minimum

• 最小、極小、最小限、最小値

関

local minimum、minimal、minimally、minimum value

　 　 　

WordNet

1. the smallest possible quantity (同)lower_limit
2. the point on a curve where the tangent changes from negative on the left to positive on the right
3. the least possible; "needed to enforce minimal standards"; "her grades were minimal"; "minimum wage"; "a minimal charge for the service" (同)minimum
4. to a minimal degree; "the cells minimally responsive to lines in this orientation will not fire"
5. thermometer that records the highest and lowest temperatures reached during a period of time
6. a British imperial capacity measure (liquid or dry) equal to 1/60th fluid dram or 0.059194 cubic centimeters
7. a United States liquid unit equal to 1/60 fluidram

PrepTutorEJDIC

1. 《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
2. 最小の,極小の,最近の
3. 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）』「2015/07/29 22:31:28」(JST)

wiki en

[Wiki en表示]

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

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

In mathematical analysis, the maxima and minima (the plural of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.

As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

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 largest (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 largest (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−^√15⁄₃, a 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

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 consists itself of functions, 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.

See also

• First derivative test
• Second derivative test
• Higher-order derivative test
• Limit superior and limit inferior
• Mechanical equilibrium
• Sample maximum and minimum
• Saddle point

References

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

• Accuracy and reproducibility of aortic annulus sizing using a dedicated three-dimensional computed tomography reconstruction tool in patients evaluated for transcatheter aortic valve replacement.

• Stortecky S, Heg D, Gloekler S, Wenaweser P, Windecker S, Buellesfeld L.Author information Department of Cardiology, Swiss Cardiovascular Centre, Bern University Hospital, Bern, Switzerland.AbstractAims: To evaluate the accuracy and reproducibility of aortic annulus sizing using a multislice computed tomography (MSCT) based aortic root reconstruction tool compared with conventional imaging among patients evaluated for transcatheter aortic valve replacement (TAVR). Methods and results: Patients referred for TAVR underwent standard preprocedural assessment of aortic annulus parameters using MSCT, angiography and transoesophageal echocardiography (TEE). Three-dimensional (3-D) reconstruction of MSCT images of the aortic root was performed using 3mensio (3mensio Medical Imaging BV, Bilthoven, The Netherlands), allowing for semi-automated delineation of the annular plane and assessment of annulus perimeter, area, maximum, minimum and virtual diameters derived from area and perimeter (aVD and pVD). A total of 177 patients were enrolled. We observed a good inter-observer variability of 3-D reconstruction assessments with concordance coefficients for agreement of 0.91 (95% CI: 0.87-0.93) and 0.91 (0.88-0.94) for annulus perimeter and area assessments, respectively. 3-D derived pVD and aVD correlated very closely with a concordance coefficient of 0.97 (0.96-0.98) with a mean difference of 0.5±0.3 mm (pVD-aVD). 3-D derived pVD showed the best, but moderate concordance with diameters obtained from coronal MSCT (0.67, 0.56-0.75; 0.3±1.8 mm), and the lowest concordance with diameters obtained from TEE (0.42, 0.31-0.52; 1.9±1.9 mm). Conclusions: MSCT-based 3-D reconstruction of the aortic annulus using the 3mensio software enables accurate and reproducible assessment of aortic annulus dimensions.
• EuroIntervention : journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology.EuroIntervention.2015 Nov 25. pii: 20130610-03. [Epub ahead of print]
• Aims: To evaluate the accuracy and reproducibility of aortic annulus sizing using a multislice computed tomography (MSCT) based aortic root reconstruction tool compared with conventional imaging among patients evaluated for transcatheter aortic valve replacement (TAVR). Methods and results: Patients
• PMID 24273249

• Efficacy and safety of incobotulinum toxin A in periocular rhytides and masseteric hypertrophy: side-by-side comparison with onabotulinum toxin A.

