text: π γ0((cid:15)) = 2niu2ν((cid:15)), (1.8a) π ∆ γ3((cid:15)) = 2niu2 (cid:15) ν((cid:15)), (1.8b) where ν((cid:15)) is the density of states (DOS), (cid:88) (cid:112) (cid:88) | | 1 − (cid:15) − − ν((cid:15)) = δ((cid:15) ηεk) = (cid:15)2 ∆2 Θ(η(cid:15) ∆) (1.9) Ω ± 2π2v3 k,η= η  with Θ(x) being the Heaviside step funct(cid:90)ion,   x 1 (x > 0), Θ(x) = dt δ(t) = −∞ 0 (x < 0). From these, the retarded Green function including the self energy is obtained as (cid:16) (cid:17) 1 · GRk((cid:15)) = DR((cid:15) + µ) gR0 ((cid:15) + µ) + ρ2gR2 (k) σ + gR3 ((cid:15) + µ)ρ3 , (1.10) k where − − − DRk((cid:15)) = ((cid:15) + iγ0((cid:15)))2 v2k2 (∆ iγ3((cid:15)))2, (1.11a) gR0 ((cid:15)) = (cid:15) + iγ0((cid:15)), (1.11b) − gR(k) = vk, (1.11c) 2 − gR3 ((cid:15)) = ∆ iγ3((cid:15)). (1.11d) O We dropped the real part of the self energy since they are just (ni) and can be absorbed into µ and ∆. To include skew scattering in the above formalism, we ﬁrst consider the self-consistent T-matrix ap- (cid:28) (cid:28) proximation, and then restrict to the case ni 1 and πuν(µ) 1. Then, the self energy reduces to the one in the Born approximation [Eq. (1.6)]. II. CALCULATION OF SPIN HALL CONDUCTIVITY 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: π

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text: π

text: where

text: is

text: the

text: density

text: of

text: states

text: (DOS),

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text: ηεk)

text: ∆2

text: Ω

text: k,η=

text: η

text: with

text: Θ(x)

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text: the

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text: step

text: x

text: (x

text: Θ(x)

text: dt

text: δ(t)

text: (x

text: From

text: these,

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text: Green

text: function

text: including

text: the

text: self

text: energy

text: is

text: obtained

text: as

text: µ)

text: µ)

text: (k)

text: σ

text: µ)ρ3

text: k

text: where

text: gR(k)

text: vk,

text: O

text: We

text: dropped

text: the

text: real

text: part

text: of

text: the

text: self

text: energy

text: since

text: they

text: are

text: just

text: (ni)

text: and

text: can

text: be

text: absorbed

text: into

text: µ

text: and

text: To

text: include

text: skew

text: scattering

text: in

text: the

text: above

text: formalism,

text: we

text: ﬁrst

text: consider

text: the

text: self-consistent

text: T-matrix

text: ap-

text: proximation,

text: and

text: then

text: restrict

text: to

text: the

text: case

text: ni

text: and

text: πuν(µ)

text: Then,

text: the

text: self

text: energy

text: reduces

text: to

text: the

text: one

text: in

text: the

text: Born

text: approximation

text: [Eq.

text: II.

text: CALCULATION

text: OF

text: SPIN

text: HALL

text: CONDUCTIVITY

text: ##LTLine##

text: ##LTLine##

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text: where U = U((cid:15) + µ), V = V((cid:15) + µ), and     (cid:88)∞  n  − −  2U 0 (1 2U) 1 0 Aˆ = n=0 0  U = 0 (1 − U)−1 , (2.18) 0 V Bˆ = − . (2.19) V 0 O Hence, the VCs up to (n ) are evaluated as i 1 V Λ1,x = 1 − 2U ρ1σx + (1 − U)(1 − 2U)ρ2σx, (2.20a) − V 1 Λ2,x = (1 − U)(1 − 2U)ρ1σx + 1 − U ρ2σx. (2.20b) ∗ ∗ O Similarly, by interchanging R and A in Eqs. (2.4), we obtain Λ and Λ up to (n ) as 1,x 2,x i ∗ 1 − V Λ1,x = 1 − 2U ρ1σx (1 − U)(1 − 2U)ρ2σx, (2.21a) ∗ V 1 Λ2,x = +(1 − U)(1 − 2U)ρ1σx + 1 − U ρ2σx, (2.21b) Note that only the sign of V is diﬀerent from Eqs. (2.20). By using the above relations and Eqs. (2.4), the terms with VCs are obtained as KVC = Kld + Ksk with surf surf surf (cid:90) (cid:32) (cid:33) ∞ Kld = − (cid:90) d(cid:15) −(cid:32)∂ fFD π(cid:33)∆ν((cid:15)) (β((cid:15)) − 1) + O(γ2), (2.22) surf −∞ ∂(cid:15) (cid:15)2 + ∆2 ∞ Ksk = 1 d(cid:15) −∂ fFD π∆ν((cid:15))β((cid:15))(U((cid:15)))2 + O(n0u). (2.23) surf niu −∞ ∂(cid:15) (cid:15) i It should be noted that Kb+ld = Kb + Kld is simply expressed as surf surf surf (cid:90) (cid:32) (cid:33) ∞ Kb+ld = − d(cid:15) −∂ fFD π∆ν((cid:15)) β((cid:15)) + O(γ2), (2.24) surf −∞ ∂(cid:15) (cid:15)2 + ∆2 and from Eqs. (2.11) and (2.24), the eﬀect of side-jump is contained in the factor 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

label: paragraph

text: U

text: µ),

text: V

text: µ),

text: and

label: equation

text: n

text: Aˆ

text: n=0

text: U

text: V

text: Bˆ

text: V

text: O

text: Hence,

text: the

text: VCs

text: up

text: to

text: (n

text: are

text: evaluated

text: as

text: i

text: V

text: ρ1σx

text: V

text: U

text: O

text: Similarly,

text: by

text: interchanging

text: R

text: and

text: A

text: in

text: Eqs.

text: we

text: obtain

text: Λ

text: and

text: Λ

text: up

text: to

text: (n

text: as

text: i

text: V

text: ρ1σx

text: V

text: U

text: Note

text: that

text: only

text: the

text: sign

text: of

text: V

text: is

text: diﬀerent

text: from

text: Eqs.

text: By

text: using

text: the

text: above

text: relations

text: and

text: Eqs.

text: the

text: terms

text: with

text: VCs

text: are

text: obtained

text: as

text: KVC

text: Kld

text: Ksk

text: with

text: surf

text: surf

text: surf

text: Kld

text: fFD

text: surf

text: ∆2

text: Ksk

text: fFD

text: surf

text: niu

text: i

text: It

text: should

text: be

text: noted

text: that

text: Kb+ld

text: Kb

text: Kld

text: is

text: simply

text: expressed

text: as

text: surf

text: surf

text: surf

text: Kb+ld

text: fFD

text: surf

text: ∆2

text: and

text: from

text: Eqs.

text: and

text: the

text: eﬀect

text: of

text: side-jump

text: is

text: contained

text: in

text: the

text: factor

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text: ≡ 1 9((cid:15)2 + ∆2)2 β((cid:15)) − − = , (2.25) (1 U((cid:15)))(1 2U((cid:15))) 2((cid:15)2 + 2∆2)((cid:15)2 + 5∆2) ≤ ≤ ≤ | | ≤ ∞ which has a value between 1 β((cid:15)) 9/2 for ∆ (cid:15) . III. RESULTS AND DISCUSSION We summarize the analytic results of the spin Hall conductivity at T = 0, σzs,yx/σ0s = σ˜ b+ld + σ˜ sk + σ˜ (sienat) (3.1) with the dimensionless spin Hall conductivities, 9(µ˜2 + 1) σ˜ b+ld = (µ˜2 + 2)(µ˜2 + 5)ν˜, (3.2a) − − ∆ (cid:90)(µ˜2 1)2 ν˜ σ˜ sk = niu (µ˜2 + 2)(µ˜2 + 5) µ˜ , (3.2b) σ˜ (sienat) = −µ˜ν˜2 − −µ∞˜ dxν˜x(x3), (3.2c) ∗ where σ0 = eg ∆/4π2v = mev/2π2 is the unit of the spin Hall conductivity, µ˜ = µ/∆ is the chemical s potential divided by the gap, ν˜(x) = (2π2v3/∆2)ν((cid:15)) with x = (cid:15)/∆ is the dimensionless DOS [Eq. (1.9)], and ν˜ = ν˜(µ˜) is that at the Fermi level. Here, σ˜ b+ld is contributed from the Fermi-surface term of the bare bubble and the ladder type VC, σ˜ sk is a contribution due to the skew scattering, and σ˜ (sienat) is the intrinsic contribution from the Fermi-sea term. We now verify that the intrinsic contributions of our results are consistent with those of Fuseya et al. [15]. Since we have integrated by parts using Eq. (2.12), σ˜ (sienat) seems to be diﬀerent from KsIyIx of Ref. [15] [Eq. (10) in Ref. [15]], but σ˜ (sienat) is further calculated as      (cid:112)   log | | 2E˜c − (|µ˜| > 1), σ˜ (sienat) = µ˜ + µ˜2 1 (3.3) | | ≤ log(2E˜c) ( µ˜ 1), 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: which

text: has

text: a

text: value

text: between

text: for

text: III.

