WordNet

the 23rd letter of the Roman alphabet (同)w, double-u
give numbers to; "You should number the pages of the thesis"
place a limit on the number of (同)keep_down
a concept of quantity involving zero and units; "every number has a unique position in the sequence"
the property possessed by a sum or total or indefinite quantity of units or individuals; "he had a number of chores to do"; "the number of parameters is small"; "the figure was about a thousand" (同)figure
a clothing measurement; "a number 13 shoe"
an item of merchandise offered for sale; "she preferred the black nylon number"; "this sweater is an all-wool number"
a numeral or string of numerals that is used for identification; "she refused to give them her Social Security number" (同)identification number
a select company of people; "I hope to become one of their number before I die"
the grammatical category for the forms of nouns and pronouns and verbs that are used depending on the number of entities involved (singular or dual or plural); "in English the subject and the verb must agree in number"
enumerate; "We must number the names of the great mathematicians" (同)list
so frightened as to be unable to move; stunned or paralyzed with terror; petrified; "too numb with fear to move"
make numb or insensitive; "The shock numbed her senses" (同)benumb, blunt, dull

PrepTutorEJDIC

Watt / West; Western
wolfram(=tungsten)の原子記号
〈U〉〈C〉(数えて得られる)『数,数量』 / 〈C〉(概念としての)『数,数字』 / 〈C〉『番号』 / 〈C〉(演奏会や演劇の)番組,出し物;曲目 / 〈C〉(雑誌の)号 / 〈U〉(文法で)数(すう) / 《複数形で》数の上の優勢 / 《複数形で》算数 / 〈C〉《単数形で》《話》(商品としての)洋服の1点;商品,売り物 / 〈C〉《単随形で》《俗》女の子 / …‘を'数える / (…の中に,…として)…‘を'含める,加える《+『among』(『with, as』)+『名』》 / …‘に'番号をつける / …‘の'数となる / 《しばしば受動態で》…‘の'数を制限する / 総計(…に)なる《+『in』+『名』〈数〉》
麻痺(まひ)した,感覚を失った,しびれた / …‘の'感覚を失わせる,‘を'しびれさせる
数学,数学 / (人員などの)数の優勢 / 《米》《the~》=numbers game / 《古》詞句,韻文

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/05/04 17:42:14」(JST)

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In knot theory, there are several competing notions of the quantity writhe, or Wr. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of "coiling" of a mathematical knot (or any closed, simple curve) in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve (or diagram) in such a way that does not change its topology, one may still change its writhe.^[1]

Writhe of link diagrams

In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings.

A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. If as you travel along a link component and cross over a crossing, the strand underneath goes from right to left, the crossing is positive; if the lower strand goes from left to right, the crossing is negative. One way of remembering this is to use a variation of the right-hand rule.


Positive crossing		Negative crossing

For a knot diagram, using the right-hand rule with either orientation gives the same result, so the writhe is well-defined on unoriented knot diagrams.

A Type I Reidemeister move changes the writhe by 1

The writhe of a knot is unaffected by two of the three Reidemeister moves: moves of Type II and Type III do not affect the writhe. Reidemeister move Type I, however, increases or decreases the writhe by 1. This implies that the writhe of a knot is not an isotopy invariant of the knot itself — only the diagram. By a series of Type I moves one can set the writhe of a diagram for a given knot to be any integer at all.

Writhe of a closed curve

Writhe is also a property of a knot represented as a curve in three-dimensional space. Strictly speaking, a knot is such a curve, defined mathematically as an embedding of a circle in three-dimensional Euclidean space, R³. By viewing the curve from different vantage points, one can obtain different projections and draw the corresponding knot diagrams. Its Wr (in the space curve sense) is equal to the average of the integral writhe values obtained from the projections from all vantage points.^[2] Hence, writhe in this situation can take on any real number as a possible value.^[1]

We can calculate Wr with an integral. Let be a smooth, simple, closed curve and let and be points on . Then the writhe is equal to

Wr=\frac{1}{4\pi}\int_{C}\int_{C}d\mathbf{r}_{1}\times d\mathbf{r}_{2}\cdot\frac{\mathbf{r}_{1}-\mathbf{r}_{2}}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|^{3}}

.^[3]

Numerically approximating the Gauss integral for writhe of a curve in space

Since writhe for a curve in space is defined as a double integral, we can approximate its value numerically by first representing our curve as a finite chain of line segments. We then calculate^[3]

