Complex Algebra Research Articles

Polarity-driven laminar pattern formation by lateral-inhibition in 2D and 3D bilayer geometries

Abstract Fine-grain patterns produced by juxtacrine signalling have previously been studied using static monolayers as cellular domains. However, analytic results are usually restricted to a few cells due to the algebraic complexity of non-linear dynamical systems. Motivated by concentric patterning of Notch expression observed in the mammary gland, we combine concepts from graph and control theory to represent cellular connectivity and behaviour. The resulting theoretical framework allows us to exploit the symmetry of multicellular bilayer structures in 2D and 3D, thereby deriving analytical conditions that drive the dynamical system to form laminar patterns, consistent with the formation of cell polarity by activator localization. Critically, the patterning conditions are independent of the precise dynamical details, thus the framework allows for generality in understanding the influence of cellular geometry and signal polarity on patterning using lateral-inhibition systems. Applying the analytic conditions to mammary organoids suggests that intense cell signalling polarity is required for the maintenance of stratified cell types within a static bilayer using a lateral-inhibition mechanism. Furthermore, by employing 2D and 3D cell-based models, we highlight that the cellular polarity conditions derived from static domains can generate laminar patterning in dynamic environments. However, they are insufficient for the maintenance of patterning when subjected to substantial morphological perturbations. In agreement with the mathematical implications of strict signalling polarity induced on the cells, we propose an adhesion-dependent Notch-Delta biological process that has the potential to initiate bilayer stratification in a developing mammary organoid.

Bit complexity for computing one point in each connected component of a smooth real algebraic set

We analyze the bit complexity of an algorithm for the computation of at least one point in each connected component of a smooth real algebraic set. This work is a continuation of our analysis of the hypersurface case (On the bit complexity of finding points in connected components of a smooth real hypersurface, ISSAC'20). In this paper, we extend the analysis to more general cases.Let F=(f1,…,fp) in Z[X1,…,Xn]p be a sequence of polynomials with V=V(F)⊂Cn a smooth and equidimensional variety and 〈F〉⊂C[X1,…,Xn] a radical ideal. To compute at least one point in each connected component of V∩Rn, our starting point is an algorithm by Safey El Din and Schost (Polar varieties and computation of one point in each connected component of a smooth real algebraic set, ISSAC'03). This algorithm uses random changes of variables that are proven to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model; here, we analyze the bit complexity and the error probability, and we provide a quantitative analysis of the genericity statements. In particular, we are led to use Lagrange systems to describe polar varieties, as they make it simpler to rely on techniques such as weak transversality and an effective Nullstellensatz.

Geometry of Linear Convolutional Networks

We study the family of functions that are represented by a linear convolutional network (LCN). These functions form a semi-algebraic subset of the set of linear maps from input space to output space. In contrast, the families of functions represented by fully connected linear networks form algebraic sets. We observe that the functions represented by LCNs can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network's architecture on the geometry of the resulting function space. We further study the optimization of an objective function over an LCN, analyzing critical points in function space and in parameter space and describing dynamical invariants for gradient descent. Overall, our theory predicts that the optimized parameters of an LCN will often correspond to repeated filters across layers, or filters that can be decomposed as repeated filters. We also conduct numerical and symbolic experiments that illustrate our results and present an in-depth analysis of the landscape for small architectures.

$K$-stability: The Recent Interaction Between Algebraic Geometry and Complex Geometry

$K$-stability: The Recent Interaction Between Algebraic Geometry and Complex Geometry

