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Weighing Matrix
In mathematics, a weighing matrix of order n and weight w is a matrix W with entries from the set \ such that: :WW^\mathsf = wI_n Where W^\mathsf is the transpose of W and I_n is the identity matrix of order n. The weight w is also called the ''degree'' of the matrix. For convenience, a weighing matrix of order n and weight w is often denoted by W(n,w). Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a balance scale, the statistical variance of the measurement can be minimized by weighing multiple objects at once, including some objects in the opposite pan of the scale where they subtract from the measurement. Properties Some properties are immediate from the definition. If W is a W(n,w), then: * The rows of W are pairwise orthogonal (that is, every pair of rows you pick from W will be orthogonal). Similarly, the columns are pairwise orthogonal. * Each row and each column o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conference Matrix
In mathematics, a conference matrix (also called a C-matrix) is a square matrix ''C'' with 0 on the diagonal and +1 and −1 off the diagonal, such that ''C''T''C'' is a multiple of the identity matrix ''I''. Thus, if the matrix has order ''n'', ''C''T''C'' = (''n''−1)''I''. Some authors use a more general definition, which requires there to be a single 0 in each row and column but not necessarily on the diagonal. Conference matrices first arose in connection with a problem in telephony.Belevitch, pp. 231-244. They were first described by Vitold Belevitch, who also gave them their name. Belevitch was interested in constructing ideal telephone conference networks from ideal transformers and discovered that such networks were represented by conference matrices, hence the name. Other applications are in statistics, and another is in elliptic geometry. For ''n'' > 1, there are two kinds of conference matrix. Let us normalize ''C'' by, first (if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Theory
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measuring Spectrum Of Light Using A Weighing Matrix Optical Mask
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to a basic reference quantity of the same kind. The scope and application of measurement are dependent on the context and discipline. In natural sciences and engineering, measurements do not apply to nominal properties of objects or events, which is consistent with the guidelines of the ''International vocabulary of metrology'' published by the International Bureau of Weights and Measures. However, in other fields such as statistics as well as the social and behavioural sciences, measurements can have multiple levels, which would include nominal, ordinal, interval and ratio scales. Measurement is a cornerstone of trade, science, technology and quantitative research in many disciplines. Historically, many measurement systems existed for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation. The most important application is in data fitting. When the problem has substantial uncertainties in the independent variable (the ''x'' variable), then simple regression and least-squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares. Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regressio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth. Definition A plane curve defined by an implicit equation :F(x,y)=0, where is a smooth function is said to be ''singular'' at a point if the Taylor series of has order at least at this point. The reason for this is that, in differential calculus, the tangent at the point of such a curve is defined by the equation :(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0)=0, whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent And Identically Distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as ''i.i.d.'', ''iid'', or ''IID''. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tare Weight
Tare weight , sometimes called unladen weight, is the weight of an empty vehicle or container. By subtracting tare weight from gross weight ( laden weight), one can determine the weight of the goods carried or contained (the net weight). Etymology The word '' tare'' originates from the Middle French word 'wastage in goods, deficiency, imperfection' (15th ), from Italian , from Arabic , lit. 'thing deducted or rejected', from 'to reject'. Usage This can be useful in computing the cost of the goods carried for purposes of taxation or for tolls related to barge, rail, road, or other traffic, especially where the toll will vary with the value of the goods carried (''e.g.'', tolls on the Erie Canal). Tare weight is often published upon the sides of railway cars and transport vehicles to facilitate the computation of the load carried. Tare weight is also used in body composition assessment when doing underwater weighing.hydrostatic_weighing/ref> Tare_weight_is_accounted_for_in_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Matrix
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The ''n''-dimensional parallelotope spanned by the rows of an ''n''×''n'' Hadamard matrix has the maximum possible ''n''-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of (t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
