Surjective Map Research Articles

Digital Image Correlation and Color Cameras

Digital Image Correlation (DIC) is a well-known experimental technique. It works by constructing a (surjective) mapping of pixel intensity from reference to target image, where the mapping parameters are identified using a Least Squares approach. Because it makes use of the luminance component of the image, Digital Image Correlation is usually implemented by assuming monochrome cameras. In this work, we will discuss its implementation when color cameras are used, focusing on pitfalls and potential advantages of this solution. Since most cameras implement color acquisition using a Color Filter Array (CFA), much of the article will focus on this technology. However, we will not limit ourselves to this aspect and will show that Three-CCD cameras can provide significant advantages over both CFA and monochrome cameras.

A remark on smooth images of Banach spaces

Let X be a non-separable super-reflexive Banach space. Then for any separable Banach space Y of dimension at least two there exists a C∞-smooth surjective mapping f:X→Y such that the restriction of f onto any separable subspace of X fails to be surjective. This solves a problem posed by Aron, Jaramillo, and Ransford (Problem 186 in the book [5]).

Continuity of Jones' function is not preserved under monotone mappings

We show two metric continua X and Z and a monotone surjective mapping f:X→Z such that the Jones' function T is continuous for X, but it is not continuous for Z. This answers a question by D. P. Bellamy.

The slice method for G-torsors

The notion of a (G,N)-slice of a G-variety was introduced by P.I. Katsylo in the early 80's for an algebraically closed base field of characteristic 0. Slices (also known under the name of relative sections) have ever since provided a fundamental tool in invariant theory, allowing reduction of rational or regular invariants of an algebraic group G to invariants of a “simpler” group. We refine this notion for a G-scheme over an arbitrary field, and use it to get reduction of structure group results for G-torsors. Namely we show that any (G,N)-slice of a versal G-scheme gives surjective maps H1(L,N)→H1(L,G) in fppf-cohomology for infinite fields L containing F. We show that every stabilizer in general position H for a geometrically irreducible G-variety V gives rise to a (G,NG(H))-slice in our sense. The combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type E7.

Maps preserving strong 2-Jordan product on some algebras

Let [Formula: see text] be a surjective map between some operator algebras such that [Formula: see text] for all [Formula: see text], where [Formula: see text] defined by [Formula: see text] and [Formula: see text] is Jordan product, i.e. [Formula: see text]. In this paper, we determine the concrete form of map [Formula: see text] on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if [Formula: see text] is a factor von Neumann algebra that satisfies [Formula: see text] for all [Formula: see text] and idempotents [Formula: see text] then there exists nonzero scalar [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]

Exceptional values of holomorphic functions. Remarks on a Nishino's Theorem

Let F(z,w)∈O(Cn+1), where (z,w)∈Cn×C. Let z′∈Cn such that F(z′,w) is not constant. If F(z′,w) is not surjective it takes all the values of C minus one π(z′) (Picard). T. Nishino studied in [8]π(z) when n=1, F(z,w) is of finite order in w and π(z) is defined in a set E⊂C with at least one accumulation point. In this work, we see that his result allows to obtain an explicit expression of such a F(z,w) when n≥1 and F(z′,w) is not a constant for any z′∈Cn, and conclude that π(z)=η(z)−1/ξ(z) for η(z) and ξ(z)∈O(Cn) when π(z) is defined on a nonempty open set U⊂Cn. Moreover, we give several applications of this fact. We show that the complement of the graph of π(z) in Cn+1 is dominated by Cn+1 via a family of surjective fiber-preserving holomorphic maps with non-vanishing Jacobian determinant, which are described in terms of the flow of a complete vector field of type C⁎. In particular, Buzzard and Lu's results in [2] applied to π(z) for n=1 can be extended for n≥2. It will allow to define new examples of Oka manifolds.