Hartogs-type theorems in real algebraic geometry, I

Abstract Let f : X → ℝ {f:X\rightarrow\mathbb{R}} be a function defined on a connected nonsingular real algebraic set X in ℝ n {\mathbb{R}^{n}} . We prove that regularity of f can be detected by controlling the restrictions of f to either algebraic curves or algebraic surfaces in X. If dim ⁡ X ≥ 2 {\operatorname{dim}X\geq 2} and k is a positive integer, then f is a regular function whenever the restriction f | C {f|_{C}} is a regular function for every algebraic curve C in X that is a 𝒞 k {\mathcal{C}^{k}} submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of C is equivalent to the plane curve singularity defined by the equation x p = y q {x^{p}=y^{q}} for some primes p < q {p<q} . If dim ⁡ X ≥ 3 {\operatorname{dim}X\geq 3} , then f is a regular function whenever the restriction f | S {f|_{S}} is a regular function for every nonsingular algebraic surface S in X that is homeomorphic to the unit 2-sphere. We also have suitable versions of these results for X not necessarily connected.

Kripke Contexts, Double Boolean Algebras with Operators and Corresponding Modal Systems

The notion of a context in formal concept analysis and that of an approximation space in rough set theory are unified in this study to define a Kripke context. For any context (G,M,I), a relation on the set G of objects and a relation on the set M of properties are included, giving a structure of the form ((G,R), (M,S), I). A Kripke context gives rise to complex algebras based on the collections of protoconcepts and semiconcepts of the underlying context. On abstraction, double Boolean algebras (dBas) with operators and topological dBas are defined. Representation results for these algebras are established in terms of the complex algebras of an appropriate Kripke context. As a natural next step, logics corresponding to classes of these algebras are formulated. A sequent calculus is proposed for contextual dBas, modal extensions of which give logics for contextual dBas with operators and topological contextual dBas. The representation theorems for the algebras result in a protoconcept-based semantics for these logics.

On the simplicity of Nekrashevych algebras of contracting self-similar groups

Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych \(C^*\)-algebras. In particular, Nekrashevych, Clark, Exel, Pardo, Sims and Starling have studied the question of simplicity of Nekrashevych algebras, in part, because non-simplicity of the complex algebra implies non-simplicity of the \(C^*\)-algebra. In this paper we give necessary and sufficient conditions for the Nekrashevych algebra of a contracting group to be simple. Nekrashevych algebras of contracting groups are finitely presented. We give an algorithm which on input the nucleus of the contracting group, outputs all characteristics of fields over which the corresponding Nekrashevych algebra is simple. Using our methods, we determine the fields over which the Nekrashevych algebras of a number of well-known contracting groups are simple including the Basilica group, Gupta–Sidki groups, GGS-groups, multi-edge spinal groups, Šunić groups associated to polynomials (this latter family includes the Grigorchuk group, Grigorchuk–Erschler group and Fabrykowski–Gupta group) and self-replicating spinal automaton groups.

Mirror Symmetry for Truncated Cluster Varieties

In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK construction in arbitrary dimension: given a truncated cluster variety, we construct a symplectic manifold and prove homological mirror symmetry for the resulting pair. We also describe how our construction can be obtained from toric geometry, and we relate our construction to various aspects of cluster theory which are known to symplectic geometers.

Efficient Local Refinement near Parametric Boundaries Using kd-Tree Data Structure and Algebraic Level Sets

In analysis of problems with parametric spline boundaries that are immersed or inserted into an underlying domain, the discretization on the underlying domain usually does not conform to the inserted boundaries. While the fixed underlying discretization is of great convenience as the immersed boundaries evolve, the field approximations near the inserted boundaries require refinement in the underlying domain, as do the quadrature cells. In this paper, a kd-tree data structure together with a sign-based and/or distance-based refinement strategy is proposed for local refinement near the inserted boundaries as well as for adaptive quadrature near the boundaries. The developed algorithms construct and utilize implicit forms of parametric Non-Uniform Rational B-Spline (NURBS) surfaces to algebraically (and non-iteratively) estimate distance as well as sign relative to the inserted boundary. The kd-tree local refinement is demonstrated to produce fewer sub-cells for the same accuracy of solution as compared to the classical quad/oct tree-based subdivision. Consistent with the kd-tree data structure, we describe a new a priori refinement algorithm based on the signed and unsigned distance from the inserted boundary. We first demonstrate the local refinement strategy coupled with the the kd-tree data structure by constructing Truncated Hierarchical B-spline (THB-spline) “meshes”. We next demonstrate the accuracy and efficiency of the developed local refinement strategy through adaptive quadrature near NURBS boundaries inserted within volumetric three-dimensional NURBS discretizations.

