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Mating takes place at different times of the year in different parts of the manta's range. Courtship is difficult to observe in this fast-swimming fish, although mating "trains" with multiple individuals swimming closely behind each other are sometimes seen in shallow water. The mating sequence may be triggered by a fu... | Wikipedia - Manta ray | null | null | null |
Once he has a tight grip, he turns upside-down and presses his ventral side against hers. He then inserts one of his claspers into her cloaca, where it remains for 60–90 seconds. The claspers form a tube and a siphon propels semen from the genital papilla into the oviduct. | Wikipedia - Manta ray | null | null | null |
: 41–42 The male continues to grip the female's pectoral fin with his teeth for a further few minutes as both continue to swim, often followed by up to 20 other males. The pair then parts, the female being left with scars on her fin. : 46 The fertilized eggs develop within the female's oviduct. | Wikipedia - Manta ray | null | null | null |
At first, they are enclosed in an egg case while the developing embryos absorb the yolk. After hatching, the pups remain in the oviduct and receive additional nutrition from milky secretions. | Wikipedia - Manta ray | null | null | null |
With no umbilical cord or placenta, the unborn pup relies on buccal pumping to obtain oxygen. Brood size is usually one or occasionally two. The gestation period is thought to be 12–13 months. | Wikipedia - Manta ray | null | null | null |
When fully developed, the pup resembles a miniature adult and is expelled from the oviduct with no further parental care. In wild populations, an interval of two years between births may be normal, but a few individuals become pregnant in consecutive years, demonstrating an annual ovulatory cycle. The Okinawa Churaumi ... | Wikipedia - Manta ray | null | null | null |
In one of these pregnancies, the gestation period was 372 days and at birth the pup had a width of 192 cm (76 in) and weight of 70 kg (150 lb). In Indonesia, M. birostris males appear to mature at 3.75 m (12 ft), while female mature around 4 m (13 ft). In the Maldives, males of M. alfredi mature at a width of 2.5 m (8 ... | Wikipedia - Manta ray | null | null | null |
In Hawaii, M. alfredi matures at a width of 2.8 m (9 ft 2 in) for males and 3.4 m (11 ft) for females. Female mantas appear to mature at 8–10 years. Manta rays may live as long as 50 years. | Wikipedia - Manta ray | null | null | null |
Matlab implementation of the uncalibrated trifocal tensor estimation and comparison to pairwise fundamental matrices C++ implementation of the calibrated trifocal tensor estimation using optimized Homotopy Continuation code. Presently includes cases of three corresponding points with lines at these points (as in featur... | Wikipedia - Trifocal tensor | null | null | null |
Matlab: The PDAF and JPDAF algorithms are implemented in the singleScanUpdate function that is part of the United States Naval Research Laboratory's free Tracker Component Library.Python: The PDAF and other data association methods are implemented in Stone-Soup. A tutorial demonstrates how the algorithms can be used. =... | Wikipedia - Probabilistic Data Association Filter | null | null | null |
Matlab: The PDAF, JPDAF, Set JPDAF, JPDAF*, GNN-JPDAF and multiple other exact and approximate variants of the JPDAF are implemented in the singleScanUpdate function that is part of the United States Naval Research Laboratory's free Tracker Component Library. The sample code in demo2DDataAssociation demonstrates how th... | Wikipedia - Joint Probabilistic Data Association Filter | null | null | null |
Matoušek, Sharir & Welzl (1996) describe an algorithm that uses an additional property of linear programs that is not always held by other LP-type problems, that all bases have the same cardinality of each other. If an LP-type problem does not have this property, it can be made to have it by adding d new dummy elements... | Wikipedia - LP-type problem | null | null | null |
At each step it maintains a basis C that may initially be the set of dummy elements. It may be described with the following pseudocode: function msw(S, f, C) is if S = C then return C choose a random element x of S \ C B = msw(S \ x, f, C) if f(B) ≠ f(B ∪ {x}) then B := basis(B ∪ {x}) B := msw(S, f, B) return B In most... | Wikipedia - LP-type problem | null | null | null |
Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is thus an essential part of linear algebra. Let V be a finite-dimensional vector space over a field F, and (v1, v2, ..., vm) be a basis of V (thus m is the dimension of V). | Wikipedia - Linear Algebra | null | null | null |
