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Sleeveface-Sleeveface-Sleeveface is an internet phenomenon wherein one or more persons obscure or augment body parts with record sleeve(s), causing an illusion. Sleeveface has become popular on social networking sites.The precise origin of the concept is unknown. A collection of photographs was posted online at Waxider... | milkshake721/2.1M-wiki-STEM |
Sleeveface-Sleeveface-Sleeveface contributors regularly hold Sleeveface parties across the world.Sleeveface contributors have helped organise Sleeveface workshops for children. One such workshop took place at the National Museum Cardiff in November 2008 as part of the city's annual Sŵn Festival.There is also a Sleevefa... | milkshake721/2.1M-wiki-STEM |
Standalone program-Standalone program-A standalone program, also known as a freestanding program, is a computer program that does not load any external module, library function or program and that is designed to boot with the bootstrap procedure of the target processor – it runs on bare metal. In early computers like t... | milkshake721/2.1M-wiki-STEM |
Standalone program-Standalone program-Later standalone programs typically were provided for utility functions such as disk formatting. Also, computers with very limited memory may use standalone programs, i.e. most computers until the mid-1950s and later still embedded processors. | milkshake721/2.1M-wiki-STEM |
Standalone program-Standalone program-Standalone programs are now mainly limited to SoC's or microcontrollers (where battery life, price, and data space are at premiums) and critical systems. In extreme cases every possible set of inputs and errors must be tested and thus every potential output known; fully independent... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Multiple endocrine neoplasia type 1-Multiple endocrine neoplasia type 1 (MEN-1) is one of a group of disorders, the multiple endocrine neoplasias, that affect the endocrine system through development of neoplastic lesions in pituitary, parathyroid gland and pancreas. Individuals suff... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Signs and symptoms-Parathyroid Hyperparathyroidism is present in ≥ 90% of patients. Asymptomatic hypercalcemia is the most common manifestation: about 25% of patients have evidence of nephrolithiasis or nephrocalcinosis. In contrast to sporadic cases of hyperparathyroidism, diffuse h... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Signs and symptoms-Pancreas Pancreatic islet cell tumors are today the major cause of death in persons with MEN-1. Tumors occur in 60-80% of persons with MEN-1 and they are usually multicentric. Multiple adenomas or diffuse islet cell hyperplasia commonly occurs. About 30% of tumors ... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Signs and symptoms-Most islet cell tumors secrete pancreatic polypeptide, the clinical significance of which is unknown. Gastrin is secreted by many non–β-cell tumors (increased gastrin secretion in MEN 1 also often originates from the duodenum). Increased gastrin secretion increases... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Signs and symptoms-A severe secretory diarrhea can develop and cause fluid and electrolyte depletion with non–β-cell tumors. This complex, referred to as the watery diarrhea, hypokalemia and achlorhydria syndrome (VIPoma) has been ascribed to vasoactive intestinal polypeptide, althou... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Signs and symptoms-Pituitary Pituitary tumors occur in 15 to 42% of MEN 1 patients. From 25 to 90% are prolactinomas. About 25% of pituitary tumors secrete growth hormone or growth hormone and prolactin. Excess prolactin may cause galactorrhea, and excess growth hormone causes acrome... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Signs and symptoms-Other manifestations Adenomas of adrenal glands occurs occasionally in MEN 1 patients. Hormone secretion is rarely altered as a result, and the significance of these abnormalities is uncertain. Carcinoid tumors, particularly those derived from the embryologic foreg... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Genetic-People with multiple endocrine neoplasia type 1 are born with one mutated copy of the MEN1 gene in each cell. Then, during their lifetime, the other copy of the gene is mutated in a small number of cells. These genetic changes result in no functional copies of the MEN1 gene i... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Diagnosis-In a diagnostic workup individuals with a combination of endocrine neoplasias suggestive of the MEN1 syndrome are recommended to have a mutational analysis of the MEN1 gene if additional diagnostic criteria are sufficiently met, mainly including: age <40 years positive fami... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Diagnosis-multifocal or recurrent neoplasia two or more organ systems affected Types Multiple endocrine neoplasia or MEN is part of a group of disorders that affect the body's network of hormone-producing glands (the endocrine system). Hormones are chemical messengers that travel thr... