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We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss–Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. Our approach allows us to solve problems that are impossible to solve in the pointwise conformal case. Moreover, we obtain a deep and more delicate information on pointwise conformal deformations. We prove new existence and nonexistence results for metrics with prescribed curvature in the conformal setting, which depend on the Euler characteristic. [ABSTRACT FROM AUTHOR]

A quiver representation assigns a vector space to each vertex, and a linear map to each arrow of a quiver. When one considers the category $\mathrm {Vect}(\mathbb {F}_1)$ of vector spaces "over $\mathbb {F}_1$ " (the field with one element), one obtains $\mathbb {F}_1$ -representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category $\mathrm {Rep}(Q,\mathbb {F}_1)$ is equivalent to the (suitably defined) category of coefficient quivers over Q. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of " $\mathbb {F}_1$ -rational points" of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated with $\mathbb {F}_1$ -representations. These techniques apply to a large class of $\mathbb {F}_1$ -representations, which we call the $\mathbb {F}_1$ -representations with finite nice length: we prove sufficient conditions for an $\mathbb {F}_1$ -representation to have finite nice length, and classify such representations for certain families of quivers. Finally, we explore the Hall algebras associated with $\mathbb {F}_1$ -representations of quivers. We answer the question of how a change in orientation affects the Hall algebra of nilpotent $\mathbb {F}_1$ -representations of a quiver with bounded representation type. We also discuss Hall algebras associated with representations with finite nice length, and compute them for certain families of quivers. [ABSTRACT FROM AUTHOR]

We prove that every triangulation of either of the torus, projective plane and Klein bottle, contains a vertex-spanning planar Laman graph as a subcomplex. Invoking a result of Király, we conclude that every 1-skeleton of a triangulation of a surface of nonnegative Euler characteristic has a rigid realization in the plane using at most 26 locations for the vertices. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for calculating the Euler characteristic in 2D binary images. The problem is addressed using the Three OrThogonal symbol chain code (3OT code), using only one symbol for the calculation of the Euler characteristic. Using this code, it is possible to introduce new geometric concepts represented by the same symbol of the 3OT alphabet and to simplify the overall equation of the Euler characteristic. This process is supported by the proof of a set of theorems and their numerical validation, using a set of binary images with a variable number of holes. Thus, this research proves that the 3OT code can be used not only for image compression as reported in the literature, but also to simplify the expression of the Euler characteristic as well as for the analysis and simplification of the shape of contours. [ABSTRACT FROM AUTHOR]

The induced polarization (IP) method holds a strong potential to better characterize the critical zone of our planet especially in areas characterized by multi‐phase flow. Power‐law relationships between the bulk, surface, and quadrature conductivities versus the pore water saturation are potentially useable to map the subsurface water content distribution. However, the saturation exponents n and p in these power‐law relationships have been observed to vary with the texture of geomaterials and the wettabilities of pore fluids. Traditional experimental setups in the laboratory do not allow to independently visualize the pore fluid distribution. Therefore, the physical interpretations of the two saturation exponents have remained unclear. We developed a novel milli‐fluidic micromodel using clay‐coated glass beads that exhibit excellent visibility and high IP response. Through laboratory experiments, we simultaneously determined the micromodel complex conductivity and acquired the corresponding pore‐scale fluid distributions generated by drainage and imbibition through such class of porous materials. Finite‐element simulations of complex conductivity based on the upscaling of the complex surface conductance of grains were conducted to determine the saturation exponents under ideal pore fluid distributions. Results indicate that saturation exponents n and p vary depending on the ganglia size of the insulating fluids. The saturation exponents n and p exhibit power‐law relationships with the change rate of pore water connectivity with saturation, which is calculated through the computation of the derivative of Euler characteristics. These findings provide a new physical explanation to the relationships between the saturation exponents and the microscopic fluid distributions within the geomaterials. Plain Language Summary: Water saturation of porous bodies can be related to the complex conductivity through power‐law relationships. The existence of these power‐law relationships has been clearly documented in the literature. They are critical in the realm of hydrogeophysics to better characterize the critical zone of the solid Earth. However, the values of the saturation exponents in these power‐laws have been observed to vary with different material textures and no underlying mechanisms have been able to explain these variations to date. This lack of physical understanding could limit the applicability of the induced polarization method to characterize the critical zone especially when immiscible fluid phases are present. To tackle this, we developed a milli‐fluidic pore model that allows investigating the saturation exponents while monitoring the pore‐scale fluid distributions. Together with numerical simulations for ideal fluid distribution cases, we found a relationship between the saturation exponents and a microscopic pore parameter called "change rate of the pore water connectivity with saturation." These findings suggest that when estimating subsurface water saturation from electrical parameters, taking the pore fluid distribution into account can significantly improve the estimation accuracy, therefore enhancing the efficiency of geo‐electrical applications for a better characterization of hydrocarbon contaminated aquifers and oil reservoirs. Key Points: A micromodel setup allows for simultaneous pore fluid visualization and complex electrical conductivity measurementThe two saturation exponents for the in‐phase and quadrature conductivities are correlated with the ganglia size of the insulating phaseA power‐law‐type relationship between the saturation exponents and the change rate of water connectivity with saturation is demonstrated [ABSTRACT FROM AUTHOR]

