Cartesian Representation Research Articles

On the Cartesian Representation of the Molecular Polarizability Tensor Surface by Polynomial Fitting to Ab Initio Data.

We describe an approach to constructing an analytic Cartesian representation of the molecular dipole polarizability tensor surface in terms of polynomials in interatomic distances with a training set of ab initio data points obtained from a molecular dynamics (MD) simulation or by any other available means. The proposed formulation is based on a perturbation treatment of the unmodified point dipole polarizability model of Applequist [ J. Am. Chem. Soc. 1972, 94, 2952] and is shown here to be, by construction (i) free of short-range or other singularities or discontinuities, (ii) symmetric and translationally invariant, and (iii) nonreliant on a body-fixed coordinate system. Permutational invariance of like nuclei is demonstrated to be readily applicable, making this approach useful for highly fluxional and reactive systems. Derivation of the method is described in detail, adding brief didactic numerical examples of H2 and H2O and concluding with an MD simulation of the Raman spectrum of H5O2+ at 300 K with the polarizability tensor fitted to CCSD(T)/aug-cc-pVTZ data obtained using the HBB-4B potential [ J. Chem. Phys. 2005, 122, 044308].

Calculation of electric quadrupole linestrengths for diatomic molecules: Application to the H2, CO, HF, and O2 molecules.

We present a unified variational treatment of the electric quadrupole (E2) matrix elements, Einstein coefficients, and linestrengths for general open-shell diatomic molecules in the general purpose diatomic code Duo. Transformation relations between the Cartesian representation (typically used in electronic structure calculations) to the tensorial representation (required for spectroscopic applications) of the electric quadrupole moment components are derived. The implementation has been validated against accurate theoretical calculations and experimental measurements of quadrupole intensities of 1H2 available in the literature. We also present accurate electronic structure calculations of the electric quadrupole moment functions for the X1Σ+ electronic states of CO and HF, as well as for the a1Δg-b1Σg + quadrupole transition moment of O2 with the MRCI level of theory. Accurate infrared E2 line lists for 12C16O and 1H19F are provided. A demonstration of spectroscopic applications is presented by simulating E2 spectra for 12C16O, H19F, and 16O2 (Noxon a1Δg-b1Σg + band).

Skyrme pseudopotentials at next-to-next-to-leading order: Construction of local densities and first symmetry-breaking calculations

There is an ongoing quest to improve on the spectroscopic quality of nuclear energy density functionals (EDFs) of the Skyrme type through extensions of its traditional form. One direction for such activities is the inclusion of terms of higher order in gradients in the EDF. We report on exploratory symmetry-breaking calculations performed for an extension of the Skyrme EDF that includes central terms with four gradients at next-to-next-to-leading order (N2LO) and for which the high-quality parametrization SN2LO1 has been constructed recently [P. Becker et al, Phys. Rev. C 96, 044330 (2017)]. Up to now, the investigation of such functionals with higher-order terms was limited to infinite matter and spherically symmetric configurations of singly- and doubly-magic nuclei. We address here nuclei and phenomena that require us to consider axial and non-axial deformation, both for reflection-symmetric and also reflection-asymmetric shapes, as well as the breaking of time-reversal invariance. Achieving these calculations demanded a number of formal developments. These all resulted from the formulation of the N2LO EDF requiring the introduction of new local densities with additional gradients that are not present in the EDF at NLO. Their choice is not unique, but can differ in the way the gradients are coupled. While designing a numerical implementation of N2LO EDFs in Cartesian 3d coordinate-space representation, we have developed a novel definition and a new unifying notation for normal and pair densities that contain gradients at arbitrary order. The resulting scheme resolves several issues with some of the choices that have been made for local densities in the past, in particular when breaking time-reversal symmetry. Guided by general practical considerations, we propose an alternative form of the N2LO contribution to the Skyrme EDF that is built from a different set of densities.

Complex-Valued Disparity: Unified Depth Model of Depth from Stereo, Depth from Focus, and Depth from Defocus Based on the Light Field Gradient.

