Cycle integrals of meromorphic modular forms and Siegel theta functions.

We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke's indefinite theta functions.

Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves.

We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order q∈[0,∞). We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if q>1/2. Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if q>3/2, whereas if q<3/2 then finite-time blowup may occur. The geodesic completeness for q>3/2 is obtained by proving metric completeness of the space of Hq-immersed curves with the distance induced by the Riemannian metric.

Torsion affects the calculation of DNA twisting number.

The topological properties of DNA have long been a focal point in biophysics. In the 1970s, White proposed that the topology of closed DNA double helix follows White's formula: Lk=Wr+Tw. However, there has been controversy in the calculation of DNA twisting number, partly due to discrepancies in the definition of torsion in differential geometry. In this paper, we delved into a detailed study of torsion, revealing that the calculation of DNA twisting number should use the curve's geodesic torsion. Furthermore, we found that the discrepancy in DNA twisting numbers calculated using different torsion is N. This study elucidated the impact of torsion on the calculation of DNA twisting numbers, aiming to resolve controversies in the calculation of DNA topology and provided accurate computational methods and theoretical foundations for related research.

Adaptive activation Functions with Deep Kronecker Neural Network optimized with Bear Smell Search Algorithm for preventing MANET Cyber security attacks.

An Adaptive activation Functions with Deep Kronecker Neural Network optimized with Bear Smell Search Algorithm (BSSA) (ADKNN-BSSA-CSMANET) is proposed for preventing MANET Cyber security attacks. The mobile users are enrolled with Trusted Authority Using a Crypto Hash Signature (SHA-256). Every mobile user uploads their finger vein biometric, user ID, latitude and longitude for confirmation. The packet analyser checks if any attack patterns are identified. It is implemented using adaptive density-based spatial clustering (ADSC) that deems information from packet header. Geodesic filtering (GF) is used as a pre-processing method for eradicating the unsolicited content and filtering pertinent data. Group Teaching Algorithm (GTA)-based feature selection is utilized for ideal collection of features and Adaptive Activation Functions along Deep Kronecker Neural Network (ADKNN) is used to categorizing normal and attack packets (DoS, Probe, U2R, and R2L). Then BSSA is utilized for optimizing the weight parameters of ADKNN classifier for optimal classification. The proposed technique is executed in python and its efficiency is evaluated by several performances metrics, such as Accuracy, Attack Detection Rate, Detection Delay, Packet Delivery Ratio, Throughput, and Energy Consumption. The proposed technique provides 36.64%, 33.06%, and 33.98% lower Detection Delay on NSL-KDD dataset compared with the existing methods.

Prioritizing replay when future goals are unknown.

Although hippocampal place cells replay nonlocal trajectories, the computational function of these events remains controversial. One hypothesis, formalized in a prominent reinforcement learning account, holds that replay plans routes to current goals. However, recent puzzling data appear to contradict this perspective by showing that replayed destinations lag current goals. These results may support an alternative hypothesis that replay updates route information to build a "cognitive map." Yet no similar theory exists to formalize this view, and it is unclear how such a map is represented or what role replay plays in computing it. We address these gaps by introducing a theory of replay that learns a map of routes to candidate goals, before reward is available or when its location may change. Our work extends the planning account to capture a general map-building function for replay, reconciling it with data, and revealing an unexpected relationship between the seemingly distinct hypotheses.

Modular geodesics and wedge domains in non-compactly causal symmetric spaces.

We continue our investigation of the interplay between causal structures on symmetric spaces and geometric aspects of Algebraic Quantum Field Theory. We adopt the perspective that the geometric implementation of the modular group is given by the flow generated by an Euler element of the Lie algebra (an element defining a 3-grading). Since any Euler element of a semisimple Lie algebra specifies a canonical non-compactly causal symmetric space M=G/H, we turn in this paper to the geometry of this flow. Our main results concern the positivity region W of the flow (the corresponding wedge region): If G has trivial center, then W is connected, it coincides with the so-called observer domain, specified by a trajectory of the modular flow which at the same time is a causal geodesic. It can also be characterized in terms of a geometric KMS condition, and it has a natural structure of an equivariant fiber bundle over a Riemannian symmetric space that exhibits it as a real form of the crown domain of G/K. Among the tools that we need for these results are two observations of independent interest: a polar decomposition of the positivity domain and a convexity theorem for G-translates of open H-orbits in the minimal flag manifold specified by the 3-grading.

Planar Pseudo-geodesics and Totally Umbilic Submanifolds.

