即使以相同的『數學方法』，因為談論不同的事，有時還是得腦筋『急轉彎』。這個腦筋『急轉彎』的由來，也許是因為得用『非常不同』的觀點來考察事物。突然發現某種不期而遇之『共通處』。若問笛卡爾如何想出『解析幾何』，使得幾何得以用『座標系』與『座標』來處理？雖說不得而知，但是這套數學『思維方法』構成了牛頓力學的骨幹。就像問著『約瑟夫‧傅立葉』 Joseph Fourier 為什麼會將函數用『正弦級數』來作展開？？難道是受到笛卡爾的啟發！或是牛頓的影響！！可能很難定論。此處只能借著科學史上的一個小故事來暗示那種概念間冥冥的『聯繫』︰

電阻的單位是『歐姆』

傅立葉分析
數學上最美麗的樂章

一八二五年德國物理學家蓋歐格‧西蒙‧歐姆 Georg Simon Ohm 是一位優秀的實驗者，很會設計與製造實驗設備，又具有良好的數學素養和嚴謹的處事態度。他使用伏打電堆為電源，繼而用安裝於扭秤 torsion balance 的磁針來測量電流產生的磁力。因為載流導線的電流所產生的磁場與電流成正比，所以只要測量在載流導線附近的磁針所感受到的磁力，就可以度量電流的大小。他將電流通過不同長度的受測電線；由於長度的不同，電流也就不同。並且基於實驗結果推導出了兩者之間的數學關係式︰

$\Delta I = C \ln\left(1+\frac{x}{a}\right)$

其中， $\Delta I$ 是受測電線造成的電流差值， $C$ 是跟實驗參數有關的係數， $x$ 是受測電線的長度， $a$ 是跟固定長度的載流導線有關的常數。不久之後，歐姆便發覺了這個關係式可能是不正確的。

法國數學家與物理學家約瑟夫‧傅立葉男爵 Joseph Fourier 以所提出的『傅立葉級數』稱名於世，對於任何一個周期為 $T$ 的函數 $f(t)$ ，都可以用『無窮級數』表示為

$f(t) = \sum _{k=-\infty}^{+\infty}a_k\cdot e^{i k 2\pi \frac{t}{T}}$

，式中係數 $a_k$ 可以依下式來計算

$a_k = \frac{1}{T}\int_{T}f(t)\cdot e^{-i k 2\pi \frac{t}{T}}dt$

。此處 $i=\sqrt{-1}$ ，而且 $f_k(t)=e^{i k 2\pi \frac{t}{T}}$ 是基本周期為 $T$ 的三角『正、餘弦』函數，這就是周期訊號 $f(t)$ 的『諧波分析』。當 $k = 0$ 時， $a_0$ 係數稱之為『直流分量』，就是訊號 $f(t)$ 在整個周期的平均值。當 $k = \pm 1$ 時， $a_{\pm 1}$ 係數是『一次諧波』頻率，隨著 $k$ 的值，而有『二次諧波』，『三次諧波』等等。並且將之應用於『熱傳導理論』與『振動理論』；據聞他也是『溫室效應』的發現者。一八二二年傅立葉發表了《熱的解析理論》 Théorie analytique de la chaleur)。依據『牛頓冷卻定律』假設了︰兩相鄰分子的熱流與其間微小的溫度差成正比。

受到傅立葉對熱傳導規律研究的啟發，歐姆認為『電流現象』與『熱傳導』相似，便猜想導線中兩點之間的電流也正比於這兩點間的某種驅動力 ── 即現在所說的『電動勢』 ──。由於當時『伏打電堆』所產生的電流並不穩定，據聞歐姆採納了《物理與化學年鑑》的總編輯約翰‧波根多夫 Johann poggendorff 的建議，改用『熱電偶』為電源。雖然『鉍和銅』所作的『溫差電池』保證了電流的穩定性。然而卻產生了測量『電流太小』的難題。歐姆先是利用『電流』的『熱效應』，想藉著『熱脹冷縮』的方法來測量電流大小，但是量測的結果並『不精確』，後來他把奧斯特『電流磁效應』的發現和『庫侖扭秤』巧妙的結合起來設計了一個『電流扭秤』︰讓導線和連接的磁針平行放置，當導線中通過電流時，磁針的偏轉角與導線中的電流成正比，這就代表了電流的大小。

一八二六年歐姆所發表兩篇重要論文中，建立了『電傳導』的『數學模型』和『表達式』。著名的『歐姆定律』發表於一八二七年的《直流電路的數學研究》 Die galvanische Kette, mathematisch bearbeitet 一書中，提出了『電路分析』中『電流』、『電壓』與『電阻』之間的基本關係

$I = \frac{V}{R}$ 。

─── 引自《【Sonic π】電聲學導引《七》》

或許能說明『正交函數族』之概念，並非一蹴而成的，所謂『正交』也不可講成『畢氏定理』自然的擴張。雖然說這些『象徵』表達概念可能的深層『融會』，提示讀者進入的門徑。究其實，依舊是例說寓言罷了。

就像『振幅調變』 $x(t) \cdot y(t)$ 之『頻譜』，可以用『頻譜摺積』來計算，然而什麼是『摺積』呢？

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated. Convolution is similar to cross-correlation. It has applications that include probability, statistics, computer vision, natural language processing, image and signal processing, engineering, and differential equations.

The convolution can be defined for functions on groups other than Euclidean space. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 10 at DTFT#Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.

Computing the inverse of the convolution operation is known as deconvolution.

