Neurobiological and molecular studies suggest a dichromatic colour vision in canine species, which appears to be similar to that of human red–green colour blindness. Here, we show that dogs exhibit a behavioural response similar to that of red–green blind human subjects when tested with a modified version of a test commonly used for the diagnosis of human deuteranopia (i.e. the Ishihara's test). Besides contributing to increasing the knowledge about the perceptual ability of dogs, the present work describes for the first time, to our knowledge, a method that can be used to assess colour vision in the animal kingdom. SpeciAlthough early behavioural studies on dogs' colour vision produced conflicting results (reviewed by ), recentet al. .The change between stimuli (stimuli presentation) was controlled by the experimenter and was dependent on the attention of the dog to the screen (inter-stimulus presentation time: 7–120 s). The stimulus presentation procedure was the same as described in §2.3. The side of appearance of the ‘moving cat’ stimuli was randomized within each session. Owners were asked not to influence their dogs' behaviour (e.g. either to indicate the screen or to force looking behaviour). If the dog was distracted during the presentation of the stimuli or if it left the starting position (despite the beep sound) it was repositioned and the stimulus was then represented. Dogs visual angle was approximately of 2° 51′ 0.85The behaviour of the dogs was video recorded continuously during stimulus. Two trained observers who were blind to the testing paradigm subsequently analysed the video footage. The video was used to score any of the following listed behaviour: ears up-forward, scanning (dog turning head from left to right), eyes wide open, forward body orientation, eye/ear directed towards the target, gaze, head slightly lowered, paw lifted, freezing, alert position and head tilt. Each performed behaviour was allocated a score of 1, and the total for each dog was used to generate a reactivity index for the ‘alerting–targeting’ behavioural category.In addition, looking time to different visual stimuli was measured, analysing dogs behaviour from the beginning of the target animation (soon after the ‘beep’ sound) until the stimuli disappeared  and calculated as percentage agreement .olorPro 5 colorimeter sensor and Sencore ColorPro 6000 software were used to calibrate the colours of the monitor to CIE Standard Illuminant D65 . The same apparatus was also used to measure the brightness of the white background and the average brightness of the tested stimuli  was tested with achromatic versions of RG-Cat stimuli in order to see if subjects use achromatic cues to perceive cat's moving animations.Two new animated cat targets were edited using eight frames (4 × each animation) in which achromatic versions of red and green shades of Ishihara plate no. 22 were used .Figure In particular, the A-Cat-2 and the A-Cat-6 stimuli were edited using respectively the achromatic versions of the RG-Cat-2 and the RG-Cat-6 animations used during the red–green colour-blind test.We predicted that if dogs use achromatic cues to perceive the cat animation most of the tested subjects should see and correctly interpret (showing orientation movements of the head and tip of predatory behaviour) more the movements of the cat obtained by using the achromatic version of the same colour shades employed to depict the number ‘2’ on the Ishihara's plate no. 22 with respect to the movements of the cat edited by using the achromatic version of the same colour shades of the number ‘6’ taken from the same plate (this prediction is based on the fact that achromatic contrasts are more apparent in the first condition than in the second).In addition, two different random achromatic animations (i.e. A-Background-1 and A-Background-2) were made using G-Background-1, and G-Background-2 stimuli without colour cues.owerPoint slideshows on a white background and were presented in pairs: the first slide showed two A-Background-1 (control animated plates), the second and the third presentation showed, respectively, the animations of A-Cat-6 versus A-Background-2 and A-Cat-2 versus A-Background-2; finally, the B-Cat animation was presented alone as a control on the same side of the A-Cat targets.Five stimuli were displayed during each trial by PThe testing procedure was identical to that described above for the red–green colour blind test.The red–green colour blind and the achromatic tests were presented separately  and the presentation order was alternated between subjects (i.e. five subjects performed the red–green blindness test first and four dogs performed the achromatic test first).3.3.1.a,b.Results for the red–green colour-blind test are shown in figure 4p < 0.001) (a). Dunn's post hoc test revealed that this main effect was owing to the time spent at looking both RG-Cat-2 and B-Cat being longer with respect to other stimuli: RG-Cat-2  versus G-Background-1 (0.09 ± 0.05 (s); m ± s.e.m.) (p < 0.01); RG-Cat-2 versus G-Background-2 (0.31 ± 0.11 (s); m ± s.e.m.) and versus RG-Cat-6 (0.63 ± 0.35 (s); m ± s.e.m.) (p < 0.05); B-Cat (1.87 ± 0.22 (s); m ± s.e.m.) versus G-Background-1 (p < 0.001); B-Cat versus G-Background-2 and RG-Cat-6 (p < 0.01). No statistical significant differences were found between the other pairwise comparisons .Friedman's ANOVA revealed a main effect of the type of visual stimulus on the mean looking time (< 0.001) a. Dunn'sp < 0.001; Friedman's ANOVA, figure 4b). Dunn's post hoc test revealed that the RG-Cat-2  elicited a higher alerting behavioural response with respect to the G-Background-1 , G-Background-2  (p < 0.01) and RG-Cat-6  (p < 0.05). Similarly, the analysis revealed higher alerting behavioural response between B-Cat and the other visual stimuli with the exception of RG-Cat-2: B-Cat  versus G-Background-1, G-Background-2 and RG-Cat-6 (p < 0.001). No other significant statistical differences were found .As to the behavioural score, the analysis revealed that there was a statistical significant difference in the score for alerting behavioural response between visual stimuli  (a). Dunn's post hoc test revealed that this main effect was owing to the time spent at looking the B-Cat being longer with respect to other stimuli: B-Cat (2.23 ± 0.13 (s); m ± s.e.m.) versus A-Cat-2 (0.55 ± 0.18 (s); m ± s.e.m.) and G-Background-1 (0.46 ± 0.13 (s); m ± s.e.m.) (p < 0.01); B-Cat versus G-Background-2 (0.31 ± 0.11 (s); m ± s.e.m.) and A-Cat-6 (0.66 ± 0.21 (s); m ± s.e.m.) (p < 0.05). No statistical significant differences were found between the other pairwise comparisons .Friedman's ANOVA revealed a main effect of the type of visual stimulus on the mean looking time (= 0.001) a. Dunn'sp < 0.01; Friedman's ANOVA, figure 5b). Dunn's post hoc test revealed that this main effect was owing to the score for alerting behavioural response to the B-Cat stimulus being higher with respect to other stimuli: B-Cat (3.75 ± 0.16 (s); m ± s.e.m.) versus G-Background-1 (0.44 ± 0.17 (s); m ± s.e.m.) (p < 0.01); B-Cat versus G-Background-2 (0.66 ± 0.11 (s); m ± s.e.m.), A-Cat-2 (0.62 ± 0.18 (s); m ± s.e.m