Isomorphism Theory for the Unitary Groups

Let D be a division ring with the center F and suppose that D* is the multiplicative group of D. D is called centrally finite if D is a finite dimensional vector space over F and D is locally centrally finite if every finite subset of D generates over F a division subring which is a finite dimensional vector space over F. We say that D is a linear division ring if every finite subset of D generates over Fa centrally finite division subring. It is obvious that every locally centrally finite division ring is linear. In this report we show that the inverse is not true by giving an example of a linear division ring which is not locally centrally finite. Further, we give some properties of subgroups in linear division rings. In particular, we show that every finitely generated subnormal subgroup in a linear ring is central. An interesting corollary is obtained as the following: If D is a linear division ring and D* is finitely generated, then D is a finite field.

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

In a recent series of short papers [2, 3, 41 I have been discussing the relationships of uniform convexity with certain other properties of normed vector spaces. In this paper I propose to discuss relationships between uniform convexity, factor spaces, and conjugate spaces. Because of the large number of special results which are needed for two dimensional spaces-and which are false in general-the paper will be divided into two parts: In Part I (??2-4) B is two dimensional; in Part II (??5-7) this restriction is removed and B may be any normed vector space. To recall the definition, B is said to be uniformly convex if there exists a function 6 such that 0 e, is a modulus of convexity for B. Note that if 51 is nowhere greater than 5, a modulus of convexity for B, then 5l is a modulus of convexity for B. Because of the pointwise nature of the definition of uniform convexity, it is clear that B is uniformly convex with modulus of convexity a if and only if all the two dimensional subspaces of B have the common modulus of convexity 6. One of the most useful results of this investigation (Theorem 5.5) is the complementary fact that B is uniformly convex if and only if all the two dimensional factor spaces of B have a common modulus of convexity. In the study of the effect of uniform convexity of B on the nature of the conjugate space B*, this result makes it possible to reduce the problem to the study of two dimensional spaces. In such a space B uniform convexity is equivalent to strict convexity: that is, a two (or finite) dimensional space is uniformly convex if and only if there does not exist a line segment of positive length all of whose points are of norm one. It has been observed [1, Footnote 13] that such a line segment on the unit sphere is equivalent to the existence of a sharp edge on the unit sphere in B*. The attempt to describe a sharp edge of the unit sphere in a finite dimensional space B in terms of the norm in that space leads to the condition that there exists a k > 0 such that for any e > 0 a pair of points bi and b2 exists such that II b -b2 11 k 11 bl-be 11 . Contradicting this suggests the following condition, a sort of dual concept to uniform convexity. A space B is said to be uniformly flattened if there exists a function v7 positive for 0 < e ? 2 such that lim.b.o 7(e) = 0 while (211 b + b2 11)/i b2 11 < 7(e) if J]b11 = JJ b1 = 1 and 11 bb2 ? || ; 375

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

The notion of the shell of aHilbert spa ce opera tor, which is auseful genera liza tion (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators in two dimensional indefinite inner product spaces. For normal operators, it is conjectured that the shell is convex and its closure is polyhedral; the conjecture is proved for indefinite inner product spaces of dimension at most three, and for finite dimensional inner product spaces with one positive eigenvalue.

Chapter 2 - Projective and Affine Geometry over Division Rings

Given a signal, whether it is a discrete vector or a continuous function, one desires to write it in terms of simpler components. Typically, these components or “building blocks” form what is called a basis. A basis is an optimal set, containing the minimal number of elements needed to uniquely represent any signal in a given space. A frame can be thought of as a redundant basis, having more elements than needed. In fact, in any finite dimensional vector space every finite spanning set is a frame. The redundancy of a frame leads to a non-unique representation, however, this makes signal representation resilient to noise and robust to transmission losses. Frames are now standard tools in signal processing and are of great interest to mathematicians and engineers alike. This chapter presents a brief introduction to frames in finite dimensional spaces, and in particular discusses a highly desirable class of frames called equiangular tight frames. Possible research ideas suitable for an undergraduate curriculum are also discussed.

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.

Continuous fields of C*-algebras over finite dimensional spaces

Elementary equivalence of infinite-dimensional classical groups

Characterization of simultaneous similarity for commuting m m - tuples of operators is an open problem even in finite-dimensional spaces; known as “A wild problem in linear algebra”. In this paper we offer a criterion for simultaneous similarity of m m -tuples of k k -cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m m -tuples of commuting square matrices.

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.

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

The notion of “exceptional family of elments (EFE)” plays a very important role in solving complementarity problems. It has been applied in finite dimensional spaces and Hilbert spaces by many authors. In this paper, by using the generalized projection defined by Alber, we extend this notion from Hilbert spaces to uniformly smooth and uniformly convex Banach spaces, and apply this extension to the study of nonlinear complementarity problems in Banach spaces.

Introduction The notion of kernel-induced vector spaces is the cornerstone of kernel-based machine learning. Generalization of the traditional definition of a similarity metric plays a vital role in facilitating the analysis of complex and big data. It is often necessary to generalize the traditional Euclidean inner product to the more flexible and nonlinear inner products characterized by properly chosen kernel functions. The new inner product leads to a new distance metric, allowing the data analysis to be effectively performed in a higher-dimensional vector space. The topics addressed in this chapter are as follows. Section 2.2 introduces Mercer's fundamental theorem stating the necessary and sufficient condition for a function be a Mercer kernel. It will examine several prominent kernel functions, including the polynomial and Gaussian kernel functions. Section 2.3 introduces the notion of intrinsic space associated with a kernel function. The intrinsic space is so named because it is independent of the training dataset. The dimension of the space is denoted by J and will be referred to as the intrinsic degree . This degree, whether finite or infinite, dictates the training efficiency and computational cost. Section 2.4 introduces a finite-dimensional kernel-induced vector space, known as empirical space , which is jointly determined by the kernel function and the training dataset. The dimension of the empirical space is equal to the data size N . With the LSP condition, both the intrinsic-space and the kernelized learning model will be at our disposal.