eigenvalue

A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity. The action of parity acting on a Dirac fermion is called chiral symmetry. An experiment on the weak decay of cobalt in 1956 showed that parity is not a symmetry of the universe. Chirality physics - Chirality. A massless fermion is L ...

Including:

• Chirality physics - Chirality
• Chirality physics - Chiral symmetry

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Data clustering is a common technique for statistical data analysis, which is used in many fields, including machine learning, data mining, pattern recognition, image analysis and bioinformatics. Clustering is the classification of similar objects into different groups, or more precisely, the partitioning of a data set into subsets (clusters), so that the data in each subset (ideally) share some common trait - often proximity according to some defined distance measure. Machine learning typically regar ...

Including:

• Data clustering - Types of clustering
• Data clustering - Hierarchical clustering
• Data clustering - Introduction
• Data clustering - Agglomerative hierarchical clustering
• Data clustering - Partitional clustering
• Data clustering - k-means and derivatives
• Data clustering - The elbow criterion
• Data clustering - Spectral clustering
• Data clustering - Applications
• Data clustering - Biology
• Data clustering - Marketing research
• Data clustering - Other applications
• Data clustering - Comparisons between data clusterings
• Data clustering - Bibliography
• Data clustering - Software implementations
• Data clustering - Free
• Data clustering - Non-free

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There is a request, submitted by DaBlade, for an audio version of this article to be created. See WikiProject Spoken Wikipedia for further information. The rationale behind the request is: "This article is very technical, and needs to be explained in an easier to understand fashion for those who aren't very technical within physics. With it being spoken, the speaker can add more general information, to make it more understandable for the not-so physics savvy.". See also: Category ...

Including:

• Zero-point energy - Introduction
• Zero-point energy - What is it?
• Zero-point energy - How do we know it exists?
• Zero-point energy - Is it controversial?
• Zero-point energy - Is it relevant?
• Zero-point energy - A few formulae
• Zero-point energy - Existence
• Zero-point energy - History
• Zero-point energy - Problems and answers
• Zero-point energy - Properties
• Zero-point energy - Cultural references

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Computational chemistry is a branch of theoretical chemistry whose major goals are to create efficient mathematical approximations and computer programs that calculate the properties of molecules (such as total energy, dipole and quadrupole moment, vibrational frequencies, reactivity and other diverse spectroscopic quantitities and cross sections for collision of molecules with diverse atomic or subatomic projectiles) and to apply these programs to concrete chemical objects. The term is also sometimes used to cover the areas of overla ...

Including:

• Computational chemistry - Introduction
• Computational chemistry - Ab initio methods
• Computational chemistry - Electronic structure
• Computational chemistry - Chemical dynamics
• Computational chemistry - Semiempirical methods
• Computational chemistry - Electronic structure
• Computational chemistry - Molecular mechanics
• Computational chemistry - Software packages

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In mathematics, a boundary value problem consists of a differential equation to be satisfied at all points in the interior of an interval or a region and a set of boundary conditions specifying the values of the solution or some of its derivatives everywhere on the boundary of the interval or region. Boundary value problems may be posed for ordinary differential equations as well as partial differential equations. Boundary value problems arise in several branches of physics. Problems involving the wave equation, such as the determination ...

Including:

• Boundary value problem - Example
• Boundary value problem - Example ordinary
• Boundary value problem - Example partial
• Boundary value problem - Bibliography

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Quantum mechanics is a fundamental physical theory that replaces Newtonian mechanics and classical electromagnetism at the atomic and subatomic levels and is the underlying framework of many fields of physics and chemistry, including condensed matter physics, quantum chemistry, and particle physics. Along with general relativity, it is one of the pillars of modern physics. Quantum mechanics - Introduction. The term quantum (Latin, "how much") refers to the discrete units that the theory assign ...