Moving level-of-detail surfaces

We present a simple, fast, and smooth scheme to approximate Algebraic Point Set Surfaces using non-compact kernels, which is particularly suited for filtering and reconstructing point sets presenting large missing parts. Our key idea is to consider a moving level-of-detail of the input point set which is adaptive w.r.t. to the evaluation location, just such as the samples weights are output sensitive in the traditional moving least squares scheme. We also introduce an adaptive progressive octree refinement scheme, driven by the resulting implicit surface, to properly capture the modeled geometry even far away from the input samples. Similarly to typical compactly-supported approximations, our operator runs in logarithmic time while defining high quality surfaces even on challenging inputs for which only global optimizations achieve reasonable results. We demonstrate our technique on a variety of point sets featuring geometric noise as well as large holes.

Efficient Knowledge Compilation Beyond Weighted Model Counting

AbstractQuantitative extensions of logic programming often require the solution of so called second level inference tasks, that is, problems that involve a third operation, such as maximization or normalization, on top of addition and multiplication, and thus go beyond the well-known weighted or algebraic model counting setting of probabilistic logic programming under the distribution semantics. We introduce Second Level Algebraic Model Counting (2AMC) as a generic framework for these kinds of problems. As 2AMC is to (algebraic) model counting what forall-exists-SAT is to propositional satisfiability, it is notoriously hard to solve. First level techniques based on Knowledge Compilation (KC) have been adapted for specific 2AMC instances by imposing variable order constraints on the resulting circuit. However, those constraints can severely increase the circuit size and thus decrease the efficiency of such approaches. We show that we can exploit the logical structure of a 2AMC problem to omit parts of these constraints, thus limiting the negative effect. Furthermore, we introduce and implement a strategy to generate a sufficient set of constraints statically, with a priori guarantees for the performance of KC. Our empirical evaluation on several benchmarks and tasks confirms that our theoretical results can translate into more efficient solving in practice.

Equations in acylindrically hyperbolic groups and verbal closedness

Let H be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system S of equations with constants from H is equivalent to a single equation. We also show that the algebraic set associated with S is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such an equation has the form w(x_1,\ldots,x_n)=h , where w\in F(X) , h\in H ). From this we deduce the following statement: Let G be an arbitrary overgroup of the above group H . Then H is verbally closed in G if and only if it is algebraically closed in G . These statements have interesting implications; here we give only two of them: If H is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from H is equivalent to a single equation. We give a positive solution to Problem 5.2 from the paper [J. Group Theory 17 (2014), 29–40] of Myasnikov and Roman’kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts. Moreover, we describe solutions of the equation x^ny^m=a^nb^m in acylindrically hyperbolic groups (AH-groups), where a , b are non-commensurable jointly special loxodromic elements and n,m are integers with sufficiently large common divisor. We also prove the existence of special test words in AH-groups and give an application to endomorphisms of AH-groups.

Hadamard product and related inequalities in the Jordan spin algebra

Due to its simple yet elegant structure, the study of an entry-wise product of matrices, called the Hadamard product, has received extensive attention from researchers and has expanded to various disciplines, including Euclidean Jordan algebras. As an ongoing effort to extend this product to Euclidean Jordan algebras, in this article, we propose a Hadamard product in the setting of Jordan spin algebra, , under the scheme of the Peirce decomposition, and show that it preserves the diagonal structure of the elements in the algebra. It is shown that this new product corresponds to the standard Hadamard product of symmetric matrices in the case of . Lastly, we prove that this novel product satisfies an analog of the Schur product theorem as well as the inequalities of Hadamard, Oppenheim, Fiedler, etc.

