Geometric Lattice Research Articles

The classification of Boolean degree 1 functions in high-dimensional finite vector spaces

We classify the Boolean degree 1 1 functions of k k -spaces in a vector space of dimension n n (also known as Cameron-Liebler classes) over the field with q q elements for n ≥ n 0 ( k , q ) n \geq n_0(k, q) . This also implies that two-intersecting sets with respect to k k -spaces do not exist for n ≥ n 0 ( k , q ) n \geq n_0(k, q) . Our main ingredient is the Ramsey theory for geometric lattices.

Twinning and homo-epitaxy cooperation in the already rich growth morphology of CaCO3 polymorphs. II. Calcite.

The two most abundant CaCO3 polymorphs, calcite and aragonite, are universally recognized for the richness of their morphology to which different twins make relevant contributions. The epitaxial transformation calcite ↔ aragonite has long been debated. While the twinning has been thoroughly treated, the homo-epitaxy occurring within each of these minerals has, inexplicably, been overlooked to date, both experimentally and theoretically. Twinning can be deceptive to the point where it can be mistaken for homo-epitaxy, thus making the proposed growth mechanism in the crystal aggregate wrong. Within the present work, the first aim is a theoretical investigation of the homo-epitaxies among the three {10.4}-cleavage, {01.2}-steep and {01.8}-flat rhombohedra of calcite. Accordingly, the specific adhesion energies were calculated between facing crystal forms, unequivocally showing that the {01.2}/{01.8} homo-epitaxy competes with the generation of both {01.2} and {01.8} contact twins. Secondly, the calculation of the specific adhesion energy was extended to consider homo-epitaxy for the {10.4} rhombohedron. The two-dimensional geometric lattice coincidence has been tried for the {00.1} pinacoidal form as well.

Multiresolution lattice-based support generation for additive manufacturing

The generation of support for 3D models toward 3D printing is a highly challenging task that is of great need in many additive manufacturing processes. In this work, we explore the use of multiresolution geometric lattices to generate support with controlled contact locations. That is, with bounds on the maximal distance between adjacent local support points. A variety of end-user controls over the synthesized support are provided, such as the angular slopes in the model that are provided with support and/or controls on the dimensions and sizes of the support lattice tiles. These controls are augmented with the option of an automated optimization via a direct link to analysis. We demonstrate this proposed lattice approach for support synthesis on several 3D models of different types.

Modular flats of oriented matroids and poset quasi-fibrations

We study the combinatorics of modular flats of oriented matroids and the topological consequences for their Salvetti complexes. We show that the natural map to the localized Salvetti complex at a modular flat of corank one is what we call a poset quasi-fibration – a notion derived from Quillen’s fundamental Theorem B from algebraic K K -theory. As a direct consequence, the Salvetti complex of an oriented matroid whose geometric lattice is supersolvable is a K ( π , 1 ) K(\pi ,1) -space – a generalization of the classical result for supersolvable hyperplane arrangements due to Falk, Randell and Terao. Furthermore, the fundamental group of the Salvetti complex of a supersolvable oriented matroid is an iterated semidirect product of finitely generated free groups – analogous to the realizable case. Our main tools are discrete Morse theory, the shellability of certain subcomplexes of the covector complex of an oriented matroid, a nice combinatorial decomposition of poset fibers of the localization map, and an isomorphism of covector posets associated to modular elements. We provide a simple construction of supersolvable oriented matroids. This gives many non-realizable supersolvable oriented matroids and by our main result aspherical CW-complexes.

Exploiting Zone-Folding Induced Quasi-Bound Modes to Achieve Highly Coherent Thermal Emissions.

Thermal emissions with high coherence, although not as high as those of lasers, still play a crucial role in many practical applications. In this work, by exploiting the geometric perturbation-induced optical lattice tripling and the associated Brillion zone folding effect, we propose and investigate thermal emissions in the mid-infrared with simultaneous high temporal and spatial coherence. In contrast with the case of period-doubling perturbation in our previous work, the steeper part of the guided mode dispersion band will be folded to the high-symmetry Γ point in the ternary grating. In this case, a specific emission wavelength corresponds to only a very small range of wavevectors. Consequently, apart from the high temporal coherence characterized by an experimental bandwidth around 30 nm, the achieved thermal emissions also feature ultrahigh spatial coherence. Calculations show that at the thermal emission wavelengths in the mid-infrared, the spatial coherence length can easily reach up to mm scale.