The Euler characteristic transform (ECT) is an integral transform used widely in topological data analysis. Previous efforts by Curry et al. and Ghrist et al. have independently shown that the ECT is injective on all compact definable sets. In this work, we first study the injectivity of the ECT on definable sets that are not necessarily compact and prove a complete classification of constructible functions that the Euler characteristic transform is not injective on. We then introduce the quadric Euler characteristic transform (QECT) as a natural generalization of the ECT by detecting definable shapes with quadric hypersurfaces rather than hyperplanes. We also discuss some criteria for the injectivity of QECT. [ABSTRACT FROM AUTHOR]

We classify in terms of Hopf-type properties mapping tori of residually finite Poincaré Duality groups with non-zero Euler characteristic. This generalises and gives a new proof of the analogous classification for fibered 3-manifolds. Various applications are given. In particular, we deduce that rigidity results for Gromov hyperbolic groups hold for the above mapping tori with trivial center. [ABSTRACT FROM AUTHOR]

This paper covers a novel micro-level application of image processing in understanding the topological and petrophysical properties of Indian Gondwana shale using X-ray computed microtomography images. The complexity and randomness in the pore system are explained through the concept of fractal dimension (FD). In this paper, a quantitative analysis of 2D and 3D fractal dimensions of pores, grains and interfaces was performed for Indian Gondwana shale, using the box-counting method. A pore network is formed by the connection of many subpore clusters, each with a different volume. Hence, an image segmentation algorithm was applied to label different subclusters, and subsequently an analysis of FD was carried out on such subclusters of pores and grains. We implemented a novel application of Betti numbers (B0, B1 and B2) and Euler characteristics on our sample and calculated the possible flow channels of the sample. The FD of grains was found to be greater than the FD of the pore–grain interfaces, while the FD of pores was found to have the smallest value. Consequently, we also observed how the FD of both pores and grains was majorly controlled by the largest subcluster, and during fluid intrusion we observed a significant decrease in the FD of pores. Finally, the pore network with a larger B0 and larger difference of B1 was proved to be best for the storage of hydrocarbons and for fluid movement along more flow channels. [ABSTRACT FROM AUTHOR]

The Möbius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the Möbius function defined on an order ideal related to the lattice of the subgroups of an irreducible subgroup G of the general linear group GL (n , q) acting on the n-dimensional vector space V = F q n , where F q is the finite field with q elements. We find a relation between this function and the Euler characteristic of two simplicial complexes Δ 1 and Δ 2 , the former raising from the lattice of the subspaces of V, the latter from the subgroup lattice of G. [ABSTRACT FROM AUTHOR]

In this study, we concentrate on the synchronizability of multilayer totally homogeneous networks. The eigenvalue spectrum of four typical multilayer networks, including multilayer nearest-neighbor regular networks, multilayer small-world networks, multilayer random networks and multilayer totally homogeneous networks, has been rigorously derived and the synchronizability of four typical multilayer networks has been compared due to the master stability function. Furthermore, numerical simulations are performed to validate the synchronizability of these multilayer networks. The results indicate that multilayer totally homogeneous networks exhibit optimal synchronizability. The Euler characteristic numbers and Betti numbers of these networks are derived, and their relationships with the synchronizability of multilayer networks are analyzed. We can observe that a negative correlation is exhibited between the Euler characteristic number and the synchronizability of multilayer networks, while a positive correlation is exhibited between the Betti number and the synchronizability of multilayer networks. [ABSTRACT FROM AUTHOR]

Let G be a finite group and X be a subgroup of G. We investigate the topological properties of the poset C X (G) of cosets Hx in G with X ≤ H < G. We show that C X (G) is non-contractible if G is solvable or N G (X) contains a Sylow 2-subgroup and a Sylow 3-subgroup of G. This result follows J. Shareshian and R. Woodroofe's work in [8] (2016). We also give some divisibility properties of the Euler characteristic of C X (G) when X is a p -group, which follows K. S. Brown's classical result in [3] (2000). [ABSTRACT FROM AUTHOR]

