関: maxima、maximal、maximum

WordNet

the greatest or most complete or best possible; "maximal expansion"; "maximum pressure" (同)maximum
relating to or applicable to or concerned with the administration of a city or town or district rather than a larger area; "local taxes"; "local authorities"
public transport consisting of a bus or train that stops at all stations or stops; "the local seemed to take forever to get to New York"
affecting only a restricted part or area of the body; "local anesthesia"
of or belonging to or characteristic of a particular locality or neighborhood; "local customs"; "local schools"; "the local citizens"; "a local point of view"; "local outbreaks of flu"; "a local bus line"
the largest possible quantity (同)upper_limit
the point on a curve where the tangent changes from positive on the left to negative on the right
a saying that is widely accepted on its own merits (同)axiom

PrepTutorEJDIC

最上限の,最高度の
maximumの複数形
『一地方の』,ある土地の,(特に)自分の住んでいる土地の;地方に特有な / (全体からみて)『部分的な』,局部の,狭い部分の / 《米》各駅停車の / 《話》《しばしば複数形で》その土地に住んでいる人々 / 《米話》(新聞の)地方記事 / 《話》(各駅停車の)列車,電車,バス / 《米》(労働組合などの)支部
低カロリーの
(量・数・程度の)『最大限』 / (数学で)極大;最大値;極大点 / 最高の,最大の
格言,金言,処世訓

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2012/05/27 20:22:30」(JST)

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For other uses, see Maxima (disambiguation) and Maximum (disambiguation). For use in statistics, see Maximum (statistics).

Local and global maxima and minima for cos(3πx)/x, 0.1≤x≤1.1

In mathematics, the maximum and minimum (plural: maxima and minima) of a function, known collectively as extrema (singular: extremum), are the largest and smallest value that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).^[1]^[2]^[3]

More generally, the maximum and minimum of a set (as defined in set theory) are the greatest and least element in the set. Unbounded infinite sets such as the set of real numbers have no minimum and maximum.

To locate extreme values is the basic objective of optimization.

Analytical definition

A real-valued function f defined on a real line is said to have a local (or relative) maximum point at the point x^∗, if there exists some ε > 0 such that f(x^∗) ≥ f(x) when |x − x^∗| < ε. The value of the function at this point is called maximum of the function. Similarly, a function has a local minimum point at x^∗, if f(x^∗) ≤ f(x) when |x − x^∗| < ε. The value of the function at this point is called minimum of the function.

A function has a global (or absolute) maximum point at x^∗ if f(x^∗) ≥ f(x) for all x. Similarly, a function has a global (or absolute) minimum point at x^∗ if f(x^∗) ≤ f(x) for all x. The global maximum and global minimum points are also known as the arg max and arg min: the argument (input) at which the maximum (respectively, minimum) occurs.

Restricted domains: There may be maxima and minima for a function whose domain does not include all real numbers. A real-valued function, whose domain is any set, can have a global maximum and minimum. There may also be local maxima and local minima points, but only at points of the domain set where the concept of neighborhood is defined. A neighborhood plays the role of the set of x such that |x − x^∗| < ε.

A continuous (real-valued) function on a compact set always takes maximum and minimum values on that set. An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above). The neighborhood requirement precludes a local maximum or minimum at an endpoint of an interval. However, an endpoint may still be a global maximum or minimum. Thus it is not always true, for finite domains, that a global maximum (minimum) must also be a local maximum (minimum).

Finding functional maxima and minima

Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one.

Local extrema can be found by Fermat's theorem, which states that they must occur at critical points. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test or second derivative test.

For any function that is defined piecewise, one finds a maxima (or minima) by finding the maximum (or minimum) of each piece separately; and then seeing which one is biggest (or smallest).

Examples

The global maximum of occurs at x = e.

The function x² has a unique global minimum at x = 0.
The function x³ has no global minima or maxima. Although the first derivative (3x²) is 0 at x = 0, this is an inflection point.
The function has a unique global maximum at x = e. (See figure at right)
The function x^-x has a unique global maximum over the positive real numbers at x = 1/e.
The function x³/3 − x has first derivative x² − 1 and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of the second derivative we can see that −1 is a local maximum and +1 is a local minimum. Note that this function has no global maximum or minimum.
The function |x| has a global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0.
The function cos(x) has infinitely many global maxima at 0, ±2π, ±4π, …, and infinitely many global minima at ±π, ±3π, ….
The function 2 cos(x) − x has infinitely many local maxima and minima, but no global maximum or minimum.
The function cos(3πx)/x with 0.1 ≤ x ≤ 1.1 has a global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. (See figure at top of page.)
The function x³ + 3x² − 2x + 1 defined over the closed interval (segment) [−4,2] has two extrema: one local maximum at x = −1−^√15⁄₃, one local minimum at x = −1+^√15⁄₃, a global maximum at x = 2 and a global minimum at x = −4.

Functions of more than one variable

Main article: Second partial derivative test

For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure at the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum because of the possibility of a saddle point. For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum.

In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by reductio ad absurdum). In two and more dimensions, this argument fails, as the function

shows. Its only critical point is at (0,0), which is a local minimum with ƒ(0,0) = 0. However, it cannot be a global one, because ƒ(4,1) = −11.

The global maximum is the point at the top

Counterexample

In relation to sets

Maxima and minima are more generally defined for sets. In general, if an ordered set S has a greatest element m, m is a maximal element. Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with respect to order induced by T, m is a least upper bound of S in T. The similar result holds for least element, minimal element and greatest lower bound.

