Duality over complete local rings

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.

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.

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.

Extended plus closure in complete local rings

Let Q be a complete local ring which has the same characteristic as its residue field P, and, for the present, let us denote by A the image of a subset A of Q under the natural homomorphism of Q onto P. Then a subfield F of Q is called a coefficient field if = P. It has been shown in [2] and in [3] that a complete equicharacteristic local ring, such as the above, always possesses at least one coefficient field; this is the embedding theorem for the equicharacteristic case.

Chapter 5 - Vector Spaces, Hilbert Spaces, and the L2 Space

Hamiltonian matrices with respect to a nondegenerate skewsymmetric or skewhermitian indefinite inner product in finite dimensional real, complex, or quaternion vector spaces are studied. Subspaces that are simultaneously invariant for the matrices and neutral in the indefinite inner product are of special interest. The dimension of maximal (by inclusion) such subspaces is identified in terms of the canonical forms and sign characteristics. Criteria for uniqueness of maximal invariant neutral subspaces are given. The important special case of invariant Lagrangian subspaces is treated separately. Comparisons are made between real, complex, and quaternion contexts; for example, for complex Hamiltonian matrices with respect to a nondegenerate skewhermitian inner product in a finite dimensional complex vector space, the (complex) dimension of (complex) maximal invariant neutral subspaces is compared to the (quaternion) dimension of (quaternion) maximal invariant neutral subspaces, and necessary and sufficient conditions are given for the two dimensions to coincide (this is not always the case).

We study the Pareto frontier for two competing norms $\|\cdot\|_X$ \nomenclature[X]$\lVert\cdot\rVert_X,\lVert\cdot\rVert_Y$norms and $\|\cdot\|_Y$ on a vector space. For a given vector $c$, the Pareto frontier describes the possible values of $(\|a\|_X,\|b\|_Y)$ for a decomposition $c=a+b$. The singular value decomposition of a matrix is closely related to the Pareto frontier for the spectral and nuclear norm. We will develop a general theory that extends the notion of singular values of a matrix to arbitrary finite dimensional Euclidean vector spaces equipped with dual norms. This also generalizes the diagonal singular value decompositions (DSVDs) for tensors introduced by the author in previous work. We can apply the results to denoising, where $c$ is a noisy signal, $a$ is a sparse signal, and $b$ is noise. Applications include 1D total variation denoising, 2D total variation Rudin--Osher--Fatemi image denoising, LASSO, basis pursuit denoising, and tensor decompositions.

Let V be a finite dimensional complex vector space, and for each positive integer n let denote the vector space whose elements are the complex valued symmetric n-linear functions on (n-copies). If , that is, f is a linear functional on V, then we define the symmetric n-linear function by for all . It is well known that the elements , known as polar elements, span . A basis for consisting entirely of polar elements is a polar basis. We present a surprisingly simple recursive algorithm for the generation of polar bases. Explicit stand alone formulas for polar bases are also presented. One such explicit basis is so natural that it should probably be called the canonical polar basis for . We introduce the notion of subspace filter and show that each subspace filter of order n provides polar bases for all of the spaces simultaneously, and we extend this result to the entire symmetric algebra .

Gabor analysis is playing an important role in time-frequency analysis for the last 60 years. The fundamental concept of Gabor frames has become a focus also in the finite dimensional setting. Here, we study which sets of discrete time frequency shifts admit a proper choice of window vector, so that they generate an orthonormal Gabor basis in the underlying finite dimensional vector space. We fully characterize such time-frequency shift sets in the case that the vector space dimension is a prime number.