• Lee JH, Park JH, Lee SK, Han KH, Kim SD, Yoon CS, Park JY, Lee JH, Yang JM, Lee JH.Author information Department of Dermatology, Samsung Medical Center, Sungkyunkwan University School of Medicine , Seoul , Republic of Korea.AbstractBackground: Incobotulinum is a newly developed botulinum toxin A in which the complexing proteins had been removed. Objective: The aim was to compare the efficacy and safety of incobotulinum with onabotulinum in treating periocular rhytides and masseteric hypertrophy. Methods: A randomized, double-blind, split-face study was planned. Fifty-six patients were treated for periocular rhytides and the other 56 patients were treated for masseteric hypertrophy. Onabotulinum was injected on one side of the face and incobotulinum was injected on the other side of the face. The degree of periocular rhytides and masseteric hypertrophy was rated using Fitzpatrick Wrinkle Classification System (FWCS) and 10-point visual analogue scale (VAS) (0: the minimum to 10: the maximum). Objective and subjective rating was performed at pretreatment and every posttreatment follow-up visit by investigators and subjects. Result: The efficacy and safety of incobotulinum were not inferior to those of onabotulinum in treating periocular rhytides and masseteric hypertrophy up to 16 weeks after injection. There were no noteworthy differences in the onset time of effect between two botulinum toxins for periocular wrinkles and masseteric hypertrophy. No adverse event was reported. Conclusion: Incobotulinum provided non-inferior efficacy and safety for the treatment of periocular rhytides and masseteric hypertrophy compared with classic onabotulinum.
• The Journal of dermatological treatment.J Dermatolog Treat.2014 Aug;25(4):326-30. doi: 10.3109/09546634.2013.769041. Epub 2013 Jun 2.
• Background: Incobotulinum is a newly developed botulinum toxin A in which the complexing proteins had been removed. Objective: The aim was to compare the efficacy and safety of incobotulinum with onabotulinum in treating periocular rhytides and masseteric hypertrophy. Methods: A randomized, double-b
• PMID 23356833

• Functional, biochemical and 3D studies of Mycobacterium tuberculosis protein peptides for an effective anti-tuberculosis vaccine.

• Ocampo M, Patarroyo MA, Vanegas M, Alba MP, Patarroyo ME.Author information Fundación Instituto de Inmunología de Colombia (FIDIC) , Bogotá , Colombia .AbstractAbstract Tuberculosis (TB) is an air-born, transmissible disease, having an estimated 9.4 million new TB cases worldwide in 2009. Eventual control of this disease by developing a safe and efficient new vaccine able to detain its spread will have an enormous impact on public health policy. Selecting potential antigens to be included in a multi-epitope, minimal subunit-based, chemically-synthesized vaccine containing the minimum sequences needed for blocking mycobacterial interaction with host cells is a complex task due to the multiple mechanisms involved in M. tuberculosis infection and the mycobacterium's immune evasion mechanisms. Our methodology, described here takes into account a highly robust, specific, sensitive and functional approach to the search for potential epitopes to be included in an anti-TB vaccine; it has been based on identifying short mycobacterial protein fragments using synthetic peptides having high affinity interaction with alveolar epithelial cells (A549) and monocyte-derived macrophages (U937) which are able to block the microorganism's entry to target cells in in vitro assays. This manuscript presents a review of the results obtained with some of the MTB H37Rv proteins studied to date, aimed at using these high activity binding peptides (HABPs) as platforms to be included in a minimal subunit-based, multiepitope, chemically-synthesized, antituberculosis vaccine.
• Critical reviews in microbiology.Crit Rev Microbiol.2014 May;40(2):117-45. doi: 10.3109/1040841X.2013.763221. Epub 2013 Feb 27.
• Abstract Tuberculosis (TB) is an air-born, transmissible disease, having an estimated 9.4 million new TB cases worldwide in 2009. Eventual control of this disease by developing a safe and efficient new vaccine able to detain its spread will have an enormous impact on public health policy. Selecting
• PMID 23445450

• Application of FTIR-ATR to Moscatel dessert wines for prediction of total phenolic and flavonoid contents and antioxidant capacity.

