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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]

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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 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]

We prove a formula, conjectured by Zagier, for the Sn-equivariant Euler characteristic of the top weight cohomology of Mg,n. [ABSTRACT FROM AUTHOR]

Topology is considered an advanced field in mathematics, and it might seem off-putting to people with no previous experience in mathematics. The classification theorem, which lies within the field of algebraic topology, is fascinating, but understanding it requires extensive mathematical knowledge. In this manuscript, we present a modular object that provides a glimpse into the construction of surfaces as defined by the classification theorem. This allows any user to explore different options and discover possible surfaces with three holes or fewer. This tactile experience of exploration grants the user intuition about the meaning of the theorem, without needing the rigorous proof; thus, it simplifies the depth of the classification theorem and its respective proof for non-mathematical audiences. We strive to accurately illustrate these esoteric concepts by giving them a form and to describe the modeling process of creating a tangible, touchable sculpture designed to intrigue and invite simple understanding. We believe that this playful object will be exciting for a variety of communities: university students, graduate students, teachers, professional mathematicians, and especially designers, who can use it to produce new tools. [ABSTRACT FROM AUTHOR]

One way to model telecommunication networks are static Boolean models. However, dynamics such as node mobility have a significant impact on the performance evaluation of such networks. Consider a Boolean model in $\mathbb {R}^d$ and a random direction movement scheme. Given a fixed time horizon $T>0$ , we model these movements via cylinders in $\mathbb {R}^d \times [0,T]$. In this work, we derive central limit theorems for functionals of the union of these cylinders. The volume and the number of isolated cylinders and the Euler characteristic of the random set are considered and give an answer to the achievable throughput, the availability of nodes, and the topological structure of the network. [ABSTRACT FROM AUTHOR]

For a class of functions (called minimal Radó functions) that arise naturally in minimal surface theory, we bound the number of interior critical points (counting multiplicity) in terms of the boundary data and the Euler characteristic of the domain of the function. [ABSTRACT FROM AUTHOR]

In this work, we study the RVB method for calculating the Hawking temperature of different black holes and find that there is an undetermined integral constant in the temperature expression. We use this method to calculate the Hawking temperature of the black hole in Einstein gravity, and massive gravity, Einstein–Gauss–Bonnet gravity, Scalar–Tensor–Vector modified gravity and f (R) gravity, respectively. By comparing with the temperature obtained by the Hawking temperature formula, we find that regardless of the gravitational theory from which the black hole solution is obtained, after the black hole metric is reduced to two dimensions, if there is no first-order term of r in g 0 0 , the integral constant is 0. If there is a first-order term of r in g 0 0 , the integral constant is determined by the coefficient in front of the first-order term of r. [ABSTRACT FROM AUTHOR]

We study constructible invariants of the moduli space M ¯ (x) of stable maps from genus zero curves to P 1 , relative to 0 and ∞ , with ramification profiles specified by x ∈ Z n . These spaces are central to the enumerative geometry of P 1 , and provide a large family of birational models of the Deligne–Mumford–Knudsen moduli space M ¯ 0 , n . For the sequence of vectors x corresponding to maps which are maximally ramified over 0 and unramified over ∞ , we prove that a generating function for the topological Euler characteristics of these spaces satisfies a differential equation which allows for its recursive calculation. We also show that the class [ M ¯ (x) ] ∈ K 0 (Var / C) of the moduli space in the Grothendieck ring of varieties is constant as x varies within a fixed chamber in the resonance decomposition of Z n . We conclude by suggesting several further directions in the study of these spaces, giving conjectures on (1) the asymptotic behavior of the Euler characteristic and (2) a potential chamber structure for the Chern numbers. [ABSTRACT FROM AUTHOR]

