Free Submodules Research Articles

Nonconvex submodule clustering via joint sliced sparse gradient and cluster-aware approach

Most existing subspace clustering methods preprocess image data by converting them into vectors, which lacks exploration of the spatial structure of high-dimensional data. Therefore, we proposes a nonconvex submodule clustering model (NSSGCA) via joint sliced sparse gradient and cluster-aware approach. NSSGCA arranges each 2D image as lateral slices of a 3rd-order tensor, and utilizes the t-product under the model of the union of free submodules to represent 3rd-order tensor samples, thereby exploring the latent spatial structure of samples. To more accurately approximate tensor rank, a nonconvex Schatten p-norm constraint is imposed on the rotated representation tensor. Under the submodule framework, a consistent gradient matrix is derived based on the δ-nearest neighbor adjacency graph to construct sliced sparse gradient (SSG) regularization, which is more conducive to clustering tasks. NSSGCA learns representation tensor with clearer block-like structure based on ℓq norm and cluster-aware attention mechanism. The convergence of the constructed sequence to the stationary Karush–Kuhn–Tucker (KKT) point is proven. Experimental results on real-world image datasets confirm the effectiveness of NSSGCA.

Equivariant K-theory of the semi-infinite flag manifold as a nil-DAHA module

The equivariant K-theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and a q-Heisenberg algebra. The action of the latter generates a free submodule of rank |W|, where W is the (finite) Weyl group. We show that this submodule is stable under the nil-DAHA, which enables one to express the nil-DAHA action in terms of |W|times |W| matrices over the q-Heisenberg algebra. Our main result gives an explicit algebraic construction of these matrices as a limit from the (non-nil) DAHA in simply-laced type. This construction reveals that multiplication by equivariant scalars, when expressed in terms of the Heisenberg algebra, is given by the nonsymmetric q-Toda system introduced by Cherednik and the author.

Linear systems of Diophantine equations

Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give a procedure to construct a basis of $L$ from a basis of $M$ assuming the invariant factors or elementary divisors of $L/M$ are known. Given a matrix $A\in M_{m,n}(R)$ of rank $r$, its nullspace $L$ in $R^n$ is a free $R$-module of rank $f=n-r$. We construct a free submodule $M$ of $L$ of rank $f$ naturally associated with $A$ and whose basis is easily computable, we determine the invariant factors of the quotient module $L/M$ and then indicate how to apply the previous procedure to build a basis of $L$ from one of $M$.

Grassmannian Optimization for Online Tensor Completion and Tracking With the t-SVD

We propose a new fast streaming algorithm for the tensor completion problem of imputing missing entries of a low-tubal-rank tensor using the tensor singular value decomposition (t-SVD) algebraic framework. We show the t-SVD is a specialization of the well-studied block-term decomposition for third-order tensors, and we present an algorithm under this model that can track changing free submodules from incomplete streaming 2-D data. The proposed algorithm uses principles from incremental gradient descent on the Grassmann manifold of subspaces to solve the tensor completion problem with linear complexity and constant memory in the number of time samples. We provide a local expected linear convergence result for our algorithm. Our empirical results are competitive in accuracy but much faster in compute time than state-of-the-art tensor completion algorithms on real applications to recover temporal chemo-sensing and MRI data under limited sampling.

How far is an extension of p-adic fields from having a normal integral basis?

Let L/K be a finite Galois extension of p-adic fields with group G. It is well-known that OL contains a free OK[G]-submodule of finite index. We study the minimal index of such a free submodule, and determine it exactly in several cases, including for any cyclic extension of degree p of p-adic fields.

Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings

An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex X features a new boundary operator and is formulated over a discrete valuation ring, R. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, Hi(X), and weighted homology, Hi,R(X), in two ways: first, via chain maps, and second, via the relative homology. We compute H0,R(X) by means of a recursive contraction procedure on a weighted spanning tree and H1,R(X) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H1,R(X). The homology module H2,R(X) is naturally obtained from H2(X) via chain maps. Furthermore, we show that all weighted homology modules Hi,R(X) are trivial for i>2. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.

Nonisotropic symplectic graphs over finite commutative rings

<abstract><p>In this paper, we study two types of nonisotropic symplectic graphs over finite commutative rings defined by nonisotropic free submodules of rank $ 2 $ and McCoy rank of matrices. We prove that the graphs are quasi-strongly regular or Deza graphs and we find their parameters. The diameter and vertex transitivity are also analyzed. Moreover, we study subconstituents of these nonisotropic symplectic graphs.</p></abstract>

The B-model connection and mirror symmetry for Grassmannians

We consider the Grassmannian X=Grn−k(Cn) and describe a ‘mirror dual’ Landau-Ginzburg model (Xˇ∘,Wq:Xˇ∘→C), where Xˇ∘ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian Xˇ, and we express W succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to one proposed for homogeneous spaces in a previous work by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990's by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to Wq a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of X. We also prove a T-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology D-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's J-function, which was conjectured in a 1998 paper by Batyrev, Ciocan-Fontanine, Kim and van Straten.

