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Macaulay Brackets
Macaulay brackets are a notation used to describe the ramp function :\ = \begin 0, & x < 0 \\ x, & x \ge 0. \end A popular alternative transcription uses angle brackets, ''viz.'' . Introduction to Aerospace Structures. Department of Aerospace Engineering Sciences, University of Colorado at Boulder Another commonly used notation is + or + for the positive part of , which avoids conflicts with for set notation [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramp Function
The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous definitions, for example "0 for negative inputs, output equals input for non-negative inputs". The term "ramp" can also be used for other functions obtained by scaling and shifting, and the function in this article is the ''unit'' ramp function (slope 1, starting at 0). In mathematics, the ramp function is also known as the positive part. In machine learning, it is commonly known as a ReLU activation function or a rectifier in analogy to half-wave rectification in electrical engineering. In statistics (when used as a likelihood function) it is known as a tobit model. This function has numerous applications in mathematics and engineering, and goes by various names, depending on the context. There are differentiable variants of the ramp function. Definitions The ramp function () may be defined analytically in several ways. Possible definitions are: * A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Number
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. In some contexts, it makes sense to distinguish between a positive and a negative zero. In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive inverse (multiplication with −1, negation), an operation which is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate binary aspects of mathematical or scientific objects, such as odd and even ( sign of a permutation), sense of orientation or rotation ( cw/ccw), one sided limits, and other concepts described in below. Sign of a number Numbers from various number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Notation
In mathematics and more specifically in set theory, set-builder notation is a notation for specifying a set by a property that characterizes its members. Specifying sets by member properties is allowed by the axiom schema of specification. This is also known as set comprehension and set abstraction. Sets defined by a predicate Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to ''true'' for an element of the set, and ''false'' otherwise. In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the separator, and a rule on the right of it. These three parts are contained in curly brackets: :\ or :\. The vertical bar (or colon) is a separator that can be read as "such that", "for which", or "with the property that". The formula is said to be the ''rule'' or the ''predicate''. All values of for which th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shear And Moment Diagram
Shear force and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear forces and bending moments at a given point of a structural element such as a beam. These diagrams can be used to easily determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure. Another application of shear and moment diagrams is that the deflection of a beam can be easily determined using either the moment area method or the conjugate beam method. Convention Although these conventions are relative and any convention can be used if stated explicitly, practicing engineers have adopted a standard convention used in design practices. Normal convention The normal convention used in most engineering applications is to label a positive shear force - one that spins an element clockwise (up on the left, and down on the right). Like ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macaulay's Method
Macaulay's method (the double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Use of Macaulay's technique is very convenient for cases of discontinuous and/or discrete loading. Typically partial uniformly distributed loads (u.d.l.) and uniformly varying loads (u.v.l.) over the span and a number of concentrated loads are conveniently handled using this technique. The first English language description of the method was by Macaulay. The actual approach appears to have been developed by Clebsch in 1862.J. T. Weissenburger, ‘Integration of discontinuous expressions arising in beam theory’, AIAA Journal, 2(1) (1964), 106–108. Macaulay's method has been generalized for Euler-Bernoulli beams with axial compression, W. H. Wittrick, "A generalization of Macaulay’s method with applications in structural mechanics", AIAA Journal, 3(2) (1965), 326–330. to Timoshenko beams,A. Yavari, S. Sarkani and J. N. Reddy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Step Function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Different conventions concerning the value are in use. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Heaviside developed the operational calculus as a tool in the analysis of telegraphic communications and represented the function as . Formulation Taking the convention that , the Heaviside function may be defined as: * a piecewise function: H(x) := \begin 1, & x \geq 0 \\ 0, & x * an indicator function: H(x) := \mathbf_=\mathbf 1_(x) For the alt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singularity Function
Singularity functions are a class of discontinuous functions that contain Mathematical singularity, singularities, i.e., they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and Distribution (mathematics), distribution theory. The functions are notated with brackets, as \langle x-a\rangle ^n where ''n'' is an integer. The "\langle \rangle" are often referred to as singularity brackets . The functions are defined as: : where: is the Dirac delta function, also called the unit impulse. The first derivative of is also called the unit doublet. The function H(x) is the Heaviside step function: for and for . The value of will depend upon the particular convention chosen for the Heaviside step function. Note that this will only be an issue for since the functions contain a multiplicative factor of for . \langle x-a\rangle^1 is also called the Ramp function. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
