Fejér kernelIn mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959). The Fejér kernel has many equivalent definitions. We outline three such definitions below: 1) The traditional definition expresses the Fejér kernel in terms of the Dirichlet kernel: where is the kth order Dirichlet kernel.
Pointwise convergenceIn mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Suppose that is a set and is a topological space, such as the real or complex numbers or a metric space, for example. A net or sequence of functions all having the same domain and codomain is said to converge pointwise to a given function often written as if (and only if) The function is said to be the pointwise limit function of the Sometimes, authors use the term bounded pointwise convergence when there is a constant such that .
Diffusion equationThe diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation, when bulk velocity is zero.
Convergence of random variablesIn probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied.
Limit of a functionAlthough the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x.
Limit inferior and limit superiorIn mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Absolute convergenceIn mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess – a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely".
Convolutional neural networkConvolutional neural network (CNN) is a regularized type of feed-forward neural network that learns feature engineering by itself via filters (or kernel) optimization. Vanishing gradients and exploding gradients, seen during backpropagation in earlier neural networks, are prevented by using regularized weights over fewer connections. For example, for each neuron in the fully-connected layer 10,000 weights would be required for processing an image sized 100 × 100 pixels.
Kernel (image processing)In , a kernel, convolution matrix, or mask is a small matrix used for blurring, sharpening, embossing, edge detection, and more. This is accomplished by doing a convolution between the kernel and an . Or more simply, when each pixel in the output image is a function of the nearby pixels (including itself) in the input image, the kernel is that function. The general expression of a convolution is where is the filtered image, is the original image, is the filter kernel. Every element of the filter kernel is considered by and .
Dirac combIn mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function with the formula for some given period . Here t is a real variable and the sum extends over all integers k. The Dirac delta function and the Dirac comb are tempered distributions. The graph of the function resembles a comb (with the s as the comb's teeth), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function. The symbol , where the period is omitted, represents a Dirac comb of unit period.
Compact convergenceIn mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Let be a topological space and be a metric space. A sequence of functions is said to converge compactly as to some function if, for every compact set , uniformly on as . This means that for all compact , If and with their usual topologies, with , then converges compactly to the constant function with value 0, but not uniformly.
Three-phase traffic theoryThree-phase traffic theory is a theory of traffic flow developed by Boris Kerner between 1996 and 2002. It focuses mainly on the explanation of the physics of traffic breakdown and resulting congested traffic on highways. Kerner describes three phases of traffic, while the classical theories based on the fundamental diagram of traffic flow have two phases: free flow and congested traffic.