label: section

text: RESULTS

text: AND

text: DISCUSSION

text: We

text: summarize

text: the

text: analytic

text: results

text: of

text: the

text: spin

text: Hall

text: conductivity

text: at

text: T

text: σ˜

text: b+ld

text: σ˜

text: sk

text: σ˜

text: (sienat)

text: with

text: the

text: dimensionless

text: spin

text: Hall

text: conductivities,

text: σ˜

text: b+ld

text: (µ˜2

text: ν˜

text: σ˜

text: sk

text: niu

text: (µ˜2

text: µ˜

text: σ˜

text: (sienat)

text: −µ˜ν˜2

text: −µ∞˜

text: where

text: σ0

text: eg

text: is

text: the

text: unit

text: of

text: the

text: spin

text: Hall

text: conductivity,

text: µ˜

text: µ/∆

text: is

text: the

text: chemical

text: s

text: potential

text: divided

text: by

text: the

text: gap,

text: ν˜(x)

text: with

text: x

text: is

text: the

text: dimensionless

text: DOS

text: [Eq.

text: and

text: ν˜

text: ν˜(µ˜)

text: is

text: that

text: at

text: the

text: Fermi

text: level.

text: Here,

text: σ˜

text: b+ld

text: is

text: contributed

text: from

text: the

text: Fermi-surface

text: term

text: of

text: the

text: bare

text: bubble

text: and

text: the

text: ladder

text: type

text: VC,

text: σ˜

text: sk

text: is

text: a

text: contribution

text: due

text: to

text: the

text: skew

text: scattering,

text: and

text: σ˜

text: (sienat)

text: is

text: the

text: intrinsic

text: contribution

text: from

text: the

text: Fermi-sea

text: term.

text: We

text: now

text: verify

text: that

text: the

text: intrinsic

text: contributions

text: of

text: our

text: results

text: are

text: consistent

text: with

text: those

text: of

text: Fuseya

text: et

text: al.

text: Since

text: we

text: have

text: integrated

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text: parts

text: using

text: Eq.

text: σ˜

text: (sienat)

text: seems

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text: diﬀerent

text: from

text: KsIyIx

text: of

text: Ref.

text: [Eq.

text: in

text: Ref.

text: but

text: σ˜

text: (sienat)

text: is

text: further

text: calculated

text: as

text: log

text: (|µ˜|

text: σ˜

text: (sienat)

text: µ˜

text: µ˜2

text: µ˜

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: formation also plays a role in human visual focus of atten- VGG16 [26] or GoogleNet [27]. Some utilize these net- tion [10]). Further, in RGB-D salient object detection, the works for extracting the low features [15, 16, 18]. Lee et effectiveness of background enclosure and of depth contrast al. incorporate high-level features based on these networks, have been demonstrated. Finally, deep CNNs have been along with low level features [15]. This approach to incor- shown to be effective for RGB salient object detection. porating top-down semantic information about objects into This paper makes three major contributions. (1) We pro- salient object detection has been effective. pose a novel learning architecture that provides the ﬁrst complete RGB-D salient object detection systems using a RGB-D Salient Object Detection Compared to RGB deep CNN, incorporating high level features, depth contrast salient object detection, fewer methods use RGB-D values and low-level features, and a novel mid-level feature. (2) for computing saliency. Peng et al. calculate a saliency map We introduce the background enclosure distribution, BED, by combining low, middle, and high level saliency infor- a novel mid-level depth feature that is suitable for learning mation [21]. Ren et al. calculate region contrast and use based on the idea of background enclosure. (3) We intro- background, depth, and orientation priors. They then pro- duce a set of low level features that are suitable for learning duce a saliency map by applying PageRank and a MRF to that incorporate the idea of depth contrast. the outputs [24]. Ju et al. calculate the saliency score us- We show that our new approach produces state-of-the- ing anisotropic center-surround difference and produce a art results for RGB-D salient object detection. Further, we saliency map by reﬁning the score applying Grabcut seg- evaluate the effectiveness of adding depth features, and of mentation and a 2D Gaussian ﬁlter [13]. Feng et al. im- adding the mid-level feature in particular. In ablation stud- prove RGB-D salient object detection results based on the ies, we show that incorporating low-level features that in- idea that salient objects are more likely to be in front of their corporate depth contrast performs better than RGB saliency surroundings for a large number of directions [6]. All the alone, and that adding our new mid-level feature, BED, im- existing RGB-D methods use hand-crafted parameters, such proves results further. as for scale and weights between metrics. However, real The rest of the paper is organized as follows: Section world scenes contain unpredictable object arrangements for 2 presents the related work on salient object detection and which ﬁxed hand coded parameters may limit generaliza- deep CNN methods for the other computer vision task. In tion. No published papers have yet presented a CNN archi- Section 3, we introduce the idea on RGB-D salient object tecture for RGB-D salient object detection. A preliminary detection with a deep CNN architecture and our algorithms paper (Arxiv only) uses only low-level color and depth fea- for salient object detection. This is followed by the detailed tures [22]. neural network structure, technical details and the training approach in Section 4. We introduce systematic and ex- Datasets Two datasets are widely used for RGB-D salient tensive experimental results in Section 5, and conclude in object detection, RGBD1000 [21] and NJUDS2000 [13]. Section 6. The RGBD1000 datasets contain 1000 RGB-D images cap- 2. Related Work tured by a standard Microsoft Kinect. The NJUDS2000 datasets contain around 2000 RGB-D images captured by Saliency detection to model eye movements began with a Fuji W3 stereo camera. low-level hand-crafted features, with classic work by Itti et al. [10] being inﬂuential. A variety of salient object detec- 3. A novel deep CNN architecture for detecting tion methods have been proposed in recent years, we focus salient objects in RGB-D images on these as more relevant to our work. RGB Salient object detection In RGB salient object de- In this section, we introduce our approach to RGB-D tection, methods often measure constrast between features salient object detection. Our novel deep CNN learning ar- of a region versus its surrounds, either locally and/or glob- chitecture is depicted in Figure 2. We combine the strengths ally [5, 10]. Contrast is mostly computed with respect to of previous approaches to high-level and low-level feature- appearance-based features such as colour, texture, and in- based deep CNN RGB salient object detection [15], with a tensity edges [4, 12]. depth channel, incorporating raw depth, low level cues to capture depth contrast, and a novel BED feature to capture background enclosure. RGB salient object detection using deep CNNs Re- 3.1. BED Feature cently, methods using deep CNNs have obtained strong re- sults for RGB salient object detection. Wang et al. [28] High-level and low-level features have been shown to combine local information and a global search. Often lead to high performance for detecting salient objects in the networks make use of deep CNN networks for object RGB images in a deep CNN framework [15]. We also classiﬁcation for a large number of classes, speciﬁcally know that the effective encoding of depth input can improve 2

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text: detection.