A note on strong skew Jordan product preserving maps on von Neumann algebras

Let \(\mathcal {A}\) be a von Neumann algebra acting on the complex Hilbert space \(\mathcal {H}\) and \(\Phi {:}\,\mathcal {A} \longrightarrow \mathcal {A}\) be a surjective map that satisfies the condition $$\begin{aligned} \Phi (T)\Phi (P)+\Phi (P)\Phi (T)^*=TP+PT^* \end{aligned}$$ for all T and all projections P in \(\mathcal {A}\). We characterize the concrete form of \(\Phi \) on selfadjoint elements of \(\mathcal {A}\). Also when \(\mathcal {A}\) is a factor von Neumann algebra, it is shown that \(\Phi \) is either of the form \(\Phi (T)=T+i\tau (T) I\) or of the form \(\Phi (T)=-T+i\tau (T)I\), where \(\tau {:}\,\mathcal {A}\longrightarrow \mathbb {R}\) is a real map.

Reduction and Integrability of Stochastic Dynamical Systems

This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction, and integrability. In particular, we show that an SDS that is diffusion-wise symmetric with respect to a proper Lie group action can be diffusion-wise reduced to an SDS on the quotient space. We also show necessary and sufficient conditions for an SDS to be projectable via a surjective map. We then introduce the notion of integrability of SDS’s, and extend the results on the existence and structure-preserving property of Liouville torus actions from the classical case to the case of integrable SDS’s. We also show how integrable SDS’s are related to compatible families of integrable Riemannian metrics on manifolds.

Preservers of radial unitary similarity functions on products of operators

Let S be a subset of the algebra L(H) of all bounded linear operators on an infinite-dimensional complex Hilbert space H, and let f:S→[0,∞) be a radial unitary similarity invariant function. Under some assumption on S and f, characterizations are obtained for surjective maps ϕ on S satisfyingf(ϕ(T)∘ϕ(S))=f(T∘S)(T,S∈S), where the binary operation ∘ stands for the product or the Jordan semi-triple product on operators. Analogous descriptions are obtained for the finite-dimensional case, without the surjectivity assumption on ϕ.

On the Surjectivity of Generalized Isometries on the Positive Definite Cone of Matrices

This short note is concerned with maps on the cone of positive definite matrices which preserve certain generalized distance measures. We mean sorts of distances which are connected with quantum divergences, relative entropies, or related to geometrical structures on the positive definite cone. The novelty here is that we do not assume that our maps, the so-called generalized isometries, are surjective. As we will see, in some cases, those transformations turn to be automatically surjective, while in other cases, we can determine the precise structure of all not necessarily surjective such maps. In spite of the presented results, a number of related questions are left open and proposed to be the objects of further research.

Immersion of Nonlinear Systems into Higher Order Systems

Two nonlinear systems having the same number of inputs, but not the same number of state variables, are considered. The problem of the existence of two invertible state feedback laws and a surjective mapping from the higher dimensional state space of one system into the lower dimensional state space of the second system is stated, such that the first system dynamics reduces exactly to the second system dynamics. This problem generalizes the linear feedback equivalence problem of two systems and is fully solved in the special case of feedback linearizable systems.

Some properties of two-dimensional surjective p-homogeneous maps

The properties of real p-homogeneous polynomial maps in R2 are examined. The relation between surjectivity and the existence of a nontrivial zero is investigated. Additionally, the relation between surjectivity and stable surjectivity is studied. Examples are discussed.

Additive preservers of Drazin invertible operators with bounded index

Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n ≥ 1, we show that an additive surjective map Φ on B(X) preserves Drazin invertible operators of index non-greater than n in both directions if and only if Φ is either of the form Φ(T) = αATA −1 or of the form Φ(T) = αBT *B −1 where α is a non-zero scalar, A: X → X and B: X* → X are two bounded invertible linear or conjugate linear operators.

