Felt Spaces

The chapter deals with the atmospheric space provided by music or sound art. The main point is that this is not Euclidian space, that is, the space of geometry, but felt space. Starting from the two classical concepts of space—topos (Aristotle) and spatium (Descartes)—the article introduces with the help of New Phenomenology—the concept of felt space (leiblicher Raum). Turning to music, in particular to Sound-Scapes (Murray Schafer) and sound art, Böhme shows that sound is tuning the felt space or better—producing a certain atmosphere. Atmospheres are spaces with a certain mood.

Triangle conics and cubics

This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also studied in detail by H.M. Cundy and C.F. Parry recently. The main task of the article was to develop an algorithm for creating curves, which pass through triangle centers. During the research, it was noticed that some different triangle centers in distinct triangles coincide. The simplest example: an incenter in a base triangle is an orthocenter in an excentral triangle. This was the key for creating an algorithm. Indeed, we can match points belonging to one curve (base curve) with other points of another triangle. Therefore, we get a new intersting geometrical object. During the research were derived number of new triangle conics and cubics, were considered their properties in Euclidian space. In addition, was discussed corollaries of the obtained theorems in projective geometry, what proves that all of the descovered results could be transfered to the projeticve plane.

Geometry-Aware Discriminative Dictionary Learning for PolSAR Image Classification

In this paper, we propose a new discriminative dictionary learning method based on Riemann geometric perception for polarimetric synthetic aperture radar (PolSAR) image classification. We made an optimization model for geometry-aware discrimination dictionary learning in which the dictionary learning (GADDL) is generalized from Euclidian space to Riemannian manifolds, and dictionary atoms are composed of manifold data. An efficient optimization algorithm based on an alternating direction multiplier method was developed to solve the model. Experiments were implemented on three public datasets: Flevoland-1989, San Francisco and Flevoland-1991. The experimental results show that the proposed method learned a discriminative dictionary with accuracies better those of comparative methods. The convergence of the model and the robustness of the initial dictionary were also verified through experiments.

Invariant quadratic operator associated with Linear Canonical Transformations

The main purpose of this work is to identify the general quadratic operator which is invariant under the action of Linear Canonical Transformations (LCTs). LCTs are known in signal processing and optics as the transformations which generalize certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this paper, LCTs corresponding to a general pseudo-Euclidian space are considered. Explicit calculations are performed for the monodimensional case to identify the corresponding LCT invariant operator then multidimensional generalizations of the obtained results are deduced. It was noticed that the introduction of a variance-covariance matrix, of coordinate and momenta operators, and a pseudo-orthogonal representation of LCTs facilitate the identification of the invariant quadratic operator. According to the calculations carried out, the LCT invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of these coordinates and momenta operators themselves. The eigenstates of the LCT invariant operator are also identified with it and some of the main possible applications of the obtained results are discussed.

Quantum SU(2|1) supersymmetric ℂN Smorodinsky-Winternitz system

We study quantum properties of SU(2|1) supersymmetric (deformed $$ \mathcal{N} $$ N = 4, d = 1 supersymmetric) extension of the superintegrable Smorodinsky-Winternitz system on a complex Euclidian space ℂN. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU(2|1) supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the “even-odd” splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU(2|1) multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU(2|1, 1) quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU(2|1, 1) ones is found.

Predicting the Fatigue Life of an AlSi9Cu3 Porous Alloy Using a Vector-Segmentation Technique for a Geometric Parameterisation of the Macro Pores

Most of the published research work related to the fatigue life of porous, high-pressure, die-cast structures is limited to a consideration of individual isolated pores. The focus of this article is on calculating the fatigue life of high-pressure, die-cast, AlSi9Cu3 parts with many clustered macro pores. The core of the presented methodology is a geometric parameterisation of the pores using a vector-segmentation technique. The input for the vector segmentation is a μ-CT scan of the porous material. After the pores are localised, they are parameterised as 3D ellipsoids with the corresponding orientations in the Euclidian space. The extracted ellipsoids together with the outer contour are then used to build a finite-element mesh of the porous structure. The stress–strain distribution is calculated using Abaqus and the fatigue life is predicted using SIMULIA fe-safe. The numerical results are compared to the experimentally determined fatigue lives to prove the applicability of the proposed approach. The outcome of this research is a usable tool for estimating the limiting quantity of a structure’s porosity that still allows for the functional performance and required durability of a product.

To the Question of Gauss’s Curvature in n-Dimensional Euclidian Space

An alternative way of calculating the Gauss curve of the surface in the Cartesian coordinates in a three-dimensional case has been proposed, and its generalization in the n- dimension case of measuring Euclidean space has been given.

Historical and Imagined GIS Borderlandscapes of the American West: Larry McMurtry's Lonesome Dove Tetralogy and LA Noirscapes

Larry McMurtry's Pulitzer Prize winning counter-western Lonesome Dove (part of a tetralogy, set between 1840 and 1900) and the works of LA Noir detective fiction writers (from the 1940s to 1997) represent the American west and urban southwest of Los Angeles as a dynamic mosaic of human and environmental borderlandscapes. McMurtry's perspective provides an Anglo-European eye, influenced by Cervantean Iberian literary tropes on the transformation of the West from indigenous and Spanish trails to American rail-road tracks. The LA Noirscapes map phenomenologically illustrates how the location of novel settings cluster in contiguous and convergent places on a street grid palimpsest of Los Angeles between 1949 and 1997. Employing HumGIS methods, this essay considers the marriage of empirical cartography and impressionistic topography; the former concerned with latitude, longitude and space, the latter with plotting literary, historical and cultural perceptions and experiences of place. By engaging the concept of Euclidian space with the phenomenology of place, geographers can contextualize field work, and other methods with literary, cartographical and GIS analysis to uncover the means to craft new avenues to study the dynamic and symbiotic formations of historical landscapes, identities, senses of place and location.