We also show that a roughness penalization can be easily incorporated into our algorithm with a potentially better fit. The appealing numerical performance of the proposed method is demonstrated by simulation studies and the analysis of a Type Ia supernova light curve dataset.

Monoidally graded manifolds

We give a generalization of the theory of Z2-graded manifolds to a theory of I-graded manifolds, where I is a commutative semi-ring with some additional properties. We prove Batchelor's theorem in this generalized setting. To our knowledge, such a proof is still missing except for some special cases.

A STUDY ON MANIFOLD KAEHLERIAIN SPACE

This work delves into the space-time theory of the 4-dimensional Kaehler manifold. Since the isotropic pressure, energy density, and energy momentum tensor all vanish in a perfect fluid Kaehler space-time manifold, we have established that this space-time manifold is an Einstein manifold and studied the Einstein equation with a cosmological constant in it. Finally, we demonstrated that, on a conformally flat, perfectly fluid Kaehler space-time manifold, the velocity vector field is infinitesimally spatially isotropic. Ideal fluid dilution to the point where only Ricci and minimal symmetry hold. We have proven that Kaehler space-time manifolds have either 0 scalar curvature or a connection between the respective rho and alpha vector fields via g(rho,alpha) = 4. To conclude, we have proved that a Kaehler space-time manifold cannot have both perfect fluidity and non-zero scalar curvature (weak Ricci symmetry).

On quantum‐induced revisiting Einstein tensor in the relativistic regime

AbstractFor a rigorous approach to reconcile principles of general relativity (GR) and quantum mechanics (QM), we suggest applying the generalized relativistic noncommutative Heisenberg algebra on spacetime coordinates and momenta of a test particle on curved Riemann geometry manifold, which in turn is extended to an eight‐dimensional manifold. The natural generalization of the four‐dimensional Riemann geometry is the Finsler geometry, in which the quadratic restriction on the length measure is relaxed. With the minimum measurable length derived from the relativistic generalized four‐dimensional uncertainty principle, the quantum‐induced corrections to the fundamental tensor could be determined in the relativistic regime. Accordingly, the affine connections could be revisited. With the quantum‐induced revisiting Riemann curvature tensor and its contractions, the Ricci curvature tensor, and then the Ricci scalar, we have been able to construct a quantum‐induced revision of the Einstein tensor, in which besides quantum‐induced corrections, additional geometric structures emerge. On the surface of the 2‐sphere, we compared the quantum‐induced revisiting and non‐revisiting Einstein tensor and concluded that the difference between both versions strongly depends on the minimum measurable length and the local geodesics of the test particle through the additional curvature.

Flexible integration of continuous sensory evidence in perceptual estimation tasks

Temporal accumulation of evidence is crucial for making accurate judgments based on noisy or ambiguous sensory input. The integration process leading to categorical decisions is thought to rely on competition between neural populations, each encoding a discrete categorical choice. How recurrent neural circuits integrate evidence for continuous perceptual judgments is unknown. Here, we show that a continuous bump attractor network can integrate a circular feature, such as stimulus direction, nearly optimally. As required by optimal integration, the population activity of the network unfolds on a two-dimensional manifold, in which the position of the network's activity bump tracks the stimulus average, and, simultaneously, the bump amplitude tracks stimulus uncertainty. Moreover, the temporal weighting of sensory evidence by the network depends on the relative strength of the stimulus compared to the internally generated bump dynamics, yielding either early (primacy), uniform, or late (recency) weighting. The model can flexibly switch between these regimes by changing a single control parameter, the global excitatory drive. We show that this mechanism can quantitatively explain individual temporal weighting profiles of human observers, and we validate the model prediction that temporal weighting impacts reaction times. Our findings point to continuous attractor dynamics as a plausible neural mechanism underlying stimulus integration in perceptual estimation tasks.

Perturbed precessing ellipses as the building blocks of spiral arms in a barred galaxy with two pattern speeds

Observations and simulations of barred spiral galaxies have shown that, in general, the spiral arms rotate at a different pattern speed to that of the bar. The main conclusion from the bibliography is that the bar rotates faster than the spiral arms with a double or even a triple value of angular velocity. The theory that prevails in explaining the formation of the spiral arms in the case of a barred spiral galaxy with two pattern speeds is the manifold theory, where the orbits that support the spiral density wave are chaotic, and are related to the manifolds emanating from the Lagrangian points L1 and L2 at the end of the bar. In the present study, we consider an alternative scenario in the case where the bar rotates fast enough in comparison with the spiral arms and the bar potential can be considered as a perturbation of the spiral potential. In this case, the stable elliptical orbits that support the spiral density wave (in the case of grand design galaxies) are transformed into quasiperiodic orbits (or 2D tori) with a certain thickness. The superposition of these perturbed preccesing ellipses for all the energy levels of the Hamiltonian creates a slightly perturbed symmetrical spiral density wave.