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text: 5 4 (a) 2 (a) 1 2 4 (b) -40 -20 0 20 40 2 4 (b) 8 (c) 2 4 -10 -5 0 5 10 t1-t2 ((cid:1)s) -80 -40 0 40 80 t1-t2 ((cid:1)s) FIG. 4: Measured second-order temporal coherence functions. (a), (b), and (c) are the results for one, two, and three RGs, FIG. 5: Measured second-order temporal coherence functions − respectively. g(2)(t t ) is the normalized second-order co- for two RGs with diﬀerent rotation speeds. (a) is measured 1 − 2 herence function. t1 t2 is the time diﬀerence between the when RG1 and RG2 are rotating at 30 and 3 Hz, respectively. two photon detection events within a two-photon coincidence (b) is measured when RG1 and RG2 are rotating at 3 and count. The squares are measured data points. The red curves 30 Hz, respectively. The squares are measured data points are theoretically ﬁttings. Two-photon bunching is observed and red curves are theoretically ﬁttings. The meanings of the in (a). Two-photon superbunching is observed in (b) and (c). symbols are similar as the ones in Fig. 4. IV. DISCUSSION In order to study the dependence of second-order co- In the last two sections, we have theoretically and ex- herence function on the coherence time of pseudother- perimentally proved that two-photon superbunching can mal light scattered by each RG, we measured the second- be observed in our scheme. In this section, we will dis- order coherence functions when two RGs rotate at very cuss why two-photon superbunching can be observed and diﬀerent speeds. Figure 5(a) shows the second-order tem- generalize it to three- and multi-photon superbunching. poral coherence function when RG and RG are rotating From quantum optical coherence point of view, the 1 2 at 30 and 3 Hz, respectively. The measured g(2)(0) equals key to have two-photon superbunching is to have more ± 3.74 0.05. The measured coherence time of pseudother- than two alternatives to trigger a two-photon coincidence ± mal light scattered by RG and RG is 1.62 0.05 and count in a HBT interferometer [32, 33]. As stated in Sec. ± 1 2 21.44 0.56 µs, respectively. The measured second-order , the premise to have Eq. (7) is that all the diﬀerent al- coherence function in Fig. 5(a) is a product of two peaks ternatives to trigger a two-photon coincidence count are with very diﬀerent widths. The results in Fig. 5(b) are in principle indistinguishable. Probability amplitudes are similar as the ones in Fig. 5(a) except the second-order summed to calculate the second-order coherence function coherence function is measured when RG and RG are in this case and there is two-photon interference [46]. 1 2 rotating at 3 and 30 Hz, respectively. The measured The constructive two-photon interference is the reason ± g(2)(0) equals 3.69 0.03. The coherence time of pseu- why two-photon bunching [41–44] and two-photon su- ± dothermal light scattered by RG and RG is 53.97 0.97 perbunching [32, 33] can be observed. The pinhole af- ± 1 2 and 4.32 0.07 µs, respectively. The reasons why the co- ter every RG is employed to ensure that photons passing herence time in Figs. 5(a) and 5(b) is diﬀerent when the through the pinhole are indistinguishable. speeds of rotation are the same are as follows. One reason Martienssen and Spiller had proved that photons is the sizes of light spot on these two RGs are diﬀerent. within the same coherence volume are in principle in- Another reason is the distances between the centers of distinguishable based on the Uncertainty Principle [45]. light spot and RG are not equal for two RGs. These two Figure 6 shows the coherence volume of thermal light, factors will inﬂuence the coherence time of pseudother- which equals the product of transverse coherence area mal light [45, 49]. and longitudinal coherence length [6]. DL is the prop- )2t-1t()2(g )2( ) t- t( g 2 1 ##LTFigure## ##LTFigure## ##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## ##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## ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: (a)

text: (a)

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text: (b)

text: (c)

text: FIG.

text: Measured

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text: (a),

text: (b),

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text: (c)

text: are

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text: two,

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text: 4 − III. PHENOMENOLOGY OF Λ Λ¯ TRANSITIONS − Now let us discuss the implications of Λ Λ oscilla- tions on the phenomenological side 7. So far that no baryon number violating process has been observed in the Standard Model (SM), thus any possible signal would be deﬁnitely exciting and eﬀorts along this direction are de- served. New physics eﬀects can be also probed at hadron- hadron colliders. As discussed in Ref. [40], BES detector with huge J/ψ data sample has been proposed to measure − the possible Λ Λ oscillation by studying the quantum − coherent ΛΛ states. Following the treatment of D0 D0 FIG. 1. We show the two quark four squarks cross section mixing [41]. To avoid the complications due to mixing, σ(y) (ﬂavor independent part) with respect to the normalized the charged B was used as an example in Ref. [42]. The s-variable y = s/Λ2 and neglecting squark masses. For y < 1, ∼ (cid:39) eigenstates emerging from the oscillations can be written σ y2; for y 1 the amplitude will saturate the unitarity as bound of string theory. For y > 1 and ﬁxing the scattering angle, the amplitude falls oﬀ exponentially with the produc- tion of the non-perturbative conﬁguration –as displayed in | (cid:105) | (cid:105) | (cid:105) this ﬁgure. ΛH = p Λ + q Λ , | (cid:105) | (cid:105) − | (cid:105) Λ = p Λ q Λ , (16) L | | | | with the normalization condition p2 + q 2 = 1. The subscript “H” (“L”) means the heavy (light) m√ass eige√n- state√s. Here √p and q can be parametrized as 1 + z/ 2 − and 1 z/ 2, respectively, where z involves the oscil- lation mass δm appearing in the oﬀ-diagonal elements ΛΛ of the eﬀective Hamiltonian used to describe the time − evolution of the Λ Λ system. For details, see Ref. [40] or the general consideration in [41]. One can deﬁne the following mass and width: FIG. 2. Diagram for a Λ − Λ¯ transition. u, d, s are right- m ≡ mH +2 mL , ∆m ≡ mH − mL handed up, down and strange quarks. Four quarks in the ≡ Γ + Γ ≡ − diagram are converted to the corresponding squarks through Γ H L , ∆Γ Γ Γ , (17) the exchange of two gaugini. The black vertex is directly 2 H L generated by an Exotic Instanton. from which the mixing parameters are deﬁned as ≡ ∆m ≡ ∆Γ of view, we expect that, with the growing of CM en- x , y . (18) Λ Γ Λ 2Γ ergy, the E2-brane starts to oscillate and its dynam- ics is described by oscillations of modulini ﬁelds. The number of modulini is a topological invariant of the ex- For the free Λ case without external magnetic ﬁelds, otic instanton solution, associated with the invariance ∆m = 2δm . For simplicity, we will conﬁne ourselves ΛΛ of intersections with physical D-branes. At the non- to this case. The inﬂuence of external magnetic ﬁeld is perturbative scale, open strings associated to modulini discussed in Ref. [40]. The dominant decay mode of Λ is → − have to reggeize. Loop-corrections to tree-level mixed- Λ pπ with branching ratio 64% [43], and correspond- → disk amplitudes at ﬁxed angle are expected to add expo- ingly Λ pπ+ assuming CP invariance. In the case that nentially suppressed contributions in the form of Eq.(15). We conjecture that this is the main contribution to the unitarization of the scattering amplitude for ﬁxed angle kinematical regime and s >> Λ2. 7 Of course, in the standard model analysis, these baryon-violating eﬀects need not to be considered, e.g., see the CP violation of → We will return to phenomenological implications in the Λc Λπ decay [35] and also the conventional NN¯ scattering next sections. [36–39]. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: III.

text: PHENOMENOLOGY

text: OF

text: Λ

text: Λ¯

text: TRANSITIONS

text: Now

text: let

text: us

text: discuss

text: the

text: implications

text: of

text: Λ

text: Λ

text: oscilla-

text: tions

text: on

text: the

text: phenomenological

text: side

text: So

text: far

text: that

text: no

text: baryon

text: number

text: violating

text: process

text: has

text: been

text: observed

text: in

text: the

text: Standard

text: Model

text: (SM),

text: thus

text: any

text: possible

text: signal

text: would

text: be

text: deﬁnitely

text: exciting

text: and

text: eﬀorts

text: along

text: this

text: direction

text: are

text: de-

text: served.

text: New

text: physics

text: eﬀects

text: can

text: be

text: also

text: probed

text: at

text: hadron-

text: hadron

text: colliders.

text: As

text: discussed

text: in

text: Ref.

text: BES

text: detector

text: with

text: huge

text: J/ψ

text: data

text: sample

text: has

text: been

text: proposed

text: to

text: measure

text: the

text: possible

text: Λ

text: Λ

text: oscillation

text: by

text: studying

text: the

text: quantum

text: coherent

text: ΛΛ

text: states.

text: Following

text: the

text: treatment

text: of

text: FIG.

text: We

text: show

text: the

text: two

text: quark

text: four

text: squarks

text: cross

text: section

text: mixing

text: To

text: avoid

text: the

text: complications

text: due

text: to

text: mixing,

text: σ(y)

text: (ﬂavor

text: independent

text: part)

text: with

text: respect

text: to

text: the

text: normalized

text: the

text: charged

text: B

text: was

text: used

text: as

text: an

text: example

text: in

text: Ref.

text: The

text: s-variable

text: y

text: and

text: neglecting

text: squark

text: masses.

text: For

text: y

text: eigenstates

text: emerging

text: from

text: the

text: oscillations

text: can

text: be

text: written

text: σ

text: for

text: y

text: the

text: amplitude

text: will

text: saturate

text: the

text: unitarity

text: as

text: bound

text: of

text: string

text: theory.

text: For

text: y

text: and

text: ﬁxing

text: the

text: scattering

text: angle,

text: the

text: amplitude

text: falls

text: oﬀ

text: exponentially

text: with

text: the

text: produc-

text: tion

text: of

text: the

text: non-perturbative

text: conﬁguration

text: –as

text: displayed

text: in

text: this

text: ﬁgure.

text: ΛH

text: p

text: Λ

text: q

text: Λ

text: Λ

text: p

text: Λ

text: q

text: Λ

text: L

text: with

text: the

text: normalization

text: condition

text: q

text: The

text: subscript

text: “H”

text: (“L”)

text: means

text: the

text: heavy

text: (light)

text: m√ass

text: eige√n-

text: state√s.

text: Here

text: √p

text: and

text: q

text: can

text: be

text: parametrized

text: as

text: z/

text: and

text: z/

text: respectively,

text: where

text: z

text: involves

text: the

text: oscil-

text: lation

text: mass

text: δm

text: appearing

text: in

text: the

text: oﬀ-diagonal

text: elements

text: ΛΛ

text: of

text: the

text: eﬀective

text: Hamiltonian

text: used

text: to

text: describe

text: the

text: time

text: evolution

text: of

text: the

text: Λ

text: Λ

text: system.

