Module Of Rank Research Articles

Projective Modules and Efficient Generation of Ideals Over Laurent Polynomial Rings

Let A be commutative Noetherian ring of dimension d. In this paper we show that every finitely generated projective \(A[X_1, X_2, \ldots , X_r]\)-module of constant rank n is generated by \(n+d\) elements. We also extend some results over the Laurent polynomial ring \(A[X,X^{-1}]\), which are true for polynomial rings.

Euler class group of certain overrings of a polynomial ring

Let $A$ be a commutative Noetherian ring of dimension~$n$ and $P$ a projective $A$-module of rank~$n$ with trivial determinant. In \cite {BR1}, Bhatwadekar and Sridharan defined the $n$th Euler class group of~$A$ and studied the obstruction to the existence of unimodular element in~$P$. For $R=A[T]$ and $R=A[T,T^{-1}]$, the $n$th Euler class groups of $R$ are defined by Das and Keshari in \cite {mkd1, MK1}, under certain assumption on $A$ in the latter case. We define the $n$th Euler class group of the ring $R=A[T,1/f(T)]$, where $f(T)\in A[T]$ is a monic polynomial and the height of the Jacobson radical of $A$ is $\geq 2$. Also, we prove results similar to those in \cite {MK1}.

Module structures on $\bm{U(\h)}$ for Kac-Moody algebras

Let $\g$ be an arbitrary Kac-Moody algebra with a Cartan subalgebra $\h$. In this paper, we determine the category of $\g$-modules that are free $U(\h)$-modules of rank 1. More precisely, this category of $\g$-modules is not empty if and only if $\g$ is of type $A_l$ or $C_l$ for any positive integer $l$.

Artin Groups and Yokonuma–Hecke Algebras

We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W. When W is finite, we prove that this algebra is a free module of finite rank which is generically semisimple. When W is the Weyl group of a Chevalley group, C_W naturally maps to the associated Yokonuma-Hecke algebra. When W = S_n this algebra can be identified with a diagram algebra called the algebra of `braids and ties'. The image of the usual braid group in this case is investigated. Finally, we generalize our construction to finite complex reflection groups, thus extending the Broue-Malle-Rouquier construction of a generalized Hecke algebra attached to these groups.

On the Auslander–Reiten conjecture for Cohen–Macaulay local rings

This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov–Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings.

Patch Reordering: A NovelWay to Achieve Rotation and Translation Invariance in Convolutional Neural Networks

Convolutional Neural Networks (CNNs) have demonstrated state-of-the-art performance on many visual recognition tasks. However, the combination of convolution and pooling operations only shows invariance to small local location changes in meaningful objects in input. Sometimes, such networks are trained using data augmentation to encode this invariance into the parameters, which restricts the capacity of the model to learn the content of these objects. A more efficient use of the parameter budget is to encode rotation or translation invariance into the model architecture, which relieves the model from the need to learn them. To enable the model to focus on learning the content of objects other than their locations, we propose to conduct patch ranking of the feature maps before feeding them into the next layer. When patch ranking is combined with convolution and pooling operations, we obtain consistent representations despite the location of meaningful objects in input. We show that the patch ranking module improves the performance of the CNN on many benchmark tasks, including MNIST digit recognition, large-scale image recognition, and image retrieval.

Loop Virasoro Lie conformal superalgebra

The loop Virasoro conformal superalgebra associated with the loop super-Virasoro algebra is constructed in the present paper. The conformal superderivation algebra of is completely determined, which is shown to consist of inner superderivations. And nontrivial free and free -graded -modules of rank two are classified. We also give a classification of irreducible free -modules of rank two and all irreducible submodules of each free -graded -module of rank two.

Existence of unimodular elements in a projective module

(1) Let R be an affine algebra over an algebraically closed field of characteristic 0 with dim(R)=n. Let P be a projective A=R[T1,⋯,Tk]-module of rank n with determinant L. Suppose I is an ideal of A of height n such that there are two surjections α:P↠I and ϕ:L⊕An−1↠I. Assume that either (a) k=1 and n≥3 or (b) k is arbitrary but n≥4 is even. Then P has a unimodular element (see 4.1, 4.3).(2) Let R be a ring containing Q of even dimension n with height of the Jacobson radical of R≥2. Let P be a projective R[T,T−1]-module of rank n with trivial determinant. Assume that there exists a surjection α:P↠I, where I⊂R[T,T−1] is an ideal of height n such that I is generated by n elements. Then P has a unimodular element (see 3.4).

