Structure Theorem Research Articles

A subclass of the Cowen–Douglas class and similarity

AbstractWe consider a subclass of the Cowen–Douglas class in which the problem of deciding whether two operators are similar becomes more manageable. A similarity criterion for Cowen–Douglas operators is known to be dependent on the trace of the curvature of the corresponding eigenvector bundles. Unless the given eignvector bundle is a line bundle, the computation of the curvature, in general, is not so simple as one might hope. By using a structure theorem on Cowen–Douglas operators, we reduce the problem of finding the trace of the curvature by looking at the curvatures of the associated line bundles. Several questions related to the similarity problem are also taken into account.

ON A TOPOLOGY DEFINED BY PRIMITIVE WORDS

Let f:X⟶X be a mapping. Consider P(f)={O⊆X:f-1(O)⊆O}. Then P(f) is an Alexandroff topology. A topological space X is called a primal space if its topology coincides with a P(f) for some mapping f:X⟶X. Let A be an alphabet and A* be the set of all finite words over A. A word is called primitive if it is not empty and not a proper power of another word. Let u be a nonempty word; then there exists a unique primitive word z and a unique integer k≥1 such that u=zk; z is called the primitive root of u; we denote by z=pA(u). It is convenient to set pA(εA)=εA, where εA is the empty word over A. By a primitive primal space we mean a space X that is homeomorphic to the subspace A+ of A* equipped with the topology P(pA) for some alphabet A. Our main result provides a structure theorem of primitive primal spaces.

One-Dimensional Dyadic Wavelets

One-Dimensional Dyadic Wavelets

Large deviations for the largest eigenvalue of Gaussian networks with constant average degree

Large deviation behavior of the largest eigenvalue \(\lambda _1\) of Wigner matrices including those arising from an Erdős-Rényi random graph \({\mathcal {G}}_{n,p}\) with i.i.d. random conductances on the edges has been the topic of considerable interest. However, despite several recent advances, not much is known when the underlying graph is sparse i.e., \(p\rightarrow 0\), except the recent works (Bhattacharya et al., Ann Probab 49(4):1847–1885, 2021and Bhattacharya and Ganguly, SIAM J Discret Math, 2020) which consider the simpler case of the graph without additional edge weights. Under sufficiently general conditions on the conductance distribution, one expects the ‘dense’ behavior as long as the average degree np is at least logarithmic in n. In this article we focus on the case of constant average degree i.e., \(p=\frac{d}{n}\) for some fixed \(d>0\) with standard Gaussian weights. Results in Bandeira and Van Handel (Ann Probab 44(4):2479–2506, 2016) about general non-homogeneous Gaussian matrices imply that in this regime \(\lambda _1\) scales like \(\sqrt{\log n}.\) We prove the following results towards a precise understanding of the large deviation behavior in this setting. 1. (Upper tail probabilities and structure theorem): For \(\delta >0,\) we pin down the exact exponent \(\psi (\delta )\) such that $$\begin{aligned} {\mathbb {P}}(\lambda _1\ge \sqrt{2(1+\delta )\log n})=n^{-\psi (\delta )+o(1)}. \end{aligned}$$ Further, we show that conditioned on the upper tail event, with high probability, a unique maximal clique emerges with a very precise \(\delta \) dependent size (takes either one or two possible values) and the Gaussian weights are uniformly high in absolute value on the edges in the clique. Finally, we also prove an optimal localization result for the leading eigenvector, showing that it allocates most of its mass on the aforementioned clique which is spread uniformly across its vertices. 2. (Lower tail probabilities): The exact stretched exponential behavior $$\begin{aligned} {\mathbb {P}}(\lambda _1\le \sqrt{2(1-\delta )\log n})=\exp \left( -n^{\ell (\delta )+o(1)}\right) \end{aligned}$$ is also established. As an immediate corollary, one obtains that \(\lambda _1\) is typically \((1+o(1))\sqrt{2\log n}\), a result which surprisingly appears to be new. A key ingredient in our proofs is an extremal spectral theory for weighted graphs obtained by an \(\ell _1-\)reduction of the standard \(\ell _2-\)variational formulation of the largest eigenvalue via the classical Motzkin-Straus theorem [37], which could be of independent interest.