<p>The goal of this thesis is to study non-commutative deformation rings of representations of algebras. The main motivation is to provide a generalization of the deformation theory over commutative local rings studied by B. Mazur, M. Schlessinger and others. The latter deformation theory has played an important role in number theory, and in particular in the proof of Fermat's Last Theorem.</p> <p>The thesis is divided into two parts.</p> <p>In the first part, <em>A</em> is an arbitrary λ-algebra for a complete local commutative Noetherian ring λ with residue field <em>k</em>. A category Ĉ is defined whose objects are complete local λ-algebras <em>R</em> with residue field <em>k</em> such that <em>R</em> is a quotient ring of a power series algebra over λ in finitely many non-commuting variables. If <em>V</em> is a finite dimensional <em>k</em>-vector space that is also a left <em>A</em>-module and that satisfies a natural finiteness condition, it is proved that <em>V</em> has a so-called versal deformation ring <em>R(A,V)</em>. More precisely, <em>R(A,V)</em> is an object in Ĉ such that the isomorphism class of every lift of <em>V</em> over an object <em>R</em> in Ĉ arises from a morphism α : <em>R(A,V)</em>→ <em>R</em> in Ĉ and α is unique if <em>R</em> is the ring of dual numbers <em>k</em>[ϵ].</p> <p>In the second part, two particular examples of λ, <em>A</em> and <em>V</em> are studied and the versal deformation ring <em>R(A,V)</em> is determined in each of these cases. In the first example, λ=<em>k</em>, <em>A</em> is a series of non-commutative <em>k</em>-algebras depending on a parameter <em>r≥2</em>, and <em>V</em> is a particular quotient module of <em>A</em>; it is shown that <em>R(A,V)</em> is isomorphic to <em>A</em>. The second example builds on the first example when <em>r=2</em> and uses that, if additionally the characteristic of <em>k</em> is 2, then <em>A</em> is isomorphic to the group ring <em>k[D<sub>8</sub>]</em> of a dihedral group <em>D<sub>8</sub></em> of order 8.</p> <p>It is shown that if <em>k</em> is perfect and <em>W</em> is the ring of infinite Witt vectors over <em>k</em>, then <em>R(W[D<sub>8</sub>],V)</em> is isomorphic to <em>W[D<sub>8</sub>]</em>.</p>

The present work investigates to which extent semantical knowledge can support the learning of basic mathematical concepts. The considered learning criteria are learning characteristic or enumerable indices for languages from positive data where the learner has to converge either syntactically (Ex) or semantically (BC). The considered classes are the classes of all monoids of a given group, all ideals of a given ring or all subspaces of a given vector space. The following is shown: (a) Learnability depends much on the amount of semantic knowledge given at the synthesis of the learner where this knowledge is represented by programs for the algebraic operations, codes for prominent elements of the algebraic structure (like 0 and 1 in fields) and certain parameters (like the dimension of finite dimensional vector spaces). For several natural examples good knowledge of the semantics may enable to keep ordinal mind change bounds while restricted knowledge may either allow only BC-convergence or even not permit learnability at all. (b) A recursive commutative ring is Noetherian iff the class of its ideals is BC-learnable. Such a BC-learner can be synthesized from programs for addition and multiplication. In many Noetherian rings, one can Ex-learn characteristic indices for the ideals with an ordinal bound on the number of mind changes. But there are also some Noetherian rings where it is impossible to Ex-learn the ideals or to learn characteristic indices for them.

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

Proof.We will prove the first part by induction on t.When r=l, u=Uy.In addition, [uk] = [u]k = [u] so u is not nilpotent.Hence, the associative algebra of all polynomials in u, denoted by F[u], is not nilpotent and thus contains an idempotent e=f(u) for fix) a polynomial.Then [e] = [/(«)] = a[u] where a=f(l) is in F. Thus a[w] = [e] = [e]2 = c72[w]2 = c72[w].Since e is not in M, a[u]^0 and a=l.Let w = Uy + u2 for r^2.Then u=w+u3+ ■ ■ ■ +ut for pairwise orthogonal idempotents [w], [u3],..., [ut].By the inductive hypothesis, there exists an idempotent e and pairwise orthogonal idempotents/ e3,...,et in Ae(l) such that

An affine algebraic group