Visual comparison of convolution, cross-correlation and autocorrelation.

摺積

在泛函分析中，捲積、疊積、摺積或旋積，是通過兩個函數f和g生成第三個函數的一種數學算子，表徵函數f與經過翻轉和平移的g的重疊部分的面積。如果將參加摺積的一個函數看作區間的指示函數，摺積還可以被看作是「滑動平均」的推廣。

圖示兩個方形脈衝波的捲積。其中函數”g”首先對 $\tau=0$ 反射，接著平移”t”，成為 $g(t-\tau)$ 。那麼重疊部份的面積就相當於”t”處的捲積，其中橫坐標代表待積變量 $\tau$ 以及新函數 $f\ast g$ 的自變量”t”。

圖示方形脈衝波和指數衰退的脈衝波的捲積（後者可能出現於RC電路中），同樣地重疊部份面積就相當於”t”處的捲積。注意到因為”g”是對稱的，所以在這兩張圖中，反射並不會改變它的形狀。

簡單介紹

摺積是分析數學中一種重要的運算。設： $f(x)$ , $g(x)$ 是 $\mathbb{R}$ 上的兩個可積函數，作積分：

\int_{-\infty}^{\infty} f(\tau) g(x - \tau)\, \mathrm{d}\tau

可以證明，關於幾乎所有的 $x \in (-\infty,\infty)$ ，上述積分是存在的。這樣，隨著 $x$ 的不同取值，這個積分就定義了一個新函數 $h(x)$ ，稱為函數 $f$ 與 $g$ 的摺積，記為 $h(x)=(f*g)(x)$ 。我們可以輕易驗證： $(f * g)(x) = (g * f)(x)$ ，並且 $(f * g)(x)$ 仍為可積函數。這就是說，把摺積代替乘法， $L^1(R^1)$ 空間是一個代數，甚至是巴拿赫代數。雖然這裡為了方便我們假設 $\textstyle f, g\in L^1(\mathbb{R})$ ，不過捲積只是運算符號，理論上並不需要對函數 $f,g$ 有特別的限制，雖然常常要求 $f,g$ 至少是可測函數（measurable function）（如果不是可測函數的話，積分可能根本沒有意義），至於生成的卷積函數性質會在運算之後討論。

摺積與傅立葉變換有著密切的關係。例如兩函數的傅立葉變換的乘積等於它們摺積後的傅立葉變換，利用此一性質，能簡化傅立葉分析中的許多問題。

由摺積得到的函數 $f*g$ 一般要比 $f$ 和 $g$ 都光滑。特別當 $g$ 為具有緊支集的光滑函數， $f$ 為局部可積時，它們的摺積 $f * g$ 也是光滑函數。利用這一性質，對於任意的可積函數 $f$ ，都可以簡單地構造出一列逼近於 $f$ 的光滑函數列 $f_s$ ，這種方法稱為函數的光滑化或正則化。

摺積的概念還可以推廣到數列、測度以及廣義函數上去。

The convolution theorem and its applications

What is a convolution?

One of the most important concepts in Fourier theory, and in crystallography, is that of a convolution. Convolutions arise in many guises, as will be shown below. Because of a mathematical property of the Fourier transform, referred to as the convolution theorem, it is convenient to carry out calculations involving convolutions.

But first we should define what a convolution is. Understanding the concept of a convolution operation is more important than understanding a proof of the convolution theorem, but it may be more difficult!

Mathematically, a convolution is defined as the integral over all space of one function at x times another function at u-x. The integration is taken over the variable x (which may be a 1D or 3D variable), typically from minus infinity to infinity over all the dimensions. So the convolution is a function of a new variable u, as shown in the following equations. The cross in a circle is used to indicate the convolution operation.

convolution equals integral of one function at x times other function at u-x

Note that it doesn’t matter which function you take first, i.e. the convolution operation is commutative. We’ll prove that below, but you should think about this in terms of the illustration below. This illustration shows how you can think about the convolution, as giving a weighted sum of shifted copies of one function: the weights are given by the function value of the second function at the shift vector. The top pair of graphs shows the original functions. The next three pairs of graphs show (on the left) the function g shifted by various values of x and, on the right, that shifted function g multiplied by f at the value of x.

illustration of convolution

The bottom pair of graphs shows, on the left, the superposition of several weighted and shifted copies of g and, on the right, the integral (i.e. the sum of all the weighted, shifted copies of g). You can see that the biggest contribution comes from the copy shifted by 3, i.e. the position of the peak of f.

If one of the functions is unimodal (has one peak), as in this illustration, the other function will be shifted by a vector equivalent to the position of the peak, and smeared out by an amount that depends on how sharp the peak is. But alternatively we could switch the roles of the two functions, and we would see that the bimodal function g has doubled the peaks of the unimodal function f.

───

它是否有所謂傳統『乘法類似物』的耶！！或者『類比推理』只該看成是『發現』的邏輯乎？？

樹莓派、樹莓派之學習、樹莓派之教育

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧周期

2015-11-24 懸鉤子

經過『九點圓』以及『留白』的洗禮，也許我們可以借著

Miller Puckette 之

1.7 Periodic Signals
A signal x[n] is said to repeat at a period τ if

x[n + τ ] = x[n]

for all n. Such a signal would also repeat at periods 2τ and so on; the smallest τ if any at which a signal repeats is called the signal’s period. In discussing periods of digital audio signals, we quickly run into the difficulty of describing signals whose “period” isn’t an integer, so that the equation above doesn’t make sense. For now we’ll effectively ignore this difficulty by supposing that the signal x[n] may somehow be interpolated between the samples so that it’s well defined whether n is an integer or not.

