出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/10/02 01:49:54」(JST)
| この項目では、数学計算ソフトウェアについて説明しています。その他のMapleについては「メイプル」をご覧ください。 |
| 開発元 | Waterloo Maple Inc. (Maplesoft) |
|---|---|
| 最新版 | 13 / 2009年4月 （日本語版：2009年6月） |
| 対応OS | クロスプラットフォーム |
| 種別 | 数式処理システム |
| ライセンス | プロプライエタリ |
| 公式サイト | 開発元サイト |
| テンプレートを表示 | |
Maple（メイプル）とは、数式処理、数値計算、グラフ作成などを行うソフトウェアのひとつである。Mapleは、1980年代前半にカナダのウォータールー大学で開発され（株式会社としてはWaterloo Maple名義。以下Maplesoft）、日本ではサイバネットシステムが販売、翻訳を行っていたが、2009年9月に、Maplesoftをサイバネットシステムが買収した。Mapleを使うと、紙と鉛筆で行う数学の計算や作図をコンピュータで行うことができる。
また、販売方法としては、アカデミックバージョンを出し、学生や、教員、研究者向けに廉価で(1ライセンス2～3万円程度)ほとんどスペックの落ちない製品を販売している。また、小学校、中学校、高校などの初等教育の現場における数学、理科の授業から、大学や企業のR&D部門などの研究機関に至るまで幅広いユーザ層が開拓されつつある。
代数演算とは加減乗除および階乗（平方根等を含む）及びはそれらを組み合わせることで作られる演算の総称である。以下、主な代数演算について説明する。
関数の定義は以下のように行う。例えば関数fをと定義したい場合には、
f:=x->x*exp(-x^2);
と入力すればよい。2変数以上の関数もほぼ同様で、例えば関数fをと定義したい場合には、
g:=(x,y)->y*exp(-x^2);
と入力すればよい。一般に関数には変数が存在する。人間は関数を見せられた場合に『何が変数であるか』を理解できるが、機械はそうではない。したがって、変数が何なのかを明示する必要がある。上記のコマンドにおいて、変数が何なのかを明示する役割を果たしている記号がそれぞれ『x->』『(x,y)->』である。もちろん
f:=x*exp(-x^2);
のように変数を明示しない定義の仕方もあるが、これだと、後々別のコマンドと組み合わせる際に何らかの不都合が生じる可能性がある。
代入に関するコマンド、つまり、文字式や関数に、文字や数字を代入するためのコマンドとしては、evalとsubsが代表的である。
evalコマンドを用いると、文字式や関数に、文字や数字を代入できる。
> eval(x^2+x+1 , x=1);
f:=(x,y,z)->y*exp(-x^2)+z;
> eval(poly,[x=2,y=3,z=t]);
同様にsubsコマンドを使えば、文字式や関数に、文字や数字を代入できる。ただし、代入する側の位置と代入する側の位置が、evalとは逆になっている。
subs( x=2, x^2+x+1 );
subs( x=A, x^2+x+1 );
h:=x*exp(-x^2); subs( x=A, f );
f:=x->x*exp(-x^2); subs( x=A, f );
（駄目）
f:=x->x*exp(-x^2): subs( x=A, f );
f
f:=x->x*exp(-x^2): subs( x=A, f(x) );
evalコマンドとsubsコマンドの違いは、次の例において顕著である。
expr := sin(x)/cos(x); subs(x=0,expr); eval(expr,x=0);
> der := diff(f(x),x) + f(x);
/ d \
der := |--- f(x)| + f(x)
\ dx /
> eval(der,x=0);
/ / d \\
|eval|--- f(x), {x = 0}|| + f(0)
\ \ dx //
> subs(x=0,der);
(diff(f(0), 0)) + f(0)
多項式等を降冪の順や昇冪の順にならべることができる。ただし、Sinの次数について並べ替えるのは難しい。
基本コマンド plot , plot3d および描画パッケージ plots(パッケージ内に描画用のコマンド群が含まれています。)
基本コマンド plot は、一変数の方程式を曲線で描画します。 例：x^2+3
基本コマンド plot3d は、二変数の方程式を曲面で描画します。 例：x+y
曲線を描画する plot コマンドの基本形は、以下となります
plot(曲線の定義式、x=a..b,options);
曲面を描画する plot3d コマンドの基本形は、以下となります
plot3d(曲面の定義式, x=a..b, y=c..d, options) ;
x,y,z で、陰関数で定義されたグラフの場合は
plots[implicitplot3d](陰関数による曲面の定義式, x=a..b, y=c..d,z=e..f, options)
である。その他に媒介変数表示等も可能だが、それは下の表に纏めます。
ただし、曲面を描画する場合には、
with(plots):
を予め読み込んでおかねばならない。具体的には、
with(plots): implicitplot3d(x^2+y^2=1, x=-1..1, y=-1..1)
のようにすればよい。ただし、この場合はzの定義域や、曲面の色やグリッドを指定するためのoptionは省略した（省略しても問題ない）。
また、with(plots):は同じシェルの上で作業する限り一度読み込めば、あとはそのシェルを終了するまで有効なので、
plot3d(x+y, x=1..2, y=1..2);
別の作業 with(plots):
implicitplot3d(x^2+y^2=1, x=-1..1, y=-1..1);
のように、一度だけ読み込んでその後は読み込む必要がないが、
plot3d(x+y, x=1..2, y=1..2);
別の作業
with(plots):
implicitplot3d(x^2+y^2=1, x=-1..1, y=-1..1);
のように、曲面を描くたびに読み込んでも別段問題がない。
基本はこれでよいのだが、現実には『複数の曲面や曲線を同じエリアに描く』あるいは 『予め定義しておいた関数に関するグラフを書く』あるいはその両方を行うことが 単独のグラフを書くことに比べ多いだろう。 そのため、比較的応用範囲が広いプログラムを一つ挙げておこう。以下のプログラムは、 『ｇ（x,y）の定義を行ったうえでそれのグラフとg(x,y)=1についての陰関数を、表示色を変えて同時に表示させる』ものである。