This paper proposes a unified depth model based on the light field gradient, in which estimated disparity is represented by the complex number. The complex-valued disparity by the proposed depth model can be represented in both the Cartesian and polar coordinates. In the Cartesian representation, the proposed depth model is represented by real and imaginary parts of the disparity. The real part can be used for disparity estimation with respect to the in-focus plane, whereas the imaginary part represents the non-Lambertian-ness. In the polar representation, the proposed depth model is expressed by the disparity magnitude and disparity angle. The disparity magnitude shows the relationship among depth from stereo, depth from focus, and depth from defocus, whereas the disparity angle shows whether or not the bundles of rays are flipped with respect to the in-focus plane. For disparity analysis, we present the real response, imaginary response, magnitude response, and angle response, which are represented by the three-dimensional volume. Experimental results on synthetic and real light field images show that the real and magnitude responses of the proposed depth model are valid for local disparity estimation.

Lessons from reinforcement learning for biological representations of space

Neuroscientists postulate 3D representations in the brain in a variety of different coordinate frames (e.g. ‘head-centred’, ‘hand-centred’ and ‘world-based’). Recent advances in reinforcement learning demonstrate a quite different approach that may provide a more promising model for biological representations underlying spatial perception and navigation. In this paper, we focus on reinforcement learning methods that reward an agent for arriving at a target image without any attempt to build up a 3D ‘map’. We test the ability of this type of representation to support geometrically consistent spatial tasks such as interpolating between learned locations using decoding of feature vectors. We introduce a hand-crafted representation that has, by design, a high degree of geometric consistency and demonstrate that, in this case, information about the persistence of features as the camera translates (e.g. distant features persist) can improve performance on the geometric tasks. These examples avoid Cartesian (in this case, 2D) representations of space. Non-Cartesian, learned representations provide an important stimulus in neuroscience to the search for alternatives to a ‘cognitive map’.

Resuscitating (and Refusing) Cartesian Representations of Daily Life: When Mobile and Grid Epistemologies of the City Meet

In community planning, the consequence of a failed or productive teaching and learning interaction could mean the preservation or destruction of someone’s house, a neighborhood school, a park, all of it. This article elucidates consistencies in how people collaborate across spatial epistemologies and power imbalances for making recommendations and decisions about communities. Holding two epistemic stances in tension—mobile and grid epistemologies—the article follows the arc of a design-based research study, beginning with ethnographic observations of participatory planning meetings, the findings from which informed the design of experimental teaching cases with youth. I focus on two vibrant and consequential instances of people walking others through a storyline of their home communities that moves in and out of epistemic commensurability. Building upon findings from interaction and multi-modal analyses, I argue that consequential learning occurred when people enacted relational attunement in which new spatial imaginaries of a community came into view and were informed by both mobile and grid epistemologies. This article serves as one instance of finding, analyzing, and designing for moments where the roles of “teaching” and “learning” are under negotiation or unknown, and how people engage with one another in politically charged and tenuous interactions.

The brain in a box: A toolbox for creating Cartesian geometric representations with isometric dimensions (Cgrids)

BackgroundThe folding of the human cortex complicates extraction of position information and recognition of patterns across the cortical surface. New MethodAs straight lines correspond better to our intuitions in spatial orientation, we developed an approach for imposing Cartesian grids on portions of the cortical surface, which can then be represented in a rectangular matrix. These functions have been implemented in the Cgrid (Cartesian Geometric Representation with Isometric Dimensions) toolbox. Cgrids can be generated based on regions of interest, or combinations thereof, according to any one of the Freesurfer’s annotation schemes. ResultsThe toolbox was evaluated using the surface reconstructions of T1-weighted images of 30 subjects, and 17 different Cgrids that in combination covered nearly the entire surface area of the brain. The vast majority of Cgrids (90.4 %) could be generated without issues. Comparison with Existing method(s)The toolbox facilitates spatial orientation and pattern recognition, in addition to allowing detailed comparison between the left and right hemisphere, and bringing existing volumetric tools to bear on surface-based data. The output of the toolbox is fully compatible with most existing fMRI/MRI analyses packages, and is immediately suitable as input for second level analysis. ConclusionsThe toolbox has the potential for broad applicability, especially when ease of data handling and representation are critical factors. The toolbox can be downloaded from: https://github.com/mathijsraemaekers/Cgrid-toolbox.

What generalizations do students achieve with respect to trigonometric functions in the transition from angles in degrees to real numbers?