Cluster algebras arising from cluster tubes II: The Caldero-Chapoton map

We continue our investigation on cluster algebras arising from cluster tubes. Let C be a cluster tube of rank n+1. For an arbitrary basic maximal rigid object T of C, one may associate a skew-symmetrizable integer matrix BT and hence a cluster algebra A(BT) to T. We define an analogue Caldero-Chapoton map XMT for each indecomposable rigid object M∈C and prove that X?T yields a bijection between the indecomposable rigid objects of C and the cluster variables of the cluster algebra A(BT). The construction of the Caldero-Chapoton map involves Grassmannians of locally free submodules over the endomorphism algebra of T. We also show that there is a non-trivial C×-action on the Grassmannians of locally free submodules, which is of independent interest.

A Remark on Structure of Projective Klingenberg Spaces over a Certain Local Algebra

This article is devoted to the projective Klingenberg spaces over a local ring, which is a linear algebra generated by one nilpotent element. In this case, subspaces of such Klingenberg spaces are described. The notion of the “degree of neighborhood” is introduced. Using this, we present the geometric description of subsets of points of a projective Klingenberg space whose arithmetical representatives need not belong to a free submodule.

Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

In this paper, it is shown that there exists a particular associative ring with unity of order 16 such that the relations between non-unimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.

Conditions for the existence of spreads in projective Hjelmslev spaces

Let R be a finite chain ring with $$|R|=q^m$$ , and $$R/\text {Rad }R\cong \mathbb {F}_q$$ . Denote by $$\varPi ={{\mathrm{PHG}}}({}_RR^n)$$ the (left) $$(n-1)$$ -dimensional projective Hjelmslev geometry over R. As in the classical case, we define a $$\lambda $$ -spread of $$\varPi $$ to be a partition of its pointset into subspaces of shape $$\lambda =(\lambda _1,\ldots ,\lambda _n)$$ . An obvious necessary condition for the existence of a $$\lambda $$ -spread $$\mathcal {S}$$ in $$\varPi $$ is that the number of points in a subspace of shape $$\lambda $$ divides the number of points in $$\varPi $$ . If the elements of $$\mathcal {S}$$ are Hjelmslev subspaces, i.e., free submodules of $${}_RR^n$$ , this necessary condition is also sufficient. If the subspaces in $$\mathcal {S}$$ are not Hjelmslev subspaces this numerical condition is not sufficient anymore. For instance, for chain rings with $$m=2$$ , there is no spread of shape $$\lambda =(2,2,1,0)$$ in $${{\mathrm{PHG}}}({}_RR^4)$$ . An important (and maybe difficult) question is to find all shapes $$\lambda $$ , for which $$\varPi $$ has a $$\lambda $$ -spread. In this paper, we present a construction which gives spreads by subspaces that are not necessarily Hjelmslev subspaces. We prove the non-existence of spreads of shape $$2^{n/2}1^a$$ [cf. (2)], $$1\le a\le n/2-1$$ , in $${{\mathrm{PHG}}}({}_RR^n)$$ , where n is even and R is a chain ring of length 2.

A Low Tensor-Rank Representation Approach for Clustering of Imaging Data

This letter proposes an algorithm for clustering of two-dimensional data. Instead of “flattening” data into vectors, the proposed algorithm keeps samples as matrices and stores them as lateral slices in a third-order tensor. It is then assumed that the samples lie near a union of free submodules and their representations under this model are obtained by imposing a low tensor-rank constraint and a structural constraint on the representation tensor. Clustering is carried out using an affinity matrix calculated from the representation tensor. Effectiveness of the proposed algorithm and its superiority over existing methods are demonstrated through experiments on two image datasets.

Generalized symplectic graphs and generalized orthogonal graphs over finite commutative rings

ABSTRACTLet R be a finite commutative ring with identity, and β a bilinear form on . In this work, we count the numbers of free submodules and totally isotropic free submodules of of rank s by using the lifting idea. We define the graph whose vertex set is the set of totally isotropic free submodules of of rank s called the generalized bilinear form graph. We study this graph when is a symplectic space and an orthogonal space. We can determine the degree of each vertex of these graphs. If R is a finite local ring, we show that these graphs are arc transitive and obtain their automorphism groups. We complete our work by proving that we can decompose the graphs over a finite commutative ring into the tensor products of graphs over finite local rings.