Validation of Immune Cell Modules in Multicellular Transcriptomic Data

Numerous gene signatures, or modules have been described to evaluate the immune cell composition in transcriptomes of multicellular tissue samples. However, significant diversity in module gene content for specific cell types is associated with heterogeneity in their performance. In order to rank modules that best reflect their purported association, we have generated the modular discrimination index (MDI) score that assesses expression of each module in the target cell type relative to other cells. We demonstrate that MDI scores predict modules that best reflect independently validated differences in cellular composition, and correlate with the covariance between cell numbers and module expression in human blood and tissue samples. Our analyses demonstrate that MDI scores provide an ordinal summary statistic that reliably ranks the accuracy of gene expression modules for deconvolution of cell type abundance in transcriptional data.

Twisting eigensystems of Drinfeld Hecke eigenforms by characters

We address some questions posed by Goss related to the modularity of Drinfeld modules of rank 1 defined over the field of rational functions in one variable with coefficients in a finite field.

New bounds of permutation codes under Hamming metric and Kendall’s $$\tau $$ τ -metric

Permutation codes are widely studied objects due to their numerous applications in various areas, such as power line communications, block ciphers, and the rank modulation scheme for flash memories. Several kinds of metrics are considered for permutation codes according to their specific applications. This paper concerns some improvements on the bounds of permutation codes under Hamming metric and Kendall’s \(\tau \)-metric respectively, using mainly a graph coloring approach. Specifically, under Hamming metric, we improve the Gilbert–Varshamov bound asymptotically by a factor n, when the minimum Hamming distance d is fixed and the code length n goes to infinity. Under Kendall’s \(\tau \)-metric, we narrow the gap between the known lower bounds and upper bounds. Besides, we also obtain some sporadic results under Kendall’s \(\tau \)-metric for small parameters.

Perfect Snake-in-the-Box Codes for Rank Modulation

For odd $n$ , the alternating group on $n$ elements is generated by the permutations that jump an element from an odd position to position 1. We prove Hamiltonicity of the associated directed Cayley graph for all odd $n\neq 5$ (a result of Rankin implies that the graph is not Hamiltonian for $n=5$ ). This solves a problem arising in rank modulation schemes for flash memory. Our result disproves a conjecture of Horovitz and Etzion, and proves another conjecture of Yehezkeally and Schwartz.

Modules over the algebra [formula omitted

For any two complex numbers a and b, Vir(a,b) is a central extension of W(a,b) which is universal in the case (a,b)≠(0,1), where W(a,b) is the Lie algebra with basis {Ln,Wn|n∈Z} and relations [Lm,Ln]=(n−m)Lm+n, [Lm,Wn]=(a+n+bm)Wm+n, [Wm,Wn]=0. In this paper, we construct and classify a class of non-weight modules over the algebra Vir(a,b) which are free U(CL0⊕CW0)-modules of rank 1. It is proved that such modules can only exist for a∈Z.

Serre dimension of monoid algebras

Let R be a commutative Noetherian ring of dimension d, M a commutative cancellative torsion-free monoid of rank r and P a finitely generated projective R[M]-module of rank t. Assume M is Φ-simplicial seminormal. If $M\in \mathcal {C}({\Phi })$ , then Serre dim R[M]≤d. If r≤3, then Serre dim R[int(M)]≤d. If $M\subset \mathbb {Z}_{+}^{2}$ is a normal monoid of rank 2, then Serre dim R[M]≤d. Assume M is c-divisible, d=1 and t≥3. Then P≅∧ t P⊕R[M] t−1. Assume R is a uni-branched affine algebra over an algebraically closed field and d=1. Then P≅∧ t P⊕R[M] t−1.