with(plots):
g:=(x,y)->y*exp(-x^2);
p1:=plot3d(g(x,y),x=-3..4,y=-2..4,axes=boxed):
p2:=implicitplot3d(g(x,y)=1,x=-3..4,y=-2..4,z=-2..4,axes=boxed,grid= [15,15,15],style=patchnogrid):
display(p1,p2);
表示するグラフを3つ以上に増やすこともできる
with(plots):
g:=(x,y)->y*exp(-x^2);
p1:=plot3d(g(x,y),x=-3..4,y=-2..4,axes=boxed):
p2:=implicitplot3d(g(x,y)=1,x=-3..4,y=-2..4,z=-2..4,axes=boxed,grid=[15,15,15],style=patchnogrid):
p3:=plot3d(x*y ,x=-3..4,y=-2..4,axes=boxed):
display(p1,p2,p3);
もちろん、1種類の曲面のみを表すことも、可能である。
with(plots):
g:=(x,y)->y*exp(-x^2); p1:=plot3d(g(x,y),x=-3..4,y=-2..4,axes=boxed): display(p1);
尚、g,p1,p2 等の変数名は好きなように変えてよい。（変更で影響を受ける部分、例えば『display(p1,p2);』等も同時に正しく変更すれば）問題ない。例えば
with(plots):
f:=(x,y)->y*exp(-x^2);
p:=plot3d(g(x,y),x=-3..4,y=-2..4,axes=boxed):
q:=implicitplot3d(f(x,y)=1,x=-3..4,y=-2..4,z=-2..4,axes=boxed,grid= [15,15,15],style=patchnogrid):
display(p,q);
のようにしても何も問題ない。
曲面の形式としては、グラフ（plot3d）、陰関数のグラフ(implicitplot3d)等がある。 その他のものは以下の表に纏める。
微分
diff(f(x),x);
不定積分
int(f(x),x);
定積分
int(f(x),x=0..100);
f(x)を定義しなかった場合は、直接x^2などを代わりに入力できる。
| この項目は、コンピュータに関連した書きかけの項目です。この項目を加筆・訂正などしてくださる協力者を求めています（PJ:コンピュータ/P:コンピュータ）。 |
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| Acer | |
|---|---|
| Acer pseudoplatanus (sycamore maple) foliage | |
| Scientific classification | |
| Kingdom: | Plantae |
| (unranked): | Angiosperms |
| (unranked): | Eudicots |
| (unranked): | Rosids |
| Order: | Sapindales |
| Family: | Sapindaceae,[1] or Aceraceae |
| Subfamily: | Hippocastanoideae |
| Genus: | Acer Linnaeus. |
| Species | |
|
See either |
|
| Distribution | |
Acer /ˈeɪsər/ is a genus of trees or shrubs commonly known as maple. There are approximately 128 species, most of which are native to Asia,[2] with a number also appearing in Europe, northern Africa, and North America. Only one species, Acer laurinum, extends to the Southern Hemisphere.[3] The type species of the genus is the sycamore maple, Acer pseudoplatanus, the most common maple species in Europe.[4]
Most maples are trees growing to 10–45 m (33–148 ft) height. Others are shrubs less than 10 metres tall with a number of small trunks originating at ground level. Most species are deciduous, and many are renowned for their autumn leaf colour, but a few in southern Asia and the Mediterranean region are evergreen. Most are shade-tolerant when young and are often riparian, understory, or pioneer species rather than climax overstory trees with a few exceptions such as Sugar Maple. Many of the root systems are typically dense and fibrous, inhibiting the growth of other vegetation underneath them. A few species, notably Acer cappadocicum, frequently produce root sprouts, which can develop into clonal colonies.[4]