Концепция верификации работы клеточных автоматов при вариации геометрических решёток для модели диффузионного процесса

The methodology of cellular automata occupies one of the key positions in the discrete dynamic simulations. The adequacy of the results of solving applied problems is largely determined by the choice of the type of geometric lattice of the cellular automaton. The article presents a formalization and numerical verification of the concept of verification of the operation of a cellular automata model of the classical diffusion process for a set of basic two-dimensional geometric lattices. The approach is based on a direct comparison of the space-time solution of the diffusion equation obtained by the finite element method with the solution of a discrete analogue of this equation based on the operation of a cellular automata algorithm. The software implementation of the cell-automation was performed with the C+ using the Unity platform while the finite element solution of the evolutionary problem of mathematical physics was carried out using the Matlab tools. Cellular automaton lattices were constructed using various cell geometric primitives traditionally used in diffusion modeling applications. Obtained findings indicate that the hexagonal cellular automaton demonstrates the minimum error in the numerical implementation.

The geometry of discrete L-algebras

Abstract The relationship of discrete L-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete L-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete L-algebra X is determined and shown to be a complete invariant. It is proved that X ∖ {1} is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality n > 3, a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated L-algebra is determined.

Proto-exact categories of modules over semirings and hyperrings

Proto-exact categories, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a non-additive setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor.In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures.In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices L has a proto-exact structure, and furthermore that the subcategory of L consisting of finite lattices is equivalent to the category of finite B-modules as proto-exact categories, where B is the Boolean semifield. We also discuss some relations between L and geometric lattices (simple matroids) from this perspective.In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the Krasner hyperfieldK, a well-known relation between finite K-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.

Cluster-Permutohedra and Submanifolds of Flag Varieties with Torus Actions

Abstract In this paper, we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardo, Del Río-Francos, López-Dudet, Oliveros, and Schulte in 2010 and toric topology of manifolds of sparse isospectral Hermitian matrices. More precisely, we recall the notion of a cluster-permutohedron, a certain finite poset defined for a simple graph $\Gamma $. This poset is build as a combination of cosets of the symmetric group, and the geometric lattice of the graphical matroid of $\Gamma $. This poset is similar to the graphicahedron of $\Gamma $, in particular, 1-skeleta of both posets are isomorphic to Cayley graphs of the symmetric group. We describe the relation between cluster-permutohedron and graphicahedron using Galois connection and the notion of a core of a finite topology. We further prove that the face poset of the natural torus action on the manifold of isospectral $\Gamma $-shaped Hermitian matrices is isomorphic to the cluster-permutohedron. Using recent results in toric topology, we show that homotopy properties of graphicahedra may serve an obstruction to equivariant formality of isospectral matrix manifolds. We introduce a generalization of a cluster-permutohedron and describe the combinatorial structure of a large family of manifolds with torus actions, including Grassmann manifolds and partial flag manifolds.

The Best Extending Cover-preserving Geometric Lattices of Semimodular Lattices

The Best Extending Cover-preserving Geometric Lattices of Semimodular Lattices

A Hilbert irreducibility theorem for Enriques surfaces

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0 0 . Bounding the rank of this object, we prove that a conjecture by Campana [Ann. Inst. Fourier (Grenoble) 54 (2004), pp. 499–630] and Corvaja–Zannier [Math. Z. 286 (2017), pp. 579–602] holds for Enriques surfaces, as well as K3 surfaces of Picard rank ≥ 6 \geq 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 0 satisfy the weak Hilbert Property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded.

Chain enumeration, partition lattices and polynomials with only real roots

The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. The chain polynomials of the partition lattices and their standard type \(B\) analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen-Macaulay posets. As a result, new families of convex polytopes whose barycentric subdivisions have real-rooted \(f\)-polynomials are presented. An application to the face enumeration of the second barycentric subdivision of the boundary complex of the simplex is also included.Mathematics Subject Classifications: 05A05, 05A18, 05E45, 06A07, 26C10Keywords: Chain polynomial, geometric lattice, partition lattice, real-rooted polynomial, flag \(h\)-vector, convex polytope, barycentric subdivision

A conformational switch in clathrin light chain regulates lattice structure and endocytosis at the plasma membrane of mammalian cells

Conformational changes in endocytic proteins are regulators of clathrin-mediated endocytosis. Three clathrin heavy chains associated with clathrin light chains (CLC) assemble into triskelia that link into a geometric lattice that curves to drive endocytosis. Structural changes in CLC have been shown to regulate triskelia assembly in solution, yet the nature of these changes, and their effects on lattice growth, curvature, and endocytosis in cells are unknown. Here, we develop a new correlative fluorescence resonance energy transfer (FRET) and platinum replica electron microscopy method, named FRET-CLEM. With FRET-CLEM, we measure conformational changes in clathrin at thousands of individual morphologically distinct clathrin-coated structures. We discover that the N-terminus of CLC repositions away from the plasma membrane and triskelia vertex as coats curve. Preventing this conformational switch with chemical tools increases lattice sizes and inhibits endocytosis. Thus, a specific conformational switch in the light chain regulates lattice curvature and endocytosis in mammalian cells.