For a C ∞ stable map f : M → Q of a compact 3-manifold with boundary into a surface without boundary, we show that f is homotopic to a stable map which has no B 2 points if the Euler characteristic of each connected component of the boundary of M is even. [ABSTRACT FROM AUTHOR]

Biomedical research now commonly integrates diverse data types or views from the same individuals to better understand the pathobiology of complex diseases, but the challenge lies in meaningfully integrating these diverse views. Existing methods often require the same type of data from all views (cross-sectional data only or longitudinal data only) or do not consider any class outcome in the integration method, which presents limitations. To overcome these limitations, we have developed a pipeline that harnesses the power of statistical and deep learning methods to integrate cross-sectional and longitudinal data from multiple sources. In addition, it identifies key variables that contribute to the association between views and the separation between classes, providing deeper biological insights. This pipeline includes variable selection/ranking using linear and nonlinear methods, feature extraction using functional principal component analysis and Euler characteristics, and joint integration and classification using dense feed-forward networks for cross-sectional data and recurrent neural networks for longitudinal data. We applied this pipeline to cross-sectional and longitudinal multiomics data (metagenomics, transcriptomics and metabolomics) from an inflammatory bowel disease (IBD) study and identified microbial pathways, metabolites and genes that discriminate by IBD status, providing information on the etiology of IBD. We conducted simulations to compare the two feature extraction methods. [ABSTRACT FROM AUTHOR]

We show that extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure. This result stems from a new proof of the fact that there is no compact complex surface which is homotopy equivalent to a closed real-hyperbolic 4-manifold. Finally, we construct infinitely many extended graph 4-manifolds with positive Euler characteristic which support almost complex structures. [ABSTRACT FROM AUTHOR]

When using traditional Euler deconvolution optimization strategies, it is difficult to distinguish between anomalies and their corresponding Euler tails (those solutions are often distributed outside the anomaly source, forming "tail"-shaped spurious solutions, i.e., misplaced Euler solutions, which must be removed or marked) with only the structural index. The nonparametric estimation method based on the normalized B-spline probability density (BSS) is used to separate the Euler solution clusters and mark different anomaly sources according to the similarity and density characteristics of the Euler solutions. For display purposes, the BSS needs to map the samples onto the estimation grid at the points where density will be estimated in order to obtain the probability density distribution. However, if the size of the samples or the estimation grid is too large, this process can lead to high levels of memory consumption and excessive computation times. To address this issue, a fast linear binning approximation algorithm is introduced in the BSS to speed up the computation process and save time. Subsequently, the sample data are quickly projected onto the estimation grid to facilitate the discrete convolution between the grid and the density function using a fast Fourier transform. A method involving multivariate B-spline probability density estimation based on the FFT (BSSFFT), in conjunction with fast linear binning appropriation, is proposed in this paper. The results of two random normal distributions show the correctness of the BSS and BSSFFT algorithms, which is verified via a comparison with the true probability density function (pdf) and Gaussian kernel smoothing estimation algorithms. Then, the Euler solutions of the two synthetic models are analyzed using the BSS and BSSFFT algorithms. The results are consistent with their theoretical values, which verify their correctness regarding Euler solutions. Finally, the BSSFFT is applied to Bishop 5X data, and the numerical results show that the comprehensive analysis of the 3D probability density distributions using the BSSFFT algorithm, derived from the Euler solution subset of  x 0 , y 0 , z 0 , can effectively separate and locate adjacent anomaly sources, demonstrating strong adaptability. [ABSTRACT FROM AUTHOR]

Processes of random tessellations of the Euclidean space $\mathbb{R}^d$ , $d\geq 1$ , are considered that are generated by subsequent division of their cells. Such processes are characterized by the laws of the life times of the cells until their division and by the laws for the random hyperplanes that divide the cells at the end of their life times. The STIT (STable with respect to ITerations) tessellation processes are a reference model. In the present paper a generalization concerning the life time distributions is introduced, a sufficient condition for the existence of such cell division tessellation processes is provided, and a construction is described. In particular, for the case that the random dividing hyperplanes have a Mondrian distribution—which means that all cells of the tessellations are cuboids—it is shown that the intrinsic volumes, except the Euler characteristic, can be used as the parameter for the exponential life time distribution of the cells. [ABSTRACT FROM AUTHOR]