By definition of a basis, the map ( a 1 , … , a m ) ↦ a 1 v 1 + ⋯ a m v m F m → V {\displaystyle {\begin{aligned}(a_{1},\ldots ,a_{m})&\mapsto a_{1}\mathbf {v} _{1}+\cdots a_{m}\mathbf {v} _{m}\\F^{m}&\to V\end{aligned}}} is a bijection from Fm, the set of the sequences of m elements of F, onto V. This is an isomorphis... | Wikipedia - Linear Algebra | null | null | null |
If W is another finite dimensional vector space (possibly the same), with a basis (w1, ..., wn), a linear map f from W to V is well defined by its values on the basis elements, that is (f(w1), ..., f(wn)). Thus, f is well represented by the list of the corresponding column matrices. That is, if f ( w j ) = a 1 , j v 1 ... | Wikipedia - Linear Algebra | null | null | null |
Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows tha... | Wikipedia - Linear Algebra | null | null | null |
It can be proved that two matrices are similar if and only if one can transform one into the other by elementary row and column operations. For a matrix representing a linear map from W to V, the row operations correspond to change of bases in V and the column operations correspond to change of bases in W. Every matrix... | Wikipedia - Linear Algebra | null | null | null |
Matrices are supported through the use of ten built-in matrices. Matrices do not support user created names or complex numbers. | Wikipedia - TI-BASIC 83 | null | null | null |
Matrices are usually denoted by boldface capital letters, e.g. A {\textstyle {\mathbf {A}}} . Column vectors are usually denoted by boldface lowercase letters, e.g. x {\textstyle {\mathbf {x}}} . The transpose operator is denoted by either a superscript T (e.g. A T {\textstyle {\mathbf {A}}^{\mathrm {T} }} ) or a prime... | Wikipedia - Notation in probability and statistics | null | null | null |
Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension is tensors, whi... | Wikipedia - Matrix operation | null | null | null |
Matrices, subject to certain requirements tend to form groups known as matrix groups. Similarly under certain conditions matrices form rings known as matrix rings. Though the product of matrices is not in general commutative yet certain matrices form fields known as matrix fields. | Wikipedia - Matrix operation | null | null | null |
Matrix Biology is abstracted and indexed in BIOBASE, Biochemistry and Biophysics Citation Index, Biological & Agricultural Index, Biological Abstracts, BIOSIS Previews, Chemical Abstracts Service, Current Advances in Ecological and Environmental Sciences, Current Awareness in Biological Sciences, Current Contents, EMBA... | Wikipedia - Matrix Biology (journal) | null | null | null |
Matrix Rectangular arrangement of numbers or other mathematical objects. | Wikipedia - Glossary of linear algebra | null | null | null |
Matrix Reference Manual, Mike Brookes, Imperial College London. Matrix Differentiation (and some other stuff), Randal J. Barnes, Department of Civil Engineering, University of Minnesota. Notes on Matrix Calculus, Paul L. Fackler, North Carolina State University. Matrix Differential Calculus (slide presentation), Zhang ... | Wikipedia - Matrix differentiation | null | null | null |
Introduction to Vector and Matrix Differentiation (notes on matrix differentiation, in the context of Econometrics), Heino Bohn Nielsen. A note on differentiating matrices (notes on matrix differentiation), Pawel Koval, from Munich Personal RePEc Archive. Vector/Matrix Calculus More notes on matrix differentiation. Mat... | Wikipedia - Matrix differentiation | null | null | null |
Matrix attachment region Matrix metalloproteinase Matrix protein | Wikipedia - Matrix (biology) | null | null | null |
Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms or iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a sequence of vectors xn converging to an eigenvector when n tends to infinity.To choose the mo... | Wikipedia - Infinite matrix | null | null | null |
As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stability. Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorit... | Wikipedia - Infinite matrix | null | null | null |
The Strassen algorithm outperforms this "naive" algorithm; it needs only n2.807 multiplications. A refined approach also incorporates specific features of the computing devices. | Wikipedia - Infinite matrix | null | null | null |
In many practical situations additional information about the matrices involved is known. An important case are sparse matrices, that is, matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient metho... | Wikipedia - Infinite matrix | null | null | null |
For example, calculating the inverse of a matrix via Laplace expansion (adj(A) denotes the adjugate matrix of A) A−1 = adj(A) / det(A)may lead to significant rounding errors if the determinant of the matrix is very small. The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as... | Wikipedia - Infinite matrix | null | null | null |