Hurwitz groups are the non-trivial finite quotients of the (2,3,7)-triangle group 〈 A , Z | A 2 = Z 3 = (A Z) 7 = 1 〉 , with the name 'Hurwitz' coming from the fact that they are the conformal automorphism groups of compact Riemann surfaces of largest possible order 84 (g − 1) = − 42 χ for given genus g ⩾ 2 (or given Euler characteristic χ − 2 − 2 g). As such, they are also the orientation-preserving automorphism groups of orientably-regular maps of largest possible order 84 (g − 1) for given genus g of the underlying hyperbolic surface, namely the 'Hurwitz maps', of type { 3 , 7 } or { 7 , 3 }. Pseudo-Hurwitz groups are the finite smooth quotients of the pseudo triangle group 〈 A , Z | A 2 = Z 3 = [ A , Z ] 4 = 1 〉 , and as such, are the automorphism groups of regular bi-oriented maps of largest possible order 48 (g − 1) = − 24 χ for given χ = 2 − 2 g ≤ − 2. In this paper we prove that the alternating group A n and the symmetric group S n are pseudo-Hurwitz groups for all but a small number positive integers n , and in particular, for all n > 23. [ABSTRACT FROM AUTHOR]

Euler characteristic (χ) , a topological invariant, helps to understand the topology of a network or complex. We demonstrate that the multi-scale topological information of dynamically evolving fluid flow systems can be crystallized into their Euler characteristic surfaces χ s (r , t). Furthermore, we demonstrate the Euler Metric (EM), introduced by the authors, can be utilized to identify the stability regime of a given flow pattern, besides distinguishing between different flow systems. The potential of the Euler characteristic surface and the Euler metric have been demonstrated first on analyzing a simulated deterministic dynamical system before being applied to analyze experimental flow patterns that develop in micrometer sized drying droplets. [ABSTRACT FROM AUTHOR]

We classify all the n -component links in the 3 -sphere that bound a Thurston norm minimizing Seifert surface Σ with Euler characteristic χ (Σ) = n - 2 and that are nearly fibered, which means that the rank of their link Floer homology group HFL ^ in the maximal (collapsed) Alexander grading s top is equal to two. In other words, such a link L satisfies s top = n - χ (Σ) 2 = 1 , and in addition rk HFL ^ ∗ (L) [ 1 ] = 2 and rk HFL ^ ∗ (L) [ s ] = 0 for every s > 1 . The proof of the main theorem is inspired by the one of a similar recent result for knots by Baldwin and Sivek, and involves techniques from sutured Floer homology. Furthermore, we also compute the group HFL ^ for each of these links. [ABSTRACT FROM AUTHOR]

Once fallen, snow settles due to the combined effects of metamorphism and deformation of the ice matrix under gravity. To understand how these coupled processes affect snow evolution, we performed oedometric compression tests and continuously monitored the snow microstructure with X-ray tomography. Centimetric samples with an initial density between 200 and 300 kg m−3 were followed during an initial sintering phase and under two different loads of 2.1 and 4.7 kPa at $-8^\circ$ C for ~1 week. The microstructure captured at a voxel size of 8.5 μm was characterized by density, specific surface area (SSA) and two metrics related to bond network, namely the Euler characteristic and the minimum cut surface. Load-induced creep of the ice matrix was observed only for sufficiently low values of initial density (<290 kg m−3 in our tests), and was shown to be associated to a significant increase of the number of bonds. Application of the load, however, did not affect the individual bond size nor the SSA, which appeared to be mainly controlled by isothermal metamorphism. The uniaxial compression did not induce any creation of anisotropy on the microstructural characteristics. Overall, our results show that, for the considered conditions, the deformation of the ice matrix mainly leads to a reduction of the pore space and an increase of the coordination number, while metamorphism mainly affects the grain and bond sizes. [ABSTRACT FROM AUTHOR]

We study arithmetic statistics for Iwasawa invariants for fine Selmer groups associated to elliptic curves. [ABSTRACT FROM AUTHOR]

