Immersion Dimension Research Articles

Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension

Abstract The s-th higher topological complexity TC s ⁡ ( X ) {\operatorname{TC}_{s}(X)} of a space X can be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when X = ℝ ⁢ P m {X=\operatorname{\mathbb{R}P}^{m}} , the real projective space of dimension m. In particular, we describe a number r ⁢ ( m ) {r(m)} , which depends on the structure of zeros and ones in the binary expansion of m, and with the property that 0 ≤ s ⁢ m - TC s ⁡ ( ℝ ⁢ P m ) ≤ δ s ⁢ ( m ) {0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)} for s ≥ r ⁢ ( m ) {s\geq r(m)} , where δ s ⁢ ( m ) = ( 0 , 1 , 0 ) {\delta_{s}(m)=(0,1,0)} for m ≡ ( 0 , 1 , 2 ) mod 4 {m\equiv(0,1,2)\bmod 4} . Such an estimation for TC s ⁡ ( ℝ ⁢ P m ) {\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} appears to be closely related to the determination of the Euclidean immersion dimension of ℝ ⁢ P m {\operatorname{\mathbb{R}P}^{m}} . We illustrate the phenomenon in the case m = 3 ⋅ 2 a {m=3\cdot 2^{a}} . In addition, we show that, for large enough s and even m, TC s ⁡ ( ℝ ⁢ P m ) {\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} is characterized as the smallest positive integer t = t ⁢ ( m , s ) {t=t(m,s)} for which there is a suitable equivariant map from Davis’ projective product space P 𝐦 s {\mathrm{P}_{\mathbf{m}_{s}}} to the ( t + 1 ) {(t+1)} -st join-power ( ( ℤ 2 ) s - 1 ) ∗ ( t + 1 ) {((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}} . This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating TC 2 {\operatorname{TC}_{2}} to the immersion dimension of real projective spaces.

Hearing Words Written

Over the past few years, I have developed a form of composition – which I callmusic–text–film –in which I explore the dynamics between sound, words and visuals. In this article I will attempt to explain how meaning is constructed in the interplay between these layers of media. Taking as an example three of my works,Subliminal: The Lucretian Picnic,Dreams of the BlindandThe Arrest, I analyse and discuss aspects of narrative, point of view, metaphor and cross-modal perception, as a way of understanding how multimedia art, specifically in the audiovisual domain, is experienced. One of the issues that arose out of these pieces was the question of location of the ‘voice’. It is as if a state of limbo is created between the narrative voice of the text and the implied voice of the music, due to the absence of a conventional focal point to pin it on – an actor or a singer. I would like to suggest that because of this vacancy and the way the projected word takes the place of the sung or spoken voice, the inner voice of the audience becomes activated. This then becomes a vital immersive dimension in the performance, as the inner voice of the audience finds its place within the space of the composition.

Visual power: The scopic regime of military drone operations

This article analyzes how visual scopic regimes of military drones configure violence as a form of man hunting. For the French philosopher Grégoire Chamayou, man hunting embodies a type of cynegetic (hunting related) violence, which military drones can execute by power surveillance. Research often focuses on the political, legal, anthropological, and ethical aspects of this type of violence; the aspects of its visual framing are often underexposed. In order to change this shortcoming, this article draws attention to the medial aspects of this violence by investigating the drone’s scopic regime. The scopic regime refers to the drone’s visual configuration, i.e. its ocular operations of capture, its optical perspective on the target, the visual sensing of the drone pilot, as well as the target’s range of vision. Three scopic dimensions of military drones, namely hypervisibility, visual immersion, and invisibility are investigated. In doing so, this article explores how drones stage, interpret, convey, mediate, and execute violence as man hunting. Excursions to the works of contemporary visual artists are conducted in order to illustrate aesthetic interventions against the drone’s visual superpower.

