Manifold Theory Research Articles

The geometry of inflationary observables: Lifts, flows, equivalence classes

The Eisenhart lift allows formulating the dynamics of a scalar field in a potential as pure geodesic motion in a curved field-space manifold involving an additional fictitious vector field. Making use of the formalism in the context of inflation, we show that the main inflationary observables can be expressed in terms of the geometrical properties of a two-dimensional uplifted field-space manifold spanned by the time derivatives of the scalar and the temporal component of the vector. This allows to abstract from specific potentials and models and describe inflation solely in terms of the flow of geometric quantities. Our findings are illustrated through several inflationary examples previously considered in the literature.

Operator Space Manifold Theory: Modeling Quantum Operators with a Riemannian Manifold

The Half-Transform Ansatz (HTA) is a proposed method to solve hyper-geometric equations in Quantum Phase Space by transforming a differential operator to an algebraic variable and including a specific exponential factor in the wave function, but the mechanism which provides this solution scheme is not known. Analysis of the HTA’s application to the Hydrogen atom suggests an underlying mechanism which the HTA is a part of. Observations of exponential factors that act on the wave function naturally suggest modeling quantum operator definitions as a point on a Riemannian manifold in the 4D Operator Space, a novel idea we call the Operator Space Manifold Theory. On this manifold, we explore the concepts of superposition, regions of unique energy eigenvalues, and translation operators. We also find the theoretical backing to derive the HTA and how Operator Space Manifold Theory can be used to describe and solve quantum systems by manipulating how a quantum state perceives position and momentum.

Mathematical Modeling of Coccidiosis Dynamics in Chickens with Some Control Strategies

Coccidiosis is an infectious disease caused by the Eimeria species. The species can infect a bird’s digestive system, severely slow down its growth, and is a serious economic burden for chickens. A mathematical model for the transmission dynamics of coccidiosis disease in chickens in the presence of control interventions has been formulated and analyzed to gain insights into the dynamics of the disease in the population. Three control interventions, namely vaccination, sanitation, and treatment, are implemented. The study intends to assess the effects of these control interventions in coccidiosis transmission dynamics. Using the theory of differential equations, the invariant set of the model was derived, and the model’s solution was found to be mathematically and biologically significant. Analytical methods are employed to establish equilibrium solutions and investigate the stability of the model system’s equilibria, while numerical simulations illustrate the analytical results. The effective reproduction number is obtained using the next-generation matrix method, and the local stability of the equilibria of the model is established. The disease-free equilibrium is proved to be locally stable when the effective reproduction number is less than unity. Also, the nature of the bifurcation and its implications for disease prevention are investigated through the application of the center manifold theory. On the other hand, sensitivity analysis is carried out to investigate the parameters that impact the transmission of coccidiosis disease using the normalized forward sensitivity index. The parameters that have a greater influence on the effective reproduction number should be targeted for control purposes to lessen the spread of disease. Furthermore, numerical simulation is performed to investigate the contribution of each control intervention.

Qualitative Analysis of the Discretization of a Continuous Fractional Order Prey-Predator Model with the Effects of Harvesting and Immigration in the Population

This study examines the discrete prey-predator model in the sense of Caputo fractional derivative by incorporating harvesting on the predator population and immigration on the prey population. We establish the topological categories of the model’s fixed points. We show analytically that a fractional order prey-predator model supports both a Neimark–Sacker (NS) bifurcation and a period-doubling (PD) bifurcation under specific parametric circumstances. Using the central manifold and bifurcation theory, we provide evidence for NS and PD bifurcations. It has been discovered that the parameter values and the initial conditions have a significant influence on the dynamical behavior of the fractional order prey-predator model. Furthermore, two chaos management strategies are applied to eliminate the chaos that objectively exists in the model. Finally, numerical simulations are used to demonstrate complicated and chaotic behavior in order to support our theoretical and analytical discussions.