Bounded trajectories of an autonomous system with an isolated positively homogeneous nonlinearity that is the gradient of a smooth function are studied. We prove the existence of nonstationary bounded trajectories lying in connected components of the set of points where the positively homogeneous function is negative and nonzero stationary points in those connected components whose closure has nonzero Euler characteristic. The existence of nonstationary bounded trajectories is substantiated using the Waűewski method; and the existence of stationary points, using methods for calculating the winding number of finite-dimensional vector fields. [ABSTRACT FROM AUTHOR]

Persistent homology is perhaps the most popular and useful tool offered by topological data analysis - with point-cloud data being the most common setup. Its older cousin, the Euler characteristic curve (ECC) is less expressive - and far easier to compute. It is particularly suitable for analyzing imaging data, and is commonly used in fields ranging from astrophysics to biomedical image analysis. These fields are embracing GPU computations to handle increasingly large datasets. We therefore propose an optimized GPU implementation of ECC computation for 2D and 3D grayscale images. The goal of this paper is twofold. First, we offer a practical tool, illustrating its performance with thorough experimentation - but also explain its inherent shortcomings. Second, this simple algorithm serves as a perfect backdrop for highlighting basic GPU programming techniques that make our implementation so efficient - and some common pitfalls we avoided. This is intended as a step towards a wider usage of GPU programming in computational geometry and topology software. We find this is particularly important as geometric and topological tools are used in conjunction with modern, GPUaccelerated machine learning frameworks. [ABSTRACT FROM AUTHOR]

We prove that semisimple four‐dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth four‐manifolds and homotopy equivalent simply connected closed oriented smooth four‐manifolds. We show that all currently known four‐dimensional field theories are semisimple, including unitary field theories, and once‐extended field theories which assign algebras or linear categories to 2‐manifolds. As an application, we compute the value of a semisimple field theory on a simply connected closed oriented 4‐manifold in terms of its Euler characteristic and signature. Moreover, we show that a semisimple four‐dimensional field theory is invariant under CP2$\mathbb {C}P^2$‐stable diffeomorphisms if and only if the Gluck twist acts trivially. This may be interpreted as the absence of fermions amongst the 'point particles' of the field theory. Such fermion‐free field theories cannot distinguish homotopy equivalent 4‐manifolds. Throughout, we illustrate our results with the Crane–Yetter–Kauffman field theory associated to a ribbon fusion category, settling in the negative the question of whether it is sensitive to smooth structure. As a purely algebraic corollary of our results applied to this field theory, we show that a ribbon fusion category contains a fermionic object if and only if its Gauss sums vanish. [ABSTRACT FROM AUTHOR]

We propose a promising model of discrete spacetime based on nonassociative geometry and complex networks. Our approach treats space as a simplicial 3-complex (or complex network), built from "atoms" of spacetime and entangled states forming n-dimensional simplices ( n = 1 , 2 , 3 ). At large scales, a highly connected network is a coarse, discrete representation of a smooth spacetime. We show that, for high temperatures, the network describes disconnected discrete space. At the Planck temperature, the system experiences phase transition, and for low temperatures, the space becomes a triangulated discrete space. We show that the cosmological constant depends on the Universe's topology. The "foamy" structure, analogous to Wheeler's "spacetime foam", significantly contributes to the effective cosmological constant, which is determined by the Euler characteristic of the Universe. [ABSTRACT FROM AUTHOR]

Real Bruhat cells give an important and well-studied stratification of such spaces as |$\operatorname {GL}_{n+1}$|⁠ , |$\operatorname {Flag}_{n+1} = \operatorname {SL}_{n+1}/B$|⁠ , |$\operatorname {SO}_{n+1}$|⁠ , and |$\operatorname {Spin}_{n+1}$|⁠. We study the intersections of a top dimensional cell with another cell (for another basis). Such an intersection is naturally identified with a subset of the lower nilpotent group |$\operatorname {Lo}_{n+1}^{1}$|⁠. We are particularly interested in the homotopy type of such intersections. In this paper, we define a stratification of such intersections. As a consequence, we obtain a finite CW complex that is homotopically equivalent to the intersection. We compute the homotopy type for several examples. It turns out that for |$n \le 4$| all connected components of such subsets of |$\operatorname {Lo}_{n+1}^1$| are contractible: we prove this by explicitly constructing the corresponding CW complexes. Conversely, for |$n \ge 5$| and the top permutation, there is always a connected component with even Euler characteristic and therefore not contractible. This follows from formulas for the number of cells per dimension of the corresponding CW complex. For instance, for the top permutation |$S_6$|⁠ , there exists a connected component with Euler characteristic equal to |$2$|⁠. We also give an example of a permutation in |$S_6$| for which there exists a connected component that is homotopically equivalent to the circle |${{\mathbb {S}}}^1$|⁠. [ABSTRACT FROM AUTHOR]

