関: minimal、minimum

WordNet

the least possible; "needed to enforce minimal standards"; "her grades were minimal"; "minimum wage"; "a minimal charge for the service" (同)minimum
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 smallest possible quantity (同)lower_limit
the point on a curve where the tangent changes from negative on the left to positive on the right
a British imperial capacity measure (liquid or dry) equal to 1/60th fluid dram or 0.059194 cubic centimeters
a United States liquid unit equal to 1/60 fluidram

PrepTutorEJDIC

最小の,極小の,最近の
『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
《通例単数形で》『最小限』,最小量 / (数学で)極小,最小値,極小値 / 《名詞の前にのみ用いて》『最小限の』,最小量の
ミニム(液量の最小単位) / 《英》二分音符(《米》half note)

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/05/08 16:03:04」(JST)

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[Wiki en表示]

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] Pierre de Fermat was one of the first mathematicians to propose a general technique (called adequality) for finding maxima and minima.

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.

Definition

A real-valued function f defined on a domain X has a global (or absolute) maximum point at x^∗ if f(x^∗) ≥ f(x) for all x in X. Similarly, the function has a global (or absolute) minimum point at x^∗ if f(x^∗) ≤ f(x) for all x in X. The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function.

If the domain X is a metric space then f 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) for all x in X within distance ε of x^∗. Similarly, the function has a local minimum point at x^∗ if f(x^∗) ≤ f(x) for all x in X within distance ε of x^∗. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods. Note that a global maximum point is always a local maximum point, and similarly for minimum points.

In both the global and local cases, the concept of a strict extremum can be defined. For example, x^∗ is a strict global maximum point if, for all x in X with x ≠ x^∗, we have f(x^∗) > f(x), and x^∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x^∗ with x ≠ x^∗, we have f(x^∗) > f(x). Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points.

A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above).

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 of differentiable functions 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, second derivative test, or higher-order derivative test, given sufficient differentiability.

For any function that is defined piecewise, one finds a maximum (or minimum) 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 a local maximum at x = −1−^√5⁄₃, a local minimum at x = −1+^√5⁄₃, a global maximum at x = 2 and a global minimum at x = −4.

Functions of more than one variable

Peano surface, a counterexample to some criteria of local maxima of the 19th century

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(2,3) = −5.

The global maximum is the point at the top

Counterexample

Maxima or minima of a functional

If the domain of a function for which an extremum is to be found is itself a function, i.e., if an extremum is to be found of a functional, the extremum is found using the calculus of variations.

In relation to sets

Maxima and minima can also be 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

Look up maxima, minima, or extremum in Wiktionary, the free dictionary.

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

Selective spinal anesthesia for outpatient transurethral prostatectomy (TURP): randomized controlled comparison of chloroprocaine with lidocaine.

Vaghadia H, Neilson G, Lennox PH.SourceDepartment of Anesthesia, Vancouver General Hospital, University of British Columbia, Vancouver, BC, Canada.
Acta anaesthesiologica Scandinavica.Acta Anaesthesiol Scand.2012 Feb;56(2):217-23. doi: 10.1111/j.1399-6576.2011.02599.x.
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Characterization of a potent antimicrobial lipopeptide via coarse-grained molecular dynamics.

Horn JN, Sengillo JD, Lin D, Romo TD, Grossfield A.AbstractThe prevalence of antibiotic-resistant pathogens is a major medical concern, prompting increased interest in the development of novel antimicrobial compounds. One such set of naturally occurring compounds, known as antimicrobial peptides (AMPs), have broad-spectrum activity, but come with many limitations for clinical use. Recent work has resulted in a set of antimicrobial lipopeptides (AMLPs) with micromolar minimum inhibitory concentrations and excellent selectivity for bacterial membranes. To characterize a potent, synthetic lipopeptide, C16-KGGK, we used multi-microsecond coarse-grained simulations with the MARTINI forcefield, with a total simulation time of nearly 46μs. These simulations show rapid binding of C16-KGGK, which forms micelles in solution, to model bacterial lipid bilayers. Furthermore, upon binding to the surface of the bilayer, these lipopeptides alter the local lipid organization by recruiting negatively charged POPG lipids to the site of binding. It is likely that this drastic reorganization of the bilayer has major effects on bilayer dynamics and cellular processes that depend on specific bilayer compositions. By contrast, the simulations revealed no association between the lipopeptides and model mammalian bilayers. These simulations provide biophysical insights into lipopeptide selectivity and suggest a possible mechanism for antimicrobial action. This article is part of a Special Issue entitled: Membrane protein structure and function.
Biochimica et biophysica acta.Biochim Biophys Acta.2012 Feb;1818(2):212-8. Epub 2011 Jul 28.
The prevalence of antibiotic-resistant pathogens is a major medical concern, prompting increased interest in the development of novel antimicrobial compounds. One such set of naturally occurring compounds, known as antimicrobial peptides (AMPs), have broad-spectrum activity, but come with many limit
PMID 21819964

Surgical technique: porous tantalum reconstruction for destructive nonprimary periacetabular tumors.

Khan FA, Rose PS, Yanagisawa M, Lewallen DG, Sim FH.SourceDepartment of Orthopedic Surgery, Mayo Clinic, 200 First Street SW, Rochester, MN, 55905, USA, sim.franklin@mayo.edu.
Clinical orthopaedics and related research.Clin Orthop Relat Res.2012 Feb;470(2):594-601. Epub 2011 Oct 12.
BACKGROUND: Large bone loss and frequently irradiated existing bone make reconstructing metastatic and other nonprimary periacetabular tumors challenging. Although existing methods are initially successful, they may fail with time. Given the low failure rates of porous tantalum acetabular implants i
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Japanese Journal

Advanced Ring-Shaped Microelectrode Assay Combined with Small Rectangular Electrode for Quasi-In vivo Measurement of Cell-to-Cell Conductance in Cardiomyocyte Network

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