Wr=\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\Omega_{ij}}{4\pi}=2\sum_{i=2}^{N}\sum_{j<i}^{N}\frac{\Omega_{ij}}{4\pi}

where is the exact evaluation of the double integral over line segments and ; note that and .^[3]

To evaluate for given segments numbered and , number the endpoints of the two segments 1, 2, 3, and 4. Let be the vector that begins at endpoint and ends at endpoint . Define the following quantities:^[3]

n_{1}=\frac{r_{13}\times r_{14}}{\left|r_{13}\times r_{14}\right|},\; n_{2}=\frac{r_{14}\times r_{24}}{\left|r_{14}\times r_{24}\right|},\; n_{3}=\frac{r_{24}\times r_{23}}{\left|r_{24}\times r_{23}\right|},\; n_{4}=\frac{r_{23}\times r_{13}}{\left|r_{23}\times r_{13}\right|}

Then we calculate^[3]

\Omega^{*}=\arcsin\left(n_{1}n_{2}\right)+\arcsin\left(n_{2}n_{3}\right)+\arcsin\left(n_{3}n_{4}\right)+\arcsin\left(n_{4}n_{1}\right).

Finally, we compensate for the possible sign difference and divide by to obtain^[3]

\frac{\Omega}{4\pi}=\frac{\Omega^{*}}{4\pi}\text{sign}\left(\left(r_{34}\times r_{12}\right)\cdot r_{13}\right).

This method, as well as other methods, are fully described mathematically and algorithmically in.^[3]

Applications in DNA topology

A simulation of an elastic rod relieving torsional stress by forming coils

DNA will coil if you twist it, just like a rubber hose or a rope will, and that is why biomathematicians use the quantity of writhe to describe the amount a piece of DNA is deformed as a result of this torsional stress. In general, this phenomenon of forming coils due to writhe is referred to as DNA supercoiling and is quite commonplace, and in fact in most organisms DNA is negatively supercoiled.^[1]

Any elastic rod, not just DNA, relieves torsional stress by coiling, an action which simultaneously untwists and bends the rod. F. Brock Fuller shows mathematically in ^[4] how the “elastic energy due to local twisting of the rod may be reduced if the central curve of the rod forms coils that increase its writhing number” .

References

^ ^a ^b ^c Bates, Andrew (2005). DNA Topology. Oxford University Press. pp. 36–37. ISBN 0198506554.
^ Cimasoni, David (2001). "Computing the Writhe of a Knot". Journal of Knot Theory and Its Ramifications 10 (387). doi:10.1142/S0218216501000913.
^ ^a ^b ^c ^d ^e ^f ^g Klenin, K; Langowski, J (2000). "Computation of writhe in modeling of supercoiled DNA". Biopolymers (54): 307–317. Cite uses deprecated parameters (help)
^ Fuller, F B (1971). "The writhing number of a space curve". Proceedings of the National Academy of Sciences 68: 815–819.

Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 0-8218-3678-1

UpToDate Contents

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Antinociceptive effects of systemic tanshinone IIA on visceral and somatic persistent nociception and pain hypersensitivity in rats.

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Japanese Journal

Tipepidine enhances the antinociceptive-like action of carbamazepine in the acetic acid writhing test

Kazuaki Kawaura,Risa Miki,Yuri Urashima,Sokichi Honda,Ahmed M. Shehata,Fumio Soeda,Tetsuya Shirasaki,Kazuo Takahama,カワウラカズアキ,ミキリサ,ウラシマユリ,ホンダソウキチ,ソエダフミオ,シラサキテツヤ,タカハマカズオ,川浦一晃,三木理沙,浦嶋優里,本田宗吉,副田二三夫,白崎哲哉,高濱和夫
European Journal of Pharmacology 651(1-3), 106-108, 2011-01-25
… In the present study, we investigated whether a combination oftipepidine and carbamazepine acts synergistically to induce an antinociceptive effect inthe acetic acid-induced writhing test in mice. … significantly reduced the number ofwrithes induced by acetic acid in mice. … alsosignificantly reduced the writhing reaction. …
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Design, Synthesis, and in-Vivo Evaluation of 4,5-Diaryloxazole as Novel Nonsteroidal Anti-inflammatory Drug

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Biological & pharmaceutical bulletin 31(5), 1017-1020, 2008-05-01
… In the acetic acid-induced writhing test, we observed that (-)-carvone-treated mice exhibited a significant decrease in the number of writhes when 100 and 200 mg/kg was administered. …
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