text: It also shows that the diﬀerence in the number of followers where T is the number of days for which we monitor the and friends is large and the purchased follower accounts do purchased follower and ucount is the number of users he i not gain a lot of followers themselves. To investigate this unfollowed on ith day. A higher value of unfollow entropy further, we look at the Follower-Friends ratio of these ac- signiﬁes that the user exhibits a suspicious unfollow pattern. counts. Figure 8 shows that a large fraction of purchased followers have a high unfollow entropy. The normalized entropy rate for 23% purchased followers is as high as 0.76 and only 8% users have a normalized unfollow entropy less than 0.21. To ﬁnd out whether the users with higher unfollow entropy have lower quality than other users, we compared their normal- ized unfollow entropy rate with Klout score. We found a strong negative correlation (Pearson correlation coeﬃcient = -0.73) indicating that users with higher unfollow entropy rate have low social reputation. Figure 6: Number of Followers vs Friends of pur- chased follower accounts We observe in Figure 7 that the follower/friends ratio ﬁts the power law (α = 1.8209, error σ = 0.029). We observe that 94% purchased followers have the follower/friends ratio as only 0.1 and none of the purchased followers had more followers than friends. This further strengthens our obser- vation that purchased accounts have a low follower count. Figure 8: Unfollow Entropy Rate for Purchased Fol- lowers 4.4 Social Engagement with Friends In this section, we explore how the purchased follower ac- counts are connected with their friends. We measure social engagement of users with their friends in form of retweets, @-mentions and favorite count. We also ﬁnd out language overlap patterns between the users and his friends. Figure 7: Follower-Followee ratio of purchased fol- 4.4.1 RTs and @-mentions lower accounts We observed in section that a large fraction of purchased accounts post only retweets instead of original content. We 4.3.2 Unfollow Entropy further explore whether these users retweet the content of We found that the purchased follower unfollowed a large their friends or not. If RTcounti is the number of tweets the number of users regularly. To quantify this behaviour, we user has retweeted of his friend ui and he has N friends, calculated the unfollow entropy of all the purchased follow- then we deﬁne ers. We observed each purchased follower over a span of 30 RTcounti days and collected his hourly followers. We deﬁne normal- RetweetRatio = ΣNi=∗1RTcounti ized unfollower entropy H for a user un as the following N RTtotal Hun = −ΣTi=1pn(fiN)log(pn(fi)) wusheerr.e TRhTitsotRaletiwsetehteRtaottiaol qnuuamnbtieﬁresofthreetwneuemtsbedronoef fbryienthdes a user has retweeted and the number of times he retweeted where, p (f ) is the probability that the user u will unfollow n i n them. at time t . The probability function is deﬁned as i ucount Similarly we deﬁne the @-mention ratio to determine whether p (f ) = i n i ΣT ucount the user engages in conversations with his friends and to i=1 i ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: the following metric features, we repeat the classiﬁcation experiment by incre- mentally adding each feature set. For evaluation, we used ΣN #spam words spam words/tweet = tweet=0 70-30 split of the training and the testing dataset. We use #tweets 10 fold cross validation to report our results. Time since last tweet:. We found that purchased follow- 5.3 Classiﬁcation Results and Evaluation ers exhibiting suspicious following behaviour have very less Table 7 shows the confusion matrix for our classiﬁcation tweeting activity and a large fraction of such users are inac- task. The confusion matrix deﬁnes the percentage of false tive. To measure time since the account has been inactive, negatives and false positives. We were able to accurately we ﬁnd the diﬀerence in time in seconds since the latest classify 82.5% users with suspicious follow behaviour and tweet with the time of our experiment. 88.3% users with legitimate behaviour. This shows that we are able to detect suspicious following behaviour to a good These are the discriminative features we use to distinguish extent. However, for the evaluation of our classiﬁcation re- between regular and suspicious following behaviour. With sult, we used the standard evaluation metrics in this clas- the help of these features, we detect users with suspicious siﬁcation task – accuracy, F-measure and Area under the follow behaviour in the following section. Curve (AUC). 5.2 Experimental Setup and Classiﬁcation For our classiﬁcation experiment, we consider the 11,760 Table 7: Confusion Matrix – Classiﬁcation Results public purchased followers as our true positive dataset of Predicted suspicious follow behaviour. For the negative class (legiti- Suspicious Legitimate mate follow behaviour), we pick random 11,760 users from Suspicious 82.5 17.5 Twitter stream using the streaming API. However, a bal- True Legitimate 11.7 88.3 anced dataset as ours may create a sample bias. Therefore, to ensure valid results and eliminate the bias, we under- sample our negative class. We draw 10 random but inde- pendent subsets from the set of 11,760 legitimate users (-ve As discussed in the previous section, we incrementally added class) and train 10 classiﬁer models based on these 10 sub- feature sets to evaluate the eﬀectiveness of all the features. sets along with the 11,760 samples of the suspicious follow Figure 13 shows the performance of our classiﬁer on Accu- behaviour users (+ve class). We then use 10 fold cross vali- racy, F1 score and AUC metrics when feature sets are incre- dation and report the average results for our prediction task. mentally added. We see that each feature set has a positive eﬀect on the performance of the classiﬁer across all metrics. We treat the detection of suspicious follow behaviour as a We also observed that adding behavioural based features two class classiﬁcation problem. In order to detect such be- suddenly increase the overall accuracy of our classiﬁcation haviour, we use several supervised learning algorithms like model. We received a maximum accuracy of 88.2%. Naive Bayes, Gradient Decent, Random Forest etc. How- ever, we achieved highest accuracy and overall best results with Support Vector Machine (SVM). The goal of a SVM is to ﬁnd the hyperplane that optimally separates the training data into two portions of an N-dimensional space where N is the total number of features used. A SVM performs clas- siﬁcation by mapping input vectors into an N-dimensional space, and checking in which side of the deﬁned hyperplane the point lies. We use a non-linear SVM with the Radial Ba- sis Function (RBF) kernel for our experiment. Table 6 gives the details of our experimental setup - dataset description and the parameter values for the SVM classiﬁcation algo- rithm. Table 6: Description of the experimental setup Dataset 23,520 ‘Suspicious’ (+ve class) 11,760 Figure 13: Classiﬁcation accuracy to predict sus- ‘Legitimate’ (-ve class) 11,760 (10 times) picious following behaviour on incremental feature Classiﬁer SVM addition C 1,000 alpha 20.0 5.4 Feature Importance Classiﬁcation Runs 10 { } { } { } { } Feature Sets A , A, B , A, B, C , A, B, C, D In this section, we look at the importance of features used Train-Test Split 70%-30% for suspicious follow behaviour detection. We found that be- Cross Validation 10-fold havioural features play an important role in detecting suspi- cious behaviour. Unfollow entropy rate plays an important To reduce the error margin, we use a large C value for the role; it is deﬁned as the frequency with which the user is un- RBF kernel of SVM. In order to assess the eﬀectiveness of following his friends over time. Some of the most informative ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: reduce

text: the

text: error

text: margin,

text: we

text: use

text: a

text: large

text: C

text: value

text: for

text: the

text: role;

text: it

text: is

text: deﬁned

text: as

text: the

text: frequency

text: with

text: which

text: the

text: user

text: is

text: un-

text: RBF

text: kernel

text: of

text: SVM.

text: In

text: order

text: to

text: assess

text: the

text: eﬀectiveness

text: of

text: following

text: his

text: friends

text: over

text: time.

text: Some

text: of

text: the

text: most

text: informative

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: VELOCITY OF RATCHETS 11 We conclude that under the hypothesis of Corollary 2.5 the limiting average drift velocity is the same for all solutions of (2.1) and is deter- mined by the periodic equation (2.7). For this reason in what follows we mostly concentrate on equation (2.7). An important class of unbiased potentials are the tilting potentials, which have the form (2.21) Ψ(x, t) = ψ(x) + H(t)x, where the base potential ψ(x) is 1-periodic in x, and H(t) is T - (cid:90) periodic in t and characterized by the property T (2.22) H(t) dt = 0. 0 Given the periodicity of ψ, equation (2.22) is equivalent to (2.3). We obtain a typical tilting potential by letting (2.23) H(t) = h(t)ω, ∈ R (cid:40) where ω characterizes the swing of the tilt, and ≤ 1 if kT < t (k + 1/2)T, (2.24) h(t) = − ≤ (k = 0, 1, . . . ). 1 if (k + 1/2)T < t (k + 1)T, The tilting potential corresponding to (2.24) periodically switches ± between the tilted potentials ψ(x) ωx. Thus the tilted potential (2.25) ψ(x) + ωx is a useful example of an obviously ‘biased’ potential (here ψ satisﬁes the same conditions as above and ω is a number). (cid:90) (cid:90) Remark 2.6. Observe that if Ψ is a tilting ratchet potential (2.21), then − 1 t0+T 1 (2.26) v = ψ (x)g(x, t) dx dt. [t0,t0+T] T x t0 0 ##LTLine##

text: VELOCITY

text: OF

text: RATCHETS

text: We

text: conclude

text: that

text: under

text: the

text: hypothesis

text: of

text: Corollary

text: the

text: limiting

text: average

text: drift

text: velocity

text: is

text: the

text: same

text: for

text: all

text: solutions

text: of

text: and

text: is

text: deter-

text: mined

text: by

text: the

text: periodic

text: equation

text: For

text: this

text: reason

text: in

text: what

text: follows

text: we

text: mostly

text: concentrate

text: on

text: equation

text: An

text: important

text: class

text: of

text: unbiased

text: potentials

text: are

text: the

text: tilting

text: potentials,

text: which

text: have

text: the

text: form

label: equation

text: Ψ(x,

text: t)

text: ψ(x)

text: H(t)x,

text: where

text: the

text: base

text: potential

text: ψ(x)

text: is

text: in

text: x,

text: and

text: H(t)

text: is

text: T

text: periodic

text: in

text: t

text: and

text: characterized

text: by

text: the

text: property

text: T

text: H(t)

text: dt

text: Given

text: the

text: periodicity

text: of

text: ψ,

text: equation

text: is

text: equivalent

text: to

text: We

text: obtain

text: a

text: typical

text: tilting

text: potential

text: by

text: letting

text: H(t)

text: h(t)ω,

text: R

text: where

text: ω

text: characterizes

text: the

text: swing

text: of

text: the

text: tilt,

text: and

text: if

text: kT

text: t

text: (k

text: h(t)

text: (k

text: if

text: (k

text: t

text: (k

text: The

text: tilting

text: potential

text: corresponding

text: to

text: periodically

text: switches

text: between

text: the

text: tilted

text: potentials

text: ψ(x)

text: ωx.