A sinusoid has a period (in samples) of 2π/ω where ω is the angular frequency. More generally, any sum of sinusoids with frequencies 2πk/ω, for integers k, will repeat after 2π/ω samples. Such a sum is called a Fourier Series:

x[n] = a0 + a1 cos (ωn + φ1 ) + a2 cos (2ωn + φ2 ) + · · · + ap cos (pωn + φp )

Moreover, if we make certain technical assumptions (in effect that signals only contain frequencies up to a finite bound), we can represent any periodic signal as such a sum. This is the discrete-time variant of Fourier analysis which will reappear in Chapter 9.

The angular frequencies of the sinusoids above are all integer multiples of ω. They are called the harmonics of ω, which in turn is called the fundamental. In terms of pitch, the harmonics ω, 2ω, . . . are at intervals of 0, 1200, 1902, 2400, 2786, 3102, 3369, 3600, …, cents above the fundamental; this sequence of pitches is sometimes called the harmonic series. The first six of these are all quite close to multiples of 100; in other words, the first six harmonics of a pitch in the Western scale land close to (but not always exactly on) other pitches of the same scale; the third and sixth miss only by 2 cents and the fifth misses by 14.

………

文本，談點『周期』概念容易『誤解』的事。

維基百科的『周期函數』Periodic function 詞條講︰

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.

An illustration of a periodic function with period $P.$

Definition

A function f is said to be periodic with period P (P being a nonzero constant) if we have

f(x+P) = f(x) \,\!

for all values of x in the domain. If there exists a least positive^[1] constant P with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.

A function that is not periodic is called aperiodic.

Properties

If a function f is periodic with period P, then for all x in the domain of f and all integers n,

f(x + nP) = f(x).

If f(x) is a function with period P, then f(ax+b), where a is a positive constant, is periodic with period P/|a|. For example, f(x)=sinx has period 2π, therefore sin(5x) will have period 2π/5.

Double-periodic functions

A function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions. (“Incommensurate” in this context means not real multiples of each other.)

Quotient spaces as domain

In signal processing you encounter the problem, that Fourier series represent periodic functions and that Fourier series satisfy convolution theorems (i.e. convolution of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a quotient space:

{\mathbb{R}/\mathbb{Z}} = \{x+\mathbb{Z} : x\in\mathbb{R}\} = \{\{y : y\in\mathbb{R}\land y-x\in\mathbb{Z}\} : x\in\mathbb{R}\}

That is, each element in ${\mathbb{R}/\mathbb{Z}}$ is an equivalence class of real numbers that share the same fractional part. Thus a function like $f : {\mathbb{R}/\mathbb{Z}}\to\mathbb{R}$ is a representation of a 1-periodic function.

也許足以澄清 Miller Puckette 文本中大多數的內容。首先一個以 $P$ 為『周期』的函數，自然也以 $n \cdot P$ 為周期【※ $n$ 非零正整數】，所以實數周期函數才會有最小周期值。

如果 $f(x)$ 以 $P_1$ 為周期， $g(x)$ 以 $P_2$ 為周期，假使 $\frac{P_1}{P_2}$ 的比值是個『有理數』，可以用『最簡分數』表示成 $\frac{n}{m}$ 。也就是說 $m \cdot P_1 \ = \ n \cdot P_2 \ = \ T$

此處的 $T$ 也就是 $f(x) + g(x)$ 的周期。然而周期不必是『有理數』，周期的『比值』當然也未必是『有理數』，因此

兩個周期函數的和，卻未必是個周期函數

！！？？

要是此時重讀《字詞網絡︰ WordNet 《一》索引》系列文本︰

Natural Language Processing with Python
— Analyzing Text with the Natural Language Toolkit

Steven Bird, Ewan Klein, and Edward Loper

一書第二章第五節《 2.5 WordNet 》之『字詞網絡』概念階層片段

※或可參考【譯著】

Figure 2-8. Fragment of WordNet concept hierarchy: Nodes correspond to synsets; edges indicate the hypernym/hyponym relation, i.e., the relation between superordinate and subordinate concepts.

『WordNet』字詞網絡計畫啟始於一九八五年，在普林斯頓大學『認知科學實驗室』由心理學教授『喬治‧A‧米勒』 George Armitage Miller 的指導下建立和維護的英語『詞彙資料庫』 lexical database 字典。因為它包含了多種『字詞』間之『語義關係』，所以別於通常意義下的『字典』。『WordNet』是什麼？也許最好先讀讀『創造者』怎麼說︰

What is WordNet?

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the creators of WordNet and do not necessarily reflect the views of any funding agency or Princeton University.When writing a paper or producing a software application, tool, or interface based on WordNet, it is necessary to properly cite the source. Citation figures are critical to WordNet funding.

About WordNet

WordNet® is a large lexical database of English. Nouns, verbs, adjectives and adverbs are grouped into sets of cognitive synonyms (synsets), each expressing a distinct concept. Synsets are interlinked by means of conceptual-semantic and lexical relations. The resulting network of meaningfully related words and concepts can be navigated with the browser. WordNet is also freely and publicly available for download. WordNet’s structure makes it a useful tool for computational linguistics and natural language processing.

WordNet superficially resembles a thesaurus, in that it groups words together based on their meanings. However, there are some important distinctions. First, WordNet interlinks not just word forms—strings of letters—but specific senses of words. As a result, words that are found in close proximity to one another in the network are semantically disambiguated. Second, WordNet labels the semantic relations among words, whereas the groupings of words in a thesaurus does not follow any explicit pattern other than meaning similarity.