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, representing a natural extrinsic generalization of both geodesics and Riemannian circles: being planar, their Cartan development in the tangent space is planar in the ordinary sense; being pseudo-geodesics, their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion ι:M↪Q is totally umbilic if and only if the extrinsic shape of every geodesic of M is planar. This extends a well-known result about surfaces in R3.

Weakly supervised segmentation of uterus by scribble labeling on endometrial cancer MR images.

Uterine segmentation of endometrial cancer MR images can be a valuable diagnostic tool for gynecologists. However, uterine segmentation based on deep learning relies on artificial pixel-level annotation, which is time-consuming, laborious and subjective. To reduce the dependence on pixel-level annotation, a method of weakly supervised uterine segmentation on endometrial cancer MRI slices is proposed, which only requires scribble label and is enhanced by pseudo-label technology, exponential geodesic distance loss and input disturbance strategy. Specifically, the limitations caused by the shortage of supervision are addressed by dynamically mixing the two outputs of the dual branch network to generate pseudo-labels, expanding supervision information and promoting mutual supervision training. On the other hand, considering the large difference of grayscale intensity between the uterus and surrounding tissues, the exponential geodesic distance loss is introduced to enhance the ability of the network to capture the edge of the uterus. Input disturbance strategies are incorporated to adapt to the flexible and variable characteristics of the uterus and further improve the segmentation performance of the network. The proposed method is evaluated on MRI images from 135 cases of endometrial cancer. Compared with other four weakly supervised segmentation methods, the performance of the proposed method is the best, whose mean DI, HD95, Recall, Precision, ADP are 92.8%, 11.632, 92.7%, 93.6%, 6.5% and increasing by 2.1%, 9.144, 0.6%, 2.4%, 2.9% respectively. The experimental results demonstrate that the proposed method is more effective than other weakly supervised methods and achieves similar performance as those fully supervised.

D'Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders.

Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about geodesic segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D'Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D'Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D'Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension.

Total torsion of three-dimensional lines of curvature.

A curve γ in a Riemannian manifold M is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when γ lies on an oriented hypersurface S of M, we say that γ is well positioned if the curve's principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that γ is three-dimensional and closed. We show that if γ is a well-positioned line of curvature of S, then its total torsion is an integer multiple of 2π; and that, conversely, if the total torsion of γ is an integer multiple of 2π, then there exists an oriented hypersurface of M in which γ is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of γ vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.

Computing geodesic paths encoding a curvature prior for curvilinear structure tracking.

In this paper, we introduce an efficient method for computing curves minimizing a variant of the Euler-Mumford elastica energy, with fixed endpoints and tangents at these endpoints, where the bending energy is enhanced with a user-defined and data-driven scalar-valued term referred to as the curvature prior. In order to guarantee that the globally optimal curve is extracted, the proposed method involves the numerical computation of the viscosity solution to a specific static Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). For that purpose, we derive the explicit Hamiltonian associated with this variant model equipped with a curvature prior, discretize the resulting HJB PDE using an adaptive finite difference scheme, and solve it in a single pass using a generalized fast-marching method. In addition, we also present a practical method for estimating the curvature prior values from image data, designed for the task of accurately tracking curvilinear structure centerlines. Numerical experiments on synthetic and real-image data illustrate the advantages of the considered variant of the elastica model with a prior curvature enhancement in complex scenarios where challenging geometric structures appear.

Harmonizing Healthy Cohorts to Support Multicenter Studies on Migraine Classification using Brain MRI Data.

Multicenter and multi-scanner imaging studies might be needed to provide sample sizes large enough for developing accurate predictive models. However, multicenter studies, which likely include confounding factors due to subtle differences in research participant characteristics, MRI scanners, and imaging acquisition protocols, might not yield generalizable machine learning models, that is, models developed using one dataset may not be applicable to a different dataset. The generalizability of classification models is key for multi-scanner and multicenter studies, and for providing reproducible results. This study developed a data harmonization strategy to identify healthy controls with similar (homogenous) characteristics from multicenter studies to validate the generalization of machine-learning techniques for classifying individual migraine patients and healthy controls using brain MRI data. The Maximum Mean Discrepancy (MMD) was used to compare the two datasets represented in Geodesic Flow Kernel (GFK) space, capturing the data variabilities for identifying a "healthy core". A set of homogeneous healthy controls can assist in overcoming some of the unwanted heterogeneity and allow for the development of classification models that have high accuracy when applied to new datasets. Extensive experimental results show the utilization of a healthy core. One dataset consists of 120 individuals (66 with migraine and 54 healthy controls) and another dataset consists of 76 (34 with migraine and 42 healthy controls) individuals. A homogeneous dataset derived from a cohort of healthy controls improves the performance of classification models by about 25% accuracy improvements for both episodic and chronic migraineurs.