text: Thus

text: the

text: tilted

text: potential

text: ψ(x)

text: ωx

text: is

text: a

text: useful

text: example

text: of

text: an

text: obviously

text: ‘biased’

text: potential

text: (here

text: ψ

text: satisﬁes

text: the

text: same

text: conditions

text: as

text: above

text: and

text: ω

text: is

text: a

text: number).

text: Remark

text: Observe

text: that

text: if

text: Ψ

text: is

text: a

text: tilting

text: ratchet

text: potential

text: then

text: v

text: ψ

text: (x)g(x,

text: t)

text: dx

text: dt.

text: T

text: x

text: ##LTLine##


text: 16 S. KONDRATYEV, J.M. URBANO, AND D. VOROTNIKOV ∈ ∈ Lemma 3.2. For any probability density g L1(S1) such that g ln g L1(S1), we have | − | ≤ (3.10) Σ [g] Σ [g−] φ (Σ−[g]), + + + | − | ≤ (3.11) Σ−[g] Σ−[g ] φ−(Σ [g]). + + (cid:90) (cid:90) Proof. A straightforward computation yields − g (cid:90)− (cid:18) g− (cid:19) Σ [g] Σ [g−] = g ln dx g− ln dx + + g g S1 + S1 + g − g− = g ln + (g g−) ln dx, g− g S1 + whence | − | ≤ M√ (cid:107) − (cid:107) Σ [g] Σ [g−] Σ−[g] + + g g− . + + L1(S1) 2 Now it remains to apply the Csisza´r–Kullback inequality (2.12) and obtain (3.10). (cid:3) Inequality (3.11) is proved by swapping g and g−. + Note that the periodic solution g∞ of (3.2) implicitly depends on T , which is a parameter of tilting. Lemma 3.3. We have · · (3.12) lim Σ [g∞( , 0)] = Σ [g−]; lim Σ−[g∞( , T/2)] = Σ−[g ]. →∞ + + →∞ + T T Proof. We only prove the ﬁrst limit in (3.12), as the proof of the second one is completely analogous. Take an arbitrary R > 0 and for any ε > 0 choose T > 0 in such a ε way that − ≤ (Σ [g−] + ε)e γ+Tε/2 R, + − ≤ − (Σ−[g ] + φ−(R))e γ−Tε/2 φ 1(ε). + + T Now take T > T and let be the resolving operator for (3.2) taking · · ε T g( , 0) to g( , T ). We claim that maps the set { ∈ ∩ { } | | − | ≤ } X = g W 1(S1) probability densities Σ [g ] Σ [g−] ε 0 2 + 0 + ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: KONDRATYEV,

text: J.M.

text: URBANO,

text: AND

text: D.

text: VOROTNIKOV

text: Lemma

text: For

text: any

text: probability

text: density

text: g

text: such

text: that

text: g

text: ln

text: g

text: we

text: have

text: Σ

text: [g]

text: Σ

text: [g−]

text: φ

text: (Σ−[g]),

text: Σ−[g]

text: Σ−[g

text: φ−(Σ

text: [g]).

text: Proof.

text: A

text: straightforward

text: computation

text: yields

text: g

text: g−

text: Σ

text: [g]

text: Σ

text: [g−]

text: g

text: ln

text: dx

text: g−

text: ln

text: dx

text: g

text: g

text: g

text: g−

text: g

text: ln

text: (g

text: g−)

text: ln

text: dx,

text: g−

text: g

text: whence

text: M√

text: Σ

text: [g]

text: Σ

text: [g−]

text: Σ−[g]

text: g

text: g−

text: Now

text: it

text: remains

text: to

text: apply

text: the

text: Csisza´r–Kullback

text: inequality

text: and

text: obtain

text: Inequality

text: is

text: proved

text: by

text: swapping

text: g

text: and

text: g−.

text: Note

text: that

text: the

text: periodic

text: solution

text: g∞

text: of

text: implicitly

text: depends

text: on

text: T

text: which

text: is

text: a

text: parameter

text: of

text: tilting.

text: Lemma

text: We

text: have

text: lim

text: Σ

text: [g∞(

text: Σ

text: [g−];

text: lim

text: Σ−[g∞(

text: Σ−[g

text: T

text: T

text: Proof.

text: We

text: only

text: prove

text: the

text: ﬁrst

text: limit

text: in

text: as

text: the

text: proof

text: of

text: the

text: second

text: one

text: is

text: completely

text: analogous.

text: Take

text: an

text: arbitrary

text: R

text: and

text: for

text: any

text: ε

text: choose

text: T

text: in

text: such

text: a

text: ε

text: way

text: that

text: (Σ

text: [g−]

text: ε)e

text: R,

text: (Σ−[g

text: φ−(R))e

text: γ−Tε/2

text: φ

text: T

text: Now

text: take

text: T

text: T

text: and

text: let

text: be

text: the

text: resolving

text: operator

text: for

text: taking

text: ε

text: T

text: g(

text: to

text: g(

text: T

text: We

text: claim

text: that

text: maps

text: the

text: set

text: X

text: g

text: W

text: probability

text: densities

text: Σ

text: [g

text: Σ

text: [g−]

text: ε

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 24 S. KONDRATYEV, J.M. URBANO, AND D. VOROTNIKOV 4. Semiadiabatic limit and the stochastic Stokes’ drift Consider the tilting ratchet with the potential (4.1) Ψ(x, t) = ψ(x) + H(t)x, (cid:40) where Ω, 0 < t < τ, (4.2) H(t) = ω, τ < t < T, and as usual ψ(x) is 1-periodic in x and H(t) is T -periodic in T . We assume that the non-bias condition (2.22) is satisﬁed, i.e., − (4.3) Ωτ + ω(T τ) = 0. We regard T , τ, and ω as the independent parameters of the tilting ratchet; Ω can be expressed via the independent parameters by means of (4.3). We study the regime (4.1) in the so-called semiadiabatic limit sup- posing that T is large, τ/T is small, and ω is constant. It follows | | from (4.3) that in the semiadiabatic limit Ω is large. As before, let v∞ = v∞(ω, T, τ) denote the eventual drift velocity ∗ (cid:90) given by (2.19). Further, let g be the 1-periodic solution of the problem 1 (4.4) g + ((ψ + ω)g) = 0, g(x) dx = 1, xx x x 0 which also satisﬁes ∗ ∗ g + (ψ + ω)g = A(ω) x x (cid:90) with A(ω) deﬁned by (2.29) (see Remark 2.10). Put − 1 ∗ ≡ − (4.5) v∞∞(ω) = ψ g dx ω A(ω). x 0 Our main result concerning the semiadiabatic tilting is that v∞∞(ω) → ∞ → is the asymptotic average drift velocity as T and τ/T 0, and its sign coincides with the sign of ω provided that ψ is nontrivial. For deﬁnitiveness we assume that ω > 0.

text: S.

text: KONDRATYEV,

text: J.M.

text: URBANO,

text: AND

text: D.

text: VOROTNIKOV

text: Semiadiabatic

text: limit

text: and

text: the

text: stochastic

text: Stokes’

text: drift

text: Consider

text: the

text: tilting

text: ratchet

text: with

text: the

text: potential

text: Ψ(x,

text: t)

text: ψ(x)

text: H(t)x,

text: where

text: Ω,

text: t

text: τ,

text: H(t)

text: ω,

text: τ

text: t

text: T,

text: and

text: as

text: usual

text: ψ(x)

text: is

text: in

text: x

text: and

text: H(t)

text: is

text: T

text: -periodic

text: in

text: T

text: We

text: assume

text: that

text: the

text: non-bias

text: condition

text: is

text: satisﬁed,

text: i.e.,

text: Ωτ

text: ω(T

text: τ)

text: We

text: regard

text: T

text: τ,

text: and

text: ω

text: as

text: the

text: independent

text: parameters

text: of

text: the

text: tilting

text: ratchet;

text: Ω

text: can

text: be

text: expressed

text: via

text: the

text: independent

text: parameters

text: by

text: means

text: of

text: We

text: study

text: the

text: regime

text: in

text: the

text: so-called

text: semiadiabatic

text: limit

text: sup-

text: posing

text: that

text: T

text: is

text: large,

text: τ/T

text: is

text: small,

text: and

text: ω

text: is

text: constant.

text: It

text: follows

text: from

text: that

text: in

text: the

text: semiadiabatic

text: limit

text: Ω

text: is

text: large.

text: As

text: before,

text: let

text: v∞

text: v∞(ω,

text: T,

text: τ)

text: denote

text: the

text: eventual

text: drift

text: velocity

text: given

text: by

text: Further,

text: let

text: g

text: be

text: the

text: solution

text: of

text: the

text: problem

text: g

text: ((ψ

text: ω)g)

text: g(x)

text: dx

text: xx

text: x

text: x

text: which

text: also

text: satisﬁes

text: g

text: (ψ

text: ω)g

text: A(ω)

text: x

text: x

text: with

text: A(ω)

text: deﬁned

text: by

text: (see

text: Remark

text: Put

text: v∞∞(ω)

text: ψ

text: g

text: dx

text: ω

text: A(ω).

text: x

text: Our

text: main

text: result

text: concerning

text: the

text: semiadiabatic

text: tilting

text: is

text: that

text: v∞∞(ω)

text: is

text: the

text: asymptotic

text: average

text: drift

text: velocity

text: as

text: T

text: and

text: τ/T

text: and

text: its

text: sign

text: coincides

text: with

text: the

text: sign

text: of

text: ω

text: provided

text: that

text: ψ

text: is

text: nontrivial.

text: For

text: deﬁnitiveness

text: we

text: assume

text: that

text: ω


text: VELOCITY OF RATCHETS 33 in the sense of distributions. The second equality is obvious if z∞ is smooth, and the general case follows from the dominated convergence theorem. (cid:90) (cid:90) But (5.5) and (5.7) yield − 1 T 1 (cid:82) (5.8) v∞ = [z∞(s(x, t))] dx dt 0 t T 0 0 − 1(z∞(x) z∞(s(x, T )) dx = 0 . T Solving (5.4) by the method of characteristics we infer that x = y(T ), where y(t) is the solution of (5.1) with the initial condition y(0) = s(x, T ). Since S1 is compact and (5.1) has no T -periodic solutions, the diﬀerence − (cid:12) d(y) = y(T ) y(0)(cid:12) is separated from zero, and its sign does not change for all solutions of (cid:54) (5.1). Consequently, v∞ = 0 (otherwise z∞ would be a constant). 0 [0,1] We conclude that v∞ and v∞ (for large n) have the same sign as d(y), 0 n (cid:3) reaching a contradiction. Remark 5.2. Observe that (5.1) has no T -periodic solutions if and only if there exists an unbounded trajectory or, in other words, if and only if the corresponding Poincar´e’s rotation number [10] is nonzero. Hence, the sign of the rotation number coincides with the sign of v∞ . A 0 related observation was made in [34] for diﬀusion-free tilting ratchets. Our conjecture is that lim → v∞ = r/T , where r is the rotation number σ 0 (cf. [34]). Remark 5.3. Existence of periodic orbits is unstable with respect to perturbations of F , thus, in a general position, v∞ is non-zero for small σ. Unfortunately, as R. Ortega and F. Zanolin pointed out in a per- sonal communication, there are no criteria of the form that ‘some set of assumptions on F implies non-existence of T -periodic solutions to (5.1)’. In [25], the non-existence of periodic solutions to (5.1) with a − tilting potential of the form F (y, t) = H(t) sin y, where H is T - sin periodic and satisﬁes (2.22), was discovered. There, it was observed that this phenomenon contrasts with the behaviour of the equation (cid:48)(cid:48) (5.9) y (t) = F (y(t), t), sin ##LTLine## ##LTLine##

text: VELOCITY

text: OF

text: RATCHETS

text: in

text: the

text: sense

text: of

text: distributions.