Structure

The main relation among words in WordNet is synonymy, as between the words shut and close or car and automobile. Synonyms–words that denote the same concept and are interchangeable in many contexts–are grouped into unordered sets (synsets). Each of WordNet’s 117 000 synsets is linked to other synsets by means of a small number of “conceptual relations.” Additionally, a synset contains a brief definition (“gloss”) and, in most cases, one or more short sentences illustrating the use of the synset members. Word forms with several distinct meanings are represented in as many distinct synsets. Thus, each form-meaning pair in WordNet is unique.

Relations

The most frequently encoded relation among synsets is the super-subordinate relation (also called hyperonymy, hyponymy or ISA relation). It links more general synsets like {furniture, piece_of_furniture} to increasingly specific ones like {bed} and {bunkbed}. Thus, WordNet states that the category furniture includes bed, which in turn includes bunkbed; conversely, concepts like bed and bunkbed make up the category furniture. All noun hierarchies ultimately go up the root node {entity}. Hyponymy relation is transitive: if an armchair is a kind of chair, and if a chair is a kind of furniture, then an armchair is a kind of furniture. WordNet distinguishes among Types (common nouns) and Instances (specific persons, countries and geographic entities). Thus, armchair is a type of chair, Barack Obama is an instance of a president. Instances are always leaf (terminal) nodes in their hierarchies.

Meronymy, the part-whole relation holds between synsets like {chair} and {back, backrest}, {seat} and {leg}. Parts are inherited from their superordinates: if a chair has legs, then an armchair has legs as well. Parts are not inherited “upward” as they may be characteristic only of specific kinds of things rather than the class as a whole: chairs and kinds of chairs have legs, but not all kinds of furniture have legs.

Verb synsets are arranged into hierarchies as well; verbs towards the bottom of the trees (troponyms) express increasingly specific manners characterizing an event, as in {communicate}-{talk}-{whisper}. The specific manner expressed depends on the semantic field; volume (as in the example above) is just one dimension along which verbs can be elaborated. Others are speed (move-jog-run) or intensity of emotion (like-love-idolize). Verbs describing events that necessarily and unidirectionally entail one another are linked: {buy}-{pay}, {succeed}-{try}, {show}-{see}, etc.

Adjectives are organized in terms of antonymy. Pairs of “direct” antonyms like wet-dry and young-old reflect the strong semantic contract of their members. Each of these polar adjectives in turn is linked to a number of “semantically similar” ones: dry is linked to parched, arid, dessicated and bone-dry and wet to soggy, waterlogged, etc. Semantically similar adjectives are “indirect antonyms” of the contral member of the opposite pole. Relational adjectives (“pertainyms”) point to the nouns they are derived from (criminal-crime).
There are only few adverbs in WordNet (hardly, mostly, really, etc.) as the majority of English adverbs are straightforwardly derived from adjectives via morphological affixation (surprisingly, strangely, etc.)

Cross-POS relations

The majority of the WordNet’s relations connect words from the same part of speech (POS). Thus, WordNet really consists of four sub-nets, one each for nouns, verbs, adjectives and adverbs, with few cross-POS pointers. Cross-POS relations include the “morphosemantic” links that hold among semantically similar words sharing a stem with the same meaning: observe (verb), observant (adjective) observation, observatory (nouns). In many of the noun-verb pairs the semantic role of the noun with respect to the verb has been specified: {sleeper, sleeping_car} is the LOCATION for {sleep} and {painter}is the AGENT of {paint}, while {painting, picture} is its RESULT.

More Information

Fellbaum, Christiane (2005). WordNet and wordnets. In: Brown, Keith et al. (eds.), Encyclopedia of Language and Linguistics, Second Edition, Oxford: Elsevier, 665-670

───

是否會有不同之體驗乎？？！！

部落格訊息

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧留白

2015-11-23 懸鉤子

若問為什麼平面上的一個一般三角形可以如下圖表示

，只用著 $a \ , b \ , \ c$ 三個參數？即使在思考過 $a$ 是『底』之『長』， $c$ 是此『底』之『高』， $b$ 是此『高』距與此『底』一端的距離。我們深信這就『確定』了那個三角形。然而若再問︰如果此三角形的三個頂點用更一般的 $A \ (x_0, y_0)$ 、 $B \ (x_1,y_1)$ 、 $C \ (x_2,y_2)$ 來表達，如是分明有六個參數。那麼這兩種『表述』當真是一樣的嗎？設想你在桌面上『移動』一個三角形，從此『位置』此『方位』到達彼『位置』彼『方位』，你會認為這個三角形『改變』了嗎？？假使『直覺』以為『不變』，這個三角形就必得有使之『不變』的『因由』，這個『因由』不必『參照』解析幾何的『座標』而確立。或可說它就是歐式幾何一個三角形的『定義』內涵而已。如此而言，一個『確定』的三角形，可由它的三個『邊長』來『確立』，所以六個參數補之以三個確定之邊長關係，豈非還是三個參數的耶？？

因為這個『歐式幾何』的『留白』，常使人懷疑『解析幾何』簡化『座標系』的『選擇』，到底『圖形』的『自由度』是幾何的了。說難道易，就請讀者思索︰平面上的『 □ 』與『 ○ 』，到底一方一圓需要幾個參數來描述的呢？

