Linear Analysis

Chapter 5 - Vector Spaces

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.

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.

The loss of orthogonality in the Gram-Schmidt orthogonalization process

We examine translation-invariant measures on Banach spaces lp, where p ∈ [1,∞]. We construct analogs of the Lebesgue measure on Borel σ-algebras generated by the topology of pointwise convergence (σ-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not σ-finite. We also study spaces of functions integrable with respect to measures constructed and prove that these spaces are not separable. We consider various dense subspaces in spaces of functions that are integrable with respect to a translation-invariant measure. We specify spaces of continuous functions, which are dense in the functional spaces considered. We discuss Borel σ-algebras corresponding to various topologies in the spaces lp, where p ∈ [1,∞]. For p ∈ [1,∞), we prove the coincidence of Borel σ-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel σ-algebra corresponding to the topology of pointwise convergence. We also verify that the space l∞ does not possess similar properties.

To compare the advantages and disadvantages of different techniques for fitting Zernike polynomials to surfaces. Two different methods, Orthogonal Projection and Gram-Schmidt orthogonalization, are compared in terms of speed and performance at fitting a complex object. Orthogonal Projection provides an extremely rapid fitting of a surface, but leaves residual high frequency noise. The Gram-Schmidt technique provides a more accurate fit of the original object, but consumes much more computing time. Orthogonal Projection, and its associated noise, may be tolerated in classifying corneal topography. Orthogonal Projection has a distinct advantage in calculation time over other methods for fitting surfaces. This advantage may be exploited in cases where accurate surface fitting is not necessary, but only general features need to be extracted for classification. If fit accuracy is needed, then slower fitting techniques, such as Gram-Schmidt, should be used.

The connected and acyclic components contained in a network are identified by the computation of its complexity, where complexity of a network refers to the total number of spanning trees present within. The article in hand deals with the enumeration of the complexity of various networks’ operations such as sum ( K 2, n + W 3 , K 2, n + n K 1 , K n + S n ), product ( K 2, n ⊠ K 2 , K 2, n ⋉ K 2 , K n × K 2 , K n ⊠ K 2 ), difference ( K 2, n ⊖ K 2 ), and the conjunction of S n with K 2 . All our computations have been concluded by implementation of the methods of linear algebra and matrix theory. Our derivations will also be highlighted with the assistance of 3D plots at the end of this article.

We prove results of existence and approximation of the solutions of the convolution equations in spaces of entire functions of exponential type on infinite dimensional spaces. In particular we obtain: let E be a complex, quasi-complete and dual nuclear locally convex space and Ω \Omega a convex balanced open subset of E; let H ( Ω ) \mathcal {H} (\Omega ) be the space of the holomorphic functions on Ω \Omega , equipped with the compact open topology and H ′ ( Ω ) \mathcal {H}’(\Omega ) its strong dual; let F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) denote the image of H ′ ( Ω ) \mathcal {H}’(\Omega ) through the Fourier-Borel transform F \mathcal {F} ; equip this space F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) with the image of the topology of H ′ ( Ω ) \mathcal {H}’(\Omega ) via the map F \mathcal {F} . Then, “every nonzero convolution operator on F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) is surjective” and “every solution of the homogeneous equation is limit of exponential-polynomial solutions". Our results are more generally valid when E is a Schwartz bornological vector space with the approximation property. Previous results in Fréchet-Schwartz and Silva spaces are thus extended to domains that are not Fréchet or D.F.-spaces.

An efficient model correction method is proposed by using the modal measurement from a structural system. The method corrects/updates the mass and stiffness matrix without imposing any parameterization. It considers the information from both the nominal finite-element model and the measurement of modal frequencies and mode shapes. The method is computationally very efficient and it does not require computation of the complete set of eigenvalues and eigenvectors of the nominal model. Instead, only the nominal eigenvalues and eigenvectors of the modes to be corrected are needed. The Gram-Schmidt orthogonalization process is used to construct a basis that satisfies the mass orthogonality condition. This basis is used to transform the eigenvectors of the nominal model so that the corrected model is compatible with the measurement. A thousand-degree-of-freedom chainlike system and a 1,440-degree-of-freedom structural frame are used to illustrate the proposed method.

Study on flexural vibration of triangular composite plates influenced by fibre orientation

Two-dimensional orthogonal polynomials for vibration of rectangular composite plates

In this paper, a reliable new scheme is presented based on combining Reproducing Kernel Method (RKM) with a practical technique for the nonlinear problem to solve the System of Singularly Perturbed Boundary Value Problems (SSPBVP). The Gram-Schmidt orthogonalization process is removed in the present RKM. However, we provide error estimation for the approximate solution and its derivative. Based on the present algorithm in this paper, can also solve linear problem. Several numerical examples demonstrate that the present algorithm does have higher precision.

An alternative approach to orthogonal graph theoretical invariants

Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of them-dimensional complex column vector space 𝔺mwith respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

Let be a linear brownian motion. Traditionally, this brownian motion is considered as a process which belongs a.s. to the space of continuous functions defined on [01], vanishing at the origin and endowed with the topology of uniform convergence. Let now be the besov space of the functions defined on [01], . They are spaces of sequences and an important fact is therefore that these Besov spaces are isomorphic to suitable Banach spaces of numerical sequences. This property is particularly well adapted to the study of brownian motion considered as a process which belongs a.s.to the spaces In this article we illustrate this in the following situations:Z. Ciesielski fias shown recently [C3] that brownian motion belongs a.s. to the space There we characterize in this first paragraph the set indexes αp, qsuch that brownian motion belongs a.s. to The classical theory of large deviations gives the behaviour, as ,of where Γ is a Borel set of the uniform topology. In the second paragraph we show that if Γ are chose...