text: The

text: second

text: equality

text: is

text: obvious

text: if

text: z∞

text: is

text: smooth,

text: and

text: the

text: general

text: case

text: follows

text: from

text: the

text: dominated

text: convergence

text: theorem.

text: But

text: and

text: yield

text: T

text: v∞

text: [z∞(s(x,

text: t))]

text: dx

text: dt

text: t

text: T

text: z∞(s(x,

text: T

text: dx

text: T

text: Solving

text: by

text: the

text: method

text: of

text: characteristics

text: we

text: infer

text: that

text: x

text: y(T

text: where

text: y(t)

text: is

text: the

text: solution

text: of

text: with

text: the

text: initial

text: condition

text: s(x,

text: T

text: Since

text: is

text: compact

text: and

text: has

text: no

text: T

text: -periodic

text: solutions,

text: the

text: diﬀerence

text: d(y)

text: y(T

text: is

text: separated

text: from

text: zero,

text: and

text: its

text: sign

text: does

text: not

text: change

text: for

text: all

text: solutions

text: of

text: Consequently,

text: v∞

text: (otherwise

text: z∞

text: would

text: be

text: a

text: constant).

text: We

text: conclude

text: that

text: v∞

text: and

text: v∞

text: (for

text: large

text: n)

text: have

text: the

text: same

text: sign

text: as

text: d(y),

text: n

text: reaching

text: a

text: contradiction.

text: Remark

text: Observe

text: that

text: has

text: no

text: T

text: -periodic

text: solutions

text: if

text: and

text: only

text: if

text: there

text: exists

text: an

text: unbounded

text: trajectory

text: or,

text: in

text: other

text: words,

text: if

text: and

text: only

text: if

text: the

text: corresponding

text: Poincar´e’s

text: rotation

text: number

text: is

text: nonzero.

text: Hence,

text: the

text: sign

text: of

text: the

text: rotation

text: number

text: coincides

text: with

text: the

text: sign

text: of

text: v∞

text: A

text: related

text: observation

text: was

text: made

text: in

text: for

text: diﬀusion-free

text: tilting

text: ratchets.

text: Our

text: conjecture

text: is

text: that

text: lim

text: v∞

text: r/T

text: where

text: r

text: is

text: the

text: rotation

text: number

text: σ

text: (cf.

text: Remark

text: Existence

text: of

text: periodic

text: orbits

text: is

text: unstable

text: with

text: respect

text: to

text: perturbations

text: of

text: F

text: thus,

text: in

text: a

text: general

text: position,

text: v∞

text: is

text: non-zero

text: for

text: small

text: σ.

text: Unfortunately,

text: as

text: R.

text: Ortega

text: and

text: F.

text: Zanolin

text: pointed

text: out

text: in

text: a

text: per-

text: sonal

text: communication,

text: there

text: are

text: no

text: criteria

text: of

text: the

text: form

text: that

text: ‘some

text: set

text: of

text: assumptions

text: on

text: F

text: implies

text: non-existence

text: of

text: T

text: -periodic

text: solutions

text: to

text: In

text: the

text: non-existence

text: of

text: periodic

text: solutions

text: to

text: with

text: a

text: tilting

text: potential

text: of

text: the

text: form

text: F

text: (y,

text: t)

text: H(t)

text: sin

text: y,

text: where

text: H

text: is

text: T

text: sin

text: periodic

text: and

text: satisﬁes

text: was

text: discovered.

text: There,

text: it

text: was

text: observed

text: that

text: this

text: phenomenon

text: contrasts

text: with

text: the

text: behaviour

text: of

text: the

text: equation

text: y

text: (t)

text: F

text: (y(t),

text: t),

text: sin

text: ##LTLine##

text: ##LTLine##


text: VELOCITY OF RATCHETS 39  (cid:16) (cid:17) (cid:82)  (cid:16) (cid:17) (cid:82)  ζ (x) = Mexp x¯ ν12(y) + ν21(y) dy /F (x), (6.21) 1 − x F2(y) F1(y) 1 ζ (x) = Mexp x¯ ν12(y) + ν21(y) dy /F (x), 2 x F2(y) F1(y) 2 where x¯ = x1+x2 . This gives that F F is negative on (x , x ). Con- (cid:82) 2 (cid:82) 1 2 1 2 sequently, x and x are zeros of the same potential — in case ii) this is 1 2 (cid:82) (cid:82) trivial — and, for deﬁniteness, let it be F . Observe that one of the in- 1 tegrals x¯ ν12(y(cid:82)) + ν21(y) dy and x¯ ν12(y) + ν21(y) dy is positive. Since F x1 F2(y) F1(y) ±∞x2 F2(y) F1(y) | ∞ 1 is C1-smooth, x2 1 dy = . Hence, x2 ζ (y) dy = . But x1 F1|(y) ≤ x1 1 (x1,x2) (6.18) implies x2 ζ (y) dy 1, and we arrive at a contradiction. x1 1 (x1,x2) Assume now that F > 0 on (x , x ). We then have to show that 1 1 2 v˜∞ < 0. One can check that, under the assumptions that we have 0 made, the only solution of (6.18), (6.19) with ﬁnite integral on (x , x ) 1 2 is  (cid:16) (cid:17) (cid:82) (cid:82)  (cid:104) (cid:16) (cid:17) (cid:105) (cid:82) (cid:82)  ζ (x) = v˜∞0 x exp s ν12(y) + ν21(y) dy ν12(s) ds, (6.22) 1 F1(x) x1− x F2(y) F1(y) F2(s) ζ (x) = v˜∞0 1 x exp s ν12(y) + ν21(y) dy ν12(s) ds . 2 F2(x) x1 x F2(y) F1(y) F2(s) ≥ | So ζ 0 and F < 0 imply v˜∞ < 0. The case F < 0 is 1 2 (x1,x2) 0 1 treated similarly. In case iii), the solution ζ is smooth on the whole circle S1, so it R may be identiﬁed with a 1-periodic function on . Moreover, it has (cid:18)(cid:90) (cid:19) the form M (cid:90) 0 ν (cid:18)((cid:90)y) ν (y) (cid:19) 12 21 (6.23) ζ (x) = exp + dy 1 F (x) F (y) F (y) 1 x 2 1 v˜∞ x s ν (y) ν (y) ν (s) 0 12 21 12 + exp + dy ds, F (x) F (y) F (y) F (s) 1 0 x 2 1 2 (cid:18)(cid:90) (cid:19) − (cid:20)M (cid:90) 0(cid:18)ν(cid:90) (y) ν (y) (cid:19) (cid:21) 12 21 (6.24) ζ (x) = exp + dy 2 F (x) F (y) F (y) 2 x 2 1 v˜∞ − x s ν (y) ν (y) ν (s) 0 12 21 12 + 1 exp + dy ds . F (x) F (y) F (y) F (s) 2 0 x 2 1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: VELOCITY

text: OF

text: RATCHETS

text: ζ

text: (x)

text: Mexp

text: x¯

text: dy

text: /F

text: (x),

text: x

text: ζ

text: (x)

text: Mexp

text: x¯

text: dy

text: /F

text: (x),

text: x

text: where

text: x¯

text: This

text: gives

text: that

text: F

text: F

text: is

text: negative

text: on

text: (x

text: x

text: Con-

text: sequently,

text: x

text: and

text: x

text: are

text: zeros

text: of

text: the

text: same

text: potential

text: in

text: case

text: ii)

text: this

text: is

text: trivial

text: and,

text: for

text: deﬁniteness,

text: let

text: it

text: be

text: F

text: Observe

text: that

text: one

text: of

text: the

text: in-

text: tegrals

text: x¯

text: dy

text: and

text: x¯

text: dy

text: is

text: positive.

text: Since

text: F

text: ±∞x2

text: is

text: dy

text: Hence,

text: ζ

text: (y)

text: dy

text: But

text: implies

text: ζ

text: (y)

text: dy

text: and

text: we

text: arrive

text: at

text: a

text: contradiction.

text: Assume

text: now

text: that

text: F

text: on

text: (x

text: x

text: We

text: then

text: have

text: to

text: show

text: that

text: v˜∞

text: One

text: can

text: check

text: that,

text: under

text: the

text: assumptions

text: that

text: we

text: have

text: made,

text: the

text: only

text: solution

text: of

text: with

text: ﬁnite

text: integral

text: on

text: (x

text: x

text: is

text: ζ

text: (x)

text: v˜∞0

text: x

text: exp

text: s

text: dy

text: ds,

text: x

text: ζ

text: (x)

text: v˜∞0

text: x

text: exp

text: s

text: dy

text: ds

text: x

text: So

text: ζ

text: and

text: F

text: imply

text: v˜∞

text: The

text: case

text: F

text: is

text: treated

text: similarly.