從物理上講，那個三角形就是『剛體』 rigid body ，它在『運動』中保持『形狀』的『不變性』。而且不同觀察者間的『座標變換』可以用

Rigid transformation

In mathematics, a rigid transformation (isometry) of a vector space preserves distances between every pair of points.^[1]^[2] Rigid transformations of the plane R², space R³, or real n-dimensional space Rⁿ are termed a Euclidean transformation because they form the basis of Euclidean geometry.^[3]

The rigid transformations include rotations, translations, reflections, or their combination. Sometimes reflections are excluded from the definition of a rigid transformation by imposing that the transformation also preserve the handedness of figures in the Euclidean space (a reflection would not preserve handedness; for instance, it would transform a left hand into a right hand). To avoid ambiguity, this smaller class of transformations is known as proper rigid transformations (informally, also known as roto-translations). In general, any proper rigid transformation can be decomposed as a rotation followed by a translation, while any rigid transformation can be decomposed as an improper rotation followed by a translation (or as a sequence of reflections).

Any object will keep the same shape and size after a proper rigid transformation.

All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of proper rigid transformation is called special Euclidean group, denoted SE(n).

In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to represent the linear and angular displacement of rigid bodies. According to Chasles’ theorem, every rigid transformation can be expressed as a screw displacement.

Formal definition

A rigid transformation is formally defined as a transformation that, when acting on any vector v, produces a transformed vector T(v) of the form

T(v) = R v + t

where R^T = R⁻¹ (i.e., R is an orthogonal transformation), and t is a vector giving the translation of the origin.

A proper rigid transformation has, in addition,

det(R) = 1

which means that R does not produce a reflection, and hence it represents a rotation (an orientation-preserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its determinant is –1.

Distance formula

A measure of distance between points, or metric, is needed in order to confirm that a transformation is rigid. The Euclidean distance formula for Rⁿ is the generalization of the Pythagorean theorem. The formula gives the distance squared between two points X and Y as the sum of the squares of the distances along the coordinate axes, that is

d(\mathbf{X},\mathbf{Y})^2 = (X_1-Y_1)^2 + (X_2-Y_2)^2 + \ldots + (X_n-Y_n)^2 = (\mathbf{X}-\mathbf{Y})\cdot(\mathbf{X}-\mathbf{Y}).

where X=(X₁, X₂, …, X_n) and Y=(Y₁, Y₂, …, Y_n), and the dot denotes the scalar product.

Using this distance formula, a rigid transformation g:Rⁿ→Rⁿ has the property,

d(g(\mathbf{X}), g(\mathbf{Y}))^2 = d(\mathbf{X}, \mathbf{Y})^2.

───

來作轉換的矣！！

如是細思二維空間上的『旋轉』︰

Two dimensions

In two dimensions, to carry out a rotation using matrices the point $(x, y)$ to be rotated (orientation from positive $x$ to $y$ ) is written as a vector, then multiplied by a matrix calculated from the angle, $θ$ :

\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

where $(x', y')$ are the coordinates of the point that after rotation, and the formulae for $x'$ and $y'$ can be seen to be

\text{[math]}

The vectors $\begin{bmatrix} x \\ y \end{bmatrix}$ and $\begin{bmatrix} x' \\ y' \end{bmatrix}$ have the same magnitude and are separated by an angle $θ$ as expected.

Points on the $R 2$ plane can be also presented as complex numbers: the point $(x, y)$ in the plane is represented by the complex number

z = x + iy

This can be rotated through an angle $θ$ by multiplying it by $e iθ$ , then expanding the product using Euler’s formula as follows:

\text{[math]}

and equating real and imaginary parts gives the same result as a two-dimensional matrix:

\text{[math]}

Since complex numbers form a commutative ring, vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only one degree of freedom, as such rotations are entirely determined by the angle of rotation.^[1]

───

那一方一圓問題的答案，是否不言而喻的哩？？！！

樹莓派、樹莓派之學習、樹莓派之教育

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧九點圓

2015-11-22 懸鉤子

人們如何發掘『現象』間的『關係』？怎麼探討事物的『性質』？因何能發現且證明隱晦的『數學定理』？！也許讀讀維基百科上『九點圓』詞條一小段文本︰

九點圓

九點圓（又稱歐拉圓、費爾巴哈圓），在平面幾何中，對任何三角形，九點圓通過三角形三邊的中點、三高的垂足、和頂點到垂心的三條線段的中點。九點圓定理指出對任何三角形，這九點必定共圓。而九點圓還具有以下性質：

九點圓的半徑是外接圓的一半，且九點圓平分垂心與外接圓上的任一點的連線。
圓心在歐拉線上，且在垂心到外心的線段的中點。
九點圓和三角形的內切圓和旁切圓相切（費爾巴哈定理）。
圓周上四點任取三點做三角形，四個三角形的九點圓圓心共圓（庫利奇－大上定理）。

九點圓

Even if the orthocenter and circumcenter fall outside of the triangle, the construction still works.