Matrix manipulation was among the earliest numerical applications of computers. The original Dartmouth BASIC had built-in commands for matrix arithmetic on arrays from its second edition implementation in 1964. As early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC... | Wikipedia - Infinite matrix | null | null | null |
Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices. As of 2023, most computers have some form of built-in matrix operations at a low-level implementing the standard BLAS specification, upon which most higher-level matrix... | Wikipedia - Infinite matrix | null | null | null |
Matrix calculus is used for deriving optimal stochastic estimators, often involving the use of Lagrange multipliers. This includes the derivation of: Kalman filter Wiener filter Expectation-maximization algorithm for Gaussian mixture Gradient descent | Wikipedia - Matrix calculus | null | null | null |
Matrix calculus refers to a number of different notations that use matrices and vectors to collect the derivative of each component of the dependent variable with respect to each component of the independent variable. In general, the independent variable can be a scalar, a vector, or a matrix while the dependent variab... | Wikipedia - Matrix derivative | null | null | null |
As a first example, consider the gradient from vector calculus. For a scalar function of three independent variables, f ( x 1 , x 2 , x 3 ) {\displaystyle f(x_{1},x_{2},x_{3})} , the gradient is given by the vector equation ∇ f = ∂ f ∂ x 1 x ^ 1 + ∂ f ∂ x 2 x ^ 2 + ∂ f ∂ x 3 x ^ 3 , {\displaystyle \nabla f={\frac {\par... | Wikipedia - Matrix derivative | null | null | null |
∇ f = ( ∂ f ∂ x ) T = T . {\displaystyle \nabla f=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}={\begin{bmatrix}{\dfrac {\partial f}{\partial x_{1}}}&{\dfrac {\partial f}{\partial x_{2}}}&{\dfrac {\partial f}{\partial x_{3}}}\\\end{bmatrix}}^{\textsf {T}}.} More complicated examples include th... | Wikipedia - Matrix derivative | null | null | null |
In that case the scalar must be a function of each of the independent variables in the matrix. As another example, if we have an n-vector of dependent variables, or functions, of m independent variables we might consider the derivative of the dependent vector with respect to the independent vector. The result could be ... | Wikipedia - Matrix derivative | null | null | null |
There are a total of nine possibilities using scalars, vectors, and matrices. Notice that as we consider higher numbers of components in each of the independent and dependent variables we can be left with a very large number of possibilities. The six kinds of derivatives that can be most neatly organized in matrix form... | Wikipedia - Matrix derivative | null | null | null |
Here, we have used the term "matrix" in its most general sense, recognizing that vectors and scalars are simply matrices with one column and one row respectively. Moreover, we have used bold letters to indicate vectors and bold capital letters for matrices. This notation is used throughout. | Wikipedia - Matrix derivative | null | null | null |
Notice that we could also talk about the derivative of a vector with respect to a matrix, or any of the other unfilled cells in our table. However, these derivatives are most naturally organized in a tensor of rank higher than 2, so that they do not fit neatly into a matrix. In the following three sections we will defi... | Wikipedia - Matrix derivative | null | null | null |
Matrix chain multiplication is a well-known example that demonstrates utility of dynamic programming. For example, engineering applications often have to multiply a chain of matrices. It is not surprising to find matrices of large dimensions, for example 100×100. Therefore, our task is to multiply matrices A 1 , A 2 , ... | Wikipedia - Dynamic Programming | null | null | null |
. A n {\displaystyle A_{1},A_{2},....A_{n}} . Matrix multiplication is not commutative, but is associative; and we can multiply only two matrices at a time. | Wikipedia - Dynamic Programming | null | null | null |
So, we can multiply this chain of matrices in many different ways, for example: ((A1 × A2) × A3) × ... AnA1×(((A2×A3)× ... ) × An)(A1 × A2) × (A3 × ... An)and so on. There are numerous ways to multiply this chain of matrices. They will all produce the same final result, however they will take more or less time to compu... | Wikipedia - Dynamic Programming | null | null | null |
If matrix A has dimensions m×n and matrix B has dimensions n×q, then matrix C=A×B will have dimensions m×q, and will require m*n*q scalar multiplications (using a simplistic matrix multiplication algorithm for purposes of illustration). For example, let us multiply matrices A, B and C. Let us assume that their dimensio... | Wikipedia - Dynamic Programming | null | null | null |