text: In

text: case

text: iii),

text: the

text: solution

text: ζ

text: is

text: smooth

text: on

text: the

text: whole

text: circle

text: so

text: it

text: R

text: may

text: be

text: identiﬁed

text: with

text: a

text: function

text: on

text: Moreover,

text: it

text: has

text: the

text: form

text: M

text: ν

text: ν

text: (y)

text: ζ

text: (x)

text: exp

text: dy

text: F

text: (x)

text: F

text: (y)

text: F

text: (y)

text: x

text: v˜∞

text: x

text: s

text: ν

text: (y)

text: ν

text: (y)

text: ν

text: (s)

text: exp

text: dy

text: ds,

text: F

text: (x)

text: F

text: (y)

text: F

text: (y)

text: F

text: (s)

text: x

text: (y)

text: ν

text: (y)

text: ζ

text: (x)

text: exp

text: dy

text: F

text: (x)

text: F

text: (y)

text: F

text: (y)

text: x

text: v˜∞

text: x

text: s

text: ν

text: (y)

text: ν

text: (y)

text: ν

text: (s)

text: exp

text: dy

text: ds

text: F

text: (x)

text: F

text: (y)

text: F

text: (y)

text: F

text: (s)

text: x

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text: E When the n-dimensional Euclidean space n is endowed with the Lorentzian inner product, we obtain the Lorentzian similarity geometry. The Lorentzian ﬂat geometry is inside the Lorentzian similarity geometry. Aristide [27] investigated the closed Lorentzian similarity manifolds. Kamishima [30] studied the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations. The geometric invariants of curves in the Lorentzian similarity geometry have not been considered so far. The theme of similarity and self-similarity will be interesting in the Lorentzian-Minkowski space. Many integrable equations, like Korteweg-de Vries (mKdV), sine-Gordon and nonlinear Schr¨odinger (NLS) equations, in soliton theory have been shown to be related to motions of inextensible curves in the Euclidean space. By using the similarity invariants of curves under the similarity motion, KS. Chou and C. Qu [12] showed that the motions of curves in two-, three- and n-dimensional (n > 3) similar- ity geometries correspond to the Burgers hierarchy, Burgers-mKdV hierarchy and a multi-component E generalization of these hierarchies in n. Moreover, to study the motion of curves in the Minkowski space also attracted researchers’ interest. Gu¨rses [19] studied the motion of curves on two-dimensional surface in Minkowski 3-space. Q. Ding and J. Inoguchi [21] showed that binormal motions of curves in Minkowski 3-space are equivalent to some integrable equations. In the 4-dimensional Minkowski space Nakayama [15] presented a formulation on the motion of curves in hyperboloids, which includes many equations integrable by means of the 1+1- dimensional AKNS inverse scattering scheme. Therefore, E the current paper can contribute to study the motion of curves with similarity invariants in n. 1 Berger [18] represented the broad content of similarity transformations in the arbitrary dimen- sional Euclidean spaces. Encheva and Georgiev [22, 23] studied the diﬀerential geometric invariants of curves according to a similarity in the ﬁnite dimensional Euclidean spaces. In the current paper, Lorentzian version of similarity transformations will be entitled by pseudo-similarity transformation deﬁned by (1) in the section 2. The main idea of this paper is to extend the fundamental theorem for a non-null curve with respect to p-similarity motion and determine non-null self-similar curves in the E Minkowski n-space n. 1 The scope of paper is as follows. First, we prove that the p-similarity transformations preserve the causal characters and angles. We introduce diﬀerential geometric invariants of a non-lightlike Frenet curve which are called p-shape curvatures according to the group of p-similarity transformations E in n. We give the uniqueness theorem which states that two non-null curves having same the 1 p-shape curvatures are equivalent according to a p-similarity. Furthermore, we obtain the existence theorem that is a process for constructing a non-null curve by its p-shape curvatures under some initial E conditions. Lastly, we study a exact description of all non-null self-similar curves in n. Especially, 1 we examine the low-dimensional cases n = 2, 3, 4 in more details. 2 Fundamental Group of the Lorentzian Similarity Geometry · · · Firstly let us give some basic notions of the Lorentzian geometry. Let x = (x , , x )T , · · · · · · (cid:112) 1 E n y = (y , , y )T and z = (z , , z )T be three arbitrary vectors in the Minkowski space n. The 1 n 1 n · ∗ ∗ − ·1· · Lorentzian inner product of x and y can be stated as x y = xT I y where I = diag( 1, 1, 1, , 1). (cid:107) (cid:107) | · | Then, the norm of the vector x is represented by x = x x , [5, 28]. E Theorem 1 Let x and y be vectors in the Minkowski n-space n. 1 E (i) If x and y are timelike vectors which are in the same timecone of n , then there is a unique ≥ · 1 − (cid:107) (cid:107) (cid:107) (cid:107) number θ 0, called the hyperbolic angle between x and y such that x y = x y cosh θ . | · | (cid:107) (cid:107) (cid:107) (cid:107) · (cid:107) (cid:107) (cid:107) (cid:107) (ii) If x and y are spacelike vectors satisfying the inequality x y < x y , then x y = x y cos θ where θ is the angle between x and y. | · | (cid:107) (cid:107) (cid:107) (cid:107) · (cid:107) (cid:107) (cid:107) (cid:107) (iii) If x and y are spacelike vectors satisfying the inequality x y > x y , then x y = x y cosh θ where θ is the hyperbolic angle between x and y. E Now, we deﬁne similarity transformation in n. A pseudo-similarity (p-similarity) of Minkowski E 1 n-space n is a composition of a dilatation (homothety) and a Lorentzian motion. Any p-similarity 1 2 ##LTLine##

text: E

text: When

text: the

text: n-dimensional

text: Euclidean

text: space

text: n

text: is

text: endowed

text: with

text: the

text: Lorentzian

text: inner

text: product,

text: we

text: obtain

text: the

text: Lorentzian

text: similarity

text: geometry.

text: The

text: Lorentzian

text: ﬂat

text: geometry

text: is

text: inside

text: the

text: Lorentzian

text: similarity

text: geometry.

text: Aristide

text: investigated

text: the

text: closed

text: Lorentzian

text: similarity

text: manifolds.

text: Kamishima

text: studied

text: the

text: properties

text: of

text: compact

text: Lorentzian

text: similarity

text: manifolds

text: using

text: developing

text: maps

text: and

text: holonomy

text: representations.

text: The

text: geometric

text: invariants

text: of

text: curves

text: in

text: the

text: Lorentzian

text: similarity

text: geometry

text: have

text: not

text: been

text: considered

text: so

text: far.

text: The

text: theme

text: of

text: similarity

text: and

text: self-similarity

text: will

text: be

text: interesting

text: in

text: the

text: Lorentzian-Minkowski

text: space.

text: Many

text: integrable

text: equations,

text: like

text: Korteweg-de

text: Vries

text: (mKdV),

text: sine-Gordon

text: and

text: nonlinear

text: Schr¨odinger

text: (NLS)

text: equations,

text: in

text: soliton

text: theory

text: have

text: been

text: shown

text: to

text: be

text: related

text: to

text: motions

text: of

text: inextensible

text: curves

text: in

text: the

text: Euclidean

text: space.

text: By

text: using

text: the

text: similarity

text: invariants

text: of

text: curves

text: under

text: the

text: similarity

text: motion,

text: KS.

text: Chou

text: and

text: C.

text: Qu

text: showed

text: that

text: the

text: motions

text: of

text: curves

text: in

text: two-,

text: three-

text: and

text: n-dimensional

text: (n

text: similar-

text: ity

text: geometries

text: correspond

text: to

text: the

text: Burgers

text: hierarchy,

text: Burgers-mKdV

text: hierarchy

text: and

text: a

text: multi-component

text: E

text: generalization

text: of

text: these

text: hierarchies

text: in

text: n.

text: Moreover,

text: to

text: study

text: the

text: motion

text: of

text: curves

text: in

text: the

text: Minkowski

text: space

text: also

text: attracted

text: researchers’

text: interest.

text: Gu¨rses

text: studied

text: the

text: motion

text: of

text: curves

text: on

text: two-dimensional

text: surface

text: in

text: Minkowski

text: Q.

text: Ding

text: and

text: J.

text: Inoguchi

text: showed

text: that

text: binormal

text: motions

text: of

text: curves

text: in

text: Minkowski

text: are

text: equivalent

text: to

text: some

text: integrable

text: equations.

text: In

text: the

text: Minkowski

text: space

text: Nakayama

text: presented

text: a

text: formulation

text: on

text: the

text: motion

text: of

text: curves

text: in

text: hyperboloids,

text: which

text: includes

text: many

text: equations

text: integrable

text: by

text: means

text: of

text: the

text: dimensional

text: AKNS

text: inverse

text: scattering

text: scheme.

text: Therefore,

text: E

text: the

text: current

text: paper

text: can

text: contribute

text: to

text: study

text: the

text: motion

text: of

text: curves

text: with

text: similarity

text: invariants

text: in

text: n.

text: Berger

text: represented

text: the

text: broad

text: content

text: of

text: similarity

text: transformations

text: in

text: the

text: arbitrary

text: dimen-

text: sional

text: Euclidean

text: spaces.

text: Encheva

text: and

text: Georgiev

text: studied

text: the

text: diﬀerential

text: geometric

text: invariants

text: of

text: curves

text: according

text: to

text: a

text: similarity

text: in

text: the

text: ﬁnite

text: dimensional

text: Euclidean

text: spaces.

text: In

text: the

text: current

text: paper,

text: Lorentzian

text: version

text: of

text: similarity

text: transformations

text: will

text: be

text: entitled

text: by

text: pseudo-similarity

text: transformation

text: deﬁned

text: by

text: in

text: the

text: section

text: The

text: main

text: idea

text: of

text: this

text: paper

text: is

text: to

text: extend

text: the

text: fundamental

text: theorem

text: for

text: a

text: non-null

text: curve

text: with

text: respect

text: to

text: p-similarity

text: motion

text: and

text: determine

text: non-null

text: self-similar

text: curves

text: in

text: the

text: E

text: Minkowski

text: n-space

text: n.

text: The

text: scope

text: of

text: paper

text: is

text: as

text: follows.

text: First,

text: we

text: prove

text: that

text: the

text: p-similarity

text: transformations

text: preserve

text: the

text: causal

text: characters

text: and

text: angles.

text: We

text: introduce

text: diﬀerential

text: geometric

text: invariants

text: of

text: a

text: non-lightlike

text: Frenet

text: curve

text: which

text: are

text: called

text: p-shape

text: curvatures

text: according

text: to

text: the

text: group

text: of

text: p-similarity

text: transformations

text: E

text: in

text: n.

text: We

text: give

text: the

text: uniqueness

text: theorem

text: which

text: states

text: that

text: two

text: non-null

text: curves

text: having

text: same

text: the

text: p-shape

text: curvatures

text: are

text: equivalent

text: according

text: to

text: a

text: p-similarity.

text: Furthermore,

text: we

text: obtain

text: the

text: existence

text: theorem

text: that

text: is

text: a

text: process

text: for

text: constructing

text: a

text: non-null

text: curve

text: by

text: its

text: p-shape

text: curvatures

text: under

text: some

text: initial

text: E

text: conditions.

text: Lastly,

text: we

text: study

text: a

text: exact

text: description

text: of

text: all

text: non-null

text: self-similar

text: curves

text: in

text: n.

text: Especially,

text: we

text: examine

text: the

text: low-dimensional

text: cases

text: n

text: in

text: more

text: details.

text: Fundamental

text: Group

text: of

text: the

text: Lorentzian

text: Similarity

text: Geometry

text: Firstly

text: let

text: us

text: give

text: some

text: basic

text: notions

text: of

text: the

text: Lorentzian

text: geometry.