Effects of intake manifold geometry in H2 & CNG fueled engine combustion

This study attempted to identify the effect of optimized intake manifold geometry on the behaviors and emission level of hydrogen compressed natural gas (H2CNG) fueled engine. For this purpose, a commercial Hyundai Sonata spark ignition engine (SIE) is modified to operate with CNG and hydrogen blend. The optimal intake pipe length was predicted using an analytical acoustic method. A new intake manifold is designed and implemented utilizing natural supercharging managed by over-pressure waves acoustic propagation. Several tests are conducted on the engine using the new manifold with a speed range from 1000 to 5000 rpm. Based on various engine speeds, the variation of brake torque (BT), in-cylinder pressure, NOx and CO emissions investigated by using gasoline, CNG and hydrogen CNG blend (HCNG) fueled engines via external mixtures. The first finding of the study is that the novel geometry improves the in-cylinder pressure by 10% at 3500 rpm. However, high engine speeds show a reduction of 14% in NOx and 40% in HC while speeds below 2000 rpm reduce CO by 40%. The second finding is that the new optimized geometry serves to get rid of both the auto-igni-tion and the backfire for high ratio of hydrogen in the blend.

Manifold alteration between major depressive disorder and healthy control subjects using dynamic mode decomposition in resting-state fMRI data.

The World Health Organization has reported that approximately 300 million individuals suffer from the mood disorder known as MDD. Non-invasive measurement techniques have been utilized to reveal the mechanism of MDD, with rsfMRI being the predominant method. The previous functional connectivity and energy landscape studies have shown the difference in the coactivation patterns between MDD and HCs. However, these studies did not consider oscillatory temporal dynamics. In this study, the dynamic mode decomposition, a method to compute a set of coherent spatial patterns associated with the oscillation frequency and temporal decay rate, was employed to investigate the alteration of the occurrence of dynamic modes between MDD and HCs. Specifically, The BOLD signals of each subject were transformed into dynamic modes representing coherent spatial patterns and discrete-time eigenvalues to capture temporal variations using dynamic mode decomposition. All the dynamic modes were disentangled into a two-dimensional manifold using t-SNE. Density estimation and density ratio estimation were applied to the two-dimensional manifolds after the two-dimensional manifold was split based on HCs and MDD. The dynamic modes that uniquely emerged in the MDD were not observed. Instead, we have found some dynamic modes that have shown increased or reduced occurrence in MDD compared with HCs. The reduced dynamic modes were associated with the visual and saliency networks while the increased dynamic modes were associated with the default mode and sensory-motor networks. To the best of our knowledge, this study showed initial evidence of the alteration of occurrence of the dynamic modes between MDD and HCs. To deepen understanding of how the alteration of the dynamic modes emerges from the structure, it is vital to investigate the relationship between the dynamic modes, cortical thickness, and surface areas.

Deep unfolding as iterative regularization for imaging inverse problems

Deep unfolding methods have gained significant popularity in the field of inverse problems as they have driven the design of deep neural networks (DNNs) using iterative algorithms. In contrast to general DNNs, unfolding methods offer improved interpretability and performance. However, their theoretical stability or regularity in solving inverse problems remains subject to certain limitations. To address this, we reevaluate unfolded DNNs and observe that their algorithmically-driven cascading structure exhibits a closer resemblance to iterative regularization. Recognizing this, we propose a modified training approach and configure termination criteria for unfolded DNNs, thereby establishing the unfolding method as an iterative regularization technique. Specifically, our method involves the joint learning of a convex penalty function using an input-convex neural network to quantify distance to a real data manifold. Then, we train a DNN unfolded from the proximal gradient descent algorithm, incorporating this learned penalty. Additionally, we introduce a new termination criterion for the unfolded DNN. Under the assumption that the real data manifold intersects the solutions of the inverse problem with a unique real solution, even when measurements contain perturbations, we provide a theoretical proof of the stable convergence of the unfolded DNN to this solution. Furthermore, we demonstrate with an example of magnetic resonance imaging reconstruction that the proposed method outperforms original unfolding methods and traditional regularization methods in terms of reconstruction quality, stability, and convergence speed.

Hopf bifurcation and normal form in a delayed oncolytic model

In this paper, we investigate the mathematical analysis of a mathematical model describing the virotherapy treatment of a cancer with logistic growth and the effect of viral cycle presented by a time delay. The cancer population size is divided into uninfected and infected compartments. Depending on time delay, we prove the positivity and boundedness and the stability of equilibria. We give conditions on which the viral cycle leads to “Jeff’s phenomenon” observed in laboratory and causes oscillations in cancer size via Hopf bifurcation theory. We establish an algorithm that determines the bifurcation elements via center manifold and normal form theories. We give conditions which lead to a supercritical or subcritical bifurcation. We end with numerical simulations illustrating our theoretical results.