text: For

text: details,

text: see

text: Ref.

text: or

text: the

text: general

text: consideration

text: in

text: One

text: can

text: deﬁne

text: the

text: following

text: mass

text: and

text: width:

text: FIG.

text: Diagram

text: for

text: a

text: Λ

text: Λ¯

text: transition.

text: u,

text: d,

text: s

text: are

text: right-

text: m

text: mH

text: mL

text: ∆m

text: mH

text: mL

text: handed

text: up,

text: down

text: and

text: strange

text: quarks.

text: Four

text: quarks

text: in

text: the

text: Γ

text: Γ

text: diagram

text: are

text: converted

text: to

text: the

text: corresponding

text: squarks

text: through

text: Γ

text: H

text: L

text: ∆Γ

text: Γ

text: Γ

text: the

text: exchange

text: of

text: two

text: gaugini.

text: The

text: black

text: vertex

text: is

text: directly

text: H

text: L

text: generated

text: by

text: an

text: Exotic

text: Instanton.

text: from

text: which

text: the

text: mixing

text: parameters

text: are

text: deﬁned

text: as

text: ∆m

text: ∆Γ

text: of

text: view,

text: we

text: expect

text: that,

text: with

text: the

text: growing

text: of

text: CM

text: en-

text: x

text: y

text: Λ

text: Γ

text: Λ

text: 2Γ

text: ergy,

text: the

text: starts

text: to

text: oscillate

text: and

text: its

text: dynam-

text: ics

text: is

text: described

text: by

text: oscillations

text: of

text: modulini

text: ﬁelds.

text: The

text: number

text: of

text: modulini

text: is

text: a

text: topological

text: invariant

text: of

text: the

text: ex-

text: For

text: the

text: free

text: Λ

text: case

text: without

text: external

text: magnetic

text: ﬁelds,

text: otic

text: instanton

text: solution,

text: associated

text: with

text: the

text: invariance

text: ∆m

text: For

text: simplicity,

text: we

text: will

text: conﬁne

text: ourselves

text: ΛΛ

text: of

text: intersections

text: with

text: physical

text: D-branes.

text: At

text: the

text: non-

text: to

text: this

text: case.

text: The

text: inﬂuence

text: of

text: external

text: magnetic

text: ﬁeld

text: is

text: perturbative

text: scale,

text: open

text: strings

text: associated

text: to

text: modulini

text: discussed

text: in

text: Ref.

text: The

text: dominant

text: decay

text: mode

text: of

text: Λ

text: is

text: have

text: to

text: reggeize.

text: Loop-corrections

text: to

text: tree-level

text: mixed-

text: Λ

text: pπ

text: with

text: branching

text: ratio

text: and

text: correspond-

text: disk

text: amplitudes

text: at

text: ﬁxed

text: angle

text: are

text: expected

text: to

text: add

text: expo-

text: ingly

text: Λ

text: pπ+

text: assuming

text: CP

text: invariance.

text: In

text: the

text: case

text: that

text: nentially

text: suppressed

text: contributions

text: in

text: the

text: form

text: of

text: We

text: conjecture

text: that

text: this

text: is

text: the

text: main

text: contribution

text: to

text: the

text: unitarization

text: of

text: the

text: scattering

text: amplitude

text: for

text: ﬁxed

text: angle

text: kinematical

text: regime

text: and

text: s

text: Of

text: course,

text: in

text: the

text: standard

text: model

text: analysis,

text: these

text: baryon-violating

text: eﬀects

text: need

text: not

text: to

text: be

text: considered,

text: e.g.,

text: see

text: the

text: CP

text: violation

text: of

text: We

text: will

text: return

text: to

text: phenomenological

text: implications

text: in

text: the

text: Λc

text: Λπ

text: decay

text: and

text: also

text: the

text: conventional

text: NN¯

text: scattering

text: next

text: sections.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTLine##

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text: 5 − → → − Λ Λ oscillation happens, the process Λ Λ pπ will be possible. This phenomenon can be probed by count- N → → − − ing the event number for J/ψ ΛΛ (pπ )(pπ ) –wrong-sign decay– while the right-sign decay would be → → − J/ψ ΛΛ (pπ )(pπ+). Time-integrated decay rate for the wrong-sign decay relative to the right sign is found to be (cid:16) (cid:17) (cid:16) (cid:17) N → → − − R J/ψ ΛΛ (pπ )(pπ ) x2 + y2 = = Λ Λ . (19) N → → − 2 J/ψ ΛΛ (pπ )(pπ+) A discussion of time-dependent observables is also pre- sented in [40], and the BES-III detector is capable to access this time information. Assuming y = 0 – it is Λ indeed a quite small quantity – one can get an estimate of the mixing mass parameter δm from Eq. (19) as ΛΛ √ √1 R δm = Γ. (20) ΛΛ 2 FIG. 3. a) Four anti-squark production in high energy col- → lisions of two quarks q q q¯˜ q¯˜ q¯˜ q¯˜ . b) Four quark f1 f2 f3 f4 f5 f6 → According to ×the designed lumi×nosity o(cid:48)f BEPC-II in production in in high energy collisions of two quarks qf1qf2 Beijing [44], 10 109 J/ψ and 3 109 ψ data samples q¯f3q¯f4q¯f5q¯f6. Black vertices are induced by exotic instantons. will be collected by the runnings per year. If ﬁnally no signal of wrong-sign decay can be detected, one should R ≤ × − put an upper limit 4 10 7 and correspondingly a more stringent bound on such new-physics (NP) sig- ≤ − δm 10 15 MeV at the 90% of conﬁdence level (C.L.) nal. However, we will show that the required energy to ΛΛ [40], inferring from the knowledge of interval estimate for generate such NP phenomenon (e.g., in exotic instanton (cid:112)× very rare signal. Currently, a sample of 1.31 109 J/ψ model below) can be accessed in the near future. Re- events has been collected [45], then the aforementioned cently, the Circular Electron Positron Collider (CEPC) ≈ upper limit will be increased by a factor of 10/1.31 has been proposed in China, which arouses great interests ≤ × − 2.8, i.e. δm 3 10 15 MeV. Consequently, the in the community [48]. According to the design agenda, ΛΛ oscillation time will be bounded by the upper limit as the electron-positron collider will be converted into a × − 2.5 10 7 s at 90% conﬁdence level. This would be the proton-proton collider, with an unprecedented center-of- ﬁrst search in the experiment. See also [40] for more mass energy of 50-90 TeV at the second phase. This is details on this analysis. the project of super proton-proton collider (SppC) [49]. Note that Υ(4S) also has the same quantum numbers At such a high energy, the new physics beyond LHC dis- − −− as J/ψ, i.e., IG(JPC) = 0 (1 ), and can also decay to cussed in the present paper would be accessible. Future coherent ΛΛ states. Belle and BaBar detectors could be measurements are expected to give valuable hints on this used to probe this process as well. Currently, there are research line. × × − 772 106 Υ(4S) data available in Belle [46] and 471 106 IV. SPACE OF PARAMETERS, n n¯ AND for BaBar[47]. Taking into account the fact that the non- PROTON-PROTON COLLIDERS BB decay mode only constitutes less than 4% (at 95% O − conﬁdence level) of the total decay rate, much less ΛΛ An operator→ΛΛ¯ generates not only Λ Λ¯ transition events are expected. Otherwise assuming that most of but also NN KK transition. As discussed in the − Υ(4S) decays into ΛΛ, the event number can increase by previous section δmΛΛ¯ can be constrained up to 10 6 e→V oanbeovoerutpwpoerolridmeirtsfolargΛer−mΛ¯oroesctilhlaatnioBnEcSou. lAdlbtheoaucgchessthede bKyKneisxtjugsetnceorantsitornainoifngexOpeΛrΛ¯imuepnttso. δHmoΛwΛ¯ev<er,10N−2N1 eV. from the beginning in the experiment, we require an ex- δmΛΛ¯ is related to a New Physics scale by tremely higher luminosity to get a comparable bound to − the n n¯ case. One can ﬁnd this point from Eq. (20) – (cid:39) Λ6 the oscillation ma−ss is proportional to the square root of δmΛΛ¯ y2 MQC5 D , (21) the÷luminosity. n n¯ oscillat−ion time is constrained to be ΛΛ¯ 107 108 smaller than the Λ Λ¯ oscillation time, and thus M (cid:39) O the luminosity should be increased by order 1015, which so that ΛΛ¯ 100 TeV scale. On t−he other hand, nn¯ is seems unrealizable in the near future. In other words, ex- actually constrained by δm < 10 23 eV, corresponding M nn¯ tremely huge data samples would be needed to pin down to > 300 TeV [50, 51]. The next generations of nn¯ ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Λ

text: Λ

text: oscillation

text: happens,

text: the

text: process

text: Λ

text: Λ

text: pπ

text: will

text: be

text: possible.