text: Let

text: x

text: (x

text: x

text: )T

text: E

text: n

text: y

text: (y

text: y

text: )T

text: and

text: z

text: (z

text: z

text: )T

text: be

text: three

text: arbitrary

text: vectors

text: in

text: the

text: Minkowski

text: space

text: n.

text: The

text: n

text: n

text: ·1·

text: Lorentzian

text: inner

text: product

text: of

text: x

text: and

text: y

text: can

text: be

text: stated

text: as

text: x

text: y

text: xT

text: I

text: y

text: where

text: I

text: diag(

text: Then,

text: the

text: norm

text: of

text: the

text: vector

text: x

text: is

text: represented

text: by

text: x

text: x

text: x

text: E

text: Theorem

text: Let

text: x

text: and

text: y

text: be

text: vectors

text: in

text: the

text: Minkowski

text: n-space

text: n.

text: E

text: (i)

text: If

text: x

text: and

text: y

text: are

text: timelike

text: vectors

text: which

text: are

text: in

text: the

text: same

text: timecone

text: of

text: n

text: then

text: there

text: is

text: a

text: unique

text: number

text: θ

text: called

text: the

text: hyperbolic

text: angle

text: between

text: x

text: and

text: y

text: such

text: that

text: x

text: y

text: x

text: y

text: cosh

text: θ

text: (ii)

text: If

text: x

text: and

text: y

text: are

text: spacelike

text: vectors

text: satisfying

text: the

text: inequality

text: x

text: y

text: x

text: y

text: then

text: x

text: y

text: x

text: y

text: cos

text: θ

text: where

text: θ

text: is

text: the

text: angle

text: between

text: x

text: and

text: y.

text: (iii)

text: If

text: x

text: and

text: y

text: are

text: spacelike

text: vectors

text: satisfying

text: the

text: inequality

text: x

text: y

text: x

text: y

text: then

text: x

text: y

text: x

text: y

text: cosh

text: θ

text: where

text: θ

text: is

text: the

text: hyperbolic

text: angle

text: between

text: x

text: and

text: y.

text: E

text: Now,

text: we

text: deﬁne

text: similarity

text: transformation

text: in

text: n.

text: A

text: pseudo-similarity

text: (p-similarity)

text: of

text: Minkowski

text: E

text: n-space

text: n

text: is

text: a

text: composition

text: of

text: a

text: dilatation

text: (homothety)

text: and

text: a

text: Lorentzian

text: motion.

text: Any

text: p-similarity

text: ##LTLine##


text: (cid:4) (cid:5) The matrix M2 is a semi skew-symmetrix matrix and has exactly k = n eigenvalues with multiplicity · · ·  2  two: λ2, λ2, , λ2. According to Corollary 3.3 in [6], the normal form of the matrix M is either 1 2 k    · · ·   0 ε λ 0 0 0 0  − 2 1 · · ·   ε λ 0 0 0 0 0   1 1 · · ·   0 0 0 ε λ 0 0   4 2   0 0 −ε λ 0 ... 0 0   3 2  ... ... ... ... ... ... ... · · · 0 0 0 0 0 ε λ · · · − n k 0 0 0 0 ε − λ 0 n 1 k in the case of even n, or the same matrix with an additional row (column) of zeros in the case of odd n. 6.1 Non-null Self-similar Curves in Even-dimensional Minkowski Space E Any non-null self-similar curve in 2k can be described by its constant p-shape curvatures with 1 respect to Theorem 7. Let’s see this via the following theorem. → E Theorem 13 Let α : I 2k be a non-null self-similar curve with the constant p-shape curva- (cid:54) (cid:54) · · · 1 (cid:54) · · · tures κ˜ = 0, κ˜ = 0, , κ˜ − = 0. Suppose that λ2, λ2, , λ2 are all diﬀerent eigenvalues of the 1 2 2k 1 1 2 k symmetric matrix M2. So, i) If the unit vector e is a timelike vector, then a parametric statement of the timelike self-similar 1 curve α according to arc-length parameter σ can be written in the form a a α (σ) = ( 1 eκ˜1σ sinh θ , 1 eκ˜1σ cosh θ , 1 1 b b 1 1 a −a · · · a −a 2 eκ˜1σ sin θ , 2 eκ˜1σ cos θ , , k eκ˜1σ sin θ , k eκ˜1σ cos θ ) (25) 2 2 k k b b b b 2 2 k (cid:32) k (cid:33) (cid:113) (cid:112) where − − − λ b1 = λ21 κ˜21, θ1 = λ1σ cosh 1 λ2 1− κ˜2 , 1 1 · · ·   and for i = 2, , k (cid:113) (cid:113)  − λ b = λ2 + κ˜2, θ = λ σ + cos 1 i . i i 1 i i λ2 + κ˜2 i 1 · · · The real diﬀerent non-zero numbers a , , a are a solution of the system of k algebraic quadratic 1 k equations · − · e e = 1 and e e = 1 i = 2, ..., k, 1 1 i i determined by the vectors 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: matrix

text: is

text: a

text: semi

text: skew-symmetrix

text: matrix

text: and

text: has

text: exactly

text: k

text: n

text: eigenvalues

text: with

text: multiplicity

text: two:

text: According

text: to

text: Corollary

text: in

text: the

text: normal

text: form

text: of

text: the

text: matrix

text: M

text: is

text: either

text: k

text: ε

text: λ

text: ε

text: λ

text: ε

text: λ

text: −ε

text: λ

text: ε

text: λ

text: n

text: k

text: ε

text: λ

text: n

text: k

text: in

text: the

text: case

text: of

text: even

text: n,

text: or

text: the

text: same

text: matrix

text: with

text: an

text: additional

text: row

text: (column)

text: of

text: zeros

text: in

text: the

text: case

text: of

text: odd

text: n.

text: Non-null

text: Self-similar

text: Curves

text: in

text: Even-dimensional

text: Minkowski

text: Space

text: E

text: Any

text: non-null

text: self-similar

text: curve

text: in

text: can

text: be

text: described

text: by

text: its

text: constant

text: p-shape

text: curvatures

text: with

text: respect

text: to

text: Theorem

text: Let’s

text: see

text: this

text: via

text: the

text: following

text: theorem.

text: E

text: Theorem

text: Let

text: α

text: I

text: be

text: a

text: non-null

text: self-similar

text: curve

text: with

text: the

text: constant

text: p-shape

text: curva-

text: tures

text: κ˜

text: κ˜

text: κ˜

text: Suppose

text: that

text: λ2

text: are

text: all

text: diﬀerent

text: eigenvalues

text: of

text: the

text: k

text: symmetric

text: matrix

text: So,

text: i)

text: If

text: the

text: unit

text: vector

text: e

text: is

text: a

text: timelike

text: vector,

text: then

text: a

text: parametric

text: statement

text: of

text: the

text: timelike

text: self-similar

text: curve

text: α

text: according

text: to

text: arc-length

text: parameter

text: σ

text: can

text: be

text: written

text: in

text: the

text: form

text: a

text: a

text: α

text: (σ)

text: eκ˜1σ

text: sinh

text: θ

text: eκ˜1σ

text: cosh

text: θ

text: b

text: b

text: a

text: −a

text: a

text: −a

text: eκ˜1σ

text: sin

text: θ

text: eκ˜1σ

text: cos

text: θ

text: k

text: eκ˜1σ

text: sin

text: θ

text: k

text: eκ˜1σ

text: cos

text: θ

text: k

text: k

text: b

text: b

text: b

text: b

text: k

text: k

text: where

text: λ

text: θ1

text: λ1σ

text: cosh

text: λ2

text: 1−

text: κ˜2

text: and

text: for

text: i

text: k

text: λ

text: b

text: λ2

text: κ˜2,

text: θ

text: λ

text: σ

text: cos

text: i

text: i

text: i

text: i

text: i

text: λ2

text: κ˜2

text: i

text: The

text: real

text: diﬀerent

text: non-zero

text: numbers

text: a

text: a

text: are

text: a

text: solution

text: of

text: the

text: system

text: of

text: k

text: algebraic

text: quadratic

text: k

text: equations

text: e

text: e

text: and

text: e

text: e

text: i

text: k,

text: i

text: i

text: determined

text: by

text: the

text: vectors

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:80) − k − where a ’s are a real constants satisfying a2 + a2 = 1. i 1 i · · · i=2 If X = (α (σ) , α (σ) , , α (σ)) is considered as the parametric equation of the timelike curve 1 2 2k α, we get the following equation by (9) d 1 X = e (29) 1 dσ κ 1 − − 1 dκ where we have κ1 = e κ˜1σ by κ˜1 = 1 . It can be obtained the following equation by (29) κ dσ 1 a − λ α = 1 eκ˜1σ cosh (λ σ) 1 α 1 1 2 κ˜ κ˜ 1 1 a − λ α = 1 eκ˜1σ sinh (λ σ) 1 α 2 1 1 κ˜ κ˜ 1 1 and a λ α − = i eκ˜1σ cosh (λ σ) + i α 2i 1 i 2i κ˜ κ˜ 1 1 a − λ α = i eκ˜1σ sinh (λ σ) i α − 2i i 2i 1 κ˜ κ˜ 1 1 · · · for i = 2, , k. The solutions of above the linear systems are a − a α = −1 eκ˜1σ (κ˜ cosh (λ σ) λ sinh (λ σ)) = 1 eκ˜1σ sinh θ 1 κ˜2 λ2 1 1 1 1 b 1 1 1 1 a − a α = −1 eκ˜1σ (κ˜ sinh (λ σ) λ cosh (λ σ)) = 1 eκ˜1σ cosh θ 2 κ˜2 λ2 1 1 1 1 b 1 1 1 1 and a a α − = i eκ˜1σ (κ˜ cos (λ σ) + λ sin (λ σ)) = i eκ˜1σ sin θ 2i 1 κ˜2 + λ2 1 i i i b i 1 i i a − −a α = −1 eκ˜1σ (κ˜ sin (λ σ) λ cos (λ σ)) = i eκ˜1σ cos θ . 2i κ˜2 λ2 1 i i i b i 1 i i We can get the unit vectors (26) by using the equations (11) for the timelike self-similar curve α. (cid:88) Thus, we can write the following equations by means of (26) k · − ⇒ − − e1 e1 = 1 (cid:88)a21 + a2i = 1 i=2 k · ⇒ e2 e2 = 1 a2i(cid:0)λ2i = 1(cid:1) (cid:88) (cid:0) (cid:1) i=1 k · ⇒ − − e e = 1 a2 1 λ2 2 + 1 + λ2 2 a2 = κ˜2 3 3 1 1 i i 2 i=2 · ⇒ · · · e e = 1 so on, i = 4, , k. i i ii) The proof is similar to the proof of i). → E Corollary 14 i) Let α : I 2k (k > 1) be a timelike self-similar curve with a parametric represen- 1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) tation (25) . Then, this curve lies on the quadratic timelike hypersurface with an equation b2 − b2 · · · b2− − b2 1 x2 + x2 + 2 x2 + x2 + + k 1 x2 − + x2 − = (k 1) k x2 − + x2 . a2 1 2 a2 1 2 a2− 2k 3 2k 2 a2 2k 1 2k 1 2 k 1 k 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: k