Geodesic theory of long association fibers arrangement in the human fetal cortex.

Association fibers connect different areas of the cerebral cortex over long distances and integrate information to achieve higher brain functions, particularly in humans. Prototyped association fibers are developed to the respective tangential direction throughout the cerebral hemispheres along the deepest border of the subplate during the fetal period. However, how guidance to remote areas is achieved is not known. Because the subplate is located below the cortical surface, the tangential direction of the fibers may be biased by the curved surface geometry due to Sylvian fissure and cortical poles. The fiber length can be minimized if the tracts follow the shortest paths (geodesics) of the curved surface. Here, we propose and examine a theory that geodesics guide the tangential direction of long association fibers by analyzing how geodesics are spatially distributed on the fetal human brains. We found that the geodesics were dense on the saddle-shaped surface of the perisylvian region and sparse on the dome-shaped cortical poles. The geodesics corresponded with the arrangement of five typical association fibers, supporting the theory. Thus, the geodesic theory provides directional guidance information for wiring remote areas and suggests that long association fibers emerge from minimizing their tangential length in fetal brains.

Hierarchical Geodesic Polynomial Model for Multilevel Analysis of Longitudinal Shape.

Longitudinal analysis is a core aspect of many medical applications for understanding the relationship between an anatomical subject's function and its trajectory of shape change over time. Whereas mixed-effects (or hierarchical) modeling is the statistical method of choice for analysis of longitudinal data, we here propose its extension as hierarchical geodesic polynomial model (HGPM) for multilevel analyses of longitudinal shape data. 3D shapes are transformed to a non-Euclidean shape space for regression analysis using geodesics on a high dimensional Riemannian manifold. At the subject-wise level, each individual trajectory of shape change is represented by a univariate geodesic polynomial model on timestamps. At the population level, multivariate polynomial expansion is applied to uni/multivariate geodesic polynomial models for both anchor points and tangent vectors. As such, the trajectory of an individual subject's shape changes over time can be modeled accurately with a reduced number of parameters, and population-level effects from multiple covariates on trajectories can be well captured. The implemented HGPM is validated on synthetic examples of points on a unit 3D sphere. Further tests on clinical 4D right ventricular data show that HGPM is capable of capturing observable effects on shapes attributed to changes in covariates, which are consistent with qualitative clinical evaluations. HGPM demonstrates its effectiveness in modeling shape changes at both subject-wise and population levels, which is promising for future studies of the relationship between shape changes over time and the level of dysfunction severity on anatomical objects associated with disease.

Cerebral cortical regions always connect with each other via the shortest paths.

In human society, the choice of transportation mode between two cities is largely influenced by the distance between the regions. Similarly, when neurons communicate with each other within the cerebral cortex, do they establish their connections based on their physical distance? In this study, we employed a data-driven approach to explore the relationships between fiber length and corresponding geodesic distance between the fiber's two endpoints on brain surface. Diffusion-MRI-derived fiber streamlines were used to represent extra-cortical axonal connections between neurons or cortical regions, while geodesic paths between cortical points were employed to simulate intra-cortical connections. The results demonstrated that the geodesic distance between two cortical regions connected by a fiber streamline was greater than the fiber length most of the time, indicating that cortical regions tend to choose the shortest path for connection; whether it be an intra-cortical or extra-cortical route, especially when intra-cortical routes within cortical regions are longer than potential extrinsic fiber routes, there is an increased probability to establish fiber routes to connect the both regions. These findings were validated in a group of human brains and may provide insights into the underlying mechanisms of neuronal growth, connection, and wiring.

The Singularity Theorems of General Relativity and Their Low Regularity Extensions.

On the occasion of Sir Roger Penrose's 2020 Nobel Prize in Physics, we review the singularity theorems of General Relativity, as well as their recent extension to Lorentzian metrics of low regularity. The latter is motivated by the quest to explore the nature of the singularities predicted by the classical theorems. Aiming at the more mathematically minded reader, we give a pedagogical introduction to the classical theorems with an emphasis on the analytical side of the arguments. We especially concentrate on focusing results for causal geodesics under appropriate geometric and initial conditions, in the smooth and in the low regularity case. The latter comprise the main technical advance that leads to the proofs of C1-singularity theorems via a regularisation approach that allows to deal with the distributional curvature. We close with an overview on related lines of research and a future outlook.