This work is part of a research program to compute the Hochschild homology groups HH ∗ (C [ x 1 , ... , x d ] / (x 1 , ... , x d) 3 ; C) in the case d = 2 through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism HH i (C [ x 1 , ... , x d ] / (x 1 , ... , x d) 3 ; C) ≅ ∑ 0 ≤ j ≤ i H C j S i + j , V ⊗ (i + j) provided by Larsen and Lindenstrauss. Here, H C ∗ denotes Coxeter cohomology, S i + j denotes the symmetric group on i + j letters, and V is the standard representation of GL d (C) on C d . We compute the Euler characteristic of the Coxeter cohomology (the alternating sum of the ranks of the Coxeter cohomology groups) of several representations of S n . In particular, the aforementioned tensor representation, and also several classes of irreducible representations of S n . Although the problem and its motivation are algebraic and topological in nature, the techniques used are largely combinatorial. [ABSTRACT FROM AUTHOR]

We study the normalized volume of toric singularities. As it turns out, there is a close relation to the notion of (nonsymmetric) Mahler volume from convex geometry. This observation allows us to use standard tools from convex geometry, such as the Blaschke–Santaló inequality and Radon's theorem to prove nontrivial facts about the normalized volume in the toric setting. For example, we prove that for every ε>0$\epsilon > 0$ there are only finitely many Q$\mathbb {Q}$‐Gorenstein toric singularities with normalized volume at least ε$\epsilon$. From this result it directly follows that there are also only finitely many toric Sasaki–Einstein manifolds of volume at least ε$\epsilon$ in each dimension. Additionally, we show that the normalized volume of every toric singularity is bounded from above by that of the rational double point of the same dimension. Finally, we discuss certain bounds of the normalized volume in terms of topological invariants of resolutions of the singularity. We establish two upper bounds in terms of the Euler characteristic and of the first Chern class, respectively. We show that a lower bound, which was conjectured earlier by He, Seong, and Yau, is closely related to the nonsymmetric Mahler conjecture in convex geometry. [ABSTRACT FROM AUTHOR]

The question of the global topology of the Universe (cosmic topology) is still open. In the Λ CDM concordance model, it is assumed that the space of the Universe possesses the trivial topology of R 3 , and thus that the Universe has an infinite volume. As an alternative, in this paper, we study one of the simplest non-trivial topologies given by a cubic 3-torus describing a universe with a finite volume. To probe cosmic topology, we analyze certain structure properties in the cosmic microwave background (CMB) using Betti functionals and the Euler characteristic evaluated on excursions sets, which possess a simple geometrical interpretation. Since the CMB temperature fluctuations δ T are observed on the sphere S 2 surrounding the observer, there are only three Betti functionals β k (ν) , k = 0 , 1 , 2 . Here, ν = δ T / σ 0 denotes the temperature threshold normalized by the standard deviation σ 0 of δ T . The analytic approximations of the Gaussian expectations for the Betti functionals and an exact formula for the Euler characteristic are given. It is shown that the amplitudes of β 0 (ν) and β 1 (ν) decrease with an increasing volume V = L 3 of the cubic 3-torus universe. Since the computation of the β k 's from observational sky maps is hindered due to the presence of masks, we suggest a method that yields lower and upper bounds for them and apply it to four Planck 2018 sky maps. It is found that the β k 's of the Planck maps lie between those of the torus universes with side-lengths L = 2.0 and L = 3.0 in units of the Hubble length and above the infinite Λ CDM case. These results give a further hint that the Universe has a non-trivial topology. [ABSTRACT FROM AUTHOR]

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include k -Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of U q (sl ˆ ℓ) -generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals. [ABSTRACT FROM AUTHOR]

We resolve the question of the existence of a finite 2‐complex with the same fundamental group and Euler characteristic as a Klein bottle with a bubble, but homotopically distinct to it. [ABSTRACT FROM AUTHOR]

Let A (ℓ , n , k) denote the number of ℓ -tuples of commuting permutations of n elements whose permutation action results in exactly k orbits or connected components. We provide a new proof of an explicit formula for A (ℓ , n , k) which is essentially due to Bryan and Fulman, in their work on orbifold higher equivariant Euler characteristics. Our proof is self-contained, elementary, and relies on the construction of an explicit bijection, in order to perform the ℓ + 1 → ℓ reduction. We also investigate a conjecture by the first author, regarding the log-concavity of A (ℓ , n , k) with respect to k. The conjecture generalizes a previous one by Heim and Neuhauser related to the Nekrasov-Okounkov formula. [ABSTRACT FROM AUTHOR]