Mathematical Modeling of Two Interacting Populations’ Dynamics of Onchocerciasis Disease Spread with Nonlinear Incidence Functions

The transmission dynamics of onchocerciasis in two interacting populations are examined using a deterministic compartmental model with nonlinear incidence functions. The model undergoes qualitative analysis to examine how it behaves near disease-free equilibrium (DFE) and endemic equilibrium. Using the Lyapunov function, it is demonstrated that the DFE is globally stable when the threshold parameter R0≤1 is taken into account. When R0>1, it suffices to show globally how asymptotically stable the endemic equilibrium is and its existence. We conduct the bifurcation analysis by looking at the possibility of the model’s equilibria coexisting at R0<1 but near R0=1 using the Center Manifold Theory. We use the sensitivity analysis method to understand how some parameters influence the R0, hence the transmission and mitigation of the disease dynamics. Furthermore, we simulate the model developed numerically to understand the population dynamics. The outcome presented in this article offers valuable understanding of the transmission dynamics of onchocerciasis, specifically in the context of two populations that interact with each other, considering the presence of nonlinear incidence.

A Grassmann manifold handbook: basic geometry and computational aspects

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.

A domain adaptation network with feature scale preservation for remaining useful life prediction of rolling bearings under variable operating conditions

Transfer learning and domain adaptation (DA) methods have been utilized in bearing prognostic and health management, but most of the current DA methods do not take into account the feature scale change of degraded features when aligning the feature distribution, and these methods are more suitable for the classification problem, which is more robust to the feature scale change. However, they perform poorly in regression problems. In addition, most of the remaining useful life (RUL) prediction methods require preprocessing such as statistical feature extraction on the signal, which makes the prediction process complicated. To solve the above problems, a DA method based on the representation subspace distance (RSD) is proposed for predicting the bearing RUL under different operating conditions. First, the proposed convolutional neural network (CNN) self-attention (SA) long short term memory network model is utilized to extract the deep features from the original signal, which overcomes the limitations of the CNN in extracting time series. Then, the RSD in the Riemannian geometry of the Grassmann manifold is proposed as a domain transfer loss to learn domain invariant features. The modified method can align the feature distribution of the source domain and the target domain without changing the feature scale. At the same time, the bases mismatch penalization is introduced to avoid destroying the semantic information of the features in the process of domain alignment. Finally, the effectiveness of the proposed method is verified by experiments on four types of transfer tasks, and its superiority is also demonstrated by comparison with other advanced methods.

THE VARIETY OF INVARIANCE IN FORMAL AND REGIONAL ONTOLOGIES

The paper examines the invariance principles proposed by the analytical and phenomenological traditions for demarcating the boundaries of formal and regional ontologies. The principle of invariance with respect to isomorphic transformations, generalizing Alfred Tarski’s criterion for logical concepts, is extended to formal ontology as the theory of manifolds in its phenomenological interpretation. Isomorphism types, which are abstract individuals of the highest order, hypostases of forms of all possible ontologies, are considered as model-theoretical analogs of manifolds. The correlativity of the phenomenological principles of demarcation of ontological regions and the criterion of invariance with respect to isomorphic transformations is demonstrated, leading to the convergence of logic and formal mathematics in both traditions and, at the same time, to the exclusion of geometry from formal ontology. Particular attention is paid to the discussion of the analytical and phenomenological traditions of the synthetic (material) a priori and the contribution that Wittgenstein’s doctrine on internal relations at different stages of its evolution makes to this discussion. The paper reveals the basis for later Wittgenstein’s criticism of his earlier project of creating a phenomenological language for expressing internal regional relations (such as the colours exclusion). The paper shows how Wittgenstein’s doubts about the possibility of an ideal notation based on the dichotomy of the logical and phenomenological led him to the study of invariants arising in language games. These invariants are not determined by the special properties of the categorical objects of formal ontology or the structures of regional subject matter, but are stable equilibria generated by the “consensus of actions.” New perspectives are outlined that are opened for logic and phenomenology by switching focus from the invariants of ontological structures to the invariants of structured interactions of various agents.