Towards causal mechanisms of consciousness through focused transcranial brain stimulation.

Conscious experience represents one of the most elusive problems of empirical science, namely neuroscience. The main objective of empirical studies of consciousness has been to describe the minimal sets of neural events necessary for a specific neuronal state to become consciously experienced. The current state of the art still does not meet this objective but rather consists of highly speculative theories based on correlates of consciousness and an ever-growing list of knowledge gaps. The current state of the art is defined by the limitations of past stimulation techniques and the emphasis on the observational approach. However, looking at the current stimulation technologies that are becoming more accurate, it is time to consider an alternative approach to studying consciousness, which builds on the methodology of causal explanations via causal alterations. The aim of this methodology is to move beyond the correlates of consciousness and focus directly on the mechanisms of consciousness with the help of the currently focused brain stimulation techniques, such as geodesic transcranial electric neuromodulation. This approach not only overcomes the limitations of the correlational methodology but will also become another firm step in the following science of consciousness.

Designing a Geodesic Faceted Acoustical Volumetric Array Using a Novel Analytical Method.

We present a novel analytical method as an efficient approach to design a geodesic-faceted array (GFA) for achieving a beam performance equivalent to that of a typical spherical array (SA). GFA is a triangle-based quasi-spherical configuration, which is conventionally created using the icosahedron method imitated from the geodesic dome roof construction process. In this conventional approach, the geodesic triangles have nonuniform geometries due to some distortions that occur during the random icosahedron division process. In this study, we took a paradigm shift from this approach and adopt a new technique to design a GFA that is based on uniform triangles. The characteristic equations that relate the geodesic triangle with a spherical platform were first developed as functions of the operating frequency and geometric parameters of the array. Then, the directional factor was derived to calculate the beam pattern associated with the array. A sample design of GFA for a given underwater sonar imaging system was synthesized through an optimization process. The GFA design was compared with that of a typical SA, and a reduction of 16.5% in the number of array elements was recorded in the GFA at a nearly equivalent performance. Both arrays were modeled, simulated, and analyzed using the finite element method (FEM) to validate the theoretical designs. Comparison of the results showed a high degree of compliance between the FEM and the theoretical method for both arrays. The proposed novel approach is faster and requires fewer computer resources than the FEM. Moreover, this approach is more flexible than the traditional icosahedron method in adjusting geometrical parameters in response to desired performance outputs.

An Adaptive Hybrid Sampling Method for Free-Form Surfaces Based on Geodesic Distance.

High precision geometric measurement of free-form surfaces has become the key to high-performance manufacturing in the manufacturing industry. By designing a reasonable sampling plan, the economic measurement of free-form surfaces can be realized. This paper proposes an adaptive hybrid sampling method for free-form surfaces based on geodesic distance. The free-form surfaces are divided into segments, and the sum of the geodesic distance of each surface segment is taken as the global fluctuation index of free-form surfaces. The number and location of the sampling points for each free-form surface segment are reasonably distributed. Compared with the common methods, this method can significantly reduce the reconstruction error under the same sampling points. This method overcomes the shortcomings of the current commonly used method of taking curvature as the local fluctuation index of free-form surfaces, and provides a new perspective for the adaptive sampling of free-form surfaces.

Organizing principles of astrocytic nanoarchitecture in the mouse cerebral cortex.

Astrocytes are increasingly understood to be important regulators of central nervous system (CNS) function in health and disease; yet, we have little quantitative understanding of their complex architecture. While broad categories of astrocytic structures are known, the discrete building blocks that compose them, along with their geometry and organizing principles, are poorly understood. Quantitative investigation of astrocytic complexity is impeded by the absence of high-resolution datasets and robust computational approaches to analyze these intricate cells. To address this, we produced four ultra-high-resolution datasets of mouse cerebral cortex using serial electron microscopy and developed astrocyte-tailored computer vision methods for accurate structural analysis. We unearthed specific anatomical building blocks, structural motifs, connectivity hubs, and hierarchical organizations of astrocytes. Furthermore, we found that astrocytes interact with discrete clusters of synapses and that astrocytic mitochondria are distributed to lie closer to larger clusters of synapses. Our findings provide a geometrically principled, quantitative understanding of astrocytic nanoarchitecture and point to an unexpected level of complexity in how astrocytes interact with CNS microanatomy.