Let |$\alpha $| and |$\beta $| be uniformly random permutations of orders |$2$| and |$3$|⁠ , respectively, in |$S_{N}$|⁠ , and consider, say, the permutation |$\alpha \beta \alpha \beta ^{-1}$|⁠. How many fixed points does this random permutation have on average? The current paper studies questions of this kind and relates them to surprising topological and algebraic invariants of elements in free products of groups. Formally, let |$\Gamma =G_{1}*\ldots *G_{k}$| be a free product of groups where each of |$G_{1},\ldots ,G_{k}$| is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element |$\gamma \in \Gamma $|⁠ , a |$\gamma $| -random permutation in the symmetric group |$S_{N}$| is the image of |$\gamma $| through a uniformly random homomorphism |$\Gamma \to S_{N}$|⁠. In this paper we study local statistics of |$\gamma $| -random permutations and their asymptotics as |$N$| grows. We first consider |$\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big)\big ]$|⁠ , the expected number of fixed points in a |$\gamma $| -random permutation in |$S_{N}$|⁠. We show that unless |$\gamma $| has finite order, the limit of |$\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big)\big ]$| as |$N\to \infty $| is an integer, and is equal to the number of subgroups |$H\le \Gamma $| containing |$\gamma $| such that |$H\cong \mathbb{Z}$| or |$H\cong C_{2}*C_{2}$|⁠. Equivalently, this is the number of subgroups |$H\le \Gamma $| containing |$\gamma $| and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for |$\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big)\big ]$| and determine the limit distribution of the number of fixed points as |$N\to \infty $|⁠. These results are then generalized to all statistics of cycles of fixed lengths. [ABSTRACT FROM AUTHOR]

We examine the dimensions of various inf-sup stable mixed finite element spaces on tetrahedral meshes in three dimensions with exact divergence constraints. More precisely, we compare the standard Scott--Vogelius elements of higher polynomial degree and low-order methods on split meshes, the Alfeld and the Worsey--Farin split. The main tool is a counting strategy to express the degrees of freedom for given polynomial degree and given split in terms of only a few mesh quantities, for which bounds and asymptotic behavior under mesh refinement is investigated. Furthermore, this is used to obtain insights on potential precursor spaces in the Stokes complex for finite element methods on the Worsey--Farin split. [ABSTRACT FROM AUTHOR]

This study aims to bridge length scales in immiscible multiphase flow simulation by connecting two published governing equations at the pore-scale and continuum-scale through a novel validation framework. We employ Niessner and Hassnaizadeh's ["A model for two-phase flow in porous media including fluid-fluid interfacial area," Water Resour. Res. 44(8), W08439 (2008)] continuum-scale model for multiphase flow in porous media, combined with the geometric equation of state of McClure et al. ["Modeling geometric state for fluids in porous media: Evolution of the Euler characteristic," Transp. Porous Med. 133(2), 229–250 (2020)]. Pore-scale fluid configurations simulated with the lattice-Boltzmann method are used to validate the continuum-scale results. We propose a mapping from the continuum-scale to pore-scale utilizing a generalized additive model to predict non-wetting phase Euler characteristics during imbibition, effectively bridging the continuum-to-pore length scale gap. Continuum-scale simulated measures of specific interfacial area, saturation, and capillary pressure are directly compared to up-scaled pore-scale simulation results. This research develops a numerical framework capable of capturing multiscale flow equations establishing a connection between pore-scale and continuum-scale simulations. [ABSTRACT FROM AUTHOR]

Two neural networks were trained to predict, respectively, the Euler characteristic and the curvature of nuclear pastas in neutron star crust conditions generated by molecular dynamics simulations of neutron star matter with 0.1 < x < 0.5, 0.040 fm−3 < ρ < 0.085 fm−3 (0.68 × 1014 g/cm3 < ρ < 1.43 × 1014 g/cm3), and 0.2 MeV < T < 4.0 MeV, where x is proton content, the density is ρ , and the temperature is T. The predictions of the two networks were combined to determine the nuclear pasta phase that is thermodynamically stable at a given x, ρ , and T, and a three-dimensional phase diagram that extrapolated slightly the regions of existing molecular dynamics data was computed. The jungle gym and anti-jungle gym structures are prevalent at high temperature and low density, while the anti-jungle gym and anti-gnocchi structures dominate at high temperature and high density. A diversity of structures exist at low temperatures and intermediate density and proton content. The trained models used in this work are open access and available at a public repository to promote comparison to pastas obtained with other models. [ABSTRACT FROM AUTHOR]

This is a survey article, in which we explore how the presence of singularities affects the geometry and topology of complex projective hypersurfaces. [ABSTRACT FROM AUTHOR]