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Diagnosis-The two major forms of multiple endocrine neoplasia are called type 1 and type 2. These two types are often confused because of their similar names. However, type 1 and type 2 are distinguished by the genes involved, the types of hormones made, and the characteristic signs ... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Diagnosis-These disorders greatly increase the risk of developing multiple cancerous and noncancerous tumors in glands such as the parathyroid, pituitary, and pancreas. Multiple endocrine neoplasia occurs when tumors are found in at least two of the three main endocrine glands (parat... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Diagnosis-Although many different types of hormone-producing tumors are associated with multiple endocrine neoplasia, tumors of the parathyroid gland, pituitary gland, and pancreas are most frequent in multiple endocrine neoplasia type 1. MEN1-associated overactivity of these three e... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Diagnosis-Neoplasia in the pituitary gland can manifest as prolactinomas whereby too much prolactin is secreted, suppressing the release of gonadotropins, causing a decrease in sex hormones such as testosterone. Pituitary tumor in MEN1 can be large and cause signs by compressing adja... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Diagnosis-Pancreatic tumors associated with MEN-1 usually form in the beta cells of the islets of Langerhans, causing over-secretion of insulin, resulting in low blood glucose levels (hypoglycemia). However, many other tumors of the pancreatic Islets of Langerhans can occur in MEN-1.... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Treatment-The treatment of choice of parathyroid tumors is open bilateral exploration with subtotal (3/4) or total parathyroidectomy. Autoimplantation may be considered in case of a total parathyroidectomy. Optimal timing for this operation has not yet been established but it should ... | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Treatment-Pituitary tumors are treated with surgery (acromegaly and Mb. Cushing) or medicine (prolactinomas). | milkshake721/2.1M-wiki-STEM |
Multiple endocrine neoplasia type 1-Culture and society-In the video game Trauma Team, Gabriel Cunningham's son, Joshua Cunningham, is diagnosed with Wermer's syndrome.
It is also mentioned in the South Korean drama "Medical Top Team", as Dr. Choi Ah Jin (Oh Yeon-seo) is diagnosed with MEN-1. | milkshake721/2.1M-wiki-STEM |
Monoidal category-Monoidal category-In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor ⊗:C×C→C that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorp... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Monoidal category-The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underl... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Monoidal category-A rather different application, of which monoidal categories can be considered an abstraction, is that of a system of data types closed under a type constructor that takes two types and builds an aggregate type; the types are the objects and ⊗ is the aggregate constructor. The assoc... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Monoidal category-Monoidal categories have numerous applications outside of category theory proper. They are used to define models for the multiplicative fragment of intuitionistic linear logic. They also form the mathematical foundation for the topological order in condensed matter physics. Braided m... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Formal definition-A monoidal category is a category C equipped with a monoidal structure. A monoidal structure consists of the following: a bifunctor ⊗:C×C→C called the monoidal product, or tensor product, an object I called the monoidal unit, unit object, or identity object, three natural isomorph... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Formal definition-The coherence conditions for these natural transformations are: for all A , B , C and D in C , the pentagon diagram commutes;for all A and B in C , the triangle diagramcommutes.A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Ev... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Examples-Any category with finite products can be regarded as monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a cartesian monoidal category. For example: Set, the category of sets with the Cartesian product, any particular one-... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Examples-Dually, any category with finite coproducts is monoidal with the coproduct as the monoidal product and the initial object as the unit. Such a monoidal category is called cocartesian monoidal R-Mod, the category of modules over a commutative ring R, is a monoidal category with the tensor produ... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Examples-Ab, the category of abelian groups, with the group of integers Z serving as the unit.