To Touch and Be Touched

In my article, I will present as a case study the collection exhibition shown at the Museum of Contemporary Art Kiasma in 2016.[1] The starting point for the exhibition was the metaphor of touch. As a concept, touch is ambivalent: it is more intimate than sight, which has been the traditional metaphor for knowledge in Western thinking. Yet touching is also about grasping or understanding, as in it we are taking hold of something. Our curatorial team, Eija Aarnio, Arja Miller, and myself, was interested in touch early on, because with it the distance to the observed is lost: when touching something we, too, are being touched. To be clear, we did not want to create an exhibition where spectators are actually able to touch. Instead, we were looking at the collections of the museum and searching for artworks that would “touch” us—works that were able get under our skin. While forming the conceptual core of the exhibition, our curatorial team recognized a tension in the way in which “touchy issues,” affects and emotions, are perceived in our society. On the one hand, we were interested in emotions and affects raised by the artworks. We wanted to focus on the immersive dimension of the art that seems to escape verbalization, a dimension that makes use of the multisensory experience and addresses the viewer in a direct manner. On the other hand, we also became aware of how society in general has been taken over by an emotional surge. If previously feelings and emotions were not meant to be shown in public, today they have become commonplace. What was emotional and affective seems no longer to be private, but shared and public.[2] In fact, strong emotions seem to be a prerequisite for success, be it a matter of reality television or politics. This is also connected to the search for extreme emotions and experiences, an aspect we felt needed to be included in the exhibition—not the least because in museums’ competition for audiences, the experience-laden works are often seen as an answer.

Motion planning in real flag manifolds

Starting from Borel's description of the mod-2 cohomology of real flag manifolds, we give a minimal presentation of the cohomology ring for semi complete flag manifolds $F_{k,m}:=F(1,\ldots,1,m)$ where $1$ is repeated $k$ times. The information is used in order to estimate Farber's topological complexity of these spaces when $m$ approaches (from below) a 2-power. In particular, we get almost sharp estimates for $F_{2,2^e-1}$ which resemble the known situation for the real projective spaces $F_{1,2^e}$. Our results indicate that the agreement between the topological complexity and the immersion dimension of real projective spaces no longer holds for other flag manifolds. More interestingly, we also get corresponding results for the $s$-th (higher) topological complexity of these spaces. Actually, we prove the surprising fact that, as $s$ increases, the estimates become stronger. Indeed, we get several full computations of the higher motion planning problem of these manifolds. This property is also shown to hold for surfaces: we get a complete computation of the higher topological complexity of all closed surfaces (orientable or not). A homotopy-obstruction explanation is included for the phenomenon of having a cohomologically accessible higher topological complexity even when the regular topological complexity is not so accessible.

Note on immersion dimension of real Grassmannians

Immersion dimension of a smooth manifold is the least integer d such that there is an immersion of that manifold into d-dimensional Euclidean space. By using the obstruction theory, we determine the exact value of the immersion dimension for Grassmann manifolds G3,n when n is a power of two.

Complex-projective and lens product spaces

Let \(t\) be a positive integer. Following work of D. M. Davis, we study the topology of complex-projective product spaces, i.e. quotients of cartesian products of odd dimensional spheres by the diagonal \(S^1\)-action, and of the \(t\)-torsion lens product spaces, i.e. the corresponding quotients when the action is restricted to the \(t\)th roots of unity. For a commutative complex-oriented cohomology theory \(h^*\), we determine the \(h^*\)-cohomology ring of these spaces (in terms of the \(t\)-series for \(h^*\), in the case of \(t\)-torsion lens product spaces). When \(h^*\) is singular cohomology with mod 2 coefficients, we also determine the action of the Steenrod algebra. We show that these spaces break apart after a suspension as a wedge of desuspensions of usual stunted complex-projective (\(t\)-torsion lens) spaces. We estimate the category and topological complexity of complex-projective and lens product spaces, showing in particular that these invariants are usually much lower than predicted by the usual dimensional bounds. We extend Davis’ analysis of manifold properties such as immersion dimension, (stable) span, and (stable) parallelizability of real-projective product spaces to the complex-projective and lens product cases.