Including:

• Quantum mechanics - Introduction
• Quantum mechanics - Description of the theory
• Quantum mechanics - Quantum mechanical effects
• Quantum mechanics - Mathematical formulation
• Quantum mechanics - Interactions with other scientific theories
• Quantum mechanics - Applications of quantum theory
• Quantum mechanics - Philosophical consequences
• Quantum mechanics - History
• Quantum mechanics - Founding experiments
• Quantum mechanics - Notes

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In Euclidean geometry, of special interest are involutions which are linear or affine transformations over the Euclidean space Rn. Such involutions are easy to characterize and they can be described geometrically. Affine involution - Linear involutions. To give a linear involution is the same as giving a square matrix A such that where I is the identity function. It is a quick check that a square matrix D that has zero off the ...

Including:

• Affine involution - Linear involutions
• Affine involution - Affine involutions
• Affine involution - Isometric involutions

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In an undirected graph, a connected component or component is a maximal connected subgraph. Two vertices are in the same connected component if and only if there exists a path between them. In a drawing of a graph, the connected components can each be drawn separately with empty space between them. A nonempty connected graph has one connected component. In an undirected graph, the existence of a path between two vertices u and v is an equivalence relation, since: There is a trivial path of leng ...

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In numerical analysis, the condition number associated with a numerical problem is a measure of that quantity's amenability to digital computation, that is, how well-posed the problem is. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. Condition number - The condition number of a matrix. For example, the condition number associated with the linear equation A ...

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• Condition number - The condition number of a matrix
• Condition number - The condition number in other contexts

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In electromagnetics and communications engineering, a waveguide is a physical structure that guides the propagation of electromagnetic waves. Waveguides can be constructed to carry waves over a wide portion of the electromagnetic spectrum, but are especially useful in the microwave and optical frequency ranges. Depending on the frequency, they can be constructed from either conductive or dielectric materials. Waveguides are used for transferring both power and communication signals. Waveguide - History. The ...

Including:

• Waveguide - History
• Waveguide - Principles of operation
• Waveguide - Hollow metallic waveguides
• Waveguide - Transmission lines
• Waveguide - Dielectric waveguides

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There are a set of quantum numbers associated with the energy states of the atom. The four quantum numbers n, l, m, and s specify the complete and unique quantum state of a single electron in an atom called its wavefunction or orbital. The wavefunction of the Schrödinger wave equation reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The magnetic quantum number arose in the solution of t ...

See also:

Magnetic quantum number, Magnetic quantum number - Derivation

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For hand-waving (imprecise, but qualitatively useful) discussion of the molecular structure, the molecular orbitals can be obtained from the "Linear combination of atomic orbitals molecular orbital method" ansatz (using eventually the concept of hybridized orbitals). In this approach, the molecular orbitals are expressed as linear combinations of atomic orbitals, as if each atom were on its own. The linear combination of atomic orbitals approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones. His g ...

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Molecular orbital, Molecular orbital - Hand-waving discussion, Molecular orbital - Examples, Molecular orbital - More quantitative approach

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In general, an object's moment of inertia depends on its shape and the distribution of mass within that shape: the greater the concentration of material away from the object's centroid, the larger the moment of inertia. It also varies depending upon the axis of rotation specified; values relative to the object's centroid are typically taken as baseline values. See the list of moments of inertia for specific examples. The parallel axes theorem can be used to determine moments o ...

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Moment of inertia, Moment of inertia - Explanation, Moment of inertia - Confusion with second moment of area, Moment of inertia - Derivation for point mass, Moment of inertia - Mathematical definition, Moment of inertia - Types of moment of inertia, Moment of inertia - Application of moment of inertia, Moment of inertia - Inertia tensor

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Let us start with a time-varying system whose impulse response is a two dimensional function and see how the condition of time-invariance helps us reduce it to one dimension. For example, suppose the input signal is x(t) where its index set is the real line, i.e., . The linear operator represents the system operating on the input signal. The appropriate ...

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LTI system theory, LTI system theory - Time invariance and linear transformation, LTI system theory - Impulse response, LTI system theory - Complex exponentials as eigenfunctions

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A marginally stable system is one that, if given an impulse of finite magnitude as input, will not "blow up" and give an unbounded output. However, oscillations in the output will persist indefinitely, and so there will, in general, be no final steady-state output. If the system is given a step as an input, the system's output will increase indefinitely, with the system effectively acting as an integrator on the input, and so a marginally stable syste ...

