On embedding numbers into quaternion orders

A generalization of the chevalley-Hasse-Noether theorem from maximal orders to arbitrary Eichler orders in quaternion algebras is given. A stability property for the numbers of orbits for unit groups in quaternion orders acting on optimal embeddings of quadratic orders is proved. The results are applied to Siegel's meanvalue of integral representations by genera of integral definite ternary quadratic forms.     

A quaternionic proof of the representation formula of a quaternary quadratic form

The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrations, Hurwitz was able to use a quaternion order to obtain the formula for the number of representations. Recently the author has been able to use certain quaternion orders to demonstrate the universality of other quaternary quadratic forms besides the sum of four squares. In this paper, we develop results analogous to Hurwitz's above mentioned work by delving into the number theory of one of these quaternion orders, and discover an alternate proof of the representation formula for the corresponding quadratic form.     

四元数矩阵实表示的基本性质及应用

在四元数实矩阵表示的基础上,给出了四元数矩阵的相同表示,利用友向量的概念,给出了这种实表示的性质,并进一步研究了四元数力学中的系列数值计算问题.     

一四元数矩阵方程组的最佳逼近解

利用四元数矩阵的广义Frobenius范数和弱圈积,建立一个关于四元数矩阵的实函数并简洁表征其极小值.再用四元数矩阵的奇异值分解和广义Frobenius范数的性质,讨论四元数矩阵方程组[AX,XB]=[C,D]的最小二乘解,得到了解的具体表达式.最后在该方程组的解集合中导出了与给定矩阵的最佳逼近解的表达式.     

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi-Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer-Severi varieties and, in the quaternion case, we show how to obtain some Brauer-Severi varieties from G-Hilbert schemes of P¹-bundles.     

四元数体上一类矩阵方程解的数值方法

建立了求解四元数体上严格对角占优矩阵方程AX=B的QJ和QSOR迭代方法,并利用四元数矩阵的右特征值最大模刻画出迭代的收敛性,给出参数的取值范围;最后运用四元数矩阵的复表示运算保结构的特性,把这两种迭代等价地转化到复数域上,从而实现了该系统的数值求解.     

Coincidence Site Modules in 3-Space

The coincidence site lattice (CSL) problem and its generalization to ℤ-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.     

四元数矩阵方程的复转化及保结构算法

给出四元数矩阵复表示运算定义及其相关性质,并运用复表示运算的保结构特性,讨论了四元数矩阵Moore-Penrose逆计算以及两类四元数矩阵方程AXB=C和AX-XB=C的数值求解方法.数值算例检验了所给算法的可行性.     

An embedding theorem for Eichler orders

Let B be a quaternion algebra over number field K. Assume that B satisfies the Eichler condition (i.e., there is at least one archimedean place which is unramified in B). Let Ω be an order in a quadratic extension L of K. The Eichler orders of B which admit an embedding of Ω are determined. This is a generalization of Chinburg and Friedman's embedding theorem for maximal orders.     

Indecomposable second-order matrix rings with a finite number of bass representations

We prove the theorem formulated in the title and its analog for orders in certain fields (in particular, in quaternion algebras).Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 601–604, November, 1972.     

Robust quaternion matrix completion with applications to image inpainting

In this paper, we study robust quaternion matrix completion and provide a rigorous analysis for provable estimation of quaternion matrix from a random subset of their corrupted entries. In order to generalize the results from real matrix completion to quaternion matrix completion, we derive some new formulas to handle noncommutativity of quaternions. We solve a convex optimization problem, which minimizes a nuclear norm of quaternion matrix that is a convex surrogate for the quaternion matrix rank, and the ?₁‐norm of sparse quaternion matrix entries. We show that, under incoherence conditions, a quaternion matrix can be recovered exactly with overwhelming probability, provided that its rank is sufficiently small and that the corrupted entries are sparsely located. The quaternion framework can be used to represent red, green, and blue channels of color images. The results of missing/noisy color image pixels as a robust quaternion matrix completion problem are given to show that the performance of the proposed approach is better than that of the testing methods, including image inpainting methods, the tensor‐based completion method, and the quaternion completion method using semidefinite programming.     