Realization of Nonlinear Time-Delay Input–Output Equations

In this letter the problem of transforming a nonlinear multi-input multi-output time-delay control system, described by a set of higher order functional differential equations relating system inputs and outputs, into a set of first order functional differential equations, is studied. The globally linearized system equations are described in terms of polynomials in a delay operator and derivative operator. The Euclidean left division of polynomials is used to compute the basis of a certain free submodule of differential 1-forms. The integrability of the submodule is proved to be necessary and sufficient for the solvability of the problem.

Quivers with Relations for Symmetrizable Cartan Matrices II: Change of Symmetrizers

For $k \ge 1$ we consider the $K$-algebra $H(k) := H(C,kD,\Omega)$ associated to a symmetrizable Cartan matrix $C$, a symmetrizer $D$, and an orientation $\Omega$ of $C$, which was defined in Part 1. We construct and analyse a reduction functor from rep$(H(k))$ to rep$(H(k-1))$. As a consequence we show that the canonical decomposition of rank vectors for $H(k)$ does not depend on $k$, and that the rigid locally free $H(k)$-modules are up to isomorphism in bijection with the rigid locally free $H(k-1)$-modules. Finally, we show that for a rigid locally free $H(k)$-module of a given rank vector the Euler characteristic of the variety of flags of locally free submodules with fixed ranks of the subfactors does not depend on the choice of $k$.

Basic Operations on Supertropical Quadratic Forms

In the case that a module V over a (commutative) supertropical semiring R is free, the R -module Quad( V ) of all quadratic forms on V is almost never a free module. Nevertheless, Quad( V ) has two free submodules, the module QL( V ) of quasilinear forms with base \frak{D}_0 and the module Rig( V ) of rigid forms with base \frak{H}_0 , such that Quad( V )=QL( V )+Rig( V ) and QL( V ) \cap Rig( V ) = \{0\} . In this paper we study endomorphisms of Quad( V ) for which each submodule Rq with q \in \frak{D}_0\cup\frak{H}_0 is invariant; these basic endomorphisms are determined by coefficients in R and do not depend on the base of V . We aim for a description of all basic endomorphisms of Quad( V ), or more generally of its submodules spanned by subsets of \frak{D}_0\cup\frak{H}_0 . But, due to complexity issues, this naive goal is highly nontrivial for an arbitrary supertropical semiring R . Our main stress is therefore on results valid under only mild conditions on R , while a complete solution is provided for the case that R is a tangible supersemifield.

Extensors and the Hilbert scheme

The Hilbert scheme $Hilb_{p(t)}^n$ parametrizes closed subschemes and families of closed subschemes in the projective space $P^n$ with a fixed Hilbert polynomial $p(t)$. It is classically realized as a closed subscheme of a Grassmannian or a product of Grassmannians. In this paper we present a method that allows to derive scheme theoretical global equations for $Hilb_{p(t)}^n$ in the Plucker coordinates of a Grassmannian $Grass_p^N$, where $p$ and $N$ depend on the dimension $n$ of the projective space and on the Hilbert polynomial $p(t)$. Using this method we obtain the already known set of equations given by Iarrobino and Kleiman in 1999, the one conjectured by Bayer in 1982 and proved by Haiman and Sturmfels in 2004, and also a new set of equations of degree lower than the previous ones. The novelties of our approach are essentially two. The first one is a local study of the Hilbert functor through special sets of open subfunctors obtained exploiting the symmetries of the Hilbert scheme and the combinatorial properties of monomial ideals, mainly the Borel-fixed ones. The second one is a generalization of the theory of extensors to the setting of free modules over any ring $A$ and the description of any exterior product of elements of a free submodule in terms of Plucker coordinates.

Generalized splines on arbitrary graphs

Let G be a graph whose edges are labeled by ideals of a commutative ring. We introduce a generalized spline, which is a vertex-labeling of G by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the corresponding edge ideal. Generalized splines arise naturally in combinatorics (em algebraic splines of Billera and others) and in algebraic topology (certain equivariant cohomology rings, described by Goresky-Kottwitz-MacPherson and others). The central question of this manuscript asks when an arbitrary edge-labeled graph has nontrivial generalized splines. The answer is `always', and we prove the stronger result that generalized splines contain a free submodule whose rank is the number of vertices in G. We describe all generalized splines when G is a tree, and give several ways to describe the ring of generalized splines as an intersection of generalized splines for simpler subgraphs of G. We also present a new tool which we call the GKM matrix, an analogue of the incidence matrix of a graph, and end with open questions.

Free Cyclic Submodules in the Context of the Projective Line

We discuss the free cyclic submodules over an associative ring R with unity. Special attention is paid to those which are generated by outliers. This paper describes all orbits of such submodules in the ring of lower triangular 3 × 3 matrices over a field F under the action of the general linear group. Besides rings with outliers generating free cyclic submodules, there are also rings with outliers generating only torsion cyclic submodules and without any outliers. We give examples of all cases.