Maples are distinguished by opposite leaf arrangement. The leaves in most species are palmate veined and lobed, with 3 to 9 (rarely to 13) veins each leading to a lobe, one of which is central or apical. A small number of species differ in having palmate compound, pinnate compound, pinnate veined or unlobed leaves. Several species, including Acer griseum (Paperbark maple); Acer mandshuricum (Manchurian maple); Acer maximowiczianum (Nikko maple); and Acer triflorum (Three-flowered maple), have trifoliate leaves. One species, Acer negundo (Box-elder), has pinnately compound leaves that may be simply trifoliate or may have five, seven, or rarely nine leaflets. A few, such as Acer laevigatum (Nepal maple) and Acer carpinifolium (Hornbeam maple), have pinnately veined simple leaves.
The flowers are regular, pentamerous, and borne in racemes, corymbs, or umbels. They have four or five sepals, four or five petals about 1 – 6 mm long (absent in some species), four to ten stamens about 6 – 10 mm long, and two pistils or a pistil with two styles. The ovary is superior and has two carpels, whose wings elongate the flowers, making it easy to tell which flowers are female. Maples flower in late winter or early spring, in most species with or just after the appearance of the leaves, but in some before the trees leaf out.[5]
Maple flowers are green, yellow, orange or red. Though individually small, the effect of an entire tree in flower can be striking in several species. Some maples are an early spring source of pollen and nectar for bees.
The distinctive fruit are called samaras, "maple keys", "helicopters", "whirlybirds" or "polynoses". These seeds occur in distinctive pairs each containing one seed enclosed in a "nutlet" attached to a flattened wing of fibrous, papery tissue. They are shaped to spin as they fall and to carry the seeds a considerable distance on the wind. People often call them "helicopters" due to the way that they spin as they fall. During World War II, the US Army developed a special air drop supply carrier that could carry up to 65 pounds of supplies and was based on the Maple seed.[6] Seed maturation is usually in a few weeks to six months after flowering, with seed dispersal shortly after maturity. However, one tree can release hundreds of thousands of seeds at a time. Depending on the species, the seeds can be small and green to orange and big with thicker seed pods. The green seeds are released in pairs, sometimes with the stems still connected. The yellow seeds are released individually and almost always without the stems. Most species require stratification in order to germinate, and some seeds can remain dormant in the soil for several years before germinating.[4]
The genus Acer together with genus Dipteronia are either classified in a family of their own, the Aceraceae, or else classified as members of the family Sapindaceae. Recent classifications, including the Angiosperm Phylogeny Group system, favour inclusion in Sapindaceae. When put in family Sapindaceae, genus Acer is put in subfamily Hippocastanoideae.