(A×B)×C This order of matrix multiplication will require mnp + mps scalar calculations.Let us assume that m = 10, n = 100, p = 10 and s = 1000. So, the first way to multiply the chain will require 1,000,000 + 1,000,000 calculations. The second way will require only 10,000+100,000 calculations. | Wikipedia - Dynamic Programming | null | null | null |
Obviously, the second way is faster, and we should multiply the matrices using that arrangement of parenthesis. Therefore, our conclusion is that the order of parenthesis matters, and that our task is to find the optimal order of parenthesis. At this point, we have several choices, one of which is to design a dynamic p... | Wikipedia - Dynamic Programming | null | null | null |
The dynamic programming solution is presented below. Let's call m the minimum number of scalar multiplications needed to multiply a chain of matrices from matrix i to matrix j (i.e. Ai × .... × Aj, i.e. i<=j). We split the chain at some matrix k, such that i <= k < j, and try to find out which combination produces mini... | Wikipedia - Dynamic Programming | null | null | null |
. . A n {\displaystyle A_{1},A_{2},...A_{n}}: function OptimalMatrixChainParenthesis(chain) n = length(chain) for i = 1, n m = 0 // Since it takes no calculations to multiply one matrix for len = 2, n for i = 1, n - len + 1 j = i + len -1 m = infinity // So that the first calculation updates for k = i, j-1 q = m + m + ... | Wikipedia - Dynamic Programming | null | null | null |
This algorithm will produce "tables" m and s that will have entries for all possible values of i and j. The final solution for the entire chain is m, with corresponding split at s. Unraveling the solution will be recursive, starting from the top and continuing until we reach the base case, i.e. multiplication of single... | Wikipedia - Dynamic Programming | null | null | null |
Matrix decomposition Cholesky decomposition LU decomposition QR decomposition Polar decomposition Reducing subspace Spectral theorem Singular value decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula | Wikipedia - Outline of linear algebra | null | null | null |
Matrix decompositions suggest a number of ways to solve the linear system r = b − Ax where we seek to minimize r, as in the regression problem. The QR algorithm solves this problem by computing the reduced QR factorization of A and rearranging to obtain R ^ x = Q ^ ∗ b {\displaystyle {\widehat {R}}x={\widehat {Q}}^{\as... | Wikipedia - Matrix computation | null | null | null |
Matrix equivalence Matrix congruence Matrix similarity Matrix consimilarity Row equivalence | Wikipedia - Outline of linear algebra | null | null | null |
Matrix inversion (Cramer's rule) and determinants can be naturally expressed in terms of the wedge product. The use of the wedge product in the solution of linear equations can be quite useful for various geometric product calculations. Traditionally, instead of using the wedge product, Cramer's rule is usually present... | Wikipedia - Comparison of vector algebra and geometric algebra | null | null | null |
{\displaystyle x={\frac {1}{|A|}}\operatorname {adj} (A)b.} This is a useful theoretic result. For numerical problems row reduction with pivots and other methods are more stable and efficient. | Wikipedia - Comparison of vector algebra and geometric algebra | null | null | null |
When the wedge product is coupled with the Clifford product and put into a natural geometric context, the fact that the determinants are used in the expression of R N {\displaystyle {\mathbb {R} }^{N}} parallelogram area and parallelepiped volumes (and higher-dimensional generalizations thereof) also comes as a nice si... | Wikipedia - Comparison of vector algebra and geometric algebra | null | null | null |
The result is then simple enough that it could be derived easily if required instead of having to remember or look up a rule. Two variables example = a x + b y = c . {\displaystyle {\begin{bmatrix}a&b\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}=ax+by=c.} | Wikipedia - Comparison of vector algebra and geometric algebra | null | null | null |
Pre- and post-multiplying by a {\displaystyle a} and b {\displaystyle b} , ( a x + b y ) ∧ b = ( a ∧ b ) x = c ∧ b {\displaystyle (ax+by)\wedge b=(a\wedge b)x=c\wedge b} a ∧ ( a x + b y ) = ( a ∧ b ) y = a ∧ c {\displaystyle a\wedge (ax+by)=(a\wedge b)y=a\wedge c} Provided a ∧ b ≠ 0 {\displaystyle a\wedge b\neq 0} the ... | Wikipedia - Comparison of vector algebra and geometric algebra | null | null | null |
Similarly, for three, or N variables, the same ideas hold = d {\displaystyle {\begin{bmatrix}a&b&c\end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}=d} = 1 a ∧ b ∧ c {\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}={\frac {1}{a\wedge b\wedge c}}{\begin{bmatrix}d\wedge b\wedge c\\a\wedge d\wedge c\\a\wedge b\we... | Wikipedia - Comparison of vector algebra and geometric algebra | null | null | null |
