LINEAR DIVISION RINGS

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

In this chapter we review the main ideas of linear algebra. The one twist is that we allow our scalars to come from any field instead of just the field of real numbers. In particular, the notion of a vector space over the field \(\mathbb{Z}_{2}\) will be essential in our study of coding theory. We will also need to look at other finite fields when we discuss Reed–Solomon codes. One benefit to working with finite dimensional vector spaces over finite fields is that all sets in question are finite, and so computers can be useful in working with them.

The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$ -vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.

The number of classes of linearly equivalent functions

A celebrated theorem of Wedderburn says that a finite associative division ring is a finite (commutative) field. An elegant proof of this, due to E. Artin [3, p. 72], reduces the problem to showing the reduced (or generic) norm has a nontrivial zero, and then applies the theorem (conjectured by Artin and proved by Chevalley) that a polynomial with coefficients in a finite field and zero constant term has a nontrivial zero in that field if its degree is less than the number of variables. The Wedderburn theorem was generalized by A. A. Albert to finite strictly power-associative division rings of characteristic # 2 in [1, p. 301 ] and [2, p. i1 ] (see also [5 ]). The proof reduced the powerassociative case to the associative case by using the properties of Jordan division algebras. It is the purpose of this note to show that Artin's method carries over directly to the power-associative case using the generic norm introduced by N. Jacobson [4]. By avoiding Jordan algebras one is able to extend Albert's result to the characteristic 2 case. The usual definition of a division ring requires that left and right multiplications by a nonzero element be bijective. If the ring is finite it is necessarily an algebra, over a finite field 4, which is algebraic (even finite-dimensional) and without zero divisors. Also, as we shall see later, the algebra necessarily has a unit element. We wish to consider something slightly more general. For our purposes we say an algebraic power-associative algebra is a division algebra if it is unital and no element xO 0 is a zero divisor in the subalgebra (D[x] it generates (since 4<[x] is finite-dimensional, this is equivalent to saying x is invertible in b [x]). Our conditions are less restrictive, since the Jordan algebra of a quadratic form can be a division algebra in our sense with zero divisors. With this definition we have

Given a symmetric bilinear form F on a finite dimensional real vector space V we attempt to find ‘nice’ bases where the matrix representation of the form is as simple as possible, in the same sort of way as we did in the Gram-Schmidt orthogonalization process for inner products. The theorem which says that this is possible and which describes all the possible matrix representations one can get is called Sylvester’s law of inertia. (Sylvester’s law of inertia has nothing to do with inertia: the story goes that Sylvester said, ‘If Isaac Newton can have a law of inertia, then so can I’, and consequently gave this theorem the name it has been known by ever since.) This ‘law’ is exactly analogous to Theorem 2.33 which classified all finite dimensional vector spaces using dimension; it turns out that symmetric bilinear forms are classified by the dimension of the underlying vector space and two further numbers, called the rank and the signature of the form.

We show that given an affine algebraic group G over a field K and a finite subgroup scheme H of G there exists a finite dimensional G-module V such that V∣H is free. The problem is raised in the recent paper by Kuzucuŏglu and Zalesskiǐ [15] which contains a treatment of the special case in which K is the algebraic closure of a finite field and H is reduced. Our treatment is divided into two parts, according to whether K has zero or positive characteristic. The essence of the characteristic 0 case is a proof that, for given n, there exists a polynomial GLn(Q)-module V of dimension 2 n Π p p ( 2 n ) , where the product is over all primes less than or equal to n+1, such that V is free as a QH-module for every finite subgroup H of GLn(Q). The module V is the tensor product of the exterior algebra Λ*(E), on the natural GLn(Q)-module E, and Steinberg modules Stp, one for each prime not exceeding n+1. The Steinberg modules also play the major role in the case in which K has characteristic p>0 and the key point in our treatment is to show that for a finite subgroup scheme H of a general linear group scheme (or universal Chevalley group scheme) G over K, the Steinberg module St p n for G is injective (and projective) on restriction to H for n≫0. A curious consequence of this is that, despite the wild behaviour of the modular representation theory of finite groups, one has the following. Let H be a finite group and V a finite dimensional vector space. Then there exists a (well-understood) faithful rational representation π GL(V)→GL(W) such that, for each faithful representation ρ: H→GL(V), the composite πορ: H→GL(W) is free, in particular all representations πορ are equivalent.

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

Application of neustadt's theory of extremals to an optimal control problem with a functional differential equation and a functional inequality constraint

Let K[?] denote the group algebra of an infinite locally finite group?. In recent years, the lattice of ideals of K[?] has been extensively studied under the assumption that? is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra K[V] stable under the action of an appropriate automorphism group of V. Specifically, in this paper, we let B be a quasi-simple group of Lie type defined over an infinite locally finite field F, and we let V be a finite-dimensional vector space over a field E of the same characteristic p. If B acts nontrivially on V by way of the homomorphism Φ: G → GL(V), and if V has no proper G-stable subgroups, then we show that the augmentation ideal ωK[V] is the unique proper G-stable ideal of K[V] when char K ¬= p. The proof of this result requires, among other things, that we study characteristic p division rings D, certain multiplicative subgroups G of D*, and the action of G on the group algebra K[A], where A is the additive group D + . In particular, properties of the quasi-simple group G come into play only in the final section of this paper.

The problem of determining whether two programs compute the same function is undecidable. We restrict the class of functions that a program can compute to be a finite dimensional vector space, enabling the problem to become tractable. We prove a theorem showing that the number of test points needed to distinguish any such programs is equal to the dimension of the vector space of functions which they compute, thus generalizing previous results of Howden. Our result unifies certain existing software testing theory and explains the successes and failures of one existing method for selecting test data called perturbation testing. We apply the theorem to obtain three results on the number of test points required to test in vector spaces of Boolean functions and polynomials. First, the Vandermonde matrix used together with our theorem yields the classical result that a polynomial of degree at most n is determined by n+1 points. Second, in vector spaces of Boolean functions we obtain useful results when the degree of the Boolean function is less than the number of inputs to the function. Third, we apply our theorem to describe a technique for testing nonpolynomial, real valued functions.

On the finite generation of additive group invariants in positive characteristic

On affine complete intersections with isolated singularities

Let P → M be a principal G-bundle with m-dimensional basis, where G is a Lie group. Let V be a finite dimensional vector space over R. We describe all canonical (\( \mathcal{P}\mathcal{B}_m \)(G)-invariant) V-valued 1-forms ϑ: TWmrP → V on the r-th order principal prolongationWmrP of P.

We give a new definition of the relative topological degree for multimaps compositions of approximable multimaps and continuous map. We apply this notion to prove an existence result for variational inequalities of Stampacchia’s type in finite dimensional vector spaces. Finally we obtain the same results also for multimaps compositions of selectionable multimaps and continuous map.