Green hydrogen, produced from surplus electricity during peak production, can be injected into subsurface reservoirs and retrieved during high-demand periods. In this study, X-ray tomography was employed to examine hysteresis resulting from repeated hydrogen injection and withdrawal. An unsteady state experiment was performed to evaluate the distribution of hydrogen and brine after drainage and imbibition cycles: images of the pore-space configuration of fluids were taken immediately once injection had stopped and after waiting for a period of 16 h with no flow. A Bentheimer sandstone sample with a length of 60 mm and diameter of 12.8 mm was used, and hydrogen was injected at ambient temperature and a pore pressure of 1 MPa. The gas flow rate was decreased from 2 ml/min to 0.08 ml/min over three cycles of gas injection followed by water flooding, while the brine injection rate was kept constant. The results showed the presence of capillary pressure hysteresis and hydrogen migration through Ostwald ripening due to the diffusion of gas dissolved in the brine. These phenomena were characterized through analysis of interfacial curvature, area, connectivity and pore occupancy. The hydrogen tended to reside in the larger pore spaces, consistent with water-wet conditions. 16 h after flow had stopped, the hydrogen aggregated into larger ganglia with a single large connected ganglion dominating the volume. Moreover, the Euler characteristic decreased after 16 h, indicating an improvement in connectivity. The work implies that Ostwald ripening – mass transport of dissolved gas – leads to less hysteresis and better connectivity than would be assumed ignoring this effect, as done in assessments of hydrocarbon flow and trapping. [Display omitted] • X-ray imaging investigated pore-scale hysteresis during hydrogen storage. • Hydrogen migration by Ostwald ripening was observed. • Ostwald ripening led to better connectivity of the gas phase. • The work implies less hysteresis for hydrogen than for hydrocarbons. [ABSTRACT FROM AUTHOR]

For a balanced sutured manifold |$(M,\gamma)$|⁠ , we construct a decomposition of |$SHI(M,\gamma)$| with respect to torsions in |$H=H_{1}(M;\mathbb {Z})$|⁠ , which generalizes the decomposition of |$I^{\sharp }(Y)$| in a previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin |$^{c}$| decompositions in |$SFH(M,\gamma)$|⁠. Based on this decomposition, we define an enhanced Euler characteristic |$\chi _{\textrm {en}}(SHI(M,\gamma))\in \mathbb {Z}[H]/\pm H$| and prove that |$\chi _{\textrm {en}}(SHI(M,\gamma))=\chi (SFH(M,\gamma))$|⁠. This provides a better lower bound on |$\dim _{\mathbb {C}}SHI(M,\gamma)$| than the graded Euler characteristic |$\chi _{\textrm {gr}}(SHI(M,\gamma))$|⁠. As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture |$KHI(Y,K)\cong \widehat {HFK}(Y,K)$| holds for all |$(1,1)$| -L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces. [ABSTRACT FROM AUTHOR]

We study the cohomology of families of |$(\varphi ,\Gamma)$| -modules with coefficients in pseudoaffinoid algebras. We prove that they have finite cohomology, and we deduce an Euler characteristic formula and Tate local duality. We classify rank- |$1$| |$(\varphi ,\Gamma)$| -modules and deduce that triangulations of pseudorigid families of |$(\varphi ,\Gamma)$| -modules can be interpolated, extending a result of [ 29 ]. We then apply this to study extended eigenvarieties at the boundary of weight space, proving in particular that the eigencurve is proper at the boundary and that Galois representations attached to certain characteristic |$p$| points are trianguline. [ABSTRACT FROM AUTHOR]

A linear forest is a disjoint union of path graphs. The linear arboricity of a graphG $G$, denoted by la(G) $\text{la}(G)$, is the least number of linear forests into which the graph can be partitioned. Clearly, la(G)≥⌈Δ(G)∕2⌉ $\text{la}(G)\ge \lceil {\rm{\Delta }}(G)\unicode{x02215}2\rceil $ for any graph of maximum degree Δ(G) ${\rm{\Delta }}(G)$. For the upper bound, the long‐standing Linear Arboricity Conjecture (LAC) due to Akiyama, Exoo, and Harary from 1981 asserts that la(G)≤⌈(Δ(G)+1)∕2⌉ $\text{la}(G)\le \lceil ({\rm{\Delta }}(G)+1)\unicode{x02215}2\rceil $. A graph is a pseudoforest if each of its components contains at most one cycle. In this paper, we prove that the union of any two pseudoforests of maximum degree up to 3 can be decomposed into three linear forests. Combining it with a recent result of Wdowinski on the minimum number of pseudoforests into which a graph can be decomposed, we prove that the LAC holds for the following simple graph classes: k $k$‐degenerate graphs with maximum degree Δ≥3k−1 ${\rm{\Delta }}\ge 3k-1$, all graphs on nonnegative Euler characteristic surfaces provided the maximum degree Δ≠7 ${\rm{\Delta }}\ne 7$, and graphs on negative Euler characteristic ϵ $\epsilon $ surfaces provided the maximum degree Δ≥35+49−24ϵ4−1 ${\rm{\Delta }}\ge 3\unicode{x02308}\frac{5+\sqrt{49-24\epsilon }}{4}\unicode{x02309}-1$, as well as graphs with no Kt ${K}_{t}$‐minor satisfying some conditions on maximum degrees. [ABSTRACT FROM AUTHOR]