Normal subgroups of big mapping class groups

Let S S be a surface and let Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) be the mapping class group of S S permuting a Cantor subset K ⊂ S K \subset S . We prove two structure theorems for normal subgroups of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) . (Purity:) if S S has finite type, every normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) either contains the kernel of the forgetful map to the mapping class group of S S , or it is ‘pure’ — i.e. it fixes the Cantor set pointwise. (Inertia:) for any n n element subset Q Q of the Cantor set, there is a forgetful map from the pure subgroup PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) to the mapping class group of ( S , Q ) (S,Q) fixing Q Q pointwise. If N N is a normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) contained in PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) , its image N Q N_Q is likewise normal. We characterize exactly which finite-type normal subgroups N Q N_Q arise this way. Several applications and numerous examples are also given.

Formal groupoids

We provide foundations for a characteristic free study of foliated varieties in terms of infinitesimal actions of formal groupoids. The ultimate goal is the bi-rational geometry of the same, and to this end we prove a cone theorem for foliations in curves, together with structure theorems for extremal rays, and, of course, a minimal model theorem for surfaces. All possible wild ramification effects of Deligne–Mumford champ are built in, along with the occasional use of Artin champ to address the mathbb Q-Gorenstein condition.

Kurosh theorem for certain Koszul Lie algebras

A. Kurosh's theorem for groups [4] provides the structure of any subgroup of a free product of groups and its proof relies on Bass-Serre theory of groups acting on trees. In the case of Lie algebras, such a general theory does not exist and the analogue of Kurosh theorem is false in general, as it was first noticed by A.I. Shirshov in [10]. However, we prove that, for a class of positively graded Lie algebras satisfying certain local properties in cohomology, such a structure theorem holds true for subalgebras generated by elements of degree 1. Such class consists of Koszul Lie algebras, in which all the subalgebras that are generated in degree 1 are Koszul.

Improved bound for improper colourings of graphs with no odd clique minor

AbstractStrengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$ -minor is properly $(t-1)$ -colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$ -minor admits a vertex $(2t-2)$ -colouring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$ . The bound on the number of colours is optimal up to a factor of $2$ , improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput.17 815–821), Kang and Oum (2019, Combin. Probab. Comput.28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc.98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$ -minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$ -minor-free graphs with some additional ideas, which make the extension to odd $K_t$ -minor-free graphs possible.

Mukai pairs and simple K-equivalence

A K-equivalent map between two smooth projective varieties is called simple if the map is resolved in both sides by single smooth blow-ups. In this paper, we will provide a structure theorem of simple K-equivalent maps, which reduces the study of such maps to that of special Fano manifolds. As applications of the structure theorem, we provide examples of simple K-equivalent maps, and classify such maps in several cases, including the case of dimension at most 8.

Topological horseshoes for surface homeomorphisms

In this work, we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of homeomorphisms of surfaces with zero genus and null topological entropy, and we obtain several applications. For homeomorphisms of the open annulus A with zero topological entropy, we show that rotation numbers exist for all points with nonempty ω-limit sets, and that if A is a generalized region of instability, then it admits a single rotation number. We also offer a new proof of a recent result of Passeggi, Potrie, and Sambarino, showing that zero entropy dissipative homeomorphisms of the annulus having a circloid as an attractor have a single rotation number. Our work also studies homeomorphisms of the sphere without horseshoes. For these maps, we present a structure theorem in terms of fixed point free invariant subannuli, as well as a very restricted description of all possible dynamical behavior in the transitive subsets. This description ensures, for instance, that transitive sets can contain at most two distinct periodic orbits and that, in many cases, the restriction of the homeomorphism to the transitive set must be an extension of an odometer. In particular, we show that any nontrivial and stable transitive subset of a dissipative diffeomorphism of the plane is always infinitely renormalizable in the sense of Bonatti et al.