This paper suggests using fractional Euler polynomials (FEPs) to solve the fractional diffusion-wave equation in Caputo's sense. We present the fundamental characteristics of Euler polynomials. The method for building FEPs is discussed. By basically converting fractional partial differential equations into a system of polynomial equations, these qualities enable us to come near to solving the original problem. A conventional numerical method is then used to solve the resulting system of equations. Theoretical analysis for our proposed strategy is also established, including the convergence theorem and error analysis. The proposed technique's error bound is confirmed for the test problems as well. The method's applicability and validity are examined using a variety of instances. The acquired solution is contrasted with other approaches' solutions described in the literature. This method is better in terms of implementation, adaptability and computing efficiency for solving other partial differential equations as a result of the comparison of the proposed method to existing methods used to solve the fractional diffusion-wave equation. [ABSTRACT FROM AUTHOR]

We consider a rotationally invariant system-bath (RISB) model in three-dimensional space that is described by a linear rigid rotor independently coupled to three harmonic-oscillator baths through functions of the rotor's Euler angles. While this model has been developed to study the dielectric relaxation of a dipolar molecule in solvation as a problem of classical Debye relaxation, here we investigate it as a problem of open quantum dynamics. Specifically, the treatment presented here is carried out as an extension of a previous work [Y. Iwamoto and Y. Tanimura, J. Chem. Phys 149, 084110 (2018)], in which we studied a two-dimensional (2D) RISB model, to a three-dimensional (3D) RISB model. As in the 2D case, due to a difference in the energy discretization of the total Hamiltonian, the dynamics described by the 3D RISB model differ significantly from those described by the rotational Caldeira-Leggett model. To illustrate the characteristic features of the quantum 3D rotor system described by angular momentum and magnetic quantum numbers, we derive a quantum master equation (QME) and hierarchical equations of motion for the 3D RISB model in the high-temperature case. Using the QME, we compute linear and 2D rotational spectra defined by the linear and nonlinear response functions of the rotor dipole, respectively. The quantum transitions between the angular momentum states and magnetic states arising from polarized Stark fields as well as the system-bath interactions can be clearly observed in 2D rotational spectroscopy. [ABSTRACT FROM AUTHOR]

Equivariant Euler characteristics are important numerical homotopy invariants for objects with group actions. They have deep connections with many other areas like modular representation theory and chromatic homotopy theory. They are also computable, especially for combinatorial objects like partition posets, buildings associated with finite groups of Lie types, etc. In this article, we make new contributions to concrete computations by determining the equivariant Euler characteristics for all subgroup complexes of symmetric groups Σ n when n is prime, twice a prime, or a power of two and several variants. There are two basic approaches to calculating equivariant Euler characteristics. One is based on a recursion formula and generating functions, and another on analyzing the fixed points of abelian subgroups. In this article, we adopt the second approach since the fixed points of abelian subgroups are simple in this case. [ABSTRACT FROM AUTHOR]