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality. [ABSTRACT FROM AUTHOR]

Let $ G $ be a graph that can be embedded in a surface $ \Sigma $ of Euler characteristic $ c'(\Sigma) $. In this paper, we proved that there exists an integer $ \Delta_0 = 4\cdot (5+\sqrt{49-24c'})\cdot [2-c'+\frac{1}{2} (5+\sqrt{49-24c'})] $ such that the total chromatic number of $ G $ is $ \Delta(G) +1 $ if $ \Delta(G) \geq \Delta_0 $.

Let f : (ℝ n , 0) → (ℝ , 0) be a definable function germ of class 2 and let (X , 0) ⊂ (ℝ n , 0) be a germ of a closed definable set. We investigate topological invariants associated with f | X . In particular, we give several topological formulae for the Euler characteristics of related sets. We also relate the topology of f | X to the topology of a definable function with isolated critical point in the stratified case. [ABSTRACT FROM AUTHOR]

Magnitude is a numerical invariant of metric spaces and graphs, analogous, in a precise sense, to Euler characteristic. Magnitude homology is an algebraic invariant constructed to categorify magnitude. Among the important features of the magnitude of graphs is its behaviour with respect to an operation known as the Whitney twist. We give a homological account of magnitude's invariance under Whitney twists, extending the previously known result to encompass a substantially wider class of gluings. As well as providing a new tool for the computation of magnitudes, this is the first new theorem about magnitude to be proved using magnitude homology. [ABSTRACT FROM AUTHOR]

Let M ⁢ (d , χ) , with (d , χ) = 1 , be the moduli space of semistable sheaves on ℙ 2 supported on curves of degree d and with Euler characteristic χ. The cohomology ring H * ⁢ (M ⁢ (d , χ) , ℤ) of M ⁢ (d , χ) is isomorphic to its Chow ring A * ⁢ (M ⁢ (d , χ)) by Markman's result. Pi and Shen have described a minimal generating set of A * ⁢ (M ⁢ (d , χ)) consisting of 3 ⁢ d - 7 generators, which they also showed to have no relation in A ≤ d - 2 ⁢ (M ⁢ (d , χ)) . We compute the two Betti numbers b 2 ⁢ (d - 1) and b 2 ⁢ d of M ⁢ (d , χ) , and as a corollary we show that the generators given by Pi and Shen have no relations in A ≤ d - 1 ⁢ (M ⁢ (d , χ)) , but do have three linearly independent relations in A d ⁢ (M ⁢ (d , χ)) . [ABSTRACT FROM AUTHOR]

Let G be a graph that can be embedded in a surface Σ of Euler characteristic ć(Σ). In this paper, we proved that there exists an integer Δ0 = 4 ·... such that the total chromatic number of G is Δ(G) + 1 if Δ(G) ≥ Δ0. [ABSTRACT FROM AUTHOR]

A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph G${\tt G}$ is the simplicial complex whose faces are the multipaths of G${\tt G}$. We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if G${\tt G}$ is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of G${\tt G}$ is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations. [ABSTRACT FROM AUTHOR]

Let M ⁢ (d , χ) , with (d , χ) = 1 , be the moduli space of semistable sheaves on ℙ 2 supported on curves of degree d and with Euler characteristic χ. The cohomology ring H * ⁢ (M ⁢ (d , χ) , ℤ) of M ⁢ (d , χ) is isomorphic to its Chow ring A * ⁢ (M ⁢ (d , χ)) by Markman's result. Pi and Shen have described a minimal generating set of A * ⁢ (M ⁢ (d , χ)) consisting of 3 ⁢ d - 7 generators, which they also showed to have no relation in A ≤ d - 2 ⁢ (M ⁢ (d , χ)) . We compute the two Betti numbers b 2 ⁢ (d - 1) and b 2 ⁢ d of M ⁢ (d , χ) , and as a corollary we show that the generators given by Pi and Shen have no relations in A ≤ d - 1 ⁢ (M ⁢ (d , χ)) , but do have three linearly independent relations in A d ⁢ (M ⁢ (d , χ)) . [ABSTRACT FROM AUTHOR]

The Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, whose k -point correlation functions (k th-order cumulants) have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-squared random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density. [ABSTRACT FROM AUTHOR]