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Marginal stability, Marginal stability - Practical Consequences

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Matrix mathematics - Sum. If two m-by-n matrices A and B are given, we may define their sum A + B as the m-by-n matrix computed by adding corresponding elements, i.e., (A + B)[i, j] = A[i, j] + B[i, j]. For example Another, much less often used notion of matrix addition is the direct sum.

Matrix mathematics, Matrix mathematics - Definitions and notations, Matrix mathematics - Example, Matrix mathematics - Adding and multiplying matrices, Matrix mathematics - Sum, Matrix mathematics - Scalar multiplication, Matrix mathematics - Multiplication, Matrix mathematics - Linear transformations ranks and transpose, Matrix mathematics - Square matrices and related definitions, Matrix mathematics - Special types of matrices, Matrix mathematics - Matrices in abstract algebra, Matrix mathematics - History

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The following picture depicts (after stereographic transformation from the sphere to the plane) the two fixed points of a Möbius transformation in the non-parabolic case: The characteristic constant can be expressed in terms of its logarithm: When expressed in this way, the real number ρ becomes an expansion factor. It indicates how repulsive the fixed point γ1 is, and how attractive γ2 i ...

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Möbius transformation, Möbius transformation - Overview, Möbius transformation - Definition, Möbius transformation - Projective matrix representations, Möbius transformation - Properties, Möbius transformation - Classification, Möbius transformation - Fixed points, Möbius transformation - Normal form, Möbius transformation - Geometric interpretation of the characteristic constant, Möbius transformation - Elliptic transformations, Möbius transformation - Hyperbolic transformations, Möbius transformation - Loxodromic transformations, Möbius transformation - Stereographic projection, Möbius transformation - Iterating a transformation, Möbius transformation - Poles of the transformation, Möbius transformation - Specifying a transformation by three points

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Let A be a square n by n matrix over a field K (for example the field R of real numbers). Then the following statements are equivalent: A is invertible. A is row-equivalent to the n by n identity matrix In. A has n pivot positions. det A ≠ 0. rank A = n. The equation Ax = 0 has only the trivial solution x = 0 (i.e. Null A = {0}). Th ...

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Invertible matrix, Invertible matrix - Properties of invertible matrices, Invertible matrix - Proof for matrix product rule, Invertible matrix - Methods of matrix inversion, Invertible matrix - Gauss-Jordan elimination, Invertible matrix - Analytic solution, Invertible matrix - Blockwise inversion, Invertible matrix - The derivative of the matrix inverse, Invertible matrix - The Moore-Penrose pseudoinverse

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Matrix mathematics - Sum. Main articles: Matrix addition, and [[]], and [[]], and See also:

Matrix mathematics, Matrix mathematics - Definitions and notations, Matrix mathematics - Example, Matrix mathematics - Adding and multiplying matrices, Matrix mathematics - Sum, Matrix mathematics - Scalar multiplication, Matrix mathematics - Multiplication, Matrix mathematics - Linear transformations ranks and transpose, Matrix mathematics - Square matrices and related definitions, Matrix mathematics - Special types of matrices, Matrix mathematics - Matrices in abstract algebra, Matrix mathematics - History

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When the canonical quantization procedure is applied to quantum field theory, the classical field variable becomes a quantum operator which acts on a quantum state of the field theory to increase or decrease the number of particles by one. In one way of viewing things, quantizing the classical theory of a fixed number of particles gave rise to a wavefunction. This wavefunction is a field variable which could then be quantized to deal with the theory of many particles. So the process of canonical quantization of a ...

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Canonical quantization, Canonical quantization - History, Canonical quantization - Quantum mechanics, Canonical quantization - Second quantization: field theory, Canonical quantization - Field operator, Canonical quantization - Condensates, Canonical quantization - Why canonical?, Canonical quantization - Mathematical quantization

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