A symplectic approach to Schrödinger equations in the infinite-dimensional unbounded setting

<p>By using the theory of analytic vectors and manifolds modeled on normed spaces, we provide a rigorous symplectic differential geometric approach to $ t $-dependent Schrödinger equations on separable (possibly infinite-dimensional) Hilbert spaces determined by families of unbounded self-adjoint Hamiltonians admitting a common domain of analytic vectors. This allows one to cope with the lack of smoothness of structures appearing in quantum mechanical problems while using differential geometric techniques. Our techniques also allow for the analysis of problems related to unbounded operators that are not self-adjoint. As an application, the Marsden-Weinstein reduction procedure was employed to map the above-mentioned $ t $-dependent Schrödinger equations onto their projective spaces. We also analyzed other physically and mathematically relevant applications, demonstrating the usefulness of our techniques.</p>

Quasinormal forms for systems of two equations with large delay

A system of two equations with delay is considered. The purpose of this work is to study the local dynamics of this system under the assumption that the delay parameter is sufficiently large. Critical cases in the problem of stability of an equilibrium state are identified and it is shown that they have infinite dimension. Methods. The research is based on the use of special methods of infinite-dimensional normalization. Classical methods based on the application of the theory of invariant integral manifolds and normal forms turn out to be directly inapplicable. Results. As the main results, special nonlinear boundary value problems are constructed, which play the role of normal forms. Their nonlocal dynamics determine the behavior of all solutions of the original system in the vicinity of the equilibrium state.

Adaptive Log-Euclidean Metrics for SPD Matrix Learning.

Symmetric Positive Definite (SPD) matrices have received wide attention in machine learning due to their intrinsic capacity to encode underlying structural correlation in data. Many successful Riemannian metrics have been proposed to reflect the non-Euclidean geometry of SPD manifolds. However, most existing metric tensors are fixed, which might lead to sub-optimal performance for SPD matrix learning, especially for deep SPD neural networks. To remedy this limitation, we leverage the commonly encountered pullback techniques and propose Adaptive Log-Euclidean Metrics (ALEMs), which extend the widely used Log-Euclidean Metric (LEM). Compared with the previous Riemannian metrics, our metrics contain learnable parameters, which can better adapt to the complex dynamics of Riemannian neural networks with minor extra computations. We also present a complete theoretical analysis to support our ALEMs, including algebraic and Riemannian properties. The experimental and theoretical results demonstrate the merit of the proposed metrics in improving the performance of SPD neural networks. The efficacy of our metrics is further showcased on a set of recently developed Riemannian building blocks, including Riemannian batch normalization, Riemannian Residual blocks, and Riemannian classifiers.

Stability and bifurcation analysis of a discrete Leslie predator-prey system via piecewise constant argument method

<abstract><p>The objective of this study was to analyze the complex dynamics of a discrete-time predator-prey system by using the piecewise constant argument technique. The existence and stability of fixed points were examined. It was shown that the system experienced period-doubling (PD) and Neimark-Sacker (NS) bifurcations at the positive fixed point by using the center manifold and bifurcation theory. The management of the system's bifurcating and fluctuating behavior may be controlled via the use of feedback and hybrid control approaches. Both methods were effective in controlling bifurcation and chaos. Furthermore, we used numerical simulations to empirically validate our theoretical findings. The chaotic behaviors of the system were recognized through bifurcation diagrams and maximum Lyapunov exponent graphs. The stability of the positive fixed point within the optimal prey growth rate range $ A_1 < a < A_2 $ was highlighted by our observations. When the value of $ a $ falls below a certain threshold $ A_1 $, it becomes challenging to effectively sustain prey populations in the face of predation, thereby affecting the survival of predators. When the growth rate surpasses a specific threshold denoted as $ A_2 $, it initiates a phase of rapid expansion. Predators initially benefit from this phase because it supplies them with sufficient food. Subsequently, resource depletion could occur, potentially resulting in long-term consequences for populations of both the predator and prey. Therefore, a moderate amount of prey's growth rate was beneficial for both predator and prey populations.</p></abstract>

Disk-band graphs in the theory of framed tangles

A disk-band graph is a smooth compact two-dimensional manifold with boundary, partitioned into handles; the partition contains only index zero and index one handles and imitates the structure of the graph. (Index zero handles are analogues of the vertices of the graph, index one handles are analogues of the graph edges.) A disk-band graph is called spatial if it is a smooth submanifold of three-dimensional Euclidean space. A tangle is usually understood as a smooth compact one-dimensional submanifold of the standard three-dimensional ball that intersects the boundary of the ball orthogonally, only along its boundary, the intersection is contained in the equator. We call a tangle framed if it is equipped with a smooth field of normal straight lines. It is well known that there is a reduction of the problem of isotopic classification of spatial disk-band graphs to the problem of isotopic classification of framed tangles. This work focuses on the application of (abstract) disk-band graphs to study the set of isotopic classes of framed tangles.