For any commutative ring R, the category of R-algebras is monoidal with the tensor product of algebras as the product and R as the unit.
The category of pointed spaces (restricted to compactly generated spac... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Examples-Monoidal preorders Monoidal preorders, also known as "preordered monoids", are special cases of monoidal categories. This sort of structure comes up in the theory of string rewriting systems, but it is plentiful in pure mathematics as well. For example, the set N of natural numbers has both ... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Examples-It's well known that a preorder can be considered as a category C, such that for every two objects c,c′∈Ob(C) , there exists at most one morphism c→c′ in C. If there happens to be a morphism from c to c' , we could write c≤c′ , but in the current section we find it more convenient to expre... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Examples-Moving forward, suppose we want to add a monoidal structure to the preorder C. To do so means we must choose an object I∈C , called the monoidal unit, and a functor C×C→C , which we will denote simply by the dot " ⋅ ", called the monoidal multiplication.Thus for any two objects c1,c2 we ha... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Examples-Note that if C is a partial order, the above description is simplified even more, because the associativity and unitality isomorphisms becomes equalities. Another simplification occurs if we assume that the set of objects is the free monoid on a generating set Σ . In this case we could write... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Examples-To return to our example, let N be the category whose objects are the natural numbers 0, 1, 2, ..., with a single morphism i→j if i≤j in the usual ordering (and no morphisms from i to j otherwise), and a monoidal structure with the monoidal unit given by 0 and the monoidal multiplication gi... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Properties and associated notions-It follows from the three defining coherence conditions that a large class of diagrams (i.e. diagrams whose morphisms are built using α , λ , ρ , identities and tensor product) commute: this is Mac Lane's "coherence theorem". It is sometimes inaccurately stated tha... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Properties and associated notions-There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid from abstract algebra. Ordinary monoids are precisely the monoid objects in the cartesian monoidal category Set. Further, any (small) strict monoidal cat... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Properties and associated notions-Monoidal functors are the functors between monoidal categories that preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product.
Every monoidal category can be... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Properties and associated notions-Free strict monoidal category For every category C, the free strict monoidal category Σ(C) can be constructed as follows: its objects are lists (finite sequences) A1, ..., An of objects of C; there are arrows between two objects A1, ..., Am and B1, ..., Bn only if m =... | milkshake721/2.1M-wiki-STEM |
Monoidal category-Specializations-If, in a monoidal category, A⊗B and B⊗A are naturally isomorphic in a manner compatible with the coherence conditions, we speak of a braided monoidal category. If, moreover, this natural isomorphism is its own inverse, we have a symmetric monoidal category.
A closed monoidal category... | milkshake721/2.1M-wiki-STEM |
Isoperimetric ratio-Isoperimetric ratio-In analytic geometry, the isoperimetric ratio of a simple closed curve in the Euclidean plane is the ratio L2/A, where L is the length of the curve and A is its area. It is a dimensionless quantity that is invariant under similarity transformations of the curve.