text: This

text: phenomenon

text: can

text: be

text: probed

text: by

text: count-

text: N

text: ing

text: the

text: event

text: number

text: for

text: J/ψ

text: ΛΛ

text: (pπ

text: )(pπ

text: –wrong-sign

text: decay–

text: while

text: the

text: right-sign

text: decay

text: would

text: be

text: J/ψ

text: ΛΛ

text: (pπ

text: )(pπ+).

text: Time-integrated

text: decay

text: rate

text: for

text: the

text: wrong-sign

text: decay

text: relative

text: to

text: the

text: right

text: sign

text: is

text: found

text: to

text: be

text: N

text: R

text: J/ψ

text: ΛΛ

text: (pπ

text: )(pπ

text: Λ

text: Λ

text: N

text: J/ψ

text: ΛΛ

text: (pπ

text: )(pπ+)

text: A

text: discussion

text: of

text: time-dependent

text: observables

text: is

text: also

text: pre-

text: sented

text: in

text: and

text: the

text: BES-III

text: detector

text: is

text: capable

text: to

text: access

text: this

text: time

text: information.

text: Assuming

text: y

text: it

text: is

text: Λ

text: indeed

text: a

text: quite

text: small

text: quantity

text: one

text: can

text: get

text: an

text: estimate

text: of

text: the

text: mixing

text: mass

text: parameter

text: δm

text: from

text: Eq.

text: as

text: ΛΛ

text: √1

text: R

text: δm

text: Γ.

text: ΛΛ

text: FIG.

text: a)

text: Four

text: anti-squark

text: production

text: in

text: high

text: energy

text: col-

text: lisions

text: of

text: two

text: quarks

text: q

text: q

text: q¯˜

text: q¯˜

text: q¯˜

text: q¯˜

text: b)

text: Four

text: quark

text: According

text: to

text: ×the

text: designed

text: lumi×nosity

text: BEPC-II

text: in

text: production

text: in

text: in

text: high

text: energy

text: collisions

text: of

text: two

text: quarks

text: Beijing

text: J/ψ

text: and

text: ψ

text: data

text: samples

text: Black

text: vertices

text: are

text: induced

text: by

text: exotic

text: instantons.

text: will

text: be

text: collected

text: by

text: the

text: runnings

text: per

text: year.

text: If

text: ﬁnally

text: no

text: signal

text: of

text: wrong-sign

text: decay

text: can

text: be

text: detected,

text: one

text: should

text: R

text: put

text: an

text: upper

text: limit

text: and

text: correspondingly

text: a

text: more

text: stringent

text: bound

text: on

text: such

text: new-physics

text: (NP)

text: sig-

text: δm

text: MeV

text: at

text: the

text: of

text: conﬁdence

text: level

text: (C.L.)

text: nal.

text: However,

text: we

text: will

text: show

text: that

text: the

text: required

text: energy

text: to

text: ΛΛ

text: inferring

text: from

text: the

text: knowledge

text: of

text: interval

text: estimate

text: for

text: generate

text: such

text: NP

text: phenomenon

text: (e.g.,

text: in

text: exotic

text: instanton

text: very

text: rare

text: signal.

text: Currently,

text: a

text: sample

text: of

text: J/ψ

text: model

text: below)

text: can

text: be

text: accessed

text: in

text: the

text: near

text: future.

text: Re-

text: events

text: has

text: been

text: collected

text: then

text: the

text: aforementioned

text: cently,

text: the

text: Circular

text: Electron

text: Positron

text: Collider

text: (CEPC)

text: upper

text: limit

text: will

text: be

text: increased

text: by

text: a

text: factor

text: of

text: has

text: been

text: proposed

text: in

text: China,

text: which

text: arouses

text: great

text: interests

text: i.e.

text: δm

text: MeV.

text: Consequently,

text: the

text: in

text: the

text: community

text: According

text: to

text: the

text: design

text: agenda,

text: ΛΛ

text: oscillation

text: time

text: will

text: be

text: bounded

text: by

text: the

text: upper

text: limit

text: as

text: the

text: electron-positron

text: collider

text: will

text: be

text: converted

text: into

text: a

text: s

text: at

text: conﬁdence

text: level.

text: This

text: would

text: be

text: the

text: proton-proton

text: collider,

text: with

text: an

text: unprecedented

text: center-of-

text: ﬁrst

text: search

text: in

text: the

text: experiment.

text: See

text: also

text: for

text: more

text: mass

text: energy

text: of

text: TeV

text: at

text: the

text: second

text: phase.

text: This

text: is

text: details

text: on

text: this

text: analysis.

text: the

text: project

text: of

text: super

text: proton-proton

text: collider

text: (SppC)

text: Note

text: that

text: also

text: has

text: the

text: same

text: quantum

text: numbers

text: At

text: such

text: a

text: high

text: energy,

text: the

text: new

text: physics

text: beyond

text: LHC

text: dis-

text: as

text: J/ψ,

text: i.e.,

text: IG(JPC)

text: and

text: can

text: also

text: decay

text: to

text: cussed

text: in

text: the

text: present

text: paper

text: would

text: be

text: accessible.

text: Future

text: coherent

text: ΛΛ

text: states.

text: Belle

text: and

text: BaBar

text: detectors

text: could

text: be

text: measurements

text: are

text: expected

text: to

text: give

text: valuable

text: hints

text: on

text: this

text: used

text: to

text: probe

text: this

text: process

text: as

text: well.

text: Currently,

text: there

text: are

text: research

text: line.

text: data

text: available

text: in

text: Belle

text: and

text: IV.

text: SPACE

text: OF

text: PARAMETERS,

text: n

text: n¯

text: AND

text: for

text: Taking

text: into

text: account

text: the

text: fact

text: that

text: the

text: non-

text: PROTON-PROTON

text: COLLIDERS

text: BB

text: decay

text: mode

text: only

text: constitutes

text: less

text: than

text: (at

text: O

text: conﬁdence

text: level)

text: of

text: the

text: total

text: decay

text: rate,

text: much

text: less

text: ΛΛ

text: An

text: operator→ΛΛ¯

text: generates

text: not

text: only

text: Λ

text: Λ¯

text: transition

text: events

text: are

text: expected.

text: Otherwise

text: assuming

text: that

text: most

text: of

text: but

text: also

text: NN

text: KK

text: transition.

text: As

text: discussed

text: in

text: the

text: decays

text: into

text: ΛΛ,

text: the

text: event

text: number

text: can

text: increase

text: by

text: previous

text: section

text: δmΛΛ¯

text: can

text: be

text: constrained

text: up

text: to

text: e→V

text: lAdlbtheoaucgchessthede

text: eV.

text: from

text: the

text: beginning

text: in

text: the

text: experiment,

text: we

text: require

text: an

text: ex-

text: δmΛΛ¯

text: is

text: related

text: to

text: a

text: New

text: Physics

text: scale

text: by

text: tremely

text: higher

text: luminosity

text: to

text: get

text: a

text: comparable

text: bound

text: to

text: the

text: n

text: n¯

text: case.

text: One

text: can

text: ﬁnd

text: this

text: point

text: from

text: Eq.

text: Λ6

text: the

text: oscillation

text: ma−ss

text: is

text: proportional

text: to

text: the

text: square

text: root

text: of

text: δmΛΛ¯

text: D

text: the÷luminosity.

text: n

text: n¯

text: oscillat−ion

text: time

text: is

text: constrained

text: to

text: be

text: ΛΛ¯

text: smaller

text: than

text: the

text: Λ

text: Λ¯

text: oscillation

text: time,

text: and

text: thus

text: M

text: O

text: the

text: luminosity

text: should

text: be

text: increased

text: by

text: order

text: which

text: so

text: that

text: ΛΛ¯

text: TeV

text: scale.

text: On

text: t−he

text: other

text: hand,

text: nn¯

text: is

text: seems

text: unrealizable

text: in

text: the

text: near

text: future.

