The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the limiting persistence diagram to have the full support. We also discuss a central limit theorem for persistent Betti numbers.
Highlights•IFN-γ and ERK/MAPK signaling activities alter upon aging in the small intestine•The balanced activities between IFN-γ and ERK/MAPK signaling maintain the ISC pool•An equilibrium between the active and the quiescent states exists in the aged ISCs•Changes in the two signaling pathways affect functions of differentiated cellsSummaryWhile the intestinal epithelium has the highest cellular turnover rates in the mammalian body, it is also considered one of the tissues most resilient to aging-related disorders. Here, we reveal an innate protective mechanism that safeguards intestinal stem cells (ISCs) from environmental conditions in the aged intestine. Using in vivo phenotypic analysis, transcriptomics, and in vitro intestinal organoid studies, we show that age-dependent activation of interferon-γ (IFN-γ) signaling and inactivation of extracellular signal-regulated kinase/mitogen-activated protein kinase (ERK/MAPK) signaling are responsible for establishing an equilibrium of Lgr5+ ISCs—between active and quiescent states—to preserve the ISC pool during aging. Furthermore, we show that differentiated cells have different sensitivities to each of the two signaling pathways, which may induce aging-related, functional, and metabolic changes in the body. Thus, our findings reveal an exquisitely balanced, age-dependent signaling mechanism that preserves stem cells at the expense of differentiated cells.Graphical abstract
Abstract The broken symmetry in the atomic-scale ordering of glassy versus crystalline solids leads to a daunting challenge to provide suitable metrics for describing the order within disorder, especially on length scales beyond the nearest neighbor that are characterized by rich structural complexity. Here, we address this challenge for silica, a canonical network-forming glass, by using hot versus cold compression to (i) systematically increase the structural ordering after densification and (ii) prepare two glasses with the same high-density but contrasting structures. The structure was measured by high-energy X-ray and neutron diffraction, and atomistic models were generated that reproduce the experimental results. The vibrational and thermodynamic properties of the glasses were probed by using inelastic neutron scattering and calorimetry, respectively. Traditional measures of amorphous structures show relatively subtle changes upon compacting the glass. The method of persistent homology identifies, however, distinct features in the network topology that change as the initially open structure of the glass is collapsed. The results for the same high-density glasses show that the nature of structural disorder does impact the heat capacity and boson peak in the low-frequency dynamical spectra. Densification is discussed in terms of the loss of locally favored tetrahedral structures comprising oxygen-decorated SiSi 4 tetrahedra.
Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.
In this study, we investigate the magnetization process at high frequencies based on the energy landscape outputted by machine learning. Employing a combination of Topological Data Analysis (TDA) and machine learning, we analyze how microstructures influence magnetization at frequencies from 1KHz up to 100KHz. Our approach uses a magneto-optical Kerr effect (MOKE) microscope for visualizing magnetic domains at various frequencies, revealing insights into their behavior and structure. Persistent homology, a method within TDA, transforms complex topological features of these domains into analyzable vectors. These vectors are then processed through Principal Component Analysis (PCA) to extract the significant information, focusing on the most impactful aspects of the data. This process allows for a detailed examination of the magnetic properties and their changes with frequency, offering an in-depth analysis of material properties under high-frequency conditions. By investigating the elements of PCA, we could analyze the energy loss and connect the topological elements to the anomalous eddy current loss. This study gives a step towards integrating advanced analytical techniques into material science, opening new pathways for innovation in high-frequency applications.
The purpose of my work is to develop a rigorous numerical technique to prove the existence of stationary solutions and to detect connecting orbits among them in dissipative PDEs. The Conley index, topological quantities dened on invariant sets in dynamical systems, is applied to these problems. We consider the cubic Swift-Hohenberg equation as an example and study how to rigorously verify the existence of stationary solutions and connecting orbits among them. An eectiv e algorithm by using FFT for this verication technique is also shown. This algorithm is applied to study the snaky bifurcation structure appearing in the quintic Swift-Hohenberg equation.
Persistence diagrams have been widely recognized as a compact descriptor for characterizing multiscale topological features in data. When many datasets are available, statistical features embedded in those persistence diagrams can be extracted by applying machine learnings. In particular, the ability for explicitly analyzing the inverse in the original data space from those statistical features of persistence diagrams is significantly important for practical applications. In this paper, we propose a unified method for the inverse analysis by combining linear machine learning models with persistence images. The method is applied to point clouds and cubical sets, showing the ability of the statistical inverse analysis and its advantages.