According to the ... | milkshake721/2.1M-wiki-STEM |
Isoperimetric ratio-Isoperimetric ratio-The curve-shortening flow decreases the isoperimetric ratio of any smooth convex curve so that, in the limit as the curve shrinks to a point, the ratio becomes 4π.For higher-dimensional bodies of dimension d, the isoperimetric ratio can similarly be defined as Bd/Vd − 1 where B i... | milkshake721/2.1M-wiki-STEM |
Security characteristic line-Security characteristic line-Security characteristic line (SCL) is a regression line, plotting performance of a particular security or portfolio against that of the market portfolio at every point in time. The SCL is plotted on a graph where the Y-axis is the excess return on a security ove... | milkshake721/2.1M-wiki-STEM |
Security characteristic line-Formula-SCL:Ri,t−Rf=αi+βi(RM,t−Rf)+ϵi,t where: αi is called the asset's alpha (abnormal return) βi(RM,t – Rf) is a nondiversifiable or systematic risk εi,t is the non-systematic or diversifiable, non-market or idiosyncratic risk RM,t is the return to market portfolio Rf is a risk-free rate | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Uncertainty principle-In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a quantum system, such ... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Uncertainty principle-Introduced first in 1927 by German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. In the published 192... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Introduction-It is vital to illustrate how the principle applies to relatively intelligible physical situations since it is indiscernible on the macroscopic scales that humans experience. Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Introduction-Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate var... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Introduction-In matrix mechanics, the mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measuremen... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Visualization-The uncertainty principle can be visualized using the position- and momentum-space wavefunctions for one spinless particle with mass in one dimension. The more localized the position-space wavefunction, the more likely the particle is to be found with the position coordinates in tha... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Visualization-Wave mechanics interpretation According to the de Broglie hypothesis, every object in the universe is associated with a wave. Thus every object, from an elementary particle to atoms, molecules and on up to planets and beyond are subject to the uncertainty principle. The time-indepen... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Visualization-On the other hand, consider a wave function that is a sum of many waves, which we may write as where An represents the relative contribution of the mode pn to the overall total. The figures to the right show how with the addition of many plane waves, the wave packet can become more l... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Visualization-The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp have an inverse relationship or are at least bounded from below. This is the uncert... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Visualization-Matrix mechanics interpretation (Ref ) In matrix mechanics, observables such as position and momentum are represented by self-adjoint operators. When considering pairs of observables, an important quantity is the commutator. For a pair of operators  and B^ , one defines their commu... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Visualization-Applying the commutator to |ψ⟩ yields where Î is the identity operator.
Suppose, for the sake of proof by contradiction, that |ψ⟩ is also a right eigenstate of momentum, with constant eigenvalue p0. If this were true, then one could write On the other hand, the above canonical comm... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Visualization-When a state is measured, it is projected onto an eigenstate in the basis of the relevant observable. For example, if a particle's position is measured, then the state amounts to a position eigenstate. This means that the state is not a momentum eigenstate, however, but rather it can... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Heisenberg limit-In quantum metrology, and especially interferometry, the Heisenberg limit is the optimal rate at which the accuracy of a measurement can scale with the energy used in the measurement. Typically, this is the measurement of a phase (applied to one arm of a beam-splitter) and the ene... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Robertson–Schrödinger uncertainty relations-The most common general form of the uncertainty principle is the Robertson uncertainty relation.For an arbitrary Hermitian operator O^ we can associate a standard deviation where the brackets ⟨O⟩ indicate an expectation value. For a pair of operators ... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Robertson–Schrödinger uncertainty relations-The Maccone–Pati uncertainty relations The Robertson–Schrödinger uncertainty relation can be trivial if the state of the system is chosen to be eigenstate of one of the observable. The stronger uncertainty relations proved by Maccone and Pati give non-tr... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Robertson–Schrödinger uncertainty relations-The second stronger uncertainty relation is given by where |Ψ¯A+B⟩ is a state orthogonal to |Ψ⟩ The form of |Ψ¯A+B⟩ implies that the right-hand side of the new uncertainty relation is nonzero unless |Ψ⟩ is an eigenstate of (A+B) . One may note that ... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Robertson–Schrödinger uncertainty relations-Improving the Robertson–Schrödinger uncertainty relation based on decompositions of the density matrix The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all components ϱk in any decomposition of the density matrix given ... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Robertson–Schrödinger uncertainty relations-The improved relation above is saturated by all single-qubit quantum states.With similar arguments, one can derive a relation with a convex roof on the right-hand side where FQ[ϱ,B] denotes the quantum Fisher information and the density matrix is decomp... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Robertson–Schrödinger uncertainty relations-Phase space In the phase space formulation of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function. Given a Wigner function W(x,p) with star product ★ and a function f, the following is... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Robertson–Schrödinger uncertainty relations-The non-negative eigenvalues then imply a corresponding non-negativity condition on the determinant, or, explicitly, after algebraic manipulation, | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Examples-Since the Robertson and Schrödinger relations are for general operators, the relations can be applied to any two observables to obtain specific uncertainty relations. A few of the most common relations found in the literature are given below. | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Examples-For position and linear momentum, the canonical commutation relation [x^,p^]=iℏ implies the Kennard inequality from above: For two orthogonal components of the total angular momentum operator of an object: where i, j, k are distinct, and Ji denotes angular momentum along the xi axis. Th... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Examples-In non-relativistic mechanics, time is privileged as an independent variable. Nevertheless, in 1945, L. I. Mandelstam and I. E. Tamm derived a non-relativistic time–energy uncertainty relation, as follows. For a quantum system in a non-stationary state ψ and an observable B represented by... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Examples-For the number of electrons in a superconductor and the phase of its Ginzburg–Landau order parameter A counterexample Suppose we consider a quantum particle on a ring, where the wave function depends on an angular variable θ , which we may take to lie in the interval [0,2π] . Define "po... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Examples-Quantum harmonic oscillators with Gaussian initial condition In a quantum harmonic oscillator of characteristic angular frequency ω, place a state that is offset from the bottom of the potential by some displacement x0 as where Ω describes the width of the initial state but need not be th... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Examples-Particle in a box Consider a particle in a one-dimensional box of length L . The eigenfunctions in position and momentum space are and where {\textstyle \omega _{n}={\frac {\pi ^{2}\hbar n^{2}}{8L^{2}m}}} and we have used the de Broglie relation p=ℏk . The variances of x and p can be... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Additional uncertainty relations-Systematic and statistical errors The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation σ . Heisenberg's original version, however, was dealing with the systematic error, a disturbance of the quantum sys... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Additional uncertainty relations-If we let εA represent the error (i.e., inaccuracy) of a measurement of an observable A and ηB the disturbance produced on a subsequent measurement of the conjugate variable B by the former measurement of A, then the inequality proposed by Ozawa — encompassing bo... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Additional uncertainty relations-Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errors σA and σB . There is increasing experimental evidence that the total quantum uncertainty cannot be described by the Heisenberg term alone, but re... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Additional uncertainty relations-It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality. By adding Robertson and Ozawa relations we obtain The f... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Additional uncertainty relations-A solution that overcomes these issues is an uncertainty based on entropic uncertainty instead of the product of variances. While formulating the many-worlds interpretation of quantum mechanics in 1957, Hugh Everett III conjectured a stronger extension of the uncer... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Additional uncertainty relations-The probability distribution functions associated with the position wave function ψ(x) and the momentum wave function φ(x) have dimensions of inverse length and momentum respectively, but the entropies may be rendered dimensionless by where x0 and p0 are some arbit... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Additional uncertainty relations-Depending on one's choice of the x0 p0 product, the expression may be written in many ways. If x0 p0 is chosen to be h, then If, instead, x0 p0 is chosen to be ħ, then If x0 and p0 are chosen to be unity in whatever system of units are being used, then where h is i... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Additional uncertainty relations-A measurement apparatus will have a finite resolution set by the discretization of its possible outputs into bins, with the probability of lying within one of the bins given by the Born rule. We will consider the most common experimental situation, in which the bin... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Uncertainty relation with three angular momentum components-For a particle of spin- j the following uncertainty relation holds where Jl are angular momentum components. The relation can be derived from and The relation can be strengthened as where FQ[ϱ,Jz] is the quantum Fisher information. | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Harmonic analysis-In the context of harmonic analysis, a branch of mathematics, the uncertainty principle implies that one cannot at the same time localize the value of a function and its Fourier transform. To wit, the following inequality holds, Further mathematical uncertainty inequalities, incl... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Harmonic analysis-Stated alternatively, "One cannot simultaneously sharply localize a signal (function f) in both the time domain and frequency domain (ƒ̂, its Fourier transform)".