Critical independent sets of König–Egerváry graphs

Let G be a simple graph with vertex set V ( G ) and edge set E ( G ) . For x ∈ V ( G ) , let N G ( x ) = { y ∈ V ( G ) | x y ∈ E ( G ) } and N G ( X ) = ⋃ x ∈ X N G ( x ) . For X ⊆ V ( G ) , the number d ( X ) ≔ | X | − | N G ( X ) | is called difference of X and critical difference of G denoted by d c ( G ) is max { d ( X ) ∣ X ⊆ V ( G ) } . A set S ⊆ V ( G ) is a critical independent set if S is an independent set and d ( S ) = d c ( G ) . Zhang (1990) proved that for every graph, there exists an independent set attaining the critical difference. Let Λ 0 ( G ) denote the family of all nonempty critical independent sets of G . If α ( G ) + μ ( G ) = | V ( G ) | , then G is called a König–Egerváry graph, where α ( G ) and μ ( G ) denote independence number and matching number of G , respectively. We define k e r 0 ( G ) to be the intersection of all sets in Λ 0 ( G ) and c o r e ( G ) to be the intersection of all maximum independent sets. In this paper, we characterize König–Egerváry graphs satisfying k e r 0 ( G ) = c o r e ( G ) in terms of Gallai–Edmonds Structure Theorem. As a corollary of this result, we solve an open problem posed by Jarden et al. (2018).

Q-Rational reduction and q-analogues of series for π

In this paper, we present a q-analogue of the polynomial reduction which was originally developed for hypergeometric terms. Using the q-Gosper representation, we describe the structure of rational functions that are summable when multiplied with a given q-hypergeometric term. The structure theorem enables us to generalize the q-polynomial reduction to the rational case, which can be used in the automatic proof and discovery of q-identities. As applications, several q-analogues of series for π are presented.

The sum of two quadratic matrices: Exceptional cases

Let p and q be polynomials with degree 2 over an arbitrary field F. A square matrix with entries in F is called a (p,q)-sum when it can be split into A+B for some pair (A,B) of square matrices such that p(A)=0 and q(B)=0.A (p,q)-sum is called regular when none of its eigenvalues is the sum of a root of p and of a root of q. A (p,q)-sum is called exceptional when each one of its eigenvalues is the sum of a root of p and of a root of q. In a previous work [7], we have shown that the study of (p,q)-sums can be entirely reduced to the one of regular (p,q)-sums and to the one of exceptional (p,q)-sums. Moreover, regular (p,q)-sums have been characterized thanks to structural theorems on quaternion algebras, giving the problem a completely unified treatment.The present article completes the study of (p,q)-sums by characterizing the exceptional ones. The new results here deal with the case where at least one of the polynomials p and q is irreducible over F.

Decomposing and colouring some locally semicomplete digraphs

A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various subclasses of locally semicomplete digraphs for which we give structural decomposition theorems. As a consequence we obtain several applications, among which an answer to a conjecture of Naserasr and the first and third authors (Aboulker et al., 2021): if an oriented graph is such that the out-neighbourhood of every vertex induces a transitive tournament, then one can partition its vertex set into two acyclic digraphs.

Fundamental Properties of Basic Slc-Trivial Fibrations I

We introduce the notion of basic slc-trivial fibrations. It is a generalization of Ambro’s lc-trivial fibrations. Then we study fundamental properties of basic slc-trivial fibrations by using the theory of variations of mixed Hodge structure on cohomology with compact support. More precisely, we prove that the moduli part of a basic slc-trivial fibration is b-potentially nef. Note that the notion of basic slc-trivial fibrations is closely related to that of normal irreducible quasi-log canonical pairs. So the results obtained in this paper will play an important role in the theory of quasi-log schemes. Here we give a structure theorem for normal irreducible quasi-log canonical pairs as an application of the main theorem. This result makes the theory of quasi-log schemes more powerful and more flexible.