歷史

1765年，萊昂哈德·歐拉證明：「垂心三角形和垂足三角形有共同的外接圓（六點圓）。」許多人誤以為九點圓是由而歐拉發現所以又稱乎此圓為歐拉圓。而第一個證明九點圓的人是彭賽列（1821年）。1822年，卡爾·威廉·費爾巴哈也發現了九點圓，並得出「九點圓和三角形的內切圓和旁切圓相切」，因此德國人稱此圓為費爾巴哈圓，並稱這四個切點為費爾巴哈點。庫利奇與大上分別於1910年與1916年發表庫利奇－大上定理「圓周上四點任取三點做三角形，四個三角形的九點圓圓心共圓。」這個圓還被稱為四邊形的九點圓，此結果還可推廣到n邊形。

九點圓證明

如圖： $D$ 、 $E$ 、 $F$ 為三邊的中點， $G$ 、 $H$ 、 $I$ 為垂足， $J$ 、 $K$ 、 $L$ 為和頂點到垂心的三條線段的中點。

容易得出 $\triangle ABS \sim \triangle AFJ$ 、 $\triangle CBS \sim \triangle CDL$ （SAS相似）

因此 $\overline{FJ} // \overline{BH} // \overline{DL}$

同樣可得出 $\triangle ABC \sim \triangle FBD$ 、 $\triangle ASC \sim \triangle JSL$ （SAS相似）

因此 $\overline{FD} // \overline{AC} // \overline{JL}$

又 $\overline{BH} \perp \overline{AC}$ ，可得出四邊形 $DFJL$ 是矩形（四點共圓）

同理可證 $FKLE$ 也是矩形（ $DKFJEL$ 共圓）

$\angle JLD = \angle JGD = 90^\circ$ ，因此可知 $G$ 也在圓上（圓周角相等）

同理可證 $H$ 、 $I$ 兩點也在圓上（九點共圓）

───

可以當成發想的起點。假使設想姑且不論到底是怎樣『發現』的，且談已經『被發現』後，是否人們就能容易了解那些『關係』、『性質』、以及『證明』呢？有人說︰閱讀『證明』容易，動手『證明』困難。似乎是講，既然都已理解了『證明』，焉有不曉『關係』與『性質』的耶！！若說條條大路通『羅馬』，就算盡觀了那些條條大路的景緻，和『羅馬』之風光能夠彼此比較的嗎？？！！更何況『始、中、終』的『學習』循環不斷，舊『終』則啟新『始』布線織網深化『閱歷』。所以『學問』浸潤良久總有所悟，宛如說今日這門古老的『幾何學』，是以前那門新創之『幾何學』的嗎！！？？

何不讓我們舉個例子從『解析幾何』的觀點，來看一個歐式幾何之『證明』呢︰

解析幾何

解析幾何（英語：Analytic geometry），又稱為坐標幾何（英語：Coordinate geometry）或卡氏幾何（英語：Cartesian geometry），早先被叫作笛卡兒幾何，是一種藉助於解析式進行圖形研究的幾何學分支。解析幾何通常使用二維的平面直角坐標系研究直線、圓、圓錐曲線、擺線、星形線等各種一般平面曲線，使用三維的空間直角坐標系來研究平面、球等各種一般空間曲面，同時研究它們的方程，並定義一些圖形的概念和參數。

在中學課本中，解析幾何被簡單地解釋為：採用數值的方法來定義幾何形狀，並從中提取數值的信息。然而，這種數值的輸出可能是一個方程或者是一種幾何形狀。

1637年，笛卡兒在《方法論》的附錄「幾何」中提出了解析幾何的基本方法。以哲學觀點寫成的這部法語著作為後來牛頓和萊布尼茨各自提出微積分學提供了基礎。

對代數幾何學者來說，解析幾何也指（實或者複）流形，或者更廣義地通過一些複變數（或實變數）的解析函數為零而定義的解析空間理論。這一理論非常接近代數幾何，特別是通過讓-皮埃爾·塞爾在《代數幾何和解析幾何》領域的工作。這是一個比代數幾何更大的領域，不過也可以使用類似的方法。

【三角形三中線交於一點的證明】

三角形的中心

形心是三角形的幾何中心，通常也稱為重心，三角形的三條中線（頂點和對邊的中點的連線）交點，此點即為重心^[1]。

三條中線共點證明

用西瓦定理逆定理可以直接證出：

\frac{BE}{EC} \cdot \frac{CF}{FA} \cdot \frac{AD}{DB}=\frac{1}{1} \cdot \frac{1}{1} \cdot \frac{1}{1}=1

因此三線共點。^[2]

【前述證明所用之定理的證明】

塞瓦定理

塞瓦線段（cevian）是各頂點與其對邊或對邊延長線上的一點連接而成的直線段。塞瓦定理指出：如果 $\triangle ABC$ 的塞瓦線段AD、BE、CF通過同一點O，則

\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB}=1

它的逆定理同樣成立：若D、E、F分別在 $\triangle ABC$ 的邊BC、CA、AB或其延長線上（都在邊上或有兩點在延長線上），且滿足

\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB}=1

，

則直線AD、BE、CF共點或彼此平行（於無限遠處共點）。當AD、BE、CF中的任意兩直線交於一點時，則三直線共點；當AD、BE、CF中的任意兩直線平行時，則三直線平行。

它最先由義大利數學家喬瓦尼·塞瓦證明。

證明

\because\quad\frac{BD}{DC}=\frac{\mathrm{S}_{\triangle ABD}}{\mathrm{S}_{\triangle ADC}}=\frac{\mathrm{S}_{\triangle OBD}}{\mathrm{S}_{\triangle ODC}}.

由等比性質,

\frac{BD}{DC}=\frac{\mathrm{S}_{\triangle ABD} - \mathrm{S}_{\triangle OBD}}{\mathrm{S}_{\triangle ADC} - \mathrm{S}_{\triangle ODC}}=\frac{\mathrm{S}_{\triangle ABO}}{\mathrm{S}_{\triangle CAO}}.

