Concept# Analytic geometry

Summary

In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results

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Robert Benny Gerber, Natalia Nagornova Boyarkine

The intrinsic structures of biomolecules in the gas phase may not reflect their native solution geometries. Microsolvation of the molecules bridges the two environments, enabling a tracking of molecular structural changes upon hydration at the atomistic level. We employ density functional calculations to compute a large pool of structures and vibrational spectra for a gas-phase complex, in which a doubly protonated decapeptide, gramicidin S, is solvated by two water molecules. Though most vibrations of this large complex are treated in a harmonic approximation, the water molecules and the vibrations of the host ion coupled to them are locally described by a quantum mechanical vibrational self-consistent field theory with second-order perturbation correction (VSCF-PT2). Guided and validated by the available cold ion spectroscopy data, the computational analysis identifies structures of the three experimentally observed conformers of the complex. They, mainly, differ by the hydration sites, of which the one at the Orn side chain is the most important for reshaping the peptide toward its native structure. The study demonstrates the ability of a quantum chemistry approach that intelligently combines the semiempirical and ab initio computations to disentangle a complex interplay of intra- and intermolecular hydrogen bonds in large molecular systems.

In the present work, we approach two key aspects of memory formation: associative memory and synaptic consolidation.
The storage of associative memory is commonly related to the medial temporal lobe in humans. Experimental evidence shows that the memories of objects, people or places are represented in this brain area by cell assemblies that respond selectively to single concepts. Neurons forming such assemblies are called concept cells. Associations between different concepts are linked to concept cells shared between assemblies: we refer to the number of shared neurons as the overlap between memory engrams. The respective assemblies of two associated concepts (e.g. Hillary and Bill Clinton) share more neurons compared to the assemblies of two unrelated concepts (e.g. Hillary Clinton and the Eiffel tower). In particular, three characteristics of assemblies of concept cells are important for this work: (a) they exhibit a very low mean activity (about 0.2% of neurons respond to each concept), (b) overlapping assemblies share about 4% of their cells, (c) non-overlapping assemblies share less then 1% of their cells. This implies that the association between two concepts induces a higher level of overlap between the relative memory engrams.
In parallel, theoretical studies have shown that overlaps between memory engrams are fundamental in the process of free recall of sequences of words. These models assume that memory engrams have such a high mean activity that all assemblies are overlapping to a certain extent. Associative memory is traditionally modeled through attractor neural networks. Memory engrams are represented by binary patterns of active/silent neurons. While there is extensive literature on independent low-activity patterns, only a few studies can be found on correlated patterns. Extending the existing theory to include correlation is a key missing point to answer questions such as: How do shared neurons encode association? Why are 4% of neurons shared and not more?
Using a mean-field approximation, we derive analytic equations for the network dynamics in the case of correlated patterns. Our results provide a theoretical framework that can explain the experimentally observed value of shared neurons. We find that for concepts represented by realistically sparse neural assemblies there are a minimal and a maximal fraction of shared neurons so that associations can be reliably coded. In the presence of a periodically modulated signal, such as hippocampal oscillations, chains of associations can be recalled analogously to theories of free recall of lists of memorized words.
Finally, we compare the predicted number of concepts a neuron responds to with experimental data. We test different ways of constructing correlated patterns and confirmed the common opinion that information in the hippocampus is non-hierarchically organised.
In the second part of the thesis, we propose a model of synaptic consolidation based on two coupled dynamical variables: the fast synaptic weights and a slow internal synaptic mechanism. In classical experiments, the consolidation of the synapse is related to the stimulation frequency and number of repetitions. We show that it is exactly the time scale separation between the dynamics of the two variables that determines which combination of stimulation amplitude and frequency are suitable to elicit synaptic consolidation.

Pascal Frossard, Renata Khasanova

Due to their wide field of view, omnidirectional cameras are frequently used by autonomous vehicles, drones and robots for navigation and other computer vision tasks. The images captured by such cameras, are often analyzed and classified with techniques designed for planar images that unfortunately fail to properly handle the native geometry of such images and therefore results in suboptimal performance. In this paper we aim at improving popular deep convolutional neural networks so that they can properly take into account the specific properties of omnidirectional data. In particular we propose an algorithm that adapts convolutional layers, which often serve as a core building block of a CNN, to the properties of omnidirectional images. Thus, our filters have a shape and size that adapt to the location on the omnidirectional image. We show that our method is not limited to spherical surfaces and is able to incorporate the knowledge about any kind of projective geometry inside the deep learning network. As depicted by our experiments, our method outperforms the existing deep neural network techniques for omnidirectional image classification and compression tasks.

2019