Reliance, Immersion, and Enjoyment: An Exploratory Socio-Psychological Analysis on Internet Involvement

The purpose of this paper is to explore the relationships between various socio-psychological variables and the level of involvement in the Internet. Two hundred thirty-four college students completed a self-administered questionnaire that contains the measures of Internet involvement, Internet usage, face-to-face interaction anxiety, computer ownership, and socio-demographic variables. A factor analysis identified three dimensions of Internet involvement: Reliance, Immersion, and Enjoyment. The results of stepwise regressions showed that face-to-face interaction anxiety is a significant indicator of both the Reliance and Immersion dimensions of Internet involvement. Age was another positive indicator of the Immersion dimension, whereas computer ownership contributed significantly to the prediction of the Reliance dimension of Internet involvement. The Enjoyment dimension was not significantly affected by any of the independent variables.

Topological complexity of motion planning in projective product spaces

We study Farber's topological complexity (TC) of Davis' projective product spaces (PPS's). We show that, in many non-trivial instances, the TC of PPS's coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in high contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of non-simply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension not exceeding 2, and now in this work for infinite families of PPS's. We discuss general bounds for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS's through generalized concepts of axial maps and, alternatively, non-singular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS's.

Projective product spaces

Let nbar=(n_1,...,n_r). The quotient space P_nbar:=(S^{n_1} x...x S^{n_r})/(x ~ -x)is what we call a projective product space. We determine the integral cohomology ring and the action of the Steenrod algebra. We give a splitting of Sigma P_nbar in terms of stunted real projective spaces, and determine when S^{n_i} is a product factor. We relate the immersion dimension and span of P_nbar to the much-studied sectioning question for multiples of the Hopf bundle over real projective spaces. We show that the immersion dimension of P_nbar depends only on min(n_i), sum n_i, and r, and determine its precise value unless all n_i exceed 9. We also determine exactly when P_nbar is parallelizable.

Symmetric topological complexity of projective and lens spaces

For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even torsion lens spaces and complex projective spaces are discussed.

Moments of Meeting: An Exploration of the Implicit Dimensions of Empathic Immersion in Adult and Child Treatment

Such staples of the child treatment process as nonverbal and noninterpretative mechanisms have recently come into focus in adult treatment. These new components of adult treatment, illuminated by the Boston Process of Change Study Group, enhance our appreciation of the nonverbal interpretive contributions of both analyst and patient. In an exploration of the work of the Boston Process of Change Study Group, this paper focuses on such key terms and concepts as implicit relational knowing, now moments, and moments of meeting. Through vignettes from adult and child treatments, the authors explore the application of these concepts. The authors suggest that (1) accounting for “implicit relational knowing” enables analysts to include the nonverbal and often elusive dimensions of the treatment process in their understanding leading to an expansion of technique and (2) allowing for the “authentic” or in the terms of the authors, the “wholehearted,” facilitates the spontaneity that is helpful in accessing and therapeutically handling those “now moments” and “moments of meeting” that often constitute the pivotal points in a treatment.

Lower bounds for the immersion dimension of homogeneous spaces

The object of the present work is to express characteristic numbers of a homogeneous space G/U which are related to the immersion dimension of G/U by Lie group invariants of G and U. New concrete nonimmersion theorems for flag manifolds and other homogeneous spaces are proved.

A Framework for Immersive Virtual Environments (FIVE): Speculations on the Role of Presence in Virtual Environments

This paper reviews the concepts of immersion and presence in virtual environments (VEs). We propose that the degree of immersion can be objectively assessed as the characteristics of a technology, and has dimensions such as the extent to which a display system can deliver an inclusive, extensive, surrounding, and vivid illusion of virtual environment to a participant. Other dimensions of immersion are concerned with the extent of body matching, and the extent to which there is a self-contained plot in which the participant can act and in which there is an autonomous response. Presence is a state of consciousness that may be concomitant with immersion, and is related to a sense of being in a place. Presence governs aspects of autonomie responses and higher-level behaviors of a participant in a VE. The paper considers single and multiparticipant shared environments, and draws on the experience of ComputerSupported Cooperative Working (CSCW) research as a guide to understanding presence in shared environments. The paper finally outlines the aims of the FIVE Working Group, and the 1995 FIVE Conference in London, UK.

Immersion dimension for real Grassmannians

Immersion dimension for real Grassmannians