同理

\frac{CE}{EA}=\frac{\mathrm{S}_{\triangle BCO}}{\mathrm{S}_{\triangle ABO}},\;\frac{AF}{FB}=\frac{\mathrm{S}_{\triangle CAO}}{\mathrm{S}_{\triangle BCO}}.

\therefore\quad\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB}=\frac{\mathrm{S}_{\triangle ABO}}{\mathrm{S}_{\triangle CAO}} \cdot \frac{\mathrm{S}_{\triangle BCO}}{\mathrm{S}_{\triangle ABO}} \cdot \frac{\mathrm{S}_{\triangle CAO}}{\mathrm{S}_{\triangle BCO}}=1.

證畢。

【解析幾何的求解證明】

相異兩點 $(x_0 , y_0) , \ (x_1, y_1)$ 決定一條線，這線的方程式可以寫成

$y \ - y_0 \ = \frac{y_1 \ - y_0}{x_1 \ - x_0} \ (x \ - x_0)$

相異三點可以決定一個三角形 $\Delta \ ABC$ ，不失一般性，假設這三個頂點座標是 $(0, 0) , \ (a, 0) , \ (b, c)$ ，那麼

線 L1 的方程式為

$y \ = \ \frac{c}{2b \ - a} \ (2x \ - a)$

線 L2 的方程式為

$y \ = \ \frac{c}{b \ - 2a} \ (x \ - a)$

線 L1 與線 L2 的交點 G ，求解聯立方程式可得

$x \ = \ \frac{a + b} {3} \ , \ y \ = \ \frac{c}{3}$

此『重心』之座標值，果然符合

幾何中心

中心分每條中線比為2:1，這就是說距一邊的距離是該邊相對頂點距該邊的1/3。如右圖所示：

如果三角形是由均勻材料做成的薄片，那麼幾何中心也就是質量中心。它的笛卡爾坐標是三個頂點的坐標算術平均值。也就是說，如果三頂點位於 $(x_a, y_a)$ ， $(x_b, y_b)$ ，和 $(x_c, y_c)$ ，那麼幾何中心位於：

\Big( \begin{matrix}\frac13\end{matrix} (x_a+x_b+x_c),\; \begin{matrix}\frac13\end{matrix} (y_a+y_b+y_c)\Big) = \begin{matrix}\frac13\end{matrix} (x_a, y_a) + \begin{matrix}\frac13\end{matrix} (x_b, y_b) + \begin{matrix}\frac13\end{matrix} (x_c, y_c)

詞條之所言。再者從向量

$\vec{FG} \ = \ \frac{1}{6} (2b \ - a , 2c)$

$\vec{FC} \ = \ \frac{1}{2} (2b \ - a , 2c) \ = 3 \ \vec{FG}$

之『關係』，故可知『重心』將『中線』分成了 2:1 之『性質』。

那麼所謂『證明』三角形三中線交於一點的事，也轉譯成了通過 A 點與 G 點的 L3 線，定然通過中點 D 。因此只需寫出 L3 的方程式，再驗證 D 點滿足那個參數方程式的耶！！如是將能夠體會不同論述之『難易煩簡』常十分不同的乎？？

順便在此介紹一下，或許你會愛上的一個『幾何學探索』工具

GeoGebra

，它在樹莓派上的安裝十分容易

sudo apt-get install geogebra-gnome openjdk-7-jre

。然後請你思考︰

兩個『週期』函數的『和』，必然是個『週期』函數的嗎？

樹莓派、樹莓派之學習、樹莓派之教育

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧調幅

2015-11-21 懸鉤子

若問『音樂』與『數學』有關嗎？無論這關係似深似淺，最終總得回到『人』的『覺知』身上！若思『天真』只屬於『童年』嗎？？當『好奇心』隨著年齡成長而漸行漸遠，彷彿必然墜入『社會』之『同化』的窠臼！！雖說總覺得，這句話

『音樂』是有聲的『數學』，『數學』是無聲的『音樂』。

是愛因斯坦講的，卻也不得不用『有人說』︰

錦帶橋的清晨

在《【Sonic π】聲波之傳播原理︰原理篇《四下》》一文中我們談到了『駐波』的形成，之後又說明了『樂器』製造通常需要『共振腔』的放大。事實上『調和序列』 $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \cdots, \frac{1}{n}, \cdots$ ，與『調和級數』 $\sum \limits_{k=1}^{k=n} \frac{1}{k}$ 的『調和』之名源於『樂理』中的『泛音』 overtone 和『泛音列』 harmonic series。一首『樂曲』從『數學』上講，就是一個個『音符』構成的頻率序列，在『時序』上『和諧』的展開，因此才有人說︰『音樂』是有聲的『數學』，『數學』是無聲的『音樂』的啊！我們如果自『遞迴關係式』所比之擬之『相空間』的『符號動力學』來看，或許它該說是『物理』中的『音符動力學』的吧！

即使由『調和平均數』一詞來說，假使 $t_k = \frac{1}{k}$ 是一個『調和序列』的第 $k$ 項，那麼任意兩項的『調和平均數』 $h$ 就是 $h = \frac{2}{\frac{1}{t_i} + \frac{1}{t_j}}$ ，所以 ${h} = \frac{2}{i + j}$ 。於是『調和序列』中任意連續的三個數 $t_{h-1}, t_h, t_{h+1)$ 『全部』都構成了『調和平均數』之『關係』 ${2h} = (h-1) + (h+1)$ 。因此這就是『為什麼』那個『調和序列』會被稱之為『調和』的了。現實裡，大概也只有大自然中，那不期而遇的『山光水影』之『漸層』的『美』，方能夠與之『匹配』的吧！！

引自《【Sonic π】電路學之補充《四》無窮小算術‧中下中‧下》

來遮掩褪色的記憶，終究無法於浩瀚書海中撈出一句話的耶？！

然而不管『抽象』以及『具體』都能觸動人『心』，引發不同的『視野』觀察宇宙人生之萬象。因此偶然的想到『振幅調變』

Amplitude modulation

Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. In amplitude modulation, the amplitude (signal strength) of the carrier wave is varied in proportion to the waveform being transmitted. That waveform may, for instance, correspond to the sounds to be reproduced by a loudspeaker, or the light intensity of television pixels. This technique contrasts with frequency modulation, in which the frequency of the carrier signal is varied, and phase modulation, in which its phase is varied.