The behavior and classification of solutions bounded from below to degenerate elliptic equations in unbounded cylinders

In this paper, we consider the behavior of solutions bounded from below to degenerate elliptic equations with lower order terms in unbounded cylinders which vanish at the boundary. After establishing the weak maximum principle, the weak Harnack inequality for supersolutions, the boundary Hölder continuity and the boundary Harnack principles under suitable assumptions regarding the lower order terms of the operator, we obtain structure theorems in unbounded cylinders.

Extremal functions for sparse minors

The notion of a graph minor, which generalizes graph subgraphs, is a central notion of modern graph theory. Classical results concerning graph minors include the Graph Minor Theorem and the Graph Structure Theorem, both due to Robertson and Seymour. The results concern properties of classes of graphs closed under taking minors; such graph classes include many important natural classes of graphs, e.g., the class of planar graphs and, more generally, the class of graphs embeddable in a fixed surface. The Graph Minor Theorem asserts that every class of graphs closed under taking minors has a finite list of forbidden minors. For example, Wagner’s Theorem, which claims that a graph is planar if and only if it does not contain or as a minor, is a particular case of this theorem. The Graph Structure Theorem asserts that graphs from a fixed class of graphs closed under taking minors can be decomposed in a tree-like fashion into graphs almost embeddable in a fixed surface. In particular, every graph in a class of graphs avoiding a fixed minor admits strongly sublinear separators (the Planar separator theorem of Lipton and Tarjan is a special case of this more general result). As the number of edges of every graph contained in a class of graphs closed under taking minors is linear in the number of its vertices, one can define to be the maximum possible density of a graph that does not contain a graph as a minor. This quantity has been a subject of very intensive research; for example, a long list of bounds concerning culminated with a result of Thomason in 2001, who precisely determined its asymptotic behavior. This paper provides bounds on when itself is from a class of sparse graphs. In particular, the authors prove an asymptotically tight bound on in terms of the number of vertices of and the ratio of the vertex cover and the number of vertices of graphs contained in a class of graphs with strongly sublinear separators.

Rectifiability of divergence-free fields along invariant 2-tori

We find conditions under which the restriction of a divergence-free vector field B to an invariant toroidal surface S is rectifiable; namely constant in a suitable global coordinate system. The main results are similar in conclusion to Arnold’s Structure Theorems but require weaker assumptions than the commutation [B,nabla times B] = 0. Relaxing the need for a first integral of B (also known as a flux function), we assume the existence of a solution u: S rightarrow {mathbb {R}} to the cohomological equation Bvert _S(u) = partial _n B on a toroidal surface S mutually invariant to B and nabla times B. The right hand side partial _n B is a normal surface derivative available to vector fields tangent to S. In this situation, we show that the field B on S is either identically zero or nowhere zero with Bvert _S/Vert BVert ^2 vert _S being rectifiable. We are calling the latter the semi-rectifiability of B (with proportionality Vert BVert ^2 vert _S). The nowhere zero property relies on Bers’ results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that Bvert _S itself is rectifiable. The rectifiability and semi-rectifiability of Bvert _S is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.

A short note on Manin–Mumford

We give a short proof of Manin–Mumford in the multiplicative group based on the pigeon-hole principle and a simple structure theorem for tori contained in subvarieties of [Formula: see text] due to Bombieri and Gubler. The arguments appear to be new and are applicable in other situations. Our proof is quite elementary and we compare it to previous proofs that rely on machinery such as intersection theory or [Formula: see text]-minimality.

On abelian points of varieties intersecting subgroups in a torus

We show, under some natural conditions, that the set of abelian points on the non-anomalous dense subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $\dim X$ in a torus is finite, generalising results of Ostafe, Sha, Shparlinski and Zannier (2017). We also generalise their structure theorem for such sets when the algebraic subgroups are not necessarily connected, and obtain a related result in the context of curves and arithmetic dynamics.