Normal forms for saddle-node bifurcations: Takens’ coefficient and applications in climate models

We show that a one-dimensional differential equation depending on a parameter μ with a saddle-node bifurcation at μ = 0 can be modelled by an extended normal form y ˙ = ν ( μ ) − y 2 + a ( μ ) y 3 , where the functions ν and a are solutions to equations that can be written down explicitly. The equivalence to the original equations is a local differentiable conjugacy on the basins of attraction and repulsion of stationary points in the parameter region for which these exist, and is a differentiable conjugacy on the whole local interval otherwise. (Recall that in standard approaches local equivalence is topological rather than differentiable.) The value a ( 0 ) is Takens’ coefficient from normal form theory. The results explain the sense in which normal forms extend away from the bifurcation point and provide a new and more detailed characterization of the saddle-node bifurcation. The one-dimensional system can be derived from higher dimensional equations using centre manifold theory. We illustrate this using two examples from climate science and show how the functions ν and a can be determined analytically in some settings and numerically in others.

Mathematical model of COVID-19 transmission dynamics incorporating booster vaccine program and environmental contamination

COVID-19 is one of the greatest human global health challenges that causes economic meltdown of many nations. In this study, we develop an SIR-type model which captures both human-to-human and environment-to-human-to-environment transmissions that allows the recruitment of corona viruses in the environment in the midst of booster vaccine program. Theoretically, we prove some basic properties of the full model as well as investigate the existence of SARS-CoV-2-free and endemic equilibria. The SARS-CoV-2-free equilibrium for the special case, where the constant inflow of corona virus into the environment by any other means, Ω is suspended (Ω=0) is globally asymptotically stable when the effective reproduction number R0c<1 and unstable if otherwise. Whereas in the presence of free-living Corona viruses in the environment (Ω>0), the endemic equilibrium using the centre manifold theory is shown to be stable globally whenever R0c>1. The model is extended into optimal control system and analyzed analytically using Pontryagin's Maximum Principle. Results from the optimal control simulations show that strategy E for implementing the public health advocacy, booster vaccine program, treatment of isolated people and disinfecting or fumigating of surfaces and dead bodies before burial is the most effective control intervention for mitigating the spread of Corona virus. Importantly, based on the available data used, the study also revealed that if at least 70% of the constituents followed the aforementioned public health policies, then herd immunity could be achieved for COVID-19 pandemic in the community.

Nontrivial quantum geometry of degenerate flat bands

The importance of the quantum metric in flat-band systems has been noticed recently in many contexts such as the superfluid stiffness, the dc electrical conductivity, and ideal Chern insulators. Both the quantum metric of degenerate and nondegenerate bands can be naturally described via the geometry of different Grassmannian manifolds, specific to the band degeneracies. Contrary to the (Abelian) Berry curvature, the quantum metric of a degenerate band resulting from the collapse of a collection of bands is not simply the sum of the individual quantum metrics. We provide a physical interpretation of this phenomenon in terms of transition dipole matrix elements between two bands. By considering a toy model, we show that the quantum metric gets enhanced, reduced, or remains unaffected depending on which bands collapse. The dc longitudinal conductivity and the superfluid stiffness are known to be proportional to the quantum metric for flat-band systems, which makes them suitable candidates for the observation of this phenomenon.

Higher Topological Complexities of Real Grassmannians and Semi-complete Real Flag Manifolds

Higher Topological Complexities of Real Grassmannians and Semi-complete Real Flag Manifolds

Null and timelike circular orbits from equivalent 2D metrics

The motion of particles on spherical dimensional spacetimes can, under some assumptions, be described by the curves on a two-dimensional manifold, the optical and Jacobi manifolds for null and timelike curves, respectively. In this paper we resort to auxiliary two-dimensional metrics to study circular geodesics of generic static, spherically symmetric, and asymptotically flat dimensional spacetimes, whose functions are at least C 2 smooth. This is done by studying the Gaussian curvature of the bidimensional equivalent manifold as well as the geodesic curvature of circular paths on these. This study considers both null and timelike circular geodesics. The study of null geodesics through the optical manifold retrieves the known result of the number of light rings on the spacetime outside a black hole and on spacetimes with horizonless compact objects. With an equivalent procedure we can formulate a similar theorem on the number of marginally stable timelike circular orbits of a given spacetime satisfying the previously mentioned assumptions.

Infection dynamics of rabbit and red fox with alternative prey

In many countries, the decline in red foxes due to Rabbit hemorrhagic disease (RHD) in primary prey European rabbits is a significant concern. We proposed a four-compartment spatiotemporal rabbits-alternative prey-red fox eco-epidemiological model with mange disease and hunting in red foxes. The essential theoretical properties, such as existence, boundedness, stability, and bifurcation analysis, are executed. We have also conducted Turing instability and Higher-order stability analysis for the spatiotemporal model. Hopf bifurcation is shown at a critical value of hunting rate \(h=h_c\) using central manifold theory. Numerical simulation reveals that the present dynamic is chaotic for a mange disease transmission rate's threshold value \(\beta=\beta_+\), the most significant factor in the present dynamics. We can control the red fox population by controlling the mange contact rate despite RHD disease in European rabbits. Also, the model does not have diffusion-driven instability due to alternative prey, and if the system is linearly stable, it remains stable for higher-order.