The aim of this study is to examine prospective mathematics teachers’ generalizations of trigonometric functions from the unit circle to the Cartesian coordinate system. The researcher developed a test that aimed to reveal students’ generalizations, as well as the possible differences between them. The test was administered to 30 students who were near completion of their university degree program. The findings showed that the students were unable to establish the link between the unit circle and the Cartesian coordinate representation system; and therefore, they were not able to interpret the outputs of trigonometric functions with input of a real number that is not a multiple of π. The researcher also found that the students had developed certain misconceptions regarding the properties of trigonometric functions. To improve the teaching of trigonometric functions an instructional sequence and an alternative definition for trigonometric functions is proposed.

A Novel 2D Standard Cartesian Representation for the Human Sensorimotor Cortex

For some experimental approaches in brain imaging, the existing normalization techniques are not always sufficient. This may be the case if the anatomical shape of the region of interest varies substantially across subjects, or if one needs to compare the left and right hemisphere in the same subject. Here we propose a new standard representation, building upon existing normalization methods: Cgrid (Cartesian geometric representation with isometric dimensions). Cgrid is based on imposing a Cartesian grid over a cortical region of interest that is bounded by anatomical (atlas-based) landmarks. We applied this new representation to the sensorimotor cortex and we evaluated its performance by studying the similarity of activation patterns for hand, foot and tongue movements between subjects, and similarity between hemispheres within subjects. The Cgrid similarities were benchmarked against the similarities of activation patterns when transformed into standard MNI space using SPM, and to similarities from FreeSurfer’s surface-based normalization. For both between-subject and between-hemisphere comparisons, similarity scores in Cgrid were high, similar to those from FreeSurfer normalization and higher than similarity scores from SPM’s MNI normalization. This indicates that Cgrid allows for a straightforward way of representing and comparing sensorimotor activity patterns across subjects and between hemispheres of the same subjects.

A Geometric Interpretation of Reference Frames and Transformations: dq0, Clarke, and Park

Transformations between abc, stationary dq0 (αβ0) and rotating dq0 reference-frames are used extensively in the analysis and control of three-phase technologies such as machines and inverters. Previous work on deriving the matrices describing these transformations follows one of two approaches. The first approach derives Clarke's matrix by modifying symmetrical components. Park's matrix can be subsequently found from a rotation matrix. The second approach derives Park's matrix using trigonometric projection by interpreting the transformation as a rotation in the plane of the cross-section of a machine. Then, Clarke's matrix can be found trivially using a reference angle of zero in Park's matrix. This paper presents a third approach to deriving the Clarke and Park transformation matrices: a geometric interpretation. The approach exploits properties of the linear transformation using the Cartesian representation. We introduce the locus diagram of a three-phase quantity and show how these diagrams have applications in power quality. We show that, unlike a phasor diagram, a single locus diagram can fully represent a three-phase system with harmonics.

Thermodynamics of Bose gases from functional renormalization with a hydrodynamic low-energy effective action

The functional renormalization group for the effective action is used to construct an effective hydrodynamic description of weakly interacting Bose gases. We employ a scale-dependent parametrization of the boson fields developed previously to start the renormalization evolution in a Cartesian representation at high momenta and interpolate to an amplitude-phase one in the low-momentum regime. This technique is applied to Bose gases in one, two and three dimensions, where we study thermodynamic quantities such as the pressure and energy per particle. The interpolation leads to a very natural description of the Goldstone modes in the physical limit, and compares well to analytic and Monte-Carlo simulations at zero temperature. The results show that our method improves aspects of the description of low-dimensional systems, with stable results for the superfluid phase in two dimensions and even in one dimension.

Can Conformational Changes of Proteins Be Represented in Torsion Angle Space? A Study with Rescaled Ridge Regression.

Torsion angles are the natural degrees of freedom of protein structures. The ability to determine torsional variations corresponding to observed changes in Cartesian coordinates is highly valuable, notably to investigate the mechanisms of functional conformational changes or to develop computational models of protein dynamics. This issue is far from trivial in practice since the impact of modifying one torsion angle strongly depends on all other angles, and the compounding effects of small variations in bond lengths and valence angles can completely disrupt a protein fold. We demonstrate that naive strategies, such as directly comparing torsion angles between structures without correcting for variations in bond lengths and valence angles or fitting torsional variations without a proper regularization scheme, fail at producing an adequate representation of conformational changes in internal coordinates. In contrast, rescaled ridge regression, a method recently introduced to regularize multidimensional regressions with correlated explanatory variables, is shown to consistently identify a minimal set of torsion angles variations that closely reproduce changes in Cartesian coordinates. This torsional representation of conformational changes is shown to be robust to the choice of experimental structures. It also provides a better agreement with theoretical models of protein dynamics than the Cartesian representation, regarding notably the predominance of low-frequency normal modes in functional motions and the presence, in predicted equilibrium dynamics, of hints of natural selection for specific functional motions. The software is available at https://github.com/ugobas/tnm .