x + y = {\displaystyle {\begin{bmatrix}1\\1\\0\end{bmatrix}}x+{\begin{bmatrix}1\\1\\1\end{bmatrix}}y={\begin{bmatrix}1\\1\\2\end{bmatrix}}} The right wedge product with ( 1 , 1 , 1 ) {\displaystyle (1,1,1)} solves for x {\displaystyle x} ∧ x = ∧ {\displaystyle {\begin{bmatrix}1\\1\\0\end{bmatrix}}\wedge {\begin{b... | Wikipedia - Comparison of vector algebra and geometric algebra | null | null | null |
Collection of results for x {\displaystyle x} and y {\displaystyle y} yields a Cramer's rule-like form: = 1 ( 1 , 1 , 0 ) ∧ ( 1 , 1 , 1 ) . {\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}}={\frac {1}{(1,1,0)\wedge (1,1,1)}}{\begin{bmatrix}(1,1,2)\wedge (1,1,1)\\(1,1,0)\wedge (1,1,2)\end{bmatrix}}.} Writing e i ∧ e j... | Wikipedia - Comparison of vector algebra and geometric algebra | null | null | null |
Matrix mechanics rapidly developed into modern quantum mechanics, and gave interesting physical results on the spectra of atoms. | Wikipedia - Matrix Mechanics | null | null | null |
Matrix population models == References == | Wikipedia - Matrix (biology) | null | null | null |
Matrix types (special types like bidiagonal/tridiagonal are not listed): Real – general (nonsymmetric) real Complex – general (nonsymmetric) complex SPD – symmetric positive definite (real) HPD – Hermitian positive definite (complex) SY – symmetric (real) HE – Hermitian (complex) BND – bandOperations: TF – triangular f... | Wikipedia - Comparison of linear algebra libraries | null | null | null |
MatrixCalculus.org, a website for evaluating matrix calculus expressions symbolically NCAlgebra, an open-source Mathematica package that has some matrix calculus functionality SymPy supports symbolic matrix derivatives in its matrix expression module, as well as symbolic tensor derivatives in its array expression modul... | Wikipedia - Matrix derivative | null | null | null |
Matroid theory developed mainly out of a deep examination of the properties of independence and dimension in vector spaces. There are two ways to present the matroids defined in this way: If E {\displaystyle E} is any finite subset of a vector space V {\displaystyle V} , then we can define a matroid M {\displaystyle M}... | Wikipedia - Matroid constraints | null | null | null |
Matroids of this kind are called vector matroids. An important example of a matroid defined in this way is the Fano matroid, a rank-three matroid derived from the Fano plane, a finite geometry with seven points (the seven elements of the matroid) and seven lines (the proper nontrivial flats of the matroid). | Wikipedia - Matroid constraints | null | null | null |
It is a linear matroid whose elements may be described as the seven nonzero points in a three-dimensional vector space over the finite field GF(2). However, it is not possible to provide a similar representation for the Fano matroid using the real numbers in place of GF(2). | Wikipedia - Matroid constraints | null | null | null |
A matrix A {\displaystyle A} with entries in a field gives rise to a matroid M {\displaystyle M} on its set of columns. The dependent sets of columns in the matroid are those that are linearly dependent as vectors. This matroid is called the column matroid of A {\displaystyle A} , and A {\displaystyle A} is said to rep... | Wikipedia - Matroid constraints | null | null | null |
For instance, the Fano matroid can be represented in this way as a 3 × 7 (0,1)-matrix. Column matroids are just vector matroids under another name, but there are often reasons to favor the matrix representation. (There is one technical difference: a column matroid can have distinct elements that are the same vector, bu... | Wikipedia - Matroid constraints | null | null | null |
Usually this difference is insignificant and can be ignored, but by letting E {\displaystyle E} be a multiset of vectors one brings the two definitions into complete agreement. )A matroid that is equivalent to a vector matroid, although it may be presented differently, is called representable or linear. If M {\displays... | Wikipedia - Matroid constraints | null | null | null |
For instance, although a graphic matroid (see below) is presented in terms of a graph, it is also representable by vectors over any field. A basic problem in matroid theory is to characterize the matroids that may be represented over a given field F {\displaystyle F} ; Rota's conjecture describes a possible characteriz... | Wikipedia - Matroid constraints | null | null | null |
Matsuhashi and Hallett's study can be summarized thus. The researchers hypothesized that, if our conscious intentions are what causes movement genesis (i.e. the start of an action), then naturally, our conscious intentions should always occur before any movement has begun. Otherwise, if we ever become aware of a moveme... | Wikipedia - Neuroscience of free will | null | null | null |