• Silva SD1, Feliciano RP2, Boas LV3, Bronze MR4.Author information 1Instituto de Biologia Experimental e Tecnológica, Apartado 12, 2780 - 901 Oeiras, Portugal; Instituto de Tecnologia Química e Biológica, Av. da República, Estação Agronómica Nacional, 2780-157 Oeiras, Portugal. Electronic address: ssilva@itqb.unl.pt.2Instituto de Biologia Experimental e Tecnológica, Apartado 12, 2780 - 901 Oeiras, Portugal.3Instituto de Biologia Experimental e Tecnológica, Apartado 12, 2780 - 901 Oeiras, Portugal; Instituto de Tecnologia Química e Biológica, Av. da República, Estação Agronómica Nacional, 2780-157 Oeiras, Portugal.4Instituto de Biologia Experimental e Tecnológica, Apartado 12, 2780 - 901 Oeiras, Portugal; Instituto de Tecnologia Química e Biológica, Av. da República, Estação Agronómica Nacional, 2780-157 Oeiras, Portugal; Faculdade de Farmácia de Lisboa, Av. das Forças Armadas, 1649 - 019 Lisboa, Portugal.AbstractFourier transform infrared spectroscopy (FTIR) attenuated total reflectance (ATR) was applied for the determination of total phenolic and flavonoid contents and antioxidant capacity (DPPH and FRAP assays) in Moscatel dessert wines (n=56). Prediction models were developed for the referred parameters using Partial Least Squares (PLS) considering the spectral region 1800-900cm(-1). The determination coefficients (r(2)) values in the calibration models ranged from 0.670 to 0.870. Cross validation (leave-one-out technique) was applied to the data. Root mean square errors of calibration (RMSEC) and cross validation (RMSECV) as well as the relative errors of prediction (REP) were calculated. Minimum errors of prediction were obtained for total flavonoid content (0.2%) and maximum values (22%) for antioxidant capacity measured by FRAP. The proposed method may be used for rapid screening of total phenolic and flavonoid contents in Moscatel dessert wines. The implemented methodologies may also be used to get rough estimates for DPPH and FRAP antioxidant capacities.
• Food chemistry.Food Chem.2014 May 1;150:489-93. doi: 10.1016/j.foodchem.2013.11.028. Epub 2013 Nov 15.
• Fourier transform infrared spectroscopy (FTIR) attenuated total reflectance (ATR) was applied for the determination of total phenolic and flavonoid contents and antioxidant capacity (DPPH and FRAP assays) in Moscatel dessert wines (n=56). Prediction models were developed for the referred parameters
• PMID 24360480

Japanese Journal

• On the equivariance criterion in statistical prediction

• Zhou Haojin,Nayak Tapan K.
• Annals of the Institute of Statistical Mathematics 67(3), 541–555, 2015-06
• NAID 40020457826

• MINIMUM-COST MAXIMUM-SATISFACTION BI-OBJECTIVE INTEGER PROGRAMMING OPTIMIZATION FOR EMPTY CONTAINERS REPOSITIONING

• LU LIHUA,ZHANG HENGZHEN,GAO XIAO-ZHI [他]
• ICIC express letters. Part B, Applications : an international journal of research and surveys 6(6), 1631-1638, 2015-06
• NAID 40020355882

• High-efficiency impurity activation by precise control of cooling rate during atmospheric pressure thermal plasma jet annealing of 4H-SiC wafer

• Maruyama Keisuke,Hanafusa Hiroaki,Ashihara Ryuhei,Hayashi Shohei,Murakami Hideki,Higashi Seiichiro
• Jpn. J. Appl. Phys. 54(6S2), 06GC01, 2015-05-27
• … A minimum resistivity of 0.0025 Ω·cm and a maximum carrier concentration of 2.9 × …
• NAID 150000110774

• Acoustic properties of co-doped AlN thin films at low temperatures studied by picosecond ultrasonics

• Nagakubo Akira,Arita Mari,Yokoyama Tsuyoshi,Matsuda Satoru,Ueda Masanori,Ogi Hirotsugu,Hirao Masahiko
• Jpn. J. Appl. Phys. 54(7S1), 07HD01, 2015-05-13
• … 0.7 GPa for pure AlN, and it largely decreases by doping Mg, Zr, and Hf, leading to a minimum values of 316.8 ± …
• NAID 150000110848

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★リンクテーブル★

リンク元	「極小」「minimal」「最小値」「最小」「最小限」
拡張検索	「minimum bactericidal concentration」「minimum inhibitory concentration」「minimum alveolar concentration」

「極小」

　　[★]

英

minimum、(数学)local minimum、minimal

関

最小、最小限、最小値

「minimal」

　　[★]

• adj.

• 最小の、極小の

関

local minimum、minimally、minimum

　 　

「最小値」

　　[★]

英

minimum value、minimum

関

極小、最小、最小限、最低値

「最小」

　　[★]

英

minimum、minimal、minimally

関

極小、最小限、最小値

「最小限」

　　[★]

英

minimum、minimally

関

極小、最小、最小値

「minimum bactericidal concentration」

　　[★]

同

MBC

同

MBC

同

MBC

「minimum inhibitory concentration」

　　[★] 最小発育阻止濃度 MIC

「minimum alveolar concentration」

　　[★] 最小肺胞内濃度 MAC