text: In

text: other

text: words,

text: ex-

text: actually

text: constrained

text: by

text: δm

text: eV,

text: corresponding

text: M

text: nn¯

text: tremely

text: huge

text: data

text: samples

text: would

text: be

text: needed

text: to

text: pin

text: down

text: to

text: TeV

text: The

text: next

text: generations

text: of

text: nn¯

text: ##LTFigure##

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4D ISIP: 4D IMPLICIT SURFACE INTEREST POINT DETECTION 3 in 4 directions (x,y,z,t). We directly calculated the 4D ISIP in (x,y,z,t) space and choose the points which simultaneously have large variations in different four directions. Utilizing a single Kinect to accurately recover 3D human action is an active and challenging research topic in recent years. Many methods [25, 2, 1] get 3D human model based on a trained human template, but those methods can’t be used to recon- struct the human body with clothes. Zhang [29] reconstructed human body with clothes. However all of those methods require the reconstructed human to stay still during ac- quisition, which is not possible in reality. Newcombe [15] proposed a real-time method to reconstruct dynamic scenes without any prior templates. But it is not capable of long term tracking, because of growing warp ﬁeld and error accumulation. Guo [7] intro- duced a novel L based motion regularizer with an iterative optimization solver, which 0 can robustly reconstruct non-rigid geometries and motions from single view depth in- put. In this paper, we combine Newcombe [15] and Guo [7] to build a system to con- struct 3D human motion dataset for 4D ISIP detection. 3 Method Before we introduce 4D-ISIP we introduce how to represent and acquire human ac- tion dataset ﬁrstly. We adopt the volumetric truncated signed distance function (TSDF) [4] to represent the 3D scene. we combine Newcombe [15] and Guo [7] to build a system to construct 3D human motion dataset for 4D ISIP detection. Then, we gave a detail about the 3D spatial temporal interest points below. Lastly, we will introduce how to extended 3D-STIP to 4D-ISIP. 3.1 Acquisition of 3D human motion dataset Upon acquisition of every input depth data, we ﬁrst estimate the ground plane using RANSAC to segment human body from the ground. This is followed by point neigh- borhood statistics to ﬁlter noise outlier data. DynamicFusion [15] can reconstruct non-rigidly deforming scenes in real-time by fusing RGBD scans acquired from commodity sensors without any template. It can gen- erate denoised, detailed and complete reconstructions. But it is not capable of long term tracking, because of growing warp ﬁeld and error accumulation. The method proposed by Guo [7] provides long term tracking using a human template. We use DynamicFu- sion to get a complete human body mesh by rotating our body in front of the Kinect as the template. Then we use Guo[20] to track human motion with partial data input. Lastly, We generated the same topology mesh for every motion frame. Then we transformed those mesh into TSDF representation. In Figure 1, we illustrate how we use TSDF to implicitly represent an arbitrary surface as zero crossings within the volume. The whole scene is represented by a 3D volume with a TSDF value in each voxel. TSDF (cid:26) is the truncated distance between spatial point and object surface. ≥ − min(1, η ) if η τ Ψ(η) − τ (1) 1 otherwise where τ is the threhold distance and η is the distance to surface, Ψ(η) is the trun- cated signed distance. 3.2 3D Spatial Temporal Interest Points ##LTLine##

text: ISIP:

text: IMPLICIT

text: SURFACE

text: INTEREST

text: POINT

text: DETECTION

text: in

text: directions

text: (x,y,z,t).

text: We

text: directly

text: calculated

text: the

text: ISIP

text: in

text: (x,y,z,t)

text: space

text: and

text: choose

text: the

text: points

text: which

text: simultaneously

text: have

text: large

text: variations

text: in

text: different

text: four

text: directions.

text: Utilizing

text: a

text: single

text: Kinect

text: to

text: accurately

text: recover

text: human

text: action

text: is

text: an

text: active

text: and

text: challenging

text: research

text: topic

text: in

text: recent

text: years.

text: Many

text: methods

text: get

text: human

text: model

text: based

text: on

text: a

text: trained

text: human

text: template,

text: but

text: those

text: methods

text: can’t

text: be

text: used

text: to

text: recon-

text: struct

text: the

text: human

text: body

text: with

text: clothes.

text: Zhang

text: reconstructed

text: human

text: body

text: with

text: clothes.

text: However

text: all

text: of

text: those

text: methods

text: require

text: the

text: reconstructed

text: human

text: to

text: stay

text: still

text: during

text: ac-

text: quisition,

text: which

text: is

text: not

text: possible

text: in

text: reality.

text: Newcombe

text: proposed

text: a

text: real-time

text: method

text: to

text: reconstruct

text: dynamic

text: scenes

text: without

text: any

text: prior

text: templates.

text: But

text: it

text: is

text: not

text: capable

text: of

text: long

text: term

text: tracking,

text: because

text: of

text: growing

text: warp

text: ﬁeld

text: and

text: error

text: accumulation.

text: Guo

text: intro-

text: duced

text: a

text: novel

text: L

text: based

text: motion

text: regularizer

text: with

text: an

text: iterative

text: optimization

text: solver,

text: which

text: can

text: robustly

text: reconstruct

text: non-rigid

text: geometries

text: and

text: motions

text: from

text: single

text: view

text: depth

text: in-

text: put.

text: In

text: this

text: paper,

text: we

text: combine

text: Newcombe

text: and

text: Guo

text: to

text: build

text: a

text: system

text: to

text: con-

text: struct

text: human

text: motion

text: dataset

text: for

text: ISIP

text: detection.

text: Method

text: Before

text: we

text: introduce

text: we

text: introduce

text: how

text: to

text: represent

text: and

text: acquire

text: human

text: ac-

text: tion

text: dataset

text: ﬁrstly.

text: We

text: adopt

text: the

text: volumetric

text: truncated

text: signed

text: distance

text: function

text: (TSDF)

text: to

text: represent

text: the

text: scene.

text: we

text: combine

text: Newcombe

text: and

text: Guo

text: to

text: build

text: a

text: system

text: to

text: construct

text: human

text: motion

text: dataset

text: for

text: ISIP

text: detection.

text: Then,

text: we

text: gave

text: a

text: detail

text: about

text: the

text: spatial

text: temporal

text: interest

text: points

text: below.

text: Lastly,

text: we

text: will

text: introduce

text: how

text: to

text: extended

text: to

text: Acquisition

text: of

text: human

text: motion

text: dataset

text: Upon

text: acquisition

text: of

text: every

text: input

text: depth

text: data,

text: we

text: ﬁrst

text: estimate

text: the

text: ground

text: plane

text: using

text: RANSAC

text: to

text: segment

text: human

text: body

text: from

text: the

text: ground.

text: This

text: is

text: followed

text: by

text: point

text: neigh-

text: borhood

text: statistics

text: to

text: ﬁlter

text: noise

text: outlier

text: data.

text: DynamicFusion

text: can

text: reconstruct

text: non-rigidly

text: deforming

text: scenes

text: in

text: real-time

text: by

text: fusing

text: RGBD

text: scans

text: acquired

text: from

text: commodity

text: sensors

text: without

text: any

text: template.

text: It

text: can

text: gen-

text: erate

text: denoised,

text: detailed

text: and

text: complete

text: reconstructions.

text: But

text: it

text: is

text: not

text: capable

text: of

text: long

text: term

text: tracking,

text: because

text: of

text: growing

text: warp

text: ﬁeld

text: and

text: error

text: accumulation.

text: The

text: method

text: proposed

text: by

text: Guo

text: provides

text: long

text: term

text: tracking

text: using

text: a

text: human

text: template.

text: We

text: use

text: DynamicFu-

text: sion

text: to

text: get

text: a

text: complete

text: human

text: body

text: mesh

text: by

text: rotating

text: our

text: body

text: in

text: front

text: of

text: the

text: Kinect

text: as

text: the

text: template.

text: Then

text: we

text: use

text: to

text: track

text: human

text: motion

text: with

text: partial

text: data

text: input.

text: Lastly,

text: We

text: generated

text: the

text: same

text: topology

text: mesh

text: for

text: every

text: motion

text: frame.

text: Then

text: we

text: transformed

text: those

text: mesh

text: into

text: TSDF

text: representation.

text: In

text: Figure

text: we

text: illustrate

text: how

text: we

text: use

text: TSDF

text: to

text: implicitly

text: represent

text: an

text: arbitrary

text: surface

text: as

text: zero

text: crossings

text: within

text: the

text: volume.

text: The

text: whole

text: scene

text: is

text: represented

text: by

text: a

text: volume

text: with

text: a

text: TSDF

text: value

text: in

text: each

text: voxel.

text: TSDF

text: is

text: the

text: truncated

text: distance

text: between

text: spatial

text: point

text: and

text: object

text: surface.

text: η

text: if

text: η

text: τ

text: Ψ(η)

text: τ

text: otherwise

text: where

text: τ

text: is

text: the

text: threhold

text: distance

text: and

text: η

text: is

text: the

text: distance

text: to

text: surface,

text: Ψ(η)

text: is

text: the

text: trun-

text: cated

text: signed

text: distance.

text: Spatial

text: Temporal

text: Interest

text: Points

text: ##LTLine##


text: 4D ISIP: 4D IMPLICIT SURFACE INTEREST POINT DETECTION 9 (a) Bowing (b) Waving (c) walking Fig. 6: 4D-ISIP detection results on dataset (a) weight-lifting (b) Clapping (c) Golf (d) Looking watch (e) Palying badminton Fig. 7: More results of 4D-ISIP detection ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: ISIP:

text: IMPLICIT

text: SURFACE

text: INTEREST

text: POINT

text: DETECTION

text: (a)

text: Bowing

text: (b)

text: Waving

text: (c)

text: walking

text: Fig.