To test this hypothesis, Matsuhashi and Hallet had volunteers perform brisk finger movements at random intervals, while not counting or planning when to make such (future) movements, but rather immediately making a movement as soon as they thought about it. An externally controlled "stop-signal" sound was played at pse... | Wikipedia - Neuroscience of free will | null | null | null |
The graph of tones before actions therefore only shows tones (a) before the subject is even aware of his "movement genesis" (or else they would have stopped or "vetoed" the movement), and (b) after it is too late to veto the action. This second set of graphed tones is of little importance here. In this work, "movement ... | Wikipedia - Neuroscience of free will | null | null | null |
By looking to see when tones started preventing actions, the researchers supposedly know the length of time (in seconds) that exists between when a subject holds a conscious intention to move and performs the action of movement. This moment of awareness is called T (the mean time of conscious intention to move). It can... | Wikipedia - Neuroscience of free will | null | null | null |
This enables the researchers to estimate the timing of the conscious intention to move without relying on the subject's knowledge or demanding them to focus on a clock. The last step of the experiment is to compare time T for each subject with their event-related potential (ERP) measures (e.g. seen in this page's lead ... | Wikipedia - Neuroscience of free will | null | null | null |
See the example of a subject's graph below on the right. Although it is not shown on the graph, the subject's readiness potentials (ERP) tells us that his actions start at −2.8 seconds, and yet this is substantially earlier than his conscious intention to move, time T (−1.8 seconds). Matsuhashi and Hallet concluded tha... | Wikipedia - Neuroscience of free will | null | null | null |
Matter composed of atoms can have different properties and appear in different forms, such as solid, liquid, superfluid, etc. These various forms of matter are often called states of matter or phases. According to condensed matter physics and the principle of emergence, the different properties of materials generally a... | Wikipedia - Topological phase | null | null | null |
It points out that different orders really correspond to different symmetries in the organizations of the constituent atoms. As a material changes from one order to another order (i.e., as the material undergoes a phase transition), what happens is that the symmetry of the organization of the atoms changes. | Wikipedia - Topological phase | null | null | null |
For example, atoms have a random distribution in a liquid, so a liquid remains the same as we displace atoms by an arbitrary distance. We say that a liquid has a continuous translation symmetry. After a phase transition, a liquid can turn into a crystal. | Wikipedia - Topological phase | null | null | null |
In a crystal, atoms organize into a regular array (a lattice). A lattice remains unchanged only when we displace it by a particular distance (integer times a lattice constant), so a crystal has only discrete translation symmetry. The phase transition between a liquid and a crystal is a transition that reduces the conti... | Wikipedia - Topological phase | null | null | null |
Such a change in symmetry is called symmetry breaking. The essence of the difference between liquids and crystals is therefore that the organizations of atoms have different symmetries in the two phases. Landau symmetry-breaking theory has been a very successful theory. For a long time, physicists believed that Landau ... | Wikipedia - Topological phase | null | null | null |
Matter propagating in a curved spacetime is similar to the electromagnetic wave propagation in a curved space and in an in homogeneous metamaterial, as stated in the previous section. Hence a black hole can possibly be simulated using electromagnetic fields and metamaterials. In July 2009 a metamaterial structure formi... | Wikipedia - Transformation optics | null | null | null |
It was constructed of a thin cylinder at 21.6 centimeters in diameter comprising 60 concentric rings of metamaterials. This structure created a gradient index of refraction, necessary for bending light in this way. However, it was characterized as being artificially inferior substitute for a real black hole. | Wikipedia - Transformation optics | null | null | null |
The characterization was justified by an absorption of only 80% in the microwave range, and that it has no internal source of energy. It is singularly a light absorber. The light absorption capability could be beneficial if it could be adapted to technologies such as solar cells. | Wikipedia - Transformation optics | null | null | null |
