Persistence and the Sheaf-Function Correspondence

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.

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.

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).

Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let S be the ring of all complex-valued polynomials on X, regarded as a G-module in the obvious way, and let J ? S be the sub-ring of all G-invariant polynomials on X.

Similarity vs unitary similarity and perturbation analysis of sign characteristics: Complex and real indefinite inner products

This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let $$\varepsilon$$ and $$\varepsilon_{+}$$ be a finite dimensional real vector space and a symmetric cone embedded in $$\varepsilon$$; examples of $$\varepsilon$$ and $$\varepsilon_{+}$$ include a pair of the N-dimensional Euclidean space and its nonnegative orthant, a pair of the N-dimensional Euclidean space and N-dimensional second-order cones, and a pair of the space of m × m real symmetric (or complex Hermitian) matrices and the cone of their positive semidefinite matrices. Sums of squares relaxations are further extended to a polynomial optimization problem over $$\varepsilon_{+}$$, i.e., a minimization of a real valued polynomial a(x) in the n-dimensional real variable vector x over a compact feasible region $$\{ {\bf x} : b({\bf x}) \in \varepsilon_{+}\}$$, where b(x) denotes an $$\varepsilon$$- valued polynomial in x. It is shown under a certain moderate assumption on the $$\varepsilon$$-valued polynomial b(x) that optimal values of a sequence of sums of squares relaxations of the problem, which are converted into a sequence of semidefinite programs when they are numerically solved, converge to the optimal value of the problem.

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

In this section some of the basic concepts of general relativity are introduced. First some necessary linear algebra is presented. Let V be a finite dimensional real vector space. A vector in V can be represented by its components vα in a given basis. The dual space Vα consists of the linear functionals on V, i.e. the linear mappings from V to the real numbers ℝ.

LetPandQbe relatively prime integers greater than 1, and letfbe a real valued discretely supported function on a finite dimensional real vector spaceV. We prove that if$f_{P}(x)=f(Px)-f(x)$and$f_{Q}(x)=f(Qx)-f(x)$are both$\Lambda $-periodic for some lattice$\Lambda \subset V$, then so isf(up to a modification at$0$). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section, we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.

The aim of this chapter is to give an account of the basic results which are needed to work with lattices, i.e. discrete subgroups of maximal rank in Euclidean vector spaces. Throughout the book, E stands for a real vector space of finite dimension n. From Section 1.2 onwards, the space is assumed to be Euclidean.

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 .

We obtain, by means of a classification of the eigenvalues, local estimates for holomorphic. functions of a class of linear operators on a finite dimensional linear vector space. We apply these methods to find new proofs of some theorems ofKreiss andMorton, and in addition we give a local estimate of the powers of the inverse of any nonsingular operator in this class.

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.

The combination of algebraic structures and graphs was carried out by investigating thoroughly the relation among the algebraic structure and the graph theoretic properties. Moreover, it needs to explore algebraic structure. The results of the combination of algebraic structures and graphs have many applications in the fields of Internet modeling, coding, etc. For example, the famous Cayley graph was constructed from groups and widely used in network models. Das introduced the subspace inclusion graph on finite-dimensional vector space over a finite filed, and studied that the graph is bipartite and some special properties if the dimension of the vector space is 3. In this paper, this bipartite inclusion graph in the case of 3-dimensional is extended to more general dimensional bipartite inclusion graph. The diameter, girth, clique number, covering number, independence number and matching number are studied and the properties are shown, such as regular, planar and Eulerian. Moreover, the authors also introduce a new results about the structure and the number of 1-dimensional and n-1-dimensional subspaces on n-dimensional vector space.