text: where

text: a

text: ’s

text: are

text: a

text: real

text: constants

text: satisfying

text: i

text: i

text: If

text: X

text: (α

text: (σ)

text: α

text: (σ)

text: α

text: (σ))

text: is

text: considered

text: as

text: the

text: parametric

text: equation

text: of

text: the

text: timelike

text: curve

text: α,

text: we

text: get

text: the

text: following

text: equation

text: by

text: d

text: X

text: e

text: dσ

text: κ

text: dκ

text: where

text: we

text: have

text: κ1

text: e

text: κ˜1σ

text: by

text: κ˜1

text: It

text: can

text: be

text: obtained

text: the

text: following

text: equation

text: by

text: κ

text: dσ

text: a

text: λ

text: α

text: eκ˜1σ

text: cosh

text: (λ

text: σ)

text: α

text: κ˜

text: κ˜

text: a

text: λ

text: α

text: eκ˜1σ

text: sinh

text: (λ

text: σ)

text: α

text: κ˜

text: κ˜

text: and

text: a

text: λ

text: α

text: i

text: eκ˜1σ

text: cosh

text: (λ

text: σ)

text: i

text: α

text: i

text: κ˜

text: κ˜

text: a

text: λ

text: α

text: i

text: eκ˜1σ

text: sinh

text: (λ

text: σ)

text: i

text: α

text: i

text: κ˜

text: κ˜

text: for

text: i

text: k.

text: The

text: solutions

text: of

text: above

text: the

text: linear

text: systems

text: are

text: a

text: a

text: α

text: −1

text: eκ˜1σ

text: (κ˜

text: cosh

text: (λ

text: σ)

text: λ

text: sinh

text: (λ

text: σ))

text: eκ˜1σ

text: sinh

text: θ

text: κ˜2

text: λ2

text: b

text: a

text: a

text: α

text: −1

text: eκ˜1σ

text: (κ˜

text: sinh

text: (λ

text: σ)

text: λ

text: cosh

text: (λ

text: σ))

text: eκ˜1σ

text: cosh

text: θ

text: κ˜2

text: λ2

text: b

text: and

text: a

text: a

text: α

text: i

text: eκ˜1σ

text: (κ˜

text: cos

text: (λ

text: σ)

text: λ

text: sin

text: (λ

text: σ))

text: i

text: eκ˜1σ

text: sin

text: θ

text: κ˜2

text: λ2

text: i

text: i

text: i

text: b

text: i

text: i

text: i

text: a

text: −a

text: α

text: −1

text: eκ˜1σ

text: (κ˜

text: sin

text: (λ

text: σ)

text: λ

text: cos

text: (λ

text: σ))

text: i

text: eκ˜1σ

text: cos

text: θ

text: κ˜2

text: λ2

text: i

text: i

text: i

text: b

text: i

text: i

text: i

text: We

text: can

text: get

text: the

text: unit

text: vectors

text: by

text: using

text: the

text: equations

text: for

text: the

text: timelike

text: self-similar

text: curve

text: α.

text: Thus,

text: we

text: can

text: write

text: the

text: following

text: equations

text: by

text: means

text: of

text: k

text: k

text: k

text: e

text: e

text: λ2

text: λ2

text: κ˜2

text: i

text: i

text: e

text: e

text: so

text: on,

text: i

text: k.

text: i

text: i

text: ii)

text: The

text: proof

text: is

text: similar

text: to

text: the

text: proof

text: of

text: i).

text: E

text: Corollary

text: i)

text: Let

text: α

text: I

text: (k

text: be

text: a

text: timelike

text: self-similar

text: curve

text: with

text: a

text: parametric

text: represen-

text: tation

text: Then,

text: this

text: curve

text: lies

text: on

text: the

text: quadratic

text: timelike

text: hypersurface

text: with

text: an

text: equation

text: k

text: (k

text: k

text: k

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: → E ii) Let α : I 2k (k > 1) be a spacelike self-similar curve with a parametric representation (27) . 1 Then, this (cid:0)curve lie(cid:1)s on th(cid:0)e quadra(cid:1)tic Lorentzian (cid:0)hypersurface w(cid:1)ith an equation(cid:0) (cid:1) b2 − b2 · · · b2− − b2 1 x2 x2 + 2 x2 + x2 + + k 1 x2 − + x2 − = (k 1) k x2 − + x2 . a2 1 2 a2 1 2 a2− 2k 3 2k 2 a2 2k 1 2k 1 2 k 1 k Now, we examine the following examples of the non-null self-similar curves in the Minkowski plane and Minkowski space-time. → E Case n = 2 (Lorentzian plane): Let α : I 2 be a timelike self-similar curve with the constant (cid:32) 2 1 (cid:33) invariant κ˜ . Then, we can write λ2 = 1 and a2λ2 = 1. Hence, a parametrization of α is 1 1(cid:112) 1 1 (cid:112) 2 1 1 (cid:18) α2 (σ)(cid:19)= − eκ˜1σ sinh θ, − eκ˜1σ cosh θ 1 κ˜2 1 κ˜2 1 1 − √ where θ = σ + cosh 1 1− . 1 κ˜2 1 → E Case n = 4 (Minkowski space-time): Let α : I 4 be a timelike self-similar curve with the constant  4 1  invariants κ˜ , κ˜ and κ˜ . Then, the semi-symmetric matrix 1 2 3      1 0 κ˜ 0  − 2 0 1 κ˜2 0 κ˜ κ˜ M2 = − 2 − − 2 3 κ˜ 0 κ˜2 κ˜2 0 2 2 3 − 0 κ˜ κ˜ 0 κ˜2 2 3 3 (cid:18) (cid:113) (cid:19) (cid:0) (cid:1) has two eigenvalues of multiplicity 2 1 − − − − − λ2 = 1 κ˜2 κ˜2 + ( 1)i 1 κ˜2 κ˜2 2 + 4κ˜2 i 2 2 3 2 3 3 for i = 1, 2. The solution of system of quadratic equations − − a2 + a2 = 1 1 2 (cid:115) a21λ21 + a22λ22 = 1 (cid:115) is given by − 1 + λ2 1 λ2 a = 2 , a = 1 1 λ2 + λ2 2 λ2 + λ2 1 2 1 2 − since we have 1 λ21 > 0(cid:18). Consequently, the spherical arc-length parametrizati(cid:19)on of the timelike self-similar curve is α given by 4 a a a −a (cid:112) α4 (σ) = (cid:112)1 eκ˜1σ sinh θ, 1 eκ˜1σ cosh θ, 2 eκ˜(cid:18)1σ sin θ, (cid:19) 2 eκ˜1σ cos θ (cid:18) (cid:19) b b b b 1 1 2 2 − − − √ − √ where b1 = λ21 κ˜21, b2 = λ22 + κ˜21 and θ1 = λ1σ cosh 1 λλ21−κ˜2 , θ2 = λ2σ+cos 1 λλ22+κ˜2 . 1 1 2 1 6.2 Non-null Self-similar Curves in Odd-dimensional Minkowski Space E It can be given non-null self-similar curves in 2k+1 with the following theorem. → E 1 Theorem 15 Let α : I 2k+1 be a non-null self-similar curve with constant p-shape curvatures (cid:54) (cid:54) · · · (cid:54) 1 · · · κ˜ = 0, κ˜ = 0, , κ˜ = 0. Suppose that λ2, λ2, , λ2 are all diﬀerent eigenvalues of the symmetric 1 2 2k 1 2 k matrix M2. So, 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: E

text: ii)

text: Let

text: α

text: I

text: (k

text: be

text: a

text: spacelike

text: self-similar

text: curve

text: with

text: a

text: parametric

text: representation

text: Then,

text: this

text: on

text: Lorentzian

text: an

text: k

text: (k

text: k

text: k

text: k

text: Now,

text: we

text: examine

text: the

text: following

text: examples

text: of

text: the

text: non-null

text: self-similar

text: curves

text: in

text: the

text: Minkowski

text: plane

text: and

text: Minkowski

text: space-time.

text: E

text: Case

text: n

text: (Lorentzian

text: plane):

text: Let

text: α

text: I

text: be

text: a

text: timelike

text: self-similar

text: curve

text: with

text: the

text: constant

text: invariant

text: κ˜

text: Then,

text: we

text: can

text: write

text: λ2

text: and

text: Hence,

text: a

text: parametrization

text: of

text: α

text: is

text: α2

text: eκ˜1σ

text: sinh

text: θ,

text: eκ˜1σ

text: cosh

text: θ

text: κ˜2

text: κ˜2

text: where

text: θ

text: σ

text: cosh

text: 1−

text: κ˜2

text: E

text: Case

text: n

text: (Minkowski

text: space-time):

text: Let

text: α

text: I

text: be

text: a

text: timelike

text: self-similar

text: curve

text: with

text: the

text: constant

text: invariants

text: κ˜

text: κ˜

text: and

text: κ˜

text: Then,

text: the

text: semi-symmetric

text: matrix

text: κ˜

text: κ˜2

text: κ˜

text: κ˜

text: κ˜

text: κ˜2

text: κ˜2

text: κ˜

text: κ˜

text: κ˜2

text: has

text: two

text: eigenvalues

text: of

text: multiplicity

text: λ2

text: κ˜2

text: κ˜2

text: κ˜2

text: κ˜2

text: 4κ˜2

text: i

text: for

text: i

text: The

text: solution

text: of

text: system

text: of

text: quadratic

text: equations

text: is

text: given

text: by

text: λ2

text: λ2

text: a

text: a

text: λ2

text: λ2

text: λ2

text: λ2

text: since

text: we

text: have

text: Consequently,

text: the

text: spherical

text: arc-length

text: of

text: the

text: timelike

text: self-similar

text: curve

text: is

text: α

text: given

text: by

text: a

text: a

text: a

text: −a

text: α4

text: (σ)

text: eκ˜1σ

text: sinh

text: θ,

text: eκ˜1σ

text: cosh

text: θ,

text: sin

text: θ,

text: eκ˜1σ

text: cos

text: θ

text: b

text: b

text: b

text: b

text: where

text: κ˜21

text: and

text: θ1

text: λ1σ

text: cosh

text: λλ21−κ˜2

text: θ2

text: λλ22+κ˜2

text: Non-null

text: Self-similar

text: Curves

text: in

text: Odd-dimensional

text: Minkowski

text: Space

text: E

text: It

text: can

text: be

text: given

text: non-null

text: self-similar

text: curves

text: in

text: with

text: the

text: following

text: theorem.