The genus is subdivided by its morphology into a multitude of sections and subsections.[7]
Fifty-four species of maples meet the International Union for Conservation of Nature criteria for being under threat of extinction in their native habitat.[3]
The leaves are used as a food plant for the larvae of a number of the Lepidoptera order (see List of Lepidoptera that feed on maples). Aphids are also very common sap-feeders on maples. In horticultural applications a dimethoate spray will solve this.
In the United States and Canada, all maple species are threatened by the Asian long-horned beetle (Anoplophora glabripennis). Infestations have resulted in the destruction of thousands of maples and other tree species in Illinois, Massachusetts, New Jersey and New York.[8]
Maples are affected by a number of fungal diseases. Several are susceptible to Verticillium wilt caused by Verticillium species, which can cause significant local mortality. Sooty bark disease, caused by Cryptostroma species, can kill trees that are under stress due to drought. Death of maples can rarely be caused by Phytophthora root rot and Ganoderma root decay. Maple leaves in late summer and autumn are commonly disfigured by "tar spot" caused by Rhytisma species and mildew caused by Uncinula species, though these diseases do not usually have an adverse effect on the trees' long-term health.[9]
A maple leaf is on the coat of arms of Canada, and is on the Canadian flag. The maple is a common symbol of strength and endurance and has been chosen as the national tree of Canada. Maple leaves are traditionally an important part of Canadian Forces military regalia, for example the military rank insignia for generals use maple leaf symbols.
In the literary world, the word maple was first published in Geoffery Chaucer's "The Knight's Tale" on line 2,065, spelled as "mapul".[10]
Some species of maple are extensively planted as ornamental trees by homeowners, businesses and municipalities due to their fall colour, relatively fast growth, ease of transplanting, and lack of hard seeds that would pose a problem for mowing lawns. Particularly popular are Norway Maple (although it is considered invasive in North America), Silver Maple, Japanese Maple, and Red Maple. Other maples, especially smaller or more unusual species, are popular as specimen trees.[4]
Numerous maple cultivars that have been selected for particular characteristics can be propagated only by asexual reproduction such as cuttings, tissue culture, budding or grafting. Acer palmatum (Japanese maple) alone has over 1,000 cultivars, most selected in Japan, and many of them no longer propagated or not in cultivation in the Western world. Some delicate cultivars are usually grown in pots and rarely reach heights of more than 50–100 cm.
Maples are a popular choice for the art of bonsai. Japanese maple (Acer palmatum), Trident maple (A. buergerianum), Amur maple (A. ginnala), Field maple (A. campestre) and Montpellier maple (A. monspessulanum) are popular choices and respond well to techniques that encourage leaf reduction and ramification, but most species can be used.[4][11]
Maple collections, sometimes called aceretums, occupy space in many gardens and arboreta around the world including the "five great W's" in England: Wakehurst Place Garden, Westonbirt Arboretum, Windsor Great Park, Winkworth Arboretum and Wisley Garden. In the United States, the aceretum at the Harvard-owned Arnold Arboretum in Boston is especially notable. In the number of species and cultivars, the Esveld Aceretum in Boskoop, Netherlands is the largest in the world.[4]
Many maples have bright autumn foliage, and many countries have leaf-watching traditions. In Japan, the custom of viewing the changing colour of maples in the autumn is called "momijigari". Nikko and Kyoto are particularly favoured destinations for this activity. In addition, in Korea, the same viewing activity is called "Danpung-Nori" and the Seoraksan and Naejang-san mountains are very famous places for it.
The Acer saccharum (sugar maple) are a contributor to seasonal fall tourism in North America, particularly in Central Ontario, Québec, and the northern tier of the United States including Wisconsin, Michigan, Vermont, New York, New Hampshire and Western Massachusetts.
Maples are important as source of syrup and wood. Dried wood is often used for the smoking of food. Charcoal from maples is an integral part of the Lincoln County Process used to make Tennessee Whiskey.[12] They are also cultivated as ornamental plants and have benefits for tourism and agriculture.