AM was the earliest modulation method used to transmit voice by radio. It was developed during the first two decades of the 20th century beginning with Roberto Landell De Moura and Reginald Fessenden‘s radiotelephone experiments in 1900.^[1] It remains in use today in many forms of communication; for example it is used in portable two way radios, VHF aircraft radio, Citizen’s Band Radio and in computer modems.^{[citation needed]} “AM” is often used to refer to mediumwave AM radio broadcasting.

Fig 1: An audio signal (top) may be carried by a carrier frequency using AM or FM methods.

Simplified analysis of standard AM

Consider a carrier wave (sine wave) of frequency f_c and amplitude A given by:

c(t) = A\cdot \sin(2 \pi f_c t)\,

Let m(t) represent the modulation waveform. For this example we shall take the modulation to be simply a sine wave of a frequency f_m, a much lower frequency (such as an audio frequency) than f_c:

m(t) = M\cdot \cos(2 \pi f_m t + \phi)\,

where M is the amplitude of the modulation. We shall insist that M<1 so that (1+m(t)) is always positive. If M>1 then overmodulation occurs and reconstruction of message signal from the transmitted signal would lead in loss of original signal. Amplitude modulation results when the carrier c(t) is multiplied by the positive quantity (1+m(t)):

$y(t)\,$	$= [1 + m(t)]\cdot c(t) \,$
	$= [1 + M\cdot \cos(2 \pi f_m t + \phi)] \cdot A \cdot \sin(2 \pi f_c t)$

In this simple case M is identical to the modulation index, discussed below. With M=0.5 the amplitude modulated signal y(t) thus corresponds to the top graph (labelled “50% Modulation”) in Figure 4.

Using prosthaphaeresis identities, y(t) can be shown to be the sum of three sine waves:

y(t) = A\cdot \sin(2 \pi f_c t) + \begin{matrix}\frac{AM}{2} \end{matrix} \left[\sin(2 \pi (f_c + f_m) t + \phi) + \sin(2 \pi (f_c - f_m) t - \phi)\right].\,

Therefore, the modulated signal has three components: the carrier wave c(t) which is unchanged, and two pure sine waves (known as sidebands) with frequencies slightly above and below the carrier frequency f_c.

Illustration of Amplitude Modulation

───

難道『概念』上不就是種『包絡產生器』嗎？？！！突然過去組裝收音機的熱情湧上心頭。心想那時哪有這時的電腦，即便說樹莓派的吧！可以用『數位』的方式來處理『類比』訊號？何不試試看的呢！！？？看看『 Pd 』⊕ 樹莓派『小盒子』到底是行還是不行的耶？！

於是本著 Pd 程式例子的『慣例』頻率

A440

A440是440赫茲的聲音音調，西方音樂上，此音為標準音高。西方樂理中，A440乃是中央C上方的A音符（參照A4）。

1939年，一個國際會議提出，把中央C上方的A定為440赫茲。到了1955年，國際標準化組織採用了這個標準，將其訂為ISO 16（1975年，此組織重新確認了這項標準）。從此，A440就成為鋼琴、小提琴以及其他樂器的頻率校準標準。

在過去，鋼琴調音就是使用標準音高，經由人耳聆聽比較來調整鋼琴的音準。調音師感受鋼琴上某一鍵所發出的聲響並與標準音高參照，然後調整鋼琴的音高（藉由調整琴弦的鬆緊）直到與標準音高相合。

一個調音師能分辨的最小的頻率偏差與受到很多因素影響，包括音量、音的長短、頻率變換的速度以及調音師的訓練程度。然而，最小可覺差常常被定義為五音分。（分的定義為：欲調整的音與相鄰兩音的頻率差的1/100）

現在的調音師多有藉助於調音器。調音器為一電子儀器，當接收到聲波震動時，會顯示其音高，調音師再依此調整鋼琴音高。

在電影Phantasm中，The Tall Man對於此頻率感到不適。

───

為主，寫了一個簡單的補丁

觀察了頻譜

玩玩弄弄中度過了一段快樂的時光。

FreeSandal

每月彙整: 2015 年 11 月

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧摺積

Convolution

摺積

簡單介紹

The convolution theorem and its applications

What is a convolution?

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧周期

Definition

Properties

Double-periodic functions

Quotient spaces as domain

What is WordNet?

About WordNet

Structure

Relations

Cross-POS relations

More Information

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧留白

Formal definition

Distance formula

Two dimensions

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧九點圓

九點圓

歷史

九點圓證明

解析幾何

三角形的中心

三條中線共點證明

塞瓦定理

證明

幾何中心

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧調幅

Amplitude modulation

Simplified analysis of standard AM

A440

輕。鬆。學。部落客

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