When applied to filters, the result implies that one cannot achieve high temporal resolution and frequency resolution... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Harmonic analysis-Alternate theorems give more precise quantitative results, and, in time–frequency analysis, rather than interpreting the (1-dimensional) time and frequency domains separately, one instead interprets the limit as a lower limit on the support of a function in the (2-dimensional) ti... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Harmonic analysis-As a result, in order to analyze signals where the transients are important, the wavelet transform is often used instead of the Fourier.
Discrete Fourier transform Let := x0,x1,…,xN−1 be a sequence of N complex numbers and := X0,X1,…,XN−1, its discrete Fourier transform. | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Harmonic analysis-Denote by ‖x‖0 the number of non-zero elements in the time sequence x0,x1,…,xN−1 and by ‖X‖0 the number of non-zero elements in the frequency sequence X0,X1,…,XN−1 . Then, This inequality is sharp, with equality achieved when x or X is a Dirac mass, or more generally when x i... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Harmonic analysis-More generally, if T and W are subsets of the integers modulo N, let LT,RW:ℓ2(Z/NN)→ℓ2(Z/NN) denote the time-limiting operator and band-limiting operators, respectively. Then where the norm is the operator norm of operators on the Hilbert space ℓ2(Z/NZ) of functions on the inte... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Harmonic analysis-Benedicks's theorem Amrein–Berthier and Benedicks's theorem intuitively says that the set of points where f is non-zero and the set of points where ƒ̂ is non-zero cannot both be small.
Specifically, it is impossible for a function f in L2(R) and its Fourier transform ƒ̂ to both b... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Harmonic analysis-Hardy's uncertainty principle The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for f and ƒ̂ to both be "very rapidly decreasing". Specifically, if f in L2(R) is such that and (C>0,N an integer), then, if ab > 1, f = 0, while if ab ... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Harmonic analysis-This result was stated in Beurling's complete works without proof and proved in Hörmander (the case d=1,N=0 ) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case ab > 1 in Hardy's Theorem while the version by Bonami–Deman... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-History-Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics. | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-History-In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad hoc old quantum theory with modern quantum mechanics. The central premise was that the classical concept of motion does not fit at the quantum level, as electrons in an ato... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-History-Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theory to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they coul... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-History-In his celebrated 1927 paper, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik" ("On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics"), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caus... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-History-Terminology and translation Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word "Ungenauigkeit" ("indeterminacy"), to describe the basic theoretical principle. Only in the endnote did he switch to the word "Unsicherheit" ("uncertainty"). When th... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-History-Heisenberg's microscope The principle is quite counter-intuitive, so the early students of quantum theory had to be reassured that naive measurements to violate it were bound always to be unworkable. One way in which Heisenberg originally illustrated the intrinsic impossibility of violatin... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-History-Problem 2 – If a large aperture is used for the microscope, the electron's location can be well resolved (see Rayleigh criterion); but by the principle of conservation of momentum, the transverse momentum of the incoming photon affects the electron's beamline momentum and hence, the new mo... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Critical reactions-The Copenhagen interpretation of quantum mechanics and Heisenberg's Uncertainty Principle were, in fact, seen as twin targets by detractors who believed in an underlying determinism and realism. According to the Copenhagen interpretation of quantum mechanics, there is no fundame... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Critical reactions-Albert Einstein believed that randomness is a reflection of our ignorance of some fundamental property of reality, while Niels Bohr believed that the probability distributions are fundamental and irreducible, and depend on which measurements we choose to perform. Einstein and Bo... | milkshake721/2.1M-wiki-STEM |
Uncertainty principle-Critical reactions-The ideal of the detached observer Wolfgang Pauli called Einstein's fundamental objection to the uncertainty principle "the ideal of the detached observer" (phrase translated from the German): "Like the moon has a definite position" Einstein said to me last winter, "whether or n... | milkshake721/2.1M-wiki-STEM |
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