We show how a novel construction of the sheaf of Cherednik algebras |$\mathscr {H}_{1, c, X, G}$| on a quotient orbifold |$Y:=X/G$| in author's prior work leads to results for |$\mathscr {H}_{1, c, X, G}$|⁠ , which until recently were viewed as intractable. First, for every orbit type stratum in |$X$|⁠ , we define a trace density map for the Hochschild chain complex of |$\mathscr {H}_{1, c, X, G}$|⁠ , which generalizes the standard Engeli–Felder's trace density construction for the sheaf of differential operators |$\mathscr {D}_X$|⁠. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of |$\mathbb {C}_{Y}\llbracket \hbar \rrbracket $| -modules, which computes the hypercohomology of the Hochschild chain complex of the sheaf of formal Cherednik algebras |$\mathscr {H}_{1, \hbar , X, G}$|⁠. We show that this hypercohomology is isomorphic to the Chen–Ruan cohomology of the orbifold |$Y$| with values in the ring of formal power series |$\mathbb {C}\llbracket \hbar \rrbracket $|⁠. We infer that the Hochschild chain complex of the sheaf of skew group algebras |$\mathscr {H}_{1, 0, X, G}$| has a well-defined Euler characteristic that is equal to the orbifold Euler characteristic of |$Y$|⁠. Finally, we prove an algebraic index theorem. [ABSTRACT FROM AUTHOR]

In this article, we investigate the geometry of 4-dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a 4-dimensional compact gradient Ricci soliton must satisfy the classical Hitchin-Thorpe inequality. In addition, some volume estimates are also obtained. [ABSTRACT FROM AUTHOR]

We construct a well-behaved Weil-étale complex for a large class of Z -constructible sheaves on a regular irreducible scheme U of finite type over Z and of dimension 1. We then give a formula for the special value at s = 0 of the L -function associated to any Z -constructible sheaf on U in terms of Euler characteristics of Weil-étale cohomology; for smooth proper curves, we obtain the formula of [GS20]. We deduce a special value formula for Artin L -functions of integral representations twisted by a singular irreducible scheme X of finite type over Z and of dimension 1. This generalizes and improves all results in [Tra16] ; as a special case, we obtain a special value formula for the arithmetic zeta function of X. [ABSTRACT FROM AUTHOR]

Let χ(F2) $\chi ({F}^{2})$ be the Euler characteristic of a surface F2 ${F}^{2}$. We characterize the set of graphs H $H$ of order at most 6 which satisfies the following: If G $G$ is a 5‐connected graph embedded in a surface F2 ${F}^{2}$ of face‐width at least 3∣H∣−2χ(F2)−5 $3| H| -2\chi ({F}^{2})-5$ and H′ $H^{\prime} $ is a subgraph of G $G$ isomorphic to H $H$ such that G−H′ $G-H^{\prime} $ has even order, then G−H′ $G-H^{\prime} $ has a perfect matching. An analogous result on near‐perfect matching is given as well. Moreover, we show the following result. Let G $G$ be a 5‐connected graph embedded in a surface F2 ${F}^{2}$ and let H1,...,Hm ${H}_{1},\ldots ,{H}_{m}$ be connected subgraphs of G $G$ of size at most k $k$ which lie pairwise sufficiently far apart in the face‐subdivision of G $G$. We prove that, if the face‐width of G $G$ is at least max{12(9km−7χ(F2)−20),6} $\max \{\frac{1}{2}(9km-7\chi ({F}^{2})-20),6\}$ and G−Hi $G-{H}_{i}$ satisfies C(1) ${\mathscr{C}}(1)$ for each i $i$ with 1≤i≤m $1\le i\le m$, then G−(H1∪...∪Hm) $G-({H}_{1}\cup \ldots \,\cup {H}_{m})$ satisfies C(1) ${\mathscr{C}}(1)$ as well, where G $G$ is said to satisfy C(1) ${\mathscr{C}}(1)$ if G−S $G-S$ has at most ∣S∣+1 $| S| +1$ components for every S⊆V(G) $S\subseteq V(G)$. It is also shown that the distance condition can be considerably weakened when G $G$ triangulates the surface. All of these results generalize previous studies on matching extension in graphs on surfaces. [ABSTRACT FROM AUTHOR]