text: E

text: Theorem

text: Let

text: α

text: I

text: be

text: a

text: non-null

text: self-similar

text: curve

text: with

text: constant

text: p-shape

text: curvatures

text: κ˜

text: κ˜

text: κ˜

text: Suppose

text: that

text: λ2

text: are

text: all

text: diﬀerent

text: eigenvalues

text: of

text: the

text: symmetric

text: k

text: matrix

text: So,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: E → E f : n n is determined by 1 1 f (x) = µAx + b, (1) × where µ is a real constant, A is a ﬁxed pseudo-orthogonal n n matrix with det(A) = 1 and b = ∈ R (b , ..., b )T n is a translation vector. When n is odd and µ is a positive real constant or n is 1 n 1 (cid:13) (cid:13) even and µ is a non-zero real constant, f is an orientation-preserv(cid:13)ing sim(cid:13)ilarity transformation. When (cid:13) (cid:13) n is odd and µ is a negative real constant, f is an orientation-reversing p-similarity transformation. Since f is a aﬃne transformation, we get f(cid:126)(u) = µAu and f(cid:126)(u) = |µ| (cid:107)u(cid:107) for any u ∈ Rn −→ −−−−−−→ 1 | | where f(cid:126)(xy) = f(x)f(y) (see [18]). The constant µ is called p-similarity ratio of the transformation f. The p-similarity transformations are a group under the composition of maps and we denote by E Sim( n). This group is a fundamental group of the Lorentzian similarity geometry. Also, the group 1 E − E of orientation-preserving (reversing) p-similarities are denoted by Sim+ ( n) (Sim ( n) , resp.). 1 1 Theorem 2 The p-similarity transformations preserve the causal characters and angles. Proof. Let f be a p-similarity. Then, since we can write the equation · · · f(cid:126)(u) f(cid:126)(u) = µ2 (Au Au) = µ2 (u u) , (2) E f preserves the causal character in n. 1 E Let u and v are timelike vectors which are in the same timecone of n. We consider θ and γ as the 1 angles between u, v and f(cid:126)(u), f(cid:126)(v), respectively. Since f(cid:126)(u) and f(cid:126)(v) have same causal characters with u and v, we can ﬁnd the following equation fr(cid:13)om th(cid:13)e(cid:13)Theo(cid:13)rem 1; (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) · − f(cid:126)(u) f(cid:126)(v) = f(cid:126)(u) f(cid:126)(v) cosh γ (3) · − (cid:107) (cid:107) (cid:107) (cid:107) µ2 (u v) = µ2 u v cosh γ − (cid:107) (cid:107) (cid:107) (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) u v cosh θ = u v cosh γ cosh θ = cosh γ. (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12) | · | (cid:107) (cid:107) (cid:107) (cid:107) (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12) From here, we have θ =(cid:13) γ. If(cid:13)u(cid:13)and v(cid:13) are spacelike vectors satisfyi(cid:12)ng the ineq(cid:12)uality u v < u v , then (cid:107) (cid:107) (cid:107) (cid:107) | · | · f(cid:126)(u) f(cid:126)(v) = µ2 u v > µ2 u v = f(cid:126)(u) f(cid:126)(v) . Therefore, it can be said from the Theorem 1 that we have θ = γ similar to (3) . It can also be found that θ is equal to γ in case of condition (iii) in the Theorem 1. As a consequence, every p-similarity transformation preserves the angle between any two vectors. 3 Geometric Invariants of Non-null Curves in Lorentzian Similarity Geometry ∈ → ∈ E { · · · } Let α : t J α (t) n be a non-null curve of class Cn and e , , e be a Frenet moving ⊂ R 1 1 n∈ E ∗ n-frame of α where J is an open interval. We denote image of α under f Sim( n) by α i.e. ∗ ◦ ∗ 1 α = f α. Then, α can be stated as ∗ ∈ α (t) = µAα (t) + b, t I. (4) 3

text: E

text: E

text: f

text: n

text: n

text: is

text: determined

text: by

text: f

text: (x)

text: µAx

text: b,

text: where

text: µ

text: is

text: a

text: real

text: constant,

text: A

text: is

text: a

text: ﬁxed

text: pseudo-orthogonal

text: n

text: n

text: matrix

text: with

text: det(A)

text: and

text: b

text: R

text: (b

text: b

text: )T

text: n

text: is

text: a

text: translation

text: vector.

text: When

text: n

text: is

text: odd

text: and

text: µ

text: is

text: a

text: positive

text: real

text: constant

text: or

text: n

text: is

text: n

text: even

text: and

text: µ

text: is

text: a

text: non-zero

text: real

text: constant,

text: f

text: is

text: an

text: transformation.

text: When

text: n

text: is

text: odd

text: and

text: µ

text: is

text: a

text: negative

text: real

text: constant,

text: f

text: is

text: an

text: orientation-reversing

text: p-similarity

text: transformation.

text: Since

text: f

text: is

text: a

text: aﬃne

text: transformation,

text: we

text: get

text: µAu

text: and

text: |µ|

text: for

text: any

text: u

text: Rn

text: where

text: f(x)f(y)

text: (see

text: The

text: constant

text: µ

text: is

text: called

text: p-similarity

text: ratio

text: of

text: the

text: transformation

text: f.

text: The

text: p-similarity

text: transformations

text: are

text: a

text: group

text: under

text: the

text: composition

text: of

text: maps

text: and

text: we

text: denote

text: by

text: E

text: Sim(

text: n).

text: This

text: group

text: is

text: a

text: fundamental

text: group

text: of

text: the

text: Lorentzian

text: similarity

text: geometry.

text: Also,

text: the

text: group

text: E

text: E

text: of

text: orientation-preserving

text: (reversing)

text: p-similarities

text: are

text: denoted

text: by

text: Sim+

text: n)

text: (Sim

text: n)

text: resp.).

text: Theorem

text: The

text: p-similarity

text: transformations

text: preserve

text: the

text: causal

text: characters

text: and

text: angles.

text: Proof.

text: Let

text: f

text: be

text: a

text: p-similarity.

text: Then,

text: since

text: we

text: can

text: write

text: the

text: equation

text: µ2

text: (Au

text: Au)

text: µ2

text: (u

text: u)

text: E

text: f

text: preserves

text: the

text: causal

text: character

text: in

text: n.

text: E

text: Let

text: u

text: and

text: v

text: are

text: timelike

text: vectors

text: which

text: are

text: in

text: the

text: same

text: timecone

text: of

text: n.

text: We

text: consider

text: θ

text: and

text: γ

text: as

text: the

text: angles

text: between

text: u,

text: v

text: and

text: respectively.

text: Since

text: and

text: have

text: same

text: causal

text: characters

text: with

text: u

text: and

text: v,

text: we

text: can

text: ﬁnd

text: the

text: following

text: equation

text: cosh

text: γ

text: µ2

text: (u

text: v)

text: µ2

text: u

text: v

text: cosh

text: γ

text: u

text: v

text: cosh

text: θ

text: u

text: v

text: cosh

text: γ

text: cosh

text: θ

text: cosh

text: γ.

text: From

text: here,

text: we

text: have

text: θ

text: γ.

text: are

text: spacelike

text: vectors

text: the

text: u

text: v

text: u

text: v

text: then

text: µ2

text: u

text: v

text: µ2

text: u

text: v

text: Therefore,

text: it

text: can

text: be

text: said

text: from

text: the

text: Theorem

text: that

text: we

text: have

text: θ

text: γ

text: similar

text: to

text: It

text: can

text: also

text: be

text: found

text: that

text: θ

text: is

text: equal

text: to

text: γ

text: in

text: case

text: of

text: condition

text: (iii)

text: in

text: the

text: Theorem

text: As

text: a

text: consequence,

text: every

text: p-similarity

text: transformation

text: preserves

text: the

text: angle

text: between

text: any

text: two

text: vectors.

text: Geometric

text: Invariants

text: of

text: Non-null

text: Curves

text: in

text: Lorentzian

text: Similarity

text: Geometry

text: E

text: Let

text: α

text: t

text: J

text: α

text: (t)

text: n

text: be

text: a

text: non-null

text: curve

text: of

text: class

text: Cn

text: and

text: e

text: e

text: be

text: a

text: Frenet

text: moving

text: R

text: n∈

text: E

text: n-frame

text: of

text: α

text: where

text: J

text: is

text: an

text: open

text: interval.

text: We

text: denote

text: image

text: of

text: α

text: under

text: f

text: Sim(

text: n)

text: by

text: α

text: i.e.

text: α

text: f

text: α.

text: Then,

text: α

text: can

text: be

text: stated

text: as

text: α

text: (t)

text: µAα

text: (t)

text: b,

text: t

text: I.


text: ∗ ∈ The arc length functions o(cid:90)f α(cid:13)(cid:13) and α(cid:13)(cid:13)starting at t0 J(cid:90)ar(cid:13)(cid:13)e (cid:13)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) t t ∗ dα (u) ∗ dα (u) | | s(t) = du, s (t) = du = µ s (t) . (5) du du t0 t0 (cid:48) In this section, we denote by a prime ” ” the diﬀerentiation with respect to s. The ith curvature κ of i non-null curve α is given by (cid:48) · κ = e e (6) i i i+1 ≤ ≤ − (cid:54) · · · − for 1 i n 1 where κ > 0 and κ − = 0, j = 1, 2, , n 2. The Frenet-Serret equations of α in j n 1 E n is 1 (cid:48) e = ε κ e 1 2 1 2 (cid:48) − ≤ ≤ − e = ε − κ − e − + ε κ e , for 2 i n 1 (7) i i 1 i 1 i 1 i+1 i i+1 (cid:48) − e = ε − κ − e − n n 1 n 1 n 1 (cid:26) where − 1 if e is the timelike vector ≤ ≤ ε = (cid:96) for 1 (cid:96) n. (cid:96) 1 if e is the spacelike vector (cid:96) Let γ (s) = e (s) be the spherical tangent indicatrix of α and σ be an arc length parameter of γ. 1 it can be given a reparametrization of α by σ → E α = α (σ) : I n, (8) 1 and the parameter σ is called a spherical arc length parameter of α. It is easily computed that dα 1 dσ = κ ds and = e . (9) 1 1 dσ κ 1   Thus, we can write        · · ·     0 ε 0 0 0     2   e1  −ε1 0 ε3 κ2 · · · 0 0   e1   κ   e2   − κ2 1 · · ·   e2  ddσ  e...3  =  0... ε2... κ1 .0.. ... 0... 0...   e...3  . (10) en−1 0 0 0 · · · 0 εn κn−1 en−1 κ en 0 0 0 (cid:110)· · · −ε − κn−1 0 1 en(cid:111) n 1 κ 1 · · · ∈ We consider the pseudo-orthogonal n-frame 1 e (σ) , 1 e (σ) , , 1 e (σ) , σ I, for the −κ1 1 κ1 2 κ1 n curve given by (8) . Let κ˜ denote the function 1 dκ1 and κ˜ denote the function κi for i = · · · − 1 κ1 dσ i κ1 2, 3, , n 1. Then, using (7) , (9) and (10) , we get 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: arc

text: length

text: functions

text: and

text: at

text: t

text: t

text: dα

text: (u)

text: dα

text: (u)

text: s(t)

text: du,

text: s

text: (t)

text: du

text: µ

text: s

text: (t)

text: du

text: du

text: In

text: this

text: section,

text: we

text: denote

text: by

text: a

text: prime

text: the

text: diﬀerentiation

text: with

text: respect

text: to

text: s.