The look: A look at non-Cartesian re-presentation

This is an attempt to elucidate non-Cartesian re-presentation (NCR) as a creative methodology allowing for the revealing of nonhuman agencies. Cartesian representation has presented a world of nonhumans ready for consumption, even exploitation. NCR works to reveal what-is and a sense of relational being through the arts encounter, negating a sense of human exceptionalism through movement, reciprocity and slippage. In this proposition I draw from the writing of Barbara Bolt to re-engage with the male philosophies of the past, to re-signify the lines of thinking, to imagine the potentialities of re-presentation outside of the predominate fictive imaginary.

Anharmonic vibrational computations with a quartic force field for curvilinear coordinates

The direct vibrational self-consistent field (VSCF) method, which combines anharmonic vibrational theory with electronic structure calculations, is a sophisticated theoretical approach to calculate the vibrational spectra of molecules from first principles. Combining the VSCF approach with the quartic force field (QFF) is a good alternative to direct VSCF, with a lower computational cost. QFF is a 4th-order Taylor expansion of the potential energy surface near an equilibrium geometry. In this study, a new strategy is proposed to derive the QFF in terms of normal coordinates; the QFF coefficients are determined through numerical differentiations of the energy by representing the normal coordinates in internal rather than Cartesian coordinates. The VSCF/QFF-internal method was implemented in the General Atomic and Molecular Electronic Structure System electronic structure program and applied to the evaluations of the fundamental vibrational frequencies of HNO2, HNO3, H2O dimer, and H2O trimer, using Møller-Plesset second order perturbation theory and the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The results are much improved, especially for the intermolecular vibrational modes, compared with the Cartesian coordinate representation of the normal coordinates in the VSCF/QFF approach.

Data Representations for Segmentation of Vascular Structures Using Convolutional Neural Networks with U-Net Architecture.

Convolutional neural networks (CNNs) produce promising results when applied to a wide range of medical imaging tasks including the segmentation of tissue structures. However, segmentation is particularly challenging when the target structures are small with respect to the complete image data and exhibit substantial curvature as in the case of coronary arteries in computed tomography angiography (CTA). Therefore, we evaluated the impact of data representation of tubular structures on the segmentation performance of CNNs with U-Net architecture in terms of the resulting Dice coefficients and Hausdorff distances. For this purpose, we considered 2D and 3D input data in cross-sectional and Cartesian representations. We found that the data representation can have a substantial impact on segmentation results with Dice coefficients ranging from 60% to 82% reaching values similar to those of human expert annotations used for training and Hausdorff distances ranging from 1.38 mm to 5.90 mm. Our results indicate that a 3D cross-sectional data representation is preferable for segmentation of thin tubular structures.

Rapidly rotating Rayleigh-Bénard convection with a tilted axis.

We numerically explore the dynamics of an incompressible fluid heated from below, bounded by free-slip horizontal plates and periodic lateral boundary conditions, subject to rapid rotation about a distant axis that is tilted with respect to the gravity vector. The angle ϕ between the rotation axis and the horizontal plane measures the tilting of the rotation axis; it can be taken as a proxy for latitude if we think of a local Cartesian representation of the convective dynamics in a rotating fluid shell. The results of the simulations indicate the existence of three different convective regimes, depending on the value of ϕ: (1) sheared, intermittent large-scale winds in the direction perpendicular to the plane defined by the gravity and rotation vectors, when rotation is "horizontal" (ϕ=0^{∘}); (2) a large-scale cyclonic vortex tilted along the rotation axis, when the angle between the rotation axis and the gravity vector is relatively small (ϕ between about 45^{∘} and 90^{∘}); and (3) a new intermediate regime characterized by vertically sheared large-scale winds perpendicular to both gravity and rotation. In this regime, the winds are organized in bands that are tilted along the rotation axis, with unit horizontal wave number in the plane defined by gravity and rotation at values of ϕ less than about 60^{∘}. This intermediate solution, studied for the first time in this work, is characterized by weaker vertical heat transport than the cases with large-scale vortices. For intermediate values of ϕ (between about 45^{∘} and 60^{∘}), the banded, sheared solution coexists with the large-scale vortex solution, with different initial conditions leading to one or the other dynamical behavior. A discussion of the possible implications of these results for the dynamics of rapidly rotating planetary atmospheres is provided.