Rank-Metric Lattices

We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on small rank-metric lattices whose characteristic polynomial we cannot compute, and provide a formula for them under a polynomiality assumption on their Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and investigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.

Prediction of Residual Deformation and Stress of Laser Powder Bed Fusion Manufactured Ti-6Al-4V Lattice Structures Based on Inherent Strain Method

The development of additive manufacturing (AM) technology provides higher feasibility for designing and manufacturing lattice structures. However, the manufacturing process usually generates residual deformation and stress, and even produces cracking, thus affecting the performance of the parts. This work establishes a simulation model of the Ti-6Al-4V lattice structures during laser powder bed fusion (LPBF) based on the inherent strain method. Effects of geometric lattice parameters (inclination angle, rod diameter, rod length) on the residual deformation and stress are analyzed. Based on the simulation results, measures for improving the quality of the lattice structures are proposed. The proposed model and simulation results can provide theoretical references for designing and manufacturing the lattice structures during practical engineering applications of LPBF.

The Poincare Polynomial of an Arrangement with the Trio Separation Property

An arrangement of hyperplanes with a modular element in its intersection lattice has a Poincaré polynomial which factors; this was proven by Stanley in the setting of geometric lattices. This note proves a factorization in the setting of hyperplane arrangements under two conditions which imply a modular element. Two well known reflection arrangements serve as motivation and their Poincaré polynomials are computed using the main theorem of this note.

Transgranular highly localized deformation and fracture mechanism in ductile CrFeNi medium-entropy alloys

The fracture evolution and damage mechanism of an engineering candidate, CrFeNi medium-entropy alloy (MEA), which exhibits superior tensile properties with the ultimate tensile strength of ∼1.03 GPa and ductility of ∼55%, were studied. This work highlights the effect of defect evolution, including microcrack, geometric necessary dislocation (GND) density, lattice rotation and dislocation configuration, on fracture evolution. The result reveals that the fracture mechanism of the studied alloy is dominated by the coalescence of microcracks, which initiate at a certain distance from the main crack tip. It is distinct compared to common ductile metals and is attributed to a transgranular highly localized deformation behavior which provides potential sites for microcrack initiation. The microstructural characteristics of the microcrack embryo are evaluated and clarified to be the localized damage accumulation with a high density of dislocations at the crystal-defect scale. It can result in elevated internal stress locally, providing the driving force for microcrack initiation. In addition, the mechanical model based on distributed dislocation technique (DDT) demonstrates that the transgranular highly localized deformation leads to a crack propagation acceleration effect and the deterioration of crack propagation resistance. By relating the defect evolution and fracture performance of the CrFeNi MEA, it provides new insight on the toughening design of ductile metals.

Polymatroids, Closure Operators and Lattices

In this article we study the closure operators of polymatroids from a lattice theoretic point of view. We show that polymatroid closure operators relate to lattices enriched with a generating set in the same way matroids relate to geometric lattices. Through this relation we define a notion of minors for lattices enriched with a generating set. For the lattice of flats of a graphic matroid, the minors of the lattice are shown to correspond to simple minors of the graph when the vertices are labeled and the edges unlabeled. This correspondence is generalized to all polymatroids.

Degenerate Whitney Numbers of First and Second Kind of Dowling Lattices

Dowling constructed Dowling lattice Qn(G), for any finite set with n elements and any finite multiplicative group G of order m, which is a finite geometric lattice. He also defined the Whitney numbers of the first and second kinds for any finite geometric lattice. These numbers for the Dowling lattice Qn(G) are the Whitney numbers of the first kind Vm(n,k) and those of the second kind Wm(n,k), which are given by Stirling number-like relations. In this paper, by `degenerating' such relations we introduce the degenerate Whitney numbers of the first kind and those of the second kind and investigate, among other things, generating functions, recurrence relations and various explicit expressions for them. As further generalizations of the degenerate Whitney numbers of both kinds, we also consider the degenerate r-Whitney numbers of both kinds.

Top-heavy phenomena for transformations

Let S be a transformation semigroup acting on a set Ω. The action of S on Ω can be naturally extended to be an action on all subsets of Ω. We say that S is ℓ-homogeneous provided it can send A to B for any two (not necessarily distinct) ℓ-subsets A and B of Ω. On the condition that k ≤ ℓ < k + ℓ ≤ |Ω|, we show that every ℓ-homogeneous transformation semigroup acting on Ω must be k-homogeneous. We report other variants of this result for Boolean semirings and affine/projective geometries. In general, any semigroup action on a poset gives rise to an automaton and we associate some sequences of integers with the phase space of this automaton. When this poset is a geometric lattice, we propose to investigate various possible regularity properties of these sequences, especially the so-called top-heavy property. In the course of this study, we are led to a conjecture about the injectivity of the incidence operator of a geometric lattice, generalizing a conjecture of Kung.