text: The

text: ith

text: curvature

text: κ

text: of

text: i

text: non-null

text: curve

text: α

text: is

text: given

text: by

text: κ

text: e

text: e

text: i

text: i

font: default

text: for

text: i

text: n

text: where

text: κ

text: and

text: κ

text: j

text: n

text: The

text: Frenet-Serret

text: equations

text: of

text: α

text: in

text: j

text: n

text: E

text: n

text: is

text: e

text: ε

text: κ

text: e

text: e

text: ε

text: κ

text: e

text: ε

text: κ

text: e

text: for

text: i

text: n

text: i

text: i

text: i

text: i

text: i

text: e

text: ε

text: κ

text: e

text: n

text: n

text: n

text: n

text: where

text: if

text: e

text: is

text: the

text: timelike

text: vector

text: ε

text: for

text: n.

text: if

text: e

text: is

text: the

text: spacelike

text: vector

text: Let

text: γ

text: (s)

text: e

text: (s)

text: be

text: the

text: spherical

text: tangent

text: indicatrix

text: of

text: α

text: and

text: σ

text: be

text: an

text: arc

text: length

text: parameter

text: of

text: γ.

text: it

text: can

text: be

text: given

text: a

text: reparametrization

text: of

text: α

text: by

text: σ

text: E

text: α

text: α

text: (σ)

text: I

text: n,

text: and

text: the

text: parameter

text: σ

text: is

text: called

text: a

text: spherical

text: arc

text: length

text: parameter

text: of

text: α.

text: It

text: is

text: easily

text: computed

text: that

text: dα

text: dσ

text: κ

text: ds

text: and

text: e

text: dσ

text: κ

text: Thus,

text: we

text: can

text: write

text: ε

text: −ε1

text: ε3

text: κ2

text: κ

text: κ2

text: ddσ

text: κ1

text: εn

text: κn−1

text: κ

text: en

text: −ε

text: κn−1

text: n

text: κ

text: We

text: consider

text: the

text: pseudo-orthogonal

text: n-frame

text: e

text: (σ)

text: e

text: (σ)

text: e

text: (σ)

text: σ

text: I,

text: for

text: the

text: −κ1

text: κ1

text: κ1

text: n

text: curve

text: given

text: by

text: Let

text: κ˜

text: denote

text: the

text: function

text: and

text: κ˜

text: denote

text: the

text: function

text: κi

text: for

text: i

text: κ1

text: dσ

text: i

text: κ1

text: n

text: Then,

text: using

text: and

text: we

text: get

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text:   with a given matrix    · · ·   0 ε 0 0 0 0  − 2 · · ·   ε 0 ε z 0 0 0   1 3 2   0 −ε z 0 ... 0 0 0   2 2  M (σ) =  0 0 −ε z ... 0 0 0  .  3 3  ... ... ... ... ... ... ... · · · − 0 0 0 ε − z − 0 ε z − · · · n 2 n 2 − n n 1 (cid:0) (cid:1) 0 0 0 0 ε − z − 0 n 1 n 1 · · · The system (17) has a unique solution W (σ) which satisﬁes the initial conditions W (σ ) = e0, e0, , e0 T ∈ 0 1 2 n for σ I. If Wt (σ) is the tra(cid:0)nsposed ma(cid:1)trix of W (σ) , then 0 d ∗ ∗ ∗ d ∗ ∗ ∗ d I WtI W = I WtI W + I WtI W (cid:0) (cid:1) dσ dσ dσ ∗ ∗ ∗ ∗ = I WtMtI W + I WtI MW (cid:2) (cid:3)∗ ∗ ∗ = I Wt MtI + I M W = 0 (cid:8) (cid:9) ∗ ∗ ∗ · · · b(cid:8)∗ecause of ∗the(cid:9)equation MtI + I M = 0 n×n where I = d·i·a·g(ε1, ε2, , εn). Also, we have I Wt (σ ) I W (σ ) = I where I is the unit matrix since e0, e0, , e0 is the pseudo-orthonormal 0 0 ∗ ∗ 1 2∈ n n-frame. As a result, we ﬁnd I Xt (σ) I X (σ) = I for any σ I. It means that the vector ﬁelds · · · e01, e02, , e→0n Eform pseudo-orthonorm(cid:90)al frame ﬁeld. Let α : I n1 be the regular non-null c(cid:82)urve given by σ ∈ α (σ) = x + e z1(σ)dσe (σ) dσ, σ I. (18) 0 1 σ0 { · · · } By the equality (17) and the linear independence of e , e , , e , we get that α (σ) is a non-null E 1 2 n · · · − space curve in n with p-shape curvatures κ˜ (σ) = z (σ) for i = 1, 2, , n 1. Also, the pseudo- 1 · · · i i orthonormal n-frame e (σ) , e (σ) , , e (σ) is a Frenet-Serret n-frame of the non-null curve α. 1 2 n By the Theorem 6 and 7, we get the following theorem. → R · · · − ∞ Theorem 8 Let z : I , i = 1, 2, , n 1, be the functions of class C . According to p-similarity i there exists a unique non-null space curve with p-shape curvatures z . i → E (cid:54) Example 9 Let p-shape curvatures (κ˜ , κ˜ ) of the α : I 3 be (0, a) , where a = 0 is real constant, (cid:18) 1 (cid:19)2 1 (cid:18) (cid:19) and the unit vector e (σ) be a timelike vector. Choose initial conditions 2 −√ 1 √ a √ a √ 1 e0 = 0, , , e0 = (1, 0, 0) , e0 = 0, , . (19) 1 2 3 1 + a2 1 + a2 1 + a2 1 + a2 (cid:18) (cid:19) (cid:16)(cid:112) (cid:17) (cid:16)(cid:112) (cid:17) Then, the system (17) describes a spacelike vector e deﬁned by 1 √ 1 −√ 1 √ a e (σ) = sinh 1 + a2σ , cosh 1 + a2σ , (20) 1 1 + a2 1 + a2 1 + a2 with e (0) = e0, in the Minkowski 3-space. Solving the equation (18) we obtain the spacelike Frenet 1 1 curve α (see ﬁgure 1) parameterized by 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: a

text: given

text: matrix

text: ε

text: ε

text: ε

text: z

text: −ε

text: z

text: M

text: (σ)

text: −ε

text: z

text: ε

text: z

text: ε

text: z

text: n

text: n

text: n

text: n

text: ε

text: z

text: n

text: n

text: The

text: system

text: has

text: a

text: unique

text: solution

text: W

text: (σ)

text: which

text: satisﬁes

text: the

text: initial

text: conditions

text: W

text: (σ

text: T

text: n

text: for

text: σ

text: I.

text: If

text: Wt

text: (σ)

text: is

text: the

text: of

text: W

text: (σ)

text: then

text: d

text: d

text: d

text: I

text: WtI

text: W

text: I

text: WtI

text: W

text: I

text: WtI

text: W

text: dσ

text: dσ

text: dσ

text: I

text: WtMtI

text: W

text: I

text: WtI

text: MW

text: I

text: Wt

text: MtI

text: I

text: M

text: W

text: of

text: MtI

text: I

text: M

text: n×n

text: where

text: I

text: εn).

text: Also,

text: we

text: have

text: I

text: Wt

text: (σ

text: I

text: W

text: (σ

text: I

text: where

text: I

text: is

text: the

text: unit

text: matrix

text: since

text: is

text: the

text: pseudo-orthonormal

text: 2∈

text: n

text: n-frame.

text: As

text: a

text: result,

text: we

text: ﬁnd

text: I

text: Xt

text: (σ)

text: I

text: X

text: (σ)

text: I

text: for

text: any

text: σ

text: I.

text: It

text: means

text: that

text: the

text: vector

text: ﬁelds

text: Eform

text: frame

text: ﬁeld.

text: Let

text: α

text: I

text: be

text: the

text: regular

text: non-null

text: given

text: by

text: σ

text: α

text: (σ)

text: x

text: e

text: (σ)

text: dσ,

text: σ

text: I.

text: σ0

text: By

text: the

text: equality

text: and

text: the

text: linear

text: independence

text: of

text: e

text: e

text: e

text: we

text: get

text: that

text: α

text: (σ)

text: is

text: a

text: non-null

text: E

text: n

text: space

text: curve

text: in

text: n

text: with

text: p-shape

text: curvatures

text: κ˜

text: (σ)

text: z

text: (σ)

text: for

text: i

text: n

text: Also,

text: the

text: pseudo-

text: i

text: i

text: orthonormal

text: n-frame

text: e

text: (σ)

text: e

text: (σ)

text: e

text: (σ)

text: is

text: a

text: Frenet-Serret

text: n-frame

text: of

text: the

text: non-null

text: curve

text: α.

text: n

text: By

text: the

text: Theorem

text: and

text: we

text: get

text: the

text: following

text: theorem.

text: R

text: Theorem

text: Let

text: z

text: I

text: i

text: n

text: be

text: the

text: functions

text: of

text: class

text: C

text: According

text: to

text: p-similarity

text: i

text: there

text: exists

text: a

text: unique

text: non-null

text: space

text: curve

text: with

text: p-shape

text: curvatures

text: z

text: i

text: E

text: Example

text: Let

text: p-shape

text: curvatures

text: (κ˜

text: κ˜

text: of

text: the

text: α

text: I

text: be

text: a)

text: where

text: a

text: is

text: real

text: constant,

text: and

text: the

text: unit

text: vector

text: e

text: (σ)

text: be

text: a

text: timelike

text: vector.

text: Choose

text: initial

text: conditions

text: a

text: a

text: Then,

text: the

text: system

text: describes

text: a

text: spacelike

text: vector

text: e

text: deﬁned

text: by

text: a

text: e

text: (σ)

text: sinh

text: cosh

text: with

text: e

text: in

text: the

text: Minkowski

text: Solving

text: the

text: equation

text: we

text: obtain

text: the

text: spacelike

text: Frenet

text: curve

text: α

text: (see

text: ﬁgure

text: parameterized

text: by

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