However, the device is limited to the microwave range.Also in 2009, transformation optics were employed to mimic a black hole of Schwarzschild form. Similar properties of photon sphere were also found numerically for the metamaterial black hole. Several reduced versions of the black hole systems were proposed for easie... | Wikipedia - Transformation optics | null | null | null |
Mattis, Daniel C. (1993). The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension. World Scientific. ISBN 978-981-02-0975-9. | Wikipedia - List of quantum-mechanical systems with analytical solutions | null | null | null |
Matula & Beck (1983) outline an algorithm to derive the degeneracy ordering of a graph G = ( V , E ) {\displaystyle G=(V,E)} with vertex set V and edge set E in O ( | V | + | E | ) {\displaystyle {\mathcal {O}}(\vert V\vert +\vert E\vert )} time and O ( | V | ) {\displaystyle {\mathcal {O}}(\vert V\vert )} words of spa... | Wikipedia - Degeneracy (graph theory) | null | null | null |
Repeat n times: Scan the array cells D, D, ... until finding an i for which D is nonempty. Set k to max(k,i) Select a vertex v from D. Add v to the beginning of L and remove it from D. For each neighbor w of v not already in L, subtract one from dw and move w to the cell of D corresponding to the new value of dw.At the... | Wikipedia - Degeneracy (graph theory) | null | null | null |
Mature R. vannielii cells are typically ovoid to spherical in shape, do not form intracellular sulfur globules, and are about 2.0-2.5 μm long and 1.2-1.5 μm wide. The cells are then attached by means of slender branched filaments, with the connecting filaments varying in length, but a uniform diameter of approximately ... | Wikipedia - Rhodomicrobium vannielii | null | null | null |
Maurotoxin is a peptide of 34 amino acids (sequence VSCTGSKDCYAPCRKQTGCPNAKCINKSCKCYGC) cross-linked by four disulfide bridges (Cys3-Cys24, Cys9-Cys29, Cys13-Cys19, Cys31-Cys34), with an atypical pattern of organization compared with other scorpion toxins; this unusual pairing of cysteine residues may be mediated by th... | Wikipedia - Maurotoxin | null | null | null |
Mauveine is a mixture of four related aromatic compounds differing in number and placement of methyl groups. Its organic synthesis involves dissolving aniline, p-toluidine, and o-toluidine in sulfuric acid and water in a roughly 1:1:2 ratio, then adding potassium dichromate.Mauveine A (C26H23N+4X−) incorporates 2 molec... | Wikipedia - Mauveine | null | null | null |
In fact, safranine is a 2,8-dimethyl phenazinium salt, whereas the parasafranine produced by Perkin is presumed to be the 1,8- (or 2,9-) dimethyl isomer. The molecular structure of mauveine proved difficult to determine, finally being identified in 1994. In 2007, two more were isolated and identified: mauveine B2, an i... | Wikipedia - Mauveine | null | null | null |
Mavalvala has also worked on the development of exotic quantum states of light, and in particular the generation of light in squeezed coherent states. By injecting such states into the kilometre-scale Michelson interferometer of the LIGO detectors, her group greatly improved the sensitivity of the detector by reducing ... | Wikipedia - Nergis Mavalvala | null | null | null |
Maven has used several algorithms for move generation, but the one that has stuck is the DAWG algorithm. The GADDAG algorithm is faster, but a DAWG for North American English is only 0.5 MB, compared to about 2.5 MB for a GADDAG. That makes a significant difference for download games, whereas the speed advantage is not... | Wikipedia - Maven (Scrabble) | null | null | null |
Maven's gameplay is sub-divided into three phases: The "mid-game" phase, the "pre-endgame" phase, and the "endgame" phase. The "mid-game" phase lasts from the beginning of the game up until there are nine or fewer tiles left in the bag. The program uses a rapid algorithm to find all possible plays from the given rack, ... | Wikipedia - Maven (Scrabble) | null | null | null |
By simulating thousands of random drawings, the program can give a very accurate quantitative evaluation of the different plays. (While a Monte Carlo search, Maven does not use Monte Carlo tree search because it evaluates game trees only 2-ply deep, rather than playing out to the end of the game, and does not reallocat... | Wikipedia - Maven (Scrabble) | null | null | null |
A true MCTS strategy is unnecessary because the endgame can be solved. The shallow search is because the Maven author argues that, due to the fast turnover of letters in one's bag, it is typically not useful to look more than 2-ply deep, because if one instead looked deeper, e.g. 4-ply, the variance of rewards will be ... | Wikipedia - Maven (Scrabble) | null | null | null |
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