Rank one quotients of vector groups

ABSTRACT

In this paper we study the possible torsion in even-dimensional higher class groups Cl _2n (Λ)(n ≥ 1) of an order Λ in a semisimple algebra A over a number field F with a ring of integers 𝒪 _F . We show that for certain orders, called generalized Eichler orders, bip-torsion in Cl _2n (Λ) can only occur for primes p dividing prime ideals ? of 𝒪 _F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders.     

Über die zentrale Picard-Gruppe und die Einheiten lokaler Quaternionenordnungen

Let M be any order of a quaternion algebra Q over a local field F. The group M^x of units and the central Picard group Picent (M) of M are investigated. The group index (L^x:M^x) and, if F is of characteristic 2, also the norms of all primitive ideals of M and the order of Picent (M) are determined explicitly, where L denotes any maximal order of Q containing M. Applications of these results to local densities of ternary quadratic forms and to class numbers and type numbers of global quaternion orders are indicated.     

Relative spinor class fields: A counterexample

It is known that classes of indefinite quadratic forms in a genus are classified by the Galois group of a spinor class field [4]. Hsia has proved the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. Spinor class fields can also be used to classify conjugacy classes of maximal orders in a central simple algebra. In [1] we left open the issue of whether for every fixed given non-maximal order in a central simple division algebra there exists a representation field L with the property that embeds into a given maximal order if and only if the corresponding element of the Galois group is trivial on L. In this work we give a negative answer to this question for central simple division algebras of dimension ≥ 3². The case of non-division algebras is also treated by replacing the phrase embeds into by is contained in a conjugate of. As a byproduct of the techniques used in this paper we compute the representation field of an Eichler order in a quaternion algebra. Received: 8 April 2008     

A Ramsey theorem for multiposets

In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result from the 1980s claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 Sokić proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of Solecki and Zhao that the class of all finite posets with several linear extensions has the Ramsey property.Using the categorical reinterpretation of the Ramsey property in this paper we prove a common generalization of all these results. We consider multiposets to be structures consisting of several partial orders and several linear orders. We allow partial orders to extend each other in an arbitrary but fixed way, and require that every partial order is extended by at least one of the linear orders. We then show that the class of all finite multiposets conforming to a fixed template has the Ramsey property.     

Quaternion normed space with isometry group ℤ2

In a finite-dimensional linear space over the quaternion field, we construct a norm with the following property. Any linear isometry is one of the two transformations x ? x and x ? ?x.     

Lanczos method for large-scale quaternion singular value decomposition

In many color image processing and recognition applications, one of the most important targets is to compute the optimal low-rank approximations to color images, which can be reconstructed with a small number of dominant singular value decomposition (SVD) triplets of quaternion matrices. All existing methods are designed to compute all SVD triplets of quaternion matrices at first and then to select the necessary dominant ones for reconstruction. This way costs quite a lot of operational flops and CPU times to compute many superfluous SVD triplets. In this paper, we propose a Lanczos-based method of computing partial (several dominant) SVD triplets of the large-scale quaternion matrices. The partial bidiagonalization of large-scale quaternion matrices is derived by using the Lanczos iteration, and the reorthogonalization and thick-restart techniques are also utilized in the implementation. An algorithm is presented to compute the partial quaternion singular value decomposition. Numerical examples, including principal component analysis, color face recognition, video compression and color image completion, illustrate that the performance of the developed Lanczos-based method for low-rank quaternion approximation is better than that of the state-of-the-art methods.

     

Uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform

ABSTRACT

In this paper, we present some new elements of harmonic analysis related to the right-sided multivariate continuous quaternion wavelet transform. The main objective of this article is to introduce the concept of the right-sided multivariate continuous quaternion wavelet transform and investigate its different properties using the machinery of multivariate quaternion Fourier transform. Last, we have proven a number of uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform.     

Fixed points of posets and clique graphs

The fixed point property for partial orders has been the object of much attention in the past twenty years. Recently, M. Roddy ([7]) proved this famous conjecture of Rival (see [6]): the class of finite orders with the fixed point property is closed under finite products.In this article, we prove that a finite order has the fixed point property if the sequence of iterated clique graphs of its comparability graph tends to the trivial graph.     

Armendariz group rings

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.