A Quantum Approach to the Discretizable Molecular Distance Geometry Problem

The Discretizable Molecular Distance Geometry Problem (DMDGP) aims to determine the three-dimensional protein structure using distance information from nuclear magnetic resonance experiments. The DMDGP has a finite number of candidate solutions and can be solved by combinatorial methods. We describe a quantum approach to the DMDGP by using Grover's algorithm with an appropriate oracle function, which is more efficient than classical methods that use brute force. We show computational results by implementing our scheme on IBM quantum computers with a small number of noisy qubits.Comment: 17 page

Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequences

The inevitable noise in real measurements motivates the problem to continuously quantify the similarity between rigid objects such as periodic time series and proteins given by ordered points and considered up to isometry maintaining inter-point distances. The past work produced many Hausdorff-like distances that have slow or approximate algorithms due to minimizations over infinitely many isometries. For finite and 1-periodic sequences under isometry in any high-dimensional Euclidean space, we introduce continuous metrics with faster algorithms. The key novelty in the periodic case is the continuity of new metrics under perturbations that change the minimum period.Comment: 16 pages, 6 figures. The second version includes extra examples to illustrate the key results. The latest version is at http://kurlin.org/projects/periodic-geometry-topology/metric1D.pd

Cycle-based formulations in Distance Geometry

The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in the given Euclidean space. Solution algorithms are generally constructed using local or global nonlinear optimization techniques. We present a new modelling technique for this problem where, instead of deciding vertex positions, formulations decide the length of the segments representing the edges in each cycle in the graph, projected in every dimension. We propose an exact formulation and a relaxation based on a Eulerian cycle. We then compare computational results from protein conformation instances obtained with stochastic global optimization techniques on the new cycle-based formulation and on the existing edge-based formulation. While edge-based formulations take less time to reach termination, cycle-based formulations are generally better on solution quality measures

Energy-Efficient Neuromorphic Architectures for Nuclear Radiation Detection Applications

A comprehensive analysis and simulation of two memristor-based neuromorphic architectures for nuclear radiation detection is presented. Both scalable architectures retrofit a locally competitive algorithm to solve overcomplete sparse approximation problems by harnessing memristor crossbar execution of vector–matrix multiplications. The proposed systems demonstrate excellent accuracy and throughput while consuming minimal energy for radionuclide detection. To ensure that the simulation results of our proposed hardware are realistic, the memristor parameters are chosen from our own fabricated memristor devices. Based on these results, we conclude that memristor-based computing is the preeminent technology for a radiation detection platform

Measurement of complicated shape parts using image processing

Bakalárska práca sa zaoberá témou merania opotrebenia pastorka REA aktuátora využívaného pri regulácii systému obtokového ventilu turbodúchadla. Prvá teoretická časť práce popisuje témy obtokových ventilov, aktuátorov, ozubených kolies a optických metód. V druhej časti je realizovaná metóda merania pastorka s využitím digitálneho mikroskopu a aplikácie na meranie šírky zubov pastorka navrhnutej v prostredí MATLAB.The bachelor's thesis deals with the topic of wear measurement of the pinion gear of the REA actuator used in the regulation of the turbocharger wastegate system. The first theoretical part of the thesis describes the topics of wastegates, actuators, gears and optical methods. In the second part, a pinion measurement method is implemented using a digital microscope and an application to measure the width of the pinion tooth thickness designed in the MATLAB environment.

An Extension of Heron's Formula to Tetrahedra, and the Projective Nature of Its Zeros

A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and three medial parallelograms, which will be referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which the tetrahedron's in-sphere touches those faces. This leads to a conjecture as to how the formula extends to $n$-dimensional simplices for all $n > 3$. Remarkably, for $n = 3$ the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios. These unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to $\mathbb Z_2^4$, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subset. The paper closes by noting that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank $3$ chirotope, aka affine oriented matroid.Comment: 51 pages, 6 sections, 5 appendices, 7 figures, 2 tables, 81 references; v7 clarifies the definitions made in the text leading up to Theorem 5.4, along with the usual miscellaneous minor corrections and improvement

Cycle-based formulations in Distance Geometry

The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in the given Euclidean space. Solution algorithms are generally constructed using local or global nonlinear optimization techniques. We present a new modelling technique for this problem where, instead of deciding vertex positions, formulations decide the length of the segments representing the edges in each cycle in the graph, projected in every dimension. We propose an exact formulation and a relaxation based on a Eulerian cycle. We then compare computational results from protein conformation instances obtained with stochastic global optimization techniques on the new cycle-based formulation and on the existing edge-based formulation. While edge-based formulations take less time to reach termination, cycle-based formulations are generally better on solution quality measures.Comment: 16 page