The Sugar maple (A. saccharum) is tapped for sap, which is then boiled to produce maple syrup or made into maple sugar or maple taffy. It takes about 40 litres (42 US qt) of sugar maple sap to make 1 litre (1.1 US qt) of syrup. While any Acer species may be tapped for syrup, many do not have sufficient quantities of sugar to be commercially useful.
Some of the larger maple species have valuable timber, particularly Sugar maple in North America, and Sycamore maple in Europe. Sugar maple wood — often known as "hard maple" — is the wood of choice for bowling pins, bowling alley lanes, pool cue shafts, and butcher's blocks. Maple wood is also used for the manufacture of wooden baseball bats, though less often than ash or hickory due to the tendency of maple bats to shatter when broken. The maple bat was introduced to Major League Baseball (MLB) in 1998 by Sam Holman of Sam Bats. Today it is the standard maple bat most in use by professional baseball.[13] Maple is also commonly used in archery as the core material in the limbs of a Recurve Bow due to its stiffness and strength.
Maple wood is often graded based on physical and aesthetic characteristics. The most common terminology includes the grading scale from common #2; which is unselected and often used for craft woods; common #1, used for commercial and residential buildings; clear; and select grade, which is sought for fine woodworking.[14]
Some maple wood has a highly decorative wood grain, known as flame maple, quilt maple, birdseye maple and burl wood. This condition occurs randomly in individual trees of several species, and often cannot be detected until the wood has been sawn, though it is sometimes visible in the standing tree as a rippled pattern in the bark.
These select decorative wood pieces also have subcategories that further filter the aesthetic looks. Crotch Wood, Bees Wing, Cats Paw, Old Growth and Mottled are some terms used to describe the look of these decorative woods.[15]
Maples have a long history of use for furniture production in the United States.[16]
Maple is considered a tonewood, or a wood that carries sound waves well, and is used in numerous musical instruments. Maple is harder and has a brighter sound than mahogany, which is another major tonewood used in instrument manufacture.[17]
The back, sides, and neck of most violins, violas, cellos, and double basses are made from maple.
Electric guitar necks are commonly made from maple, having good dimensional stability. The necks of the Fender Stratocaster and Telecaster were originally an entirely maple one piece neck, but later were also available with rosewood fingerboards. Les Paul desired an all maple guitar, but due to the weight of maple, only the tops of Gibson's Les Paul guitars are made from carved maple, often using quilted or flamed maple tops. Due to its weight, very few solid body guitars are made entirely from maple, but many guitars have maple necks, tops or veneers.
Maple is also often used to make bassoons and sometimes for other woodwind instruments like maple recorders.
Many drums are made from maple. From the 70s to the 90s, maple drum kits were a vast majority of all drum kits made, but in recent years, birch has become popular for drums once again. Some of the best drum-building companies use maple extensively throughout their mid-pro range.[4] Maple drums are favored for their bright resonant sound. [18] Certain types of drum sticks are also made from maple.
As they are a major source of pollen in early spring before many other plants have flowered, maples are important to the survival of honeybees that play a commercially important role later in the spring and summer.
Maple is used as pulpwood. The fibers have relatively thick walls that prevent collapsing upon drying. This gives good bulk and opacity in paper. Maple also gives paper good printing properties.
Acer cappadocicum (Cappadocian maple)
Acer carpinifolium leaves
Acer freemanii 'Autumn Blaze' (a cross between Acer rubrum and Acer saccharinum
Acer macrophyllum flowers and young leaves
Acer laevigatum leaves and fruit
Acer sempervirens foliage
Acer ginnala foliage
Acer palmatum trees and bamboo in Japan
Acer grandidentatum (Bigtooth Maple) in autumn colour
Acer platanoides leaf
Acer palmatum leaf in autumn
Acer platanoides (Norway maple) samaras
Acer griseum (Paperbark maple)
| Wikimedia Commons has media related to Acer. |
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| リンク元 | 「Acer」「カエデ」「Aceraceae」 |
| 拡張検索 | 「maple syrup urine disease」 |
| 関連記事 | 「map」 |