On a closed Riemannian surface (M , g ¯) with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume A > 0 and the property that their Gauss curvatures f λ = f + λ are given as the sum of a prescribed function f ∈ C ∞ (M) and an additive constant λ . Our main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local well-posedness and global compactness results. In contrast to previous work, our approach does not require any sign conditions on f. Moreover, we exhibit conditions under which the function f λ is sign changing and the standard prescribed Gauss curvature flow is not applicable. [ABSTRACT FROM AUTHOR]

As opposed to the 3D cubic grid, the body-centered cubic (BCC) grid has some favorable topological properties: each set of voxels in the grid is a 3-manifold, with 2-manifold boundary. Thus, the Euler characteristic of an object O in this grid can be computed as half of the Euler characteristic of its boundary ∂ O . We propose three new algorithms to compute the Euler characteristic in the BCC grid with this surface-based approach: one based on (critical point) Morse theory and two based on the discrete Gauss–Bonnet theorem. We provide a comparison between the three new algorithms and the classic approach based on counting the number of cells, either of the 3D object or of its 2D boundary surface. [ABSTRACT FROM AUTHOR]

Numerical solutions of the reaction–diffusion system of equations in three dimensions provide a systematic procedure to generate a variety of porous media structures of a chosen porosity. A way to characterize such geometries is by their Minkowski functional densities and their average pore diameter. We generated multiple porous geometries of the granular and foam-like types. Hydraulic flow through these series of geometries is simulated to study the dependence of the permeability k and the Forchheimer parameter β on the chosen geometrical identifiers. Both k and β exhibit two distinct behaviors aligned with the two distinct generated types of geometries when scaled with the Euler characteristic. A good correlation for dimensionless k as a function of the Minkowski functional densities and the average pore diameter was found for both types of geometries with an accuracy of 15% and 18%, respectively, while no good correlation was found for β , implying that more geometrical information is needed beyond the proposed one. From the correlations, it was found that, together with the porosity, the mean curvature is also a good parameter to characterize the permeability. Article Highlights: Porous media generation based on solutions to reaction-diffusion equations Model porous structures characterized through Minkowski functionals and average pore diameter Scaling based on the Euler characteristic reveals two distinct flow behaviors. [ABSTRACT FROM AUTHOR]

The Singer–Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer–Hopf conjecture, formulated in terms of the Euler–Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi‐positivity properties of nef vector bundles together with a topological version of the Riemann–Roch theorem, proved by Kashiwara. [ABSTRACT FROM AUTHOR]

We explain the Buchsbaum-Rim multiplicity as the Euler-Poincaré characteristic of an "ordinary" Koszul complex. This provides a generalization of Serre's formula for the Hilbert-Samuel multiplicity in terms of the length of the Koszul homologies. [ABSTRACT FROM AUTHOR]

We study the quantum self-organization of a few interacting particles with strong short-range interactions. The physical system is modeled via a 2D Hubbard square lattice model, with a nearest-neighbor interaction term of strength U and a second nearest-neighbor hopping t. For t=0, the energy of the system is determined by the number of bonds between particles that lie on adjacent sites in the Hubbard lattice. We find that this bond order persists for the ground and some of the excited states of the system, for strong interaction strength, at different fillings of the system. For our analysis, we use the Euler characteristic of the network/graph grid structures formed by the particles in real space (Fock states), which helps to quantify the energetical(bond) ordering. We find multiple ground and excited states, with integer Euler numbers, whose values persist from the t = 0 case, for strong interaction U > > t . The corresponding quantum phases for the ground state contain either density-wave-order(DWO) for low fillings, where the particles stay apart form each other, or clustering-order(CO) for high fillings, where the particles form various structures as they condense into clusters. In addition, we find various excited states containing superpositions of Fock states, whose probability amplitudes are self-tuned in a way that preserves the integer value of the Euler characteristic from the t = 0 limit. [ABSTRACT FROM AUTHOR]

We give two proofs that the Euler characteristic is multiplicative, for fiber sequences of finitely dominated spaces. This is equivalent to proving that the Becker-Gottlieb transfer is functorial on \pi _0. [ABSTRACT FROM AUTHOR]

A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G. A graph G is acyclically k-choosable if for any list assignment L = {L(v): v ∈ V(G)} with ∣L(v)∣ ≥ k for each vertex v ∈ V(G), there exists an acyclic proper vertex coloring ϕ of G such that ϕ(v) ∈ L(v) for each vertex v ∈ V(G). In this paper, we prove that every graph G embedded on the surface with Euler characteristic number ε = −1 is acyclically 11-choosable. [ABSTRACT FROM AUTHOR]