Semi-classical Locality for the Non-relativistic Path Integral in Configuration Space

In a previous paper, we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe $M$, existing on timeless configuration space $\mathcal{M}$. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. To deal with the challenges gauge symmetries may pose to a good definition of locality, I start by demanding symmetries to have an action on $\mathcal{M}$ so that the quotient wrt the symmetries respects certain factorizations of $\mathcal{M}$. These factorizations are algebraic splits of $\mathcal{M}$ into sub-spaces $\mathcal{M}=\bigoplus_i \mathcal{M}_{O_i}$-- each factor corresponding to a physical sub-region $O_i$. This deals with kinematic locality, but locality in full can only emerge dynamically, and is not postulated. I describe conditions under which it can be said to have emerged. The dynamics of $O$ is independent of its complement, $M-O$, if the projection of extremal curves on $M$ onto the space of extremal curves intrinsic to $O$ is a surjective map. This roughly corresponds to $e^{i\hat{H}t}\circ \mathsf{pr}_{\mbox{O}}= \mathsf{pr}_{\mbox{ O}}\circ e^{i\hat{H}t}$, where $\mathsf{pr}_{\mbox{ O}}:\mathcal{M}\rightarrow \mathcal{M}_O^{\partial O}$ is a linear projection. This criterion for locality can be made approximate -- an impossible feat had it been already postulated -- and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are \emph{mutually independent} according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is what one would like to obtain from any definition of locality.

Conformally natural extensions of continuous circle maps: II. The general case

We introduce a procedure for extending continuous circle maps in a conformally natural way to continuous maps from the closed disk, bounded by the circle to itself which generalizes Douady-Earle’s method of constructing conformally natural extensions of circle homeomorphisms. We also provide a criterion for the extensions to be surjective maps from the closed disk to itself.

Smooth surjections and surjective restrictions

Given a surjective mapping f : E -> F between Banach spaces, we investigate the existence of a subspace G of E, with the same density character as F, such that the restriction of f to G remains surjective. We obtain a positive answer whenever f is continuous and uniformly open. In the smooth case, we deduce a positive answer when f is a C-1-smooth surjection whose set of critical values is countable. Finally we show that, when f takes values in the Euclidean space R-n, in order to obtain this result it is not sufficient to assume that the set of critical values of f has zero-measure.

Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

In this paper, we will discuss some applications of almost surjective epsilon-isometry mapping, one of them is in Lorentz space ( L_(p,q)-space). Furthermore, using some classical theorems of w star-topology and concept of closed subspace -complemented, for every almost surjective epsilon-isometry mapping f : X to Y, where Y is a reflexive Banach space, then there exists a bounded linear operator T : Y to X with such that for every x in X.

Completely square-zero preserving maps on

Let be a complex Hilbert space with and be a surjective map on the real linear space of all bounded self-adjoint operators on . For each , let be defined byIn this paper, we show that if preserves square-zero operators in both directions, then there exist and a unitary or an antiunitary operator U on such that , for all . We also give a description of all continuous surjective maps , such that preserves square-zero operators in both directions.

FINAL STRUCTURED SUBSPACES

Structured spaces or differential spaces are a generalization of the concept of smooth manifolds. Let (𝑀,𝜏,𝐶) be a structured space in the sense of Mostow and let 𝑓∶ (𝑀,𝜏,𝐶 ) → 𝑁 where 𝑁 is arbitrary, be a surjective function. There is a unique differential structure 𝐷 on 𝑁 determined by 𝑓 called the final, or identification differential structure, and the space 𝑁 then called the final structured space . In this paper, we will study structured subspaces of the final structured space (𝑁,𝐷). The case when the subspace of the space (𝑁,𝐷) is open is studied; and we prove that this subspace is also final. Some related concepts are defined and important properties are proved.KEYWORDS Diffeomorphism, Differential Structure, Smooth map, Final Structured Space, Structured Subspace.2010 Mathematics Subject Classification: 57R10, 58A40

Finitely presented groups and the Whitehead nightmare

We define a "nice representation" of a finitely presented group \Gamma as being a non-degenerate essentially surjective simplicial map f from a „nice" space X into a 3-complex associated to a presentation of \Gamma , with a strong control over the singularities of f , and such that X is WGSC ( weakly geometrically simply connected ), meaning that it admits a filtration by simply connected and compact subcomplexes. In this paper we study such representations for a very large class of groups, namely QSF ( quasi-simply filtered ) groups, where QSF is a topological tameness condition of groups that is similar to, but weaker than, WGSC. In particular, we prove that any QSF group admits a WGSC representation which is locally finite, equivariant and whose double point set is closed.