Neural representational geometry underlies few-shot concept learning

Understanding the neural basis of the remarkable human cognitive capacity to learn novel concepts from just one or a few sensory experiences constitutes a fundamental problem. We propose a simple, biologically plausible, mathematically tractable, and computationally powerful neural mechanism for few-shot learning of naturalistic concepts. We posit that the concepts that can be learned from few examples are defined by tightly circumscribed manifolds in the neural firing-rate space of higher-order sensory areas. We further posit that a single plastic downstream readout neuron learns to discriminate new concepts based on few examples using a simple plasticity rule. We demonstrate the computational power of our proposal by showing that it can achieve high few-shot learning accuracy on natural visual concepts using both macaque inferotemporal cortex representations and deep neural network (DNN) models of these representations and can even learn novel visual concepts specified only through linguistic descriptors. Moreover, we develop a mathematical theory of few-shot learning that links neurophysiology to predictions about behavioral outcomes by delineating several fundamental and measurable geometric properties of neural representations that can accurately predict the few-shot learning performance of naturalistic concepts across all our numerical simulations. This theory reveals, for instance, that high-dimensional manifolds enhance the ability to learn new concepts from few examples. Intriguingly, we observe striking mismatches between the geometry of manifolds in the primate visual pathway and in trained DNNs. We discuss testable predictions of our theory for psychophysics and neurophysiological experiments.

Machine-learning model predicting postoperative delirium in older patients using intraoperative frontal electroencephalographic signatures.

ObjectiveIn older patients receiving general anesthesia, postoperative delirium (POD) is the most frequent form of cerebral dysfunction. Early identification of patients at higher risk to develop POD could provide the opportunity to adapt intraoperative and postoperative therapy. We, therefore, propose a machine learning approach to predict the risk of POD in elderly patients, using routine intraoperative electroencephalography (EEG) and clinical data that are readily available in the operating room.MethodsWe conducted a retrospective analysis of the data of a single-center study at the Charité-Universitätsmedizin Berlin, Department of Anesthesiology [ISRCTN 36437985], including 1,277 patients, older than 60 years with planned surgery and general anesthesia. To deal with the class imbalance, we used balanced ensemble methods, specifically Bagging and Random Forests and as a performance measure, the area under the ROC curve (AUC-ROC). We trained our models including basic clinical parameters and intraoperative EEG features in particular classical spectral and burst suppression signatures as well as multi-band covariance matrices, which were classified, taking advantage of the geometry of a Riemannian manifold. The models were validated with 10 repeats of a 10-fold cross-validation.ResultsIncluding EEG data in the classification resulted in a robust and reliable risk evaluation for POD. The clinical parameters alone achieved an AUC-ROC score of 0.75. Including EEG signatures improved the classification when the patients were grouped by anesthetic agents and evaluated separately for each group. The spectral features alone showed an AUC-ROC score of 0.66; the covariance features showed an AUC-ROC score of 0.68. The AUC-ROC scores of EEG features relative to patient data differed by anesthetic group. The best performance was reached, combining both the EEG features and the clinical parameters. Overall, the AUC-ROC score was 0.77, for patients receiving Propofol it was 0.78, for those receiving Sevoflurane it was 0.8 and for those receiving Desflurane 0.73. Applying the trained prediction model to an independent data set of a different clinical study confirmed these results for the combined classification, while the classifier on clinical parameters alone did not generalize.ConclusionA machine learning approach combining intraoperative frontal EEG signatures with clinical parameters could be an easily applicable tool to early identify patients at risk to develop POD.

A PDE-regularized smoothing method for space-time data over manifolds with application to medical data.

We propose an innovative statistical-numerical method to model spatio-temporal data, observed over a generic two-dimensional Riemanian manifold. The proposed approach consists of a regression model completed with a regularizing term based on the heat equation. The model is discretized through a finite element scheme set on the manifold, and solved by resorting to a fixed point-based iterative algorithm. This choice leads to a procedure which is highly efficient when compared with a monolithic approach, and which allows us to deal with massive datasets. After a preliminary assessment on simulation study cases, we investigate the performance of the new estimation tool in practical contexts, by dealing with neuroimaging and hemodynamic data.

Bifurcation dynamics of a delayed chemostat system with spatial diffusion

The resource’s diffusion and the populations’ migration may alter the ecosystem’s structure such that the species’ dynamics are changed. Moreover, the delay phenomenon in population behaviors, such as digestion or maturation, will inevitably affect the species’ dynamics. We, in this work, investigate a chemostat system with delay and spatial diffusion. The existence conditions of the Hopf bifurcation from the time lag and diffusive terms are determined. The concentration of population in the chemostat approaches a positive value when the bifurcation parameter’s value does not cross the critical point. The microorganisms’ concentration will fluctuate periodically as the value of the bifurcation parameter passes through the critical point. By the theory of norm form and center manifold, we further talked about the direction of the Hopf bifurcation and the stability of the periodic solutions. Several numerical examples are provided to support the theoretical results in this work.