text: detection

text: results

text: on

text: dataset

text: (a)

text: weight-lifting

text: (b)

text: Clapping

text: (c)

text: Golf

text: (d)

text: Looking

text: watch

text: (e)

text: Palying

text: badminton

text: Fig.

text: More

text: results

text: of

text: detection

text: ##LTFigure##

label: figure

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 20 E. HOFFBECK AND I. MOERDIJK

text: E.

text: HOFFBECK

text: AND

text: I.

text: MOERDIJK


text: 6 E. HOFFBECK AND I. MOERDIJK 2. Characterizations of shuffles of trees In the examples presented in the previous section, one observes that each branch of a shuﬄe of two trees S and T is given by a classical shuﬄe of a branch of S and one in T. We will start this section by showing that in fact this provides an equivalent deﬁnition of the notion of shuﬄe. Proposition 2.1. Let S and T be two trees. A shuﬄe of S and T is a tree A satisfying the conditions (1)-(3) of Deﬁnition 1.5 (on the labelling of the edges, the root and the leaves), as well as, (cid:48) (4 ) For any two leaves s of S and t of T, the branch from the leaf (s, t) of A down to its root is a classical shuﬄe, of the two branches from s down to the root in S and from t down to the root in T. Proof. First note that for linear trees, both Deﬁnition 1.5 and Proposition 2.1 are equivalent descriptions of classical shuﬄes. For general trees, given the other conditions, it is then clear that Condition (4) (cid:48) implies Condition (4 ) . For the converse implication, let us consider an edge (s, t) in A, where the edges immediately above s are denoted s ’s and the ones above t are denoted t ’s, as in i (cid:48) j the picture below Deﬁnition 1.5. Because of Condition (4 ), all edges above (s, t) must be of the form (s , t) or (s, t ). We have to show that no two diﬀerent types i j occur, and all of the same type occur. For the ﬁrst property, suppose that some (s , t) and some (s, t ) are successors of (s, t). Let (cid:96) be a leaf above s in S, and i j i i (cid:96) a leaf above t in T. Then by Condition (3), ((cid:96) , (cid:96) ) is in A. But then A is not j j i j a treelike order. For the second property, suppose now that one has only edges of type (s , t) above (s, t). Choose a leaf (cid:96) above each s , and a leaf (cid:96) above t. Then i i i (cid:48) ((cid:96) , (cid:96)) are all in A by Condition (3), so by Condition (4 ), all (s , t) occur above i i (cid:3) (s, t) (since as we have just seen no (s, t ) can). j Remark 2.2. Conversely, given two trees S and T, and two branches b and b S T inside them, it is not diﬃcult to see that any classical shuﬄe of b and b can be S T extended to a shuﬄe of S and T. The proof is left to the reader. × The set of edges of a shuﬄe A is a subset of the cartesian product E(S) E(T) of the set of edges of S and of T, and the partial order induced by the tree structure of A is the one induced by the partial orders on the edges of S and T, respectively. These two partial orders are all treelike, of course. The notion of shuﬄe can also be described in these terms. This way of deﬁning shuﬄes generalizes the classical notion of a shuﬄe of two linear trees L and M as a maximal linear suborder in × L M, cf. Subsection above. Proposition 2.3. Let S and T be two trees. A shuﬄe of S and T is a subset ⊂ × A E(S) E(T) for which × (1) The partial order on E(S) E(T) restricts to a treelike partial order on A. (2) The largest elements in this partial order on A are in bijective correspon- dence with the pairs (s, t) where s and t range over the leaves of S and T, respectively. (3) The set A is maximal with these properties (1)-(2).

text: E.

text: HOFFBECK

text: AND

text: I.

text: MOERDIJK

text: Characterizations

text: of

text: shuffles

text: of

text: trees

text: In

text: the

text: examples

text: presented

text: in

text: the

text: previous

text: section,

text: one

text: observes

text: that

text: each

text: branch

text: of

text: a

text: shuﬄe

text: of

text: two

text: trees

text: S

text: and

text: T

text: is

text: given

text: by

text: a

text: classical

text: shuﬄe

text: of

text: a

text: branch

text: of

text: S

text: and

text: one

text: in

text: T.

text: We

text: will

text: start

text: this

text: section

text: by

text: showing

text: that

text: in

text: fact

text: this

text: provides

text: an

text: equivalent

text: deﬁnition

text: of

text: the

text: notion

text: of

text: shuﬄe.

text: Proposition

text: Let

text: S

text: and

text: T

text: be

text: two

text: trees.

text: A

text: shuﬄe

text: of

text: S

text: and

text: T

text: is

text: a

text: tree

text: A

text: satisfying

text: the

text: conditions

text: of

text: Deﬁnition

text: (on

text: the

text: labelling

text: of

text: the

text: edges,

text: the

text: root

text: and

text: the

text: leaves),

text: as

text: well

text: as,

text: For

text: any

text: two

text: leaves

text: s

text: of

text: S

text: and

text: t

text: of

text: T,

text: the

text: branch

text: from

text: the

text: leaf

text: (s,

text: t)

text: of

text: A

text: down

text: to

text: its

text: root

text: is

text: a

text: classical

text: shuﬄe,

text: of

text: the

text: two

text: branches

text: from

text: s

text: down

text: to

text: the

text: root

text: in

text: S

text: and

text: from

text: t

text: down

text: to

text: the

text: root

text: in

text: T.

text: Proof.

text: First

text: note

text: that

text: for

text: linear

text: trees,

text: both

text: Deﬁnition

text: and

text: Proposition

text: are

text: equivalent

text: descriptions

text: of

text: classical

text: shuﬄes.

text: For

text: general

text: trees,

text: given

text: the

text: other

text: conditions,

text: it

text: is

text: then

text: clear

text: that

text: Condition

text: implies

text: Condition

text: For

text: the

text: converse

text: implication,

text: let

text: us

text: consider

text: an

text: edge

text: (s,

text: t)

text: in

text: A,

text: where

text: the

text: edges

text: immediately

text: above

text: s

text: are

text: denoted

text: s

text: ’s

text: and

text: the

text: ones

text: above

text: t

text: are

text: denoted

text: t

text: ’s,

text: as

text: in

text: i

text: j

text: the

text: picture

text: below

text: Deﬁnition

text: Because

text: of

text: Condition

text: all

text: edges

text: above

text: (s,

text: t)

text: must

text: be

text: of

text: the

text: form

text: (s

text: t)

text: or

text: (s,

text: t

text: We

text: have

text: to

text: show

text: that

text: no

text: two

text: diﬀerent

text: types

text: i

text: j

text: occur,

text: and

text: all

text: of

text: the

text: same

text: type

text: occur.

text: For

text: the

text: ﬁrst

text: property,

text: suppose

text: that

text: some

text: (s

text: t)

text: and

text: some

text: (s,

text: t

text: are

text: successors

text: of

text: (s,

text: t).

text: Let

text: be

text: a

text: leaf

text: above

text: s

text: in

text: S,

text: and

text: i

text: j

text: i

text: i

text: a

text: leaf

text: above

text: t

text: in

text: T.

text: Then

text: by

text: Condition

text: is

text: in

text: A.

text: But

text: then

text: A

text: is

text: not

text: j

text: j

text: i

text: j

text: a

text: treelike

text: order.

text: For

text: the

text: second

text: property,

text: suppose

text: now

text: that

text: one

text: has

text: only

text: edges

text: of

text: type

text: (s

text: t)

text: above

text: (s,

text: t).

text: Choose

text: a

text: leaf

text: above

text: each

text: s

text: and

text: a

text: leaf

text: above

text: t.

text: Then

text: i

text: i

text: i

text: are

text: all

text: in

text: A

text: by

text: Condition

text: so

text: by

text: Condition

text: all

text: (s

text: t)

text: occur

text: above

text: i

text: i

text: (s,

text: t)

text: (since

text: as

text: we

text: have

text: just

text: seen

text: no

text: (s,

text: t

text: can).

text: j

text: Remark

text: Conversely,

text: given

text: two

text: trees

text: S

text: and

text: T,

text: and

text: two

text: branches

text: b

text: and

text: b

text: S

text: T

text: inside

text: them,

text: it

text: is

text: not

text: diﬃcult

text: to

text: see

text: that

text: any

text: classical

text: shuﬄe

text: of

text: b

text: and

text: b

text: can

text: be

text: S

text: T

text: extended

text: to

text: a

text: shuﬄe

text: of

text: S

text: and

text: T.

text: The

text: proof

text: is

text: left

text: to

text: the

text: reader.

text: The

text: set

text: of

text: edges

text: of

text: a

text: shuﬄe

text: A

text: is

text: a

text: subset

text: of

text: the

text: cartesian

text: product

text: E(S)

text: E(T)

text: of

text: the

text: set

text: of

text: edges

text: of

text: S

text: and

text: of

text: T,

text: and

text: the

text: partial

text: order

text: induced

text: by

text: the

text: tree

text: structure

text: of

text: A

text: is

text: the

text: one

text: induced

text: by

text: the

text: partial

text: orders

text: on

text: the

text: edges

text: of

text: S

text: and

text: T,

text: respectively.

