Hamilton's Turns for the Lorentz Group

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by ``turns'', which are equivalence classes of directed great circle arcs on the unit sphere $S^2$, in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram law for the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group $SU(1,1) = Sp(2, R) = SL(2,R)$, the double cover of SO(2,1). The present work develops a theory of turns for $SL(2,C)$, the double and universal cover of SO(3,1) and $SO(3,C)$, rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.Comment: 13 pages, Late

The Image of the City Out of the Underlying Scaling of City Artifacts or Locations

Two fundamental issues surrounding research on the image of the city respectively focus on the city's external and internal representations. The external representation in the context of this paper refers to the city itself, external to human minds, while the internal representation concerns how the city is represented in human minds internally. This paper deals with the first issue, i.e., what trait the city has that make it imageable? We develop an argument that the image of the city arises from the underlying scaling of city artifacts or locations. This scaling refers to the fact that, in an imageable city (a city that can easily be imaged in human minds), small city artifacts are far more common than large ones; or alternatively low dense locations are far more common than high dense locations. The sizes of city artifacts in a rank-size plot exhibit a heavy tailed distribution consisting of the head, which is composed of a minority of unique artifacts (vital and very important), and the tail, which is composed of redundant other artifacts (trivial and less important). Eventually, those extremely unique and vital artifacts in the top head, i.e., what Lynch called city elements, make up the image of the city. We argue that the ever-increasing amount of geographic information on cities, in particular obtained from social media such as Flickr and Twitter, can turn research on the image of the city, or cognitive mapping in general, into a quantitative manner. The scaling property might be formulated as a law of geography. Keywords: Scaling of geographic space, face of the city, cognitive maps, power law, and heavy tailed distributions.Comment: 13 pages, 9 figures, 2 table

Symmetries of the Dirac operators associated with covariantly constant Killing-Yano tensors

The continuous and discrete symmetries of the Dirac-type operators produced by particular Killing-Yano tensors are studied in manifolds of arbitrary dimensions. The Killing-Yano tensors considered are covariantly constant and realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. The Dirac operators are related among themselves through continuous or discrete transformations. It is shown that the groups of the continuous symmetry can be only U(1) and SU(2), specific to (hyper-)Kahler spaces, but arising even in cases when the requirements for these special geometries are not fulfilled. The discrete symmetries are also studied obtaining the discrete groups Z_4 and Q. The briefly presented examples are the Euclidean Taub-NUT space and the Minkowski spacetime.Comment: 27 pages, latex, no figures, final version to be published in Class. Quantum Gravit

Scaling of Geographic Space as a Universal Rule for Map Generalization

Map generalization is a process of producing maps at different levels of detail by retaining essential properties of the underlying geographic space. In this paper, we explore how the map generalization process can be guided by the underlying scaling of geographic space. The scaling of geographic space refers to the fact that in a geographic space small things are far more common than large ones. In the corresponding rank-size distribution, this scaling property is characterized by a heavy tailed distribution such as a power law, lognormal, or exponential function. In essence, any heavy tailed distribution consists of the head of the distribution (with a low percentage of vital or large things) and the tail of the distribution (with a high percentage of trivial or small things). Importantly, the low and high percentages constitute an imbalanced contrast, e.g., 20 versus 80. We suggest that map generalization is to retain the objects in the head and to eliminate or aggregate those in the tail. We applied this selection rule or principle to three generalization experiments, and found that the scaling of geographic space indeed underlies map generalization. We further relate the universal rule to T\"opfer's radical law (or trained cartographers' decision making in general), and illustrate several advantages of the universal rule. Keywords: Head/tail division rule, head/tail breaks, heavy tailed distributions, power law, and principles of selectionComment: 12 pages, 9 figures, 4 table

Matrix representations of a special polynomial sequence in arbitrary dimension

This paper provides an insight into different structures of a special polynomial sequence of binomial type in higher dimensions with values in a Clifford algebra. The elements of the special polynomial sequence are homogeneous hypercomplex differentiable (monogenic) functions of different degrees and their matrix representation allows to prove their recursive construction in analogy to the complex power functions. This property can somehow be considered as a compensation for the loss of multiplicativity caused by the non-commutativity of the underlying algebra.Fundação para a Ciência e a Tecnologia (FCT

At the Crossroads of Sustainability: The Natural Recompositioning of Architecture

It is widely acknowledged that the mantra of sustainability has triggered a fundamental reversal in the core of design practice: If the original purpose of architecture was to protect humans from the destructive actions of nature,today it should protect nature from the damaging actions of humans. But sustainable design is far from being a coherent body of fully totalized ideas:it has a broad spectrum of disputing interpretations that oscillate between the deterministic models of energy control and technological efficiencies, and the moralistic and romantic approaches that attempt to see in nature and natural processes a fundamental way to de-escalate the global urban footprint and its associated patterns of consumption. However, mainstream green design has been evolving by progressively absorbing the narrative of deep ecology. Nature has been being integrated into architecture literally, by inserting vegetation onto buildings; digitally, by bringing environmental data into the design process (climate records, wind streams, sun rotation and air flows are computed, modelled and effectually shape architectures), and transcendentally, by claiming that sustainable architecture nurtures “the existing and evolving connections between spiritual and material consciousness.” The acknowledgement of the inexorable affiliation between architecture and the environment is, of course, not exactly new. What is distinctive today is the reification of the role of nature in architecture as an ideological stance, now totally intertwined with state-of-art data processing and the market-driven tools brought by Natural Capitalism. This paper will examine emblematic “green” buildings produced by leading architects such as Pelli Clarke Pelli, William McDonough, Stefano Boeri, Norman Foster and BIG in the light of Tim Morton’s, Slavoj Zizek and Bruno Latour’s critique of nature. It will illustrate how, despite being able to successfully forge new creative freedoms by exploring hybridizations between the domains of design and science, sustainability’s self-righteous “naturalistic” narrative is enabling a vision of the architect as an “expert manager” focused on producing projects of ecologic “beautification” while assumed to be “saving the world,” effectively depoliticizing the architectural practice. Nevertheless, these examples attest that there is a vast and fertile field of ideas to be explored and in this regard it is important to underline that we are still in the embryonic outset of the engagement of architecture with sustainability

The Use of Patterns in Service Composition

E-services are services that can be invoked over the Internet. One likely use of e-services would be to build business applications that can employ e-services from different service providers. This potential use of composite services in business settings highlights the issues of payment mechanisms, reliability, trust, inter-operability and service guarantees between different service providers. It also becomes essential to choose e-services that best fulfil the requirements of a particular business application. We investigate how patterns can be used in service composition to help in the development of business applications based on e-services