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Delay Differential Equation
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with a functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs: # Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition, actuators, sensors, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Equation (calculus)
In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of degree upon which depends the solution of a given th- order differential equation or difference equation. The characteristic equation can only be formed when the differential equation is linear and homogeneous, and has constant coefficients. Such a differential equation, with as the dependent variable, superscript denoting ''n''th-derivative, and as constants, :a_y^ + a_y^ + \cdots + a_y' + a_y = 0, will have a characteristic equation of the form :a_r^ + a_r^ + \cdots + a_r + a_ = 0 whose solutions are the roots from which the general solution can be formed. Analogously, a linear difference equation of the form :y_ = b_1y_ + \cdots + b_ny_ has characteristic equation :r^n - b_1r^ - \cdots - b_n =0, discussed in more detail at Linear recurrence with constant coefficients. The characteristic roots (roots of the characteristic equation) also provide qualitative information abo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Population Dynamics
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. Population dynamics is a branch of mathematical biology, and uses mathematical techniques such as differential equations to model behaviour. Population dynamics is also closely related to other mathematical biology fields such as epidemiology, and also uses techniques from evolutionary game theory in its modelling. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 220 years,Malthus, Thomas Robert. An Essay on the Principle of Population: Library of Economics although over the last century the scope of mathematical biology has greatly expanded. The beginning of population dynamics is widely regarded as the work of Malthus, formulated as the Malthusian growth model. According to Malthus, assuming that the conditions (the environment) remain constant ('' ceteris pari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and Risk factor (epidemiology), determinants of health and disease conditions in a defined population, and application of this knowledge to prevent diseases. It is a cornerstone of public health, and shapes policy decisions and evidence-based practice by identifying Risk factor (epidemiology), risk factors for disease and targets for preventive healthcare. Epidemiologists help with study design, collection, and statistical analysis of data, amend interpretation and dissemination of results (including peer review and occasional systematic review). Epidemiology has helped develop methodology used in clinical research, public health studies, and, to a lesser extent, basic research in the biological sciences. Major areas of epidemiological study include disease causation, transmission (medicine), transmission, outbreak investigation, disease surveillance, environmental epidemiology, forensic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diabetes
Diabetes mellitus, commonly known as diabetes, is a group of common endocrine diseases characterized by sustained high blood sugar levels. Diabetes is due to either the pancreas not producing enough of the hormone insulin, or the cells of the body becoming unresponsive to insulin's effects. Classic symptoms include polydipsia (excessive thirst), polyuria (excessive urination), polyphagia (excessive hunger), weight loss, and blurred vision. If left untreated, the disease can lead to various health complications, including disorders of the cardiovascular system, eye, kidney, and nerves. Diabetes accounts for approximately 4.2 million deaths every year, with an estimated 1.5 million caused by either untreated or poorly treated diabetes. The major types of diabetes are type 1 and type 2. The most common treatment for type 1 is insulin replacement therapy (insulin injections), while anti-diabetic medications (such as metformin and semaglutide) and lifestyle modificatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fabius Function
In mathematics, the Fabius function is an example of an smoothness, infinitely differentiable function that is nowhere analytic function, analytic, found by . This function satisfies the initial condition f(0) = 0, the symmetry condition f(1-x) = 1 - f(x) for 0 \le x \le 1, and the functional differential equation :f'(x) = 2 f(2 x) for 0 \le x \le 1/2. It follows that f(x) is monotone increasing for 0 \le x \le 1, with f(1/2)=1/2 and f(1)=1 and f'(1-x)=f'(x) and f'(x)+f'(\tfrac12-x)=2. It was also written down as the Fourier transform of : \hat(z) = \prod_^\infty \left(\cos\frac\right)^m by . The Fabius function is defined on the unit interval, and is given by the cumulative distribution function of :\sum_^\infty2^\xi_n, where the are independence (probability), independent uniform distribution (continuous), uniformly distributed random variables on the unit interval. That distribution has an expectation of \tfrac and a variance of \tfrac. There is a unique extens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambert W Function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the exponential function. The function is named after Johann Heinrich Lambert, Johann Lambert, who considered a related problem in 1758. Building on Lambert's work, Leonhard Euler described the function per se in 1783. For each integer there is one branch, denoted by , which is a complex-valued function of one complex argument. is known as the principal branch. These functions have the following property: if and are any complex numbers, then : w e^ = z holds if and only if : w=W_k(z) \ \ \text k. When dealing with real numbers only, the two branches and suffice: for real numbers and the equation : y e^ = x can be solved for only if ; yields if and the two values and if . The Lambert function's branches cannot be expressed in terms o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Eigenproblem
In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form : M (\lambda) x = 0 , where x\neq0 is a vector, and ''M'' is a matrix-valued function of the number \lambda. The number \lambda is known as the (nonlinear) eigenvalue, the vector x as the (nonlinear) eigenvector, and (\lambda,x) as the eigenpair. The matrix M (\lambda) is singular at an eigenvalue \lambda. Definition In the discipline of numerical linear algebra the following definition is typically used. Let \Omega \subseteq \Complex, and let M : \Omega \rightarrow \Complex^ be a function that maps scalars to matrices. A scalar \lambda \in \Complex is called an ''eigenvalue'', and a nonzero vector x \in \Complex^n is called a ''right eigenvector'' if M (\lambda) x = 0. Moreover, a nonzero vector y \in \Complex^n is called a ''left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of System of linear equations, systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. Mathematical background The name ''spectral theory'' was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on Principal axis theorem, principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum Of A Matrix
In mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if T\colon V \to V is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars \lambda such that T-\lambda I is not invertible. The determinant of the matrix equals the product of its eigenvalues. Similarly, the trace of the matrix equals the sum of its eigenvalues. From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of its nonzero eigenvalues (the density of multivariate normal distribution will need this quantity). In many applications, such as PageRank, one is interested in the dominant eigenvalue, i.e. that which is largest in absolute value. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix. Definition Let ''V'' be a finite-dimensional vector space over some field ''K'' and suppose ''T'' : ''V'' → ''V'' is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Polynomial
In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function. Definition In fields An exponential polynomial generally has both a variable ''x'' and some kind of exponential function ''E''(''x''). In the complex numbers there is already a canonical exponential function, the function that maps ''x'' to '' e''''x''. In this setting the term exponential polynomial is often used to mean polynomials of the form ''P''(''x'', ''e''''x'') where ''P'' ∈ C 'x'', ''y''is a polynomial in two variables. There is nothing particularly special about C here; exponential polynomials may also refer to such a polynomial on any exponential field or exponential ring with its exponential function taking the place of ''e''''x'' above. Similarly, there is no reason to have one variable, and an exponential polynomial in ''n'' variables would be of the form ''P''(''x''1, ..., ''x''''n' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