Convex Relaxation Methods for Unified Near-Field and Far-Field TDOA-Based Localization

This paper develops two convex relaxation solutions for the unified localization of a signal source using time difference of arrival measurements, regardless of whether the source is in the near field for coordinate positioning or in the far field for the direction of arrival estimation. The previous study on unified estimation only derived an iterative solution, which is sensitive to initialization. Albeit a coarse initialization was supplied to start the iteration, it may not be sufficient to ensure convergence to the global solution especially when the source is close to the sensors. The proposed solutions come from two novel formulations for optimization, one using the weighted least squares with the modified polar representation of the source position as variable and the other applying fractional programming with the Cartesian coordinate representation instead. Both the optimization problems are solved by first performing semidefinite relaxation and then tightening the relaxed problem by including a set of second-order cone constraints. The two formulations are created from different approaches. Nevertheless, we are able to prove that both the formulations reduce to solving exactly the same mixed semidefinite/second-order cone program and thus establish their equivalence. Furthermore, the proposed solution method is extended to the more practical scenario when sensor position errors are present. The results from both the simulated and real experiments show that the proposed method achieves almost the same performance of the iterative maximum likelihood estimator under ideal initialization.

The importance of conceptualization in the use of Wolfram Mathematica 10 for a course in differential calculus

With the continuous advance in the technological tools, both Hardware and Software, a massive utilization has started in different fields of society. One of the fields in which this type of tool is used is the academic one, given the importance in the generation of spaces for discussion in teacher-student mediation, using the computer as a tool that supports the process in the classroom. In particular, the use of specialized software has made the teaching work change depending on the support in the resource, however, you cannot depend exclusively on the resource, but the teacher must be able to use the tool recognizing the benefits and limitations that the software You can submit. The objective of this work is to socialize a classroom experience developed with the students of Differential Calculus in the thematic axis of inverse functions using Wolfram Mathematica 10 software, in which it is possible to show a program limitation with respect to the Cartesian representation of certain functions inverse, which can lead to misinterpretations about their behavior

Sobre el mapa: estrategias de gestión de la información en la cartografía actual

El mapa se ha transformado en el soporte ideal sobre el que gestionar múltiples las capas de información que afectan al territorio. Históricamente, las estrategias gráficas empleadas para la representación y comunicación de toda esa información, merece nuestra consideración en la medida en la que pueden extrapolarse estrategias proyectuales aplicables sobre el territorio. Tomando como punto de partida una serie de cartografías que, a lo largo de los años, han supuesto un avance en materia de comunicación gráfica, se hace un repaso de la contribución cultural que supone este soporte. Desde unos inicios que evidencian un tránsito desde el espacio topológico al de representación cartesiana, pasando por la búsqueda de herramientas de gestión gráfica de elementos cambiantes, así como la incorporación del Big Data en cartografías más próximas al diagrama que al mapa tradicional, podemos observar una necesaria confluencia de todos esos niveles de información. Todos estos recursos, sirven en la actualidad a los arquitectos y urbanistas, como punto de partida que permite no solo representar los aspectos más intangibles del proyecto, sino también explorar nuevas formas de comunicación multidimensional. En definitiva, nos otorga la capacidad de cartografiar no solo lo real, también lo imaginado.

Application of the functional renormalization group to Bose gases: From linear to hydrodynamic fluctuations

We study weakly interacting Bose gases using the functional renormalization group with a hydrodynamic effective action. We use a scale-dependent parametrization of the boson fields that interpolates between a Cartesian representation at high momenta and an amplitude-phase one for low momenta. We apply this to Bose gases in two and three dimensions near the superfluid phase transition where they can be described by statistical O(2) models. We are able to give consistent physical descriptions of the infrared regime in both two and three dimensions. In particular, and in contrast to previous studies using the functional renormalization group, we find a stable superfluid phase at finite temperatures in two dimensions. We compare our results for the superfluid and boson densities with Monte-Carlo simulations, and we find they are in reasonable agreement.