text: These

text: two

text: partial

text: orders

text: are

text: all

text: treelike,

text: of

text: course.

text: The

text: notion

text: of

text: shuﬄe

text: can

text: also

text: be

text: described

text: in

text: these

text: terms.

text: This

text: way

text: of

text: deﬁning

text: shuﬄes

text: generalizes

text: the

text: classical

text: notion

text: of

text: a

text: shuﬄe

text: of

text: two

text: linear

text: trees

text: L

text: and

text: M

text: as

text: a

text: maximal

text: linear

text: suborder

text: in

text: L

text: M,

text: cf.

text: Subsection

text: above.

text: Proposition

text: Let

text: S

text: and

text: T

text: be

text: two

text: trees.

text: A

text: shuﬄe

text: of

text: S

text: and

text: T

text: is

text: a

text: subset

text: A

text: E(S)

text: E(T)

text: for

text: which

text: The

text: partial

text: order

text: on

text: E(S)

text: E(T)

text: restricts

text: to

text: a

text: treelike

text: partial

text: order

text: on

text: A.

text: The

text: largest

text: elements

text: in

text: this

text: partial

text: order

text: on

text: A

text: are

text: in

text: bijective

text: correspon-

text: dence

text: with

text: the

text: pairs

text: (s,

text: t)

text: where

text: s

text: and

text: t

text: range

text: over

text: the

text: leaves

text: of

text: S

text: and

text: T,

text: respectively.

text: The

text: set

text: A

text: is

text: maximal

text: with

text: these

text: properties


text: SHUFFLES OF TREES 9 Remark 2.8. In the literature (cf. [6], for example) one also considers shuﬄes of trees with stumps (also referred to as bald vertices or vertices of valence zero). From our point of view, these can be described as follows: for two trees S and T, possibly with stumps, let S0 and T0 be the subtrees obtained by pruning the stumps away. Then the shuﬄes of S and T are in bijective correspondence with those of S0 and T0. In fact, a shuﬄe of the former two as considered in loc. cit. is obtained from a shuﬄe A of the latter two by reintroducing a stump above each pair of leaves of S0 and T0 which aren’t both leaves in S and in T, respectively. Here is an example, with the three shuﬄes obtained from the following two trees: ◦ • ◦ 2 3 • S : 1◦ and T : b c a 0 ◦ • (cid:12) ◦ • (cid:12) • ◦ (cid:12) ◦ ◦ ◦ ◦ • • b2 b3 c2 c3 b2 b3 c2 c3 b2 c2 b3 c3 b1◦ ◦c1 • ◦ , b1 c1 and a2 a3 • a1◦ a1◦ b0 c0 a0 a0 a0 3. The number of shuffles For two trees S and T, we will write sh(S, T) for the number of shuﬄes of S and T, in contrast with the set of shuﬄes Sh(S, T). This number grows very fast in the size of the trees S and T, and it seems diﬃcult to give some uniform closed formulas giving precise information about this number as a function of S and T, except for some very special types of trees. In this section, we will present some simple observations concerning the number sh(S, T). To begin with, from Proposition 2.5 one immediately obtains the following. Proposition 3.1. The set of shuﬄes Sh(S, T) of two trees S and T satisﬁes the following three properties: (1) sh(S, T) = sh(T, S). (2) If T is the unit tree η, sh(S, η) = 1. (3) If S = C [S , . . . , S ] an(cid:89)d T = C [T , .(cid:89). . , T ], then m 1 m n 1 n m n sh(S, T) = sh(S , T) + sh(S, T ). i j i=1 j=1 Remark 3.2. For the linear trees L and L with p and q vertices respectively, let us (cid:0) (cid:1) (cid:0) p (cid:1) (cid:0)q (cid:1) write λ(p, q) for sh(L , L ). Then the proposition states that λ(p, 0) = 1 = λ(0, q) − p q − (cid:80) and λ(p, q) = λ(p 1, q)+λ(p, q 1), which is the familiar inductive relation deﬁning − − the binomial coeﬃcient p+q = p −1+q + p+q 1 . It corresponds to the generating p p 1 p function deﬁned by Λ(x, y) = (x + y)Λ(x, y) for which Λ(x, y) = λ(p, q)xpyq. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SHUFFLES

text: OF

text: TREES

text: Remark

text: In

text: the

text: literature

text: (cf.

text: for

text: example)

text: one

text: also

text: considers

text: shuﬄes

text: of

text: trees

text: with

text: stumps

text: (also

text: referred

text: to

text: as

text: bald

text: vertices

text: or

text: vertices

text: of

text: valence

text: zero).

text: From

text: our

text: point

text: of

text: view,

text: these

text: can

text: be

text: described

text: as

text: follows:

text: for

text: two

text: trees

text: S

text: and

text: T,

text: possibly

text: with

text: stumps,

text: let

text: and

text: be

text: the

text: subtrees

text: obtained

text: by

text: pruning

text: the

text: stumps

text: away.

text: Then

text: the

text: shuﬄes

text: of

text: S

text: and

text: T

text: are

text: in

text: bijective

text: correspondence

text: with

text: those

text: of

text: and

text: In

text: fact,

text: a

text: shuﬄe

text: of

text: the

text: former

text: two

text: as

text: considered

text: in

text: loc.

text: cit.

text: is

text: obtained

text: from

text: a

text: shuﬄe

text: A

text: of

text: the

text: latter

text: two

text: by

text: reintroducing

text: a

text: stump

text: above

text: each

text: pair

text: of

text: leaves

text: of

text: and

text: which

text: aren’t

text: both

text: leaves

text: in

text: S

text: and

text: in

text: T,

text: respectively.

text: Here

text: is

text: an

text: example,

text: with

text: the

text: three

text: shuﬄes

text: obtained

text: from

text: the

text: following

text: two

text: trees:

text: S

text: 1◦

text: and

text: T

text: b

text: c

text: a

text: and

text: The

text: number

text: of

text: shuffles

text: For

text: two

text: trees

text: S

text: and

text: T,

text: we

text: will

text: write

text: sh(S,

text: T)

text: for

text: the

text: number

text: of

text: shuﬄes

text: of

text: S

text: and

text: T,

text: in

text: contrast

text: with

text: the

text: set

text: of

text: shuﬄes

text: Sh(S,

text: T).

text: This

text: number

text: grows

text: very

text: fast

text: in

text: the

text: size

text: of

text: the

text: trees

text: S

text: and

text: T,

text: and

text: it

text: seems

text: diﬃcult

text: to

text: give

text: some

text: uniform

text: closed

text: formulas

text: giving

text: precise

text: information

text: about

text: this

text: number

text: as

text: a

text: function

text: of

text: S

text: and

text: T,

text: except

text: for

text: some

text: very

text: special

text: types

text: of

text: trees.

text: In

text: this

text: section,

text: we

text: will

text: present

text: some

text: simple

text: observations

text: concerning

text: the

text: number

text: sh(S,

text: T).

text: To

text: begin

text: with,

text: from

text: Proposition

text: one

text: immediately

text: obtains

text: the

text: following.

text: Proposition

text: The

text: set

text: of

text: shuﬄes

text: Sh(S,

text: T)

text: of

text: two

text: trees

text: S

text: and

text: T

text: satisﬁes

text: the

text: following

text: three

text: properties:

text: sh(S,

text: T)

text: sh(T,

text: S).

text: If

text: T

text: is

text: the

text: unit

text: tree

text: η,

text: sh(S,

text: η)

text: If

text: S

text: C

text: [S

text: S

text: T

text: C

text: [T

text: T

text: then

text: m

text: m

text: n

text: n

text: m

text: n

text: sh(S,

text: T)

text: sh(S

text: T)

text: sh(S,

text: T

text: i

text: j

text: Remark

text: For

text: the

text: linear

text: trees

text: L

text: and

text: L

text: with

text: p

text: and

text: q

text: vertices

text: respectively,

text: let

text: us

text: p

text: write

text: λ(p,

text: q)

text: for

text: sh(L

text: L

text: Then

text: the

text: proposition

text: states

text: that

text: λ(p,

text: q)

text: p

text: q

text: and

text: λ(p,

text: q)

text: λ(p

text: q)+λ(p,

text: q

text: which

text: is

text: the

text: familiar

text: inductive

text: relation

text: deﬁning

text: the

text: binomial

text: coeﬃcient

text: p+q

text: p

text: p+q

text: It

text: corresponds

text: to

text: the

text: generating

text: p

text: p

text: p

text: function

text: deﬁned

text: by

text: Λ(x,

text: y)

text: (x

text: y)Λ(x,

text: y)

text: for

text: which

text: Λ(x,

text: y)

text: λ(p,

text: q)xpyq.

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: ##LTLine##

text: ##LTLine##