Smooth Rough Paths, Their Geometry and Algebraic Renormalization

We introduce the class of “smooth rough paths” and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer–Cartan perspective is the key to a purely algebraic form of Lyons’ extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of “sum of rough paths”. We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.

Local derivations of semisimple Leibniz algebras

We prove that every local derivation on a complex semisimple finite-dimensional Leibniz algebra is a derivation.

Higher multi-Courant algebroids

The binary bracket of a Courant algebroid structure on (E,〈⋅,⋅〉) can be extended to a n-ary bracket on Γ(E), yielding a multi-Courant algebroid. These n-ary brackets form a Poisson algebra and were defined, in an algebraic setting, by Keller and Waldmann. We construct a higher geometric version of Keller-Waldmann Poisson algebra and define higher multi-Courant algebroids. As Courant algebroid structures can be seen as degree 3 functions on a graded symplectic manifold of degree 2, higher multi-Courant structures can be seen as functions of degree n≥3 on that graded symplectic manifold.

On the Fukui–Kurdyka–Paunescu conjecture

In this paper, we prove the Fukui–Kurdyka–Paunescu conjecture, which says that sub-analytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (respectively, degree) of real and complex analytic (respectively, algebraic) sets. For instance, still in the real case, we prove a global version of the Fukui–Kurdyka–Paunescu conjecture. In the complex case, one of the results that we prove is the following: if$(X,0)\subset (\mathbb {C}^{n},0)$,$(Y,0)\subset (\mathbb {C}^{m},0)$are germs of analytic sets and$h\colon (X,0)\to (Y,0)$is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs$(X,0)$and$(Y,0)$have the same multiplicity. One of the results that we prove in the global case is the following: if$X\subset \mathbb {C}^{n}$,$Y\subset \mathbb {C}^{m}$are algebraic sets and$\phi \colon X\to Y$is a semi-algebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in$\mathbb {P}^{n+m}(\mathbb {C})$is an orientable homological cycle, then${\rm deg}(X)={\rm deg}(Y)$.

Comments on Charges on the Boolean Algebra of Regular Open Sets

This paper has three main results. The first result is an extension theorem for charges on the Boolean algebra of regular open sets of a topological space. The second result is an abstract form of the first result, which we call an abstract extension theorem for charges on Boolean algebras. The third result is a characterization of regular open sets in $R$. The first two results are useful for integration of functions on topological spaces with respect to charges on Boolean algebras of regular open sets. The third result is useful in proving that the Lebesgue measure is not a charge when restricted to the Boolean algebra of regular open sets (see [3])

Measure and Integration on Boolean Algebras of Regular Open Subsets in a Topological Space

The regular open subsets of a topological space form a Boolean algebra, where the join of two regular open sets is the interior of the closure of their union. A content is a finitely additive measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such contents. We then explain the relationship between contents, residual charges, and Borel measures. We show that a content can be represented by a normal Borel measure, augmented with a liminal structure, which specifies how two or more regular open sets share the measure of their common boundary. In particular, a content on a locally compact Hausdorff space can be represented by a normal Borel measure and a liminal structure on the Stone-Čech compactification of that space. We also show how contents can be represented by Borel measures on the Stone space of the underlying Boolean algebra of regular open sets.

Towards signature-based gröbner basis algorithms for computing the nondegenerate locus of a polynomial system

Problem statement. Let K be a field and K be an algebraic closure of K. Consider the polynomial ring R = K[ x 1 ,..., x n ] over K and a finite sequence of polynomials f 1 ,..., f c in R with c ≤ n. Let V ⊂ K n be the algebraic set defined by the simultaneous vanishing of the f i 's. Recall that V can be decomposed into finitely many irreducible components, whose codimension cannot be greater than c. The set V c which is the union of all these irreducible components of codimension exactly c is named further the nondegenerate locus of f 1 ,..., f c .