In this paper we survey some results on the symmetric semi-perfect obstruction theory on a Deligne—Mumford stack X constructed by Chang—Li, and Behrend's theorem equating the weighted Euler characteristic of X and the virtual count of X by symmetric semi-perfect obstruction theories. As an application, we prove that Joyce's d-critical scheme admits a symmetric semi-perfect obstruction theory, which can be applied to the virtual Euler characteristics by Jiang—Thomas. [ABSTRACT FROM AUTHOR]

Partial generalizations of the theory of virtual polyhedra (sometimes under different names) appeared recently in the theory of torus manifolds look very different from the original theory of virtual polyhedra. Such generalizations are based on simple arguments from homotopy theory while the original theory is based on integration over the Euler characteristic. We explain how these generalizations are related to the classical theory of convex bodies and the original theory of virtual polyhedra. The paper basically contains no proofs: all proofs and details can be found in the cited literature. [ABSTRACT FROM AUTHOR]

The geometric structure variation of the nonwetting phase (NWP) in porous media plays a critical role in numerous energy-related processes. This work conducted a series of gas/liquid displacements in three rock cores using high resolution x-ray microcomputed tomography (micro-CT), including two sandstones and a Berea core with different pore architectures. The morphological and topological characteristics of NWP were investigated at the pore scale in a full cycle of primary drainage (PD), main imbibition (MI), and main drainage (MD). It is found that roof and distal snap-off occurred during drainage, resulting in nonwetting (NW) singlets and ganglia. These NW bubbles occupied 10%–20% NWP volume. Both throat snap-off and pore snap-off were observed in MI, which generally occur at a single-pore scale. The effect of viscous force is significant even at a capillary dominant displacement. The trapped NWP during PD can be partially or totally displaced in MI through further snap-off events or viscous remobilization. Due to the temporal and spatial scale diversion in pore scale events, the amplitude of pressure drop fluctuations among PD, MI, and MD is different. The significant difference in NWP characteristics between PD and MD is connectivity, thus leading to hysteresis. [ABSTRACT FROM AUTHOR]

We consider the Euler characteristics χ(M) of closed orientable topological 2n-manifolds with (n-1)-connected universal cover and a given fundamental group G of type Fn. We define q2n(G), a generalized version of the Hausmann-Weinberger invariant for 4-manifolds, as the minimal value of (-1)nχ(M). For all n≥2, we establish a strengthened and extended version of their estimates, in terms of explicit cohomological invariants of G. As an application we obtain new restrictions for non-abelian finite groups arising as fundamental groups of rational homology 4-spheres. [ABSTRACT FROM AUTHOR]

Tools of topological data analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well-studied data summary, suffers a number of limitations; its computations are hard to distribute, and it is hard to generalize to multifiltrations and is computationally prohibitive for big datasets. In this article, we study the concept of Euler characteristics curves for 1-parameter filtrations and Euler characteristic profiles for multiparameter filtrations. While being a weaker invariant in one dimension, we show that Euler characteristic–based approaches do not possess some handicaps of persistent homology; we show efficient algorithms to compute them in a distributed way, their generalization to multifiltrations, and practical applicability for big data problems. In addition, we show that the Euler curves and profiles enjoy a certain type of stability, which makes them robust tools for data analysis. Lastly, to show their practical applicability, multiple use cases are considered. [ABSTRACT FROM AUTHOR]

Let {Xi(t): t ∈ S ⊂ Rd}i=1,2,...,n be independent copies of a stationary centered Gaussian field with almost surely smooth sample paths. In this paper, we are interested in the con- junction probability defined as P (∃t ∈ S: Xi(t) ≥ u, ∀i = 1, 2, . . ., n) for a given threshold level u. As u → ∞, we will provide an asymptotic formula for the conjunction probability. This asymptotic formula is derived from the behaviour of the volume of the set of local maximum points. The proof relies on a result of Azaïs and Wschebor (2014) describing the shape of the excursion set of a stationary centered Gaussian field. Our result partially confirms the validity of the Euler characteristic method. [ABSTRACT FROM AUTHOR]