Nonlinear systemIn mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Ordinary differential equationIn mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .
Nonlinear opticsNonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (when the electric field of the light is >108 V/m and thus comparable to the atomic electric field of ~1011 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear.
Cardiac electrophysiologyCardiac electrophysiology is a branch of cardiology and basic science focusing on the electrical activities of the heart. The term is usually used in clinical context, to describe studies of such phenomena by invasive (intracardiac) catheter recording of spontaneous activity as well as of cardiac responses to programmed electrical stimulation - clinical cardiac electrophysiology. However, cardiac electrophysiology also encompasses basic research and translational research components.
Clinical cardiac electrophysiologyClinical cardiac electrophysiology (also referred to as cardiac electrophysiology, arrhythmia services, or electrophysiology), is a branch of the medical specialty of cardiology and is concerned with the study and treatment of rhythm disorders of the heart. Cardiologists with expertise in this area are usually referred to as electrophysiologists. Electrophysiologists are trained in the mechanism, function, and performance of the electrical activities of the heart.
K-means clusteringk-means clustering is a method of vector quantization, originally from signal processing, that aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean (cluster centers or cluster centroid), serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells. k-means clustering minimizes within-cluster variances (squared Euclidean distances), but not regular Euclidean distances, which would be the more difficult Weber problem: the mean optimizes squared errors, whereas only the geometric median minimizes Euclidean distances.
Electrophysiology studyA cardiac electrophysiology study (EP test or EP study) is a minimally invasive procedure using catheters introduced through a vein or artery to record electrical activity from within the heart. This electrical activity is recorded when the heart is in a normal rhythm (sinus rhythm) to assess the conduction system of the heart and to look for additional electrical connections (accessory pathways), and during any abnormal heart rhythms that can be induced.
Cluster analysisCluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for statistical data analysis, used in many fields, including pattern recognition, , information retrieval, bioinformatics, data compression, computer graphics and machine learning.
Phase spaceIn dynamical systems theory and control theory, a phase space or state space is a space in which all possible "states" of a dynamical system or a control system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the direct product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.
Homogeneous differential equationA differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives.
Cardiac catheterizationCardiac catheterization (heart cath) is the insertion of a catheter into a chamber or vessel of the heart. This is done both for diagnostic and interventional purposes. A common example of cardiac catheterization is coronary catheterization that involves catheterization of the coronary arteries for coronary artery disease and myocardial infarctions ("heart attacks"). Catheterization is most often performed in special laboratories with fluoroscopy and highly maneuverable tables.
Determining the number of clusters in a data setDetermining the number of clusters in a data set, a quantity often labelled k as in the k-means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain class of clustering algorithms (in particular k-means, k-medoids and expectation–maximization algorithm), there is a parameter commonly referred to as k that specifies the number of clusters to detect.
State space (physics)In physics, a state space is an abstract space in which different "positions" represent, not literal locations, but rather states of some physical system. This makes it a type of phase space. Specifically, in quantum mechanics a state space is a complex Hilbert space in which each unit vector represents a different state that could come out of a measurement. Each unit vector specifies a different dimension, so the numbers of dimensions in this Hilbert space depends on the system we choose to describe.
Finite element methodThe finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems).
Variation of parametersIn mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations.
Hierarchical clusteringIn data mining and statistics, hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis that seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two categories: Agglomerative: This is a "bottom-up" approach: Each observation starts in its own cluster, and pairs of clusters are merged as one moves up the hierarchy. Divisive: This is a "top-down" approach: All observations start in one cluster, and splits are performed recursively as one moves down the hierarchy.
ElectrophysiologyElectrophysiology (from Greek ἥλεκτ, ēlektron, "amber" [see the etymology of "electron"]; φύσις, physis, "nature, origin"; and -λογία, -logia) is the branch of physiology that studies the electrical properties of biological cells and tissues. It involves measurements of voltage changes or electric current or manipulations on a wide variety of scales from single ion channel proteins to whole organs like the heart. In neuroscience, it includes measurements of the electrical activity of neurons, and, in particular, action potential activity.
State-space representationIn control engineering, model based fault detection and system identification a state-space representation is a mathematical model of a physical system specified as a set of input, output and variables related by first-order (not involving second derivatives) differential equations or difference equations. Such variables, called state variables, evolve over time in a way that depends on the values they have at any given instant and on the externally imposed values of input variables.
Clique problemIn computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique (a clique with the largest possible number of vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem of testing whether a graph contains a clique larger than a given size.
Cardiac stress testA cardiac stress test (also referred to as a cardiac diagnostic test, cardiopulmonary exercise test, or abbreviated CPX test) is a cardiological test that measures the heart's ability to respond to external stress in a controlled clinical environment. The stress response is induced by exercise or by intravenous pharmacological stimulation. Cardiac stress tests compare the coronary circulation while the patient is at rest with the same patient's circulation during maximum cardiac exertion, showing any abnormal blood flow to the myocardium (heart muscle tissue).