Generalized Gibbons-Werner method for stationary spacetimes

The Gibbons-Werner (GW) method is a powerful approach in studying the gravitational deflection of particles moving in curved spacetimes. The application of the Gauss-Bonnet theorem (GBT) to integral regions constructed in a two-dimensional manifold enables the deflection angle to be expressed and calculated from the perspective of geometry. However, different techniques are required for different scenarios in the practical implementation which leads to different GW methods. For the GW method for stationary axially symmetric (SAS) spacetimes, we identify two problems: (a) the integral region is generally infinite, which is ill-defined for some asymptotically nonflat spacetimes whose metric possesses singular behavior, and (b) the intricate double and single integrals bring about complicated calculation, especially for highly accurate results and complex spacetimes. To address these issues, a generalized GW method is proposed in which the infinite region is replaced by a flexible region to avoid the singularity, and a simplified formula involving only a single integral of a simple integrand is derived by discovering a significant relationship between the integrals in conventional methods. Our method provides a comprehensive framework for describing the GW method for various scenarios. Additionally, the generalized GW method and simplified calculation formula are applied to three different kinds of spacetimes — Kerr spacetime, Kerr-like black hole in bumblebee gravity, and rotating solution in conformal Weyl gravity. The first two cases have been previously computed by other researchers, affirming the effectiveness and superiority of our approach. Remarkably, the third case is newly examined, yielding a innovative result for the first time.

Bifurcation and optimal harvesting analysis of a discrete-time predator–prey model with fear and prey refuge effects

<p>In this contribution, the complicated dynamical behaviors and optimal harvesting policy of a discrete-time predator–prey model with fear and refuge effects are formulated. Both the fear and prey refuge effects refer to an interaction between predator and prey. In the first place, the existence and local stability of three fixed points of proposed model are investigated by virtue of our methodology, that is, the eigenvalues of the Jacobian matrix. One step further, it is worth mentioning that the model undergoes flip bifurcation (i.e., period–doubling bifurcation) and Neimark–Sacker bifurcation at the interior fixed point by the utilization of bifurcation theory and center manifold theory. Also, optimal harvesting strategy is investigated, and the expressions of optimal harvesting efforts are determined. Two examples, in the end, are put forward to prove that they are consistent with the previous theoretical results.</p>

Global stability and optimal vaccination control of SVIR models

<abstract> <p>Vaccination is widely acknowledged as an affordable and cost-effective approach to guard against infectious diseases. It is important to take vaccination rate, vaccine effectiveness, and vaccine-induced immune decline into account in epidemic dynamical modeling. In this paper, an epidemic dynamical model of vaccination is developed. This model provides a framework of the infectious disease transmission dynamics model through qualitative and quantitative analysis. The result shows that the system may have multiple equilibria. We used the next-generation operator approach to calculate the maximum spectral radius, that is, basic reproduction number $ {R_{vac}} $. Next, by dividing the model into infected and uninfected subjects, we can prove that the disease-free equilibrium is globally asymptotically stable when $ {R_{vac}} < 1 $, provided certain assumptions are satisfied. When $ {R_{vac}} > 1 $, there exists a unique endemic equilibrium. Using geometric methods, we calculate the second compound matrix and demonstrate the Lozinskii measure $ \bar q \leqslant 0 $, which is equivalent to the unique endemic equilibrium, which is globally asymptotically stable. Then, using center manifold theory, we justify the existence of forward bifurcation. As the vaccination rate decreases, the likelihood of forward bifurcation increases. We also theoretically show the presence of Hopf bifurcation. Then, we performed sensitivity analysis and found that increasing the vaccine effectiveness rate can curb the propagation of disease effectively. To examine the influence of vaccination on disease control, we chose the vaccination rate as the optimal vaccination control parameter, using the Pontryagin maximum principle, and we found that increasing vaccination rates reduces the number of infected individuals. Finally, we ran a numerical simulation to finalize the theoretical results.</p> </abstract>