In the case of a general partial order, the least element (smaller than all other) should not be confused with a minimal element (nothing is smaller). Likewise, a greatest element of a partially ordered set (poset) is an upper bound of the set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A) then m = b. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.

In a totally ordered set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element and the maximal element will also be the greatest element. Thus in a totally ordered set we can simply use the terms minimum and maximum. If a chain is finite then it will always have a maximum and a minimum. If a chain is infinite then it need not have a maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum. If an infinite chain S is bounded, then the closure Cl(S) of the set occasionally has a minimum and a maximum, in such case they are called the greatest lower bound and the least upper bound of the set S, respectively..

References

^ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5.
^ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 0-547-16702-4.
^ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 0-321-58876-2.

External links

Maxima and Minima From MathWorld--A Wolfram Web Resource.
Thomas Simpson's work on Maxima and Minima at Convergence
Application of Maxima and Minima with sub pages of solved problems

UpToDate Contents

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English Journal

A note on the phase locking value and its properties.

Aydore S, Pantazis D, Leahy RM.SourceSignal and Image Processing Institute, University of Southern California, Los Angeles, CA 90089-2564, USA.
NeuroImage.Neuroimage.2013 Jul 1;74:231-44. doi: 10.1016/j.neuroimage.2013.02.008. Epub 2013 Feb 19.
We investigate the properties of the Phase Locking Value (PLV) and the Phase Lag Index (PLI) as metrics for quantifying interactions in bivariate local field potential (LFP), electroencephalography (EEG) and magnetoencephalography (MEG) data. In particular we describe the relationship between nonpar
PMID 23435210

Discriminant subspace learning constrained by locally statistical uncorrelation for face recognition.

Chen Y, Zheng WS, Xu XH, Lai JH.SourceDepartment of Applied Mathematics, South China Agricultural University, Guangzhou, Guangdong, 510642, China; School of Mathematics and Computational Science, SunYat-Sen University, Guangzhou, Guangdong, 510275, China.
Neural networks : the official journal of the International Neural Network Society.Neural Netw.2013 Jun;42:28-43. doi: 10.1016/j.neunet.2013.01.009. Epub 2013 Jan 24.
High-dimensionality of data and the small sample size problem are two significant limitations for applying subspace methods which are favored by face recognition. In this paper, a new linear dimension reduction method called locally uncorrelated discriminant projections (LUDP) is proposed, which add
PMID 23416750

Older people's views on what they need to successfully adjust to life with a hearing aid.

Kelly TB, Tolson D, Day T, McColgan G, Kroll T, Maclaren W.SourceSchool of Education, Social Work and Community Education, University of Dundee, Dundee, UK School of Health, Glasgow Caledonian University, Glasgow, UK Audiology Department, National Health Service Grampian, Aberdeen, UK School of Applied Social Science, University of Stirling, Stirling, UK School of Nursing and Midwifery, University of Dundee, Dundee, UK School of Engineering and Computing, Glasgow Caledonian University, Glasgow, UK.
Health & social care in the community.Health Soc Care Community.2013 May;21(3):293-302. doi: 10.1111/hsc.12016. Epub 2013 Feb 2.
This article reports a study exploring what older people believe would enable them to adjust to and gain maximum benefit from wearing a hearing aid. A mixed methods approach was employed during 2006 involving interviews with key stakeholders, a survey across three Scottish health board areas and foc
PMID 23373520

Japanese Journal

Bayesian maximum likelihood estimator of phase retardation for quantitative polarization-sensitive optical coherence tomography

Kasaragod Deepa,Makita Shuichi,Fukuda Shinichi,Beheregaray Simone,Oshika Tetsuro,Yasuno Yoshiaki
Optics express 22(13), 16472-16492, 2014-06
… This paper presents the theory and numerical implementation of a maximum likelihood estimator for local phase retardation (i.e., birefringence) measured using Jones-matrix-based polarization sensitive optical coherence tomography. …
NAID 120005462711

Recovery effects of exposure to normobaric hyperoxia on central and/or peripheral factors in muscle fatigue

Yokoi Yuka,Yanagihashi Ryuya,Morishita Katsuyuki,Goto Noboru,Fujiwara Takayuki,Abe Koji
日本保健科学学会誌 16(4), 190-200, 2014-03-25
… This study aimed to identify the effects and the mechanisms of normobaric hyperoxia on recovery of local muscle fatigue. … Methods : 12 male subjects performed 3×3×greater than 30 seconds of isometric quadriceps exercise at 70% of maximum voluntary isometric contraction (MVIC) separated by two 15-minutes recovery sessions under one of two different oxygen conditions, one in normoxia (NOX; …
NAID 110009809401

雑種成犬における腕神経叢ブロックが上肢血流量に及ぼす影響について

藤岡照久,池田知史,藤井宏一,石川和由
Dokkyo journal of medical sciences 41(1), 87-93, 2014-03-25
雑種成犬の腕神経叢ブロック(BPB)モデルを作製し,ブロック後の上肢血流量の変化について調べた.【方法】雑種雄性成犬を対象とし,全身麻酔下にBPB モデルを作製し,生理食塩水1 mL(生理食塩水群:8 例)と1%メピバカイン1 mL(局所麻酔薬群:8 例)投与後の正中動脈血流量(MABF)の変化を記録,比較検討した.BPB は,超音波下に21 ゲージ針を腕神経叢内に進め,電気刺激装置にて刺激を確認 …
NAID 110009783349