Simple Jordan superalgebras with associative nil-semisimple even part

Under study are the simple infinite-dimensional abelian Jordan superalgebras not isomorphic to the superalgebra of a bilinear form. We prove that the even part of such superalgebra is a differentially simple associative commutative algebra, and the odd part is a finitely generated projective module of rank 1. We describe unital simple Jordan superalgebras with associative nil-semisimple even part possessing two even elements which induce a nonzero derivation.

Construction of simple non-weight [formula omitted]-modules of arbitrary rank

We study simple non-weight sl(2)-modules which are finitely generated as C[z]-modules. We show that they are described in terms of semilinear endomorphisms and prove that the Smith type induces a stratification on the set of these sl(2)-modules, providing thus new invariants. Moreover, we show that there is a notion of duality for these type of sl(2)-modules. Finally, we show that there are simple non-weight sl(2)-modules of arbitrary rank by constructing a whole new family of them.

An alternating matrix and a vector, with application to Aluffi algebras

Let X be a generic alternating matrix, t be a generic row vector, and J be the ideal Pf4(X)+I1(tX). We prove that J is a perfect Gorenstein ideal of grade equal to the grade of Pf4(X) plus two. This result is used by Ramos and Simis in their calculation of the Aluffi algebra of the module of derivations of the homogeneous coordinate ring of a smooth projective hypersurface. We also prove that J defines a domain, or a normal ring, or a unique factorization domain if and only if the base ring has the same property. The main object of study in the present paper is the module N which is equal to the column space of X, calculated mod Pf4(X). The module N is a self-dual maximal Cohen–Macaulay module of rank two; furthermore, J is a Bourbaki ideal for N. The ideals which define the homogeneous coordinate rings of the Plücker embeddings of the Schubert subvarieties of the Grassmannian of planes are used in the study of the module N.

A hermitian analogue of the Morita theorems

ABSTRACTLet F1 be an equivalence of type I between two categories of modules with hermitian forms. In other words, suppose F is an equivalence between the two underlying categories of modules, and F1 is an equivalence, given by and F1(f) = F(f), such that the non-singularity of a free hyperbolic module of hyperbolic rank 1 is preserved by F1. Then every equivalence generated by a set of hermitian Morita equivalence data is of type I and every equivalence of type I arises from a set of hermitian Morita equivalence data.

On Schrödinger-Virasoro type Lie conformal algebras

ABSTRACTIn this paper, two classes of Schrödinger-Virasoro type Lie conformal algebras TSV(a,b) and TSV(c) which are non-simple are introduced for some a, b, c∈ℂ. Moreover, central extensions, conformal derivations and free conformal modules of rank 1 of TSV(a,b) and TSV(c) are determined.

Snake-in-the-Box Codes for Rank Modulation under Kendall’s <inline-formula> <tex-math notation="LaTeX">$\tau $ </tex-math> </inline-formula>-Metric in <inline-formula> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math> </inline-formula>

Snake-in-the-box codes under Kendall’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> -metric are studied in the rank modulation scheme for flash memories, where codewords are a subset of permutations in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{n}$ </tex-math></inline-formula> with minimal Kendall’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> -distance two, and two cyclically consecutive codewords are connected via a push-to-the-top operation. Studies so far restrict the push-to-the-top operations only on odd indices, resulting in a snake consisting of permutations with the same parity, and thus, the minimal distance constraint is easily satisfied. Asymptotically optimal snake codes have been constructed this way in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+1}$ </tex-math></inline-formula> . As for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> , this framework keeps the last element fixed, and thus, a snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> is equivalent to a snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+1}$ </tex-math></inline-formula> , which is rather trivial. If one wants to do better, then it is inevitable to have some push-to-the-top operations on even indices, resulting in a combination of odd and even permutations in the snake, which increases the difficulty to guarantee the minimal Kendall’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> -distance constraint. Thus, Horovitz and Etzion pose the open problem to prove or disprove that the size of the largest snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> is not larger than the size of the largest snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+1}$ </tex-math></inline-formula> . A first step toward this problem is a negative answer by Wang and Fu, who construct a snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> with exactly one more permutation than an optimal snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+1}$ </tex-math></inline-formula> . In this paper, we give an explicit construction of a snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> with size asymptotically approaching <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$({1}/{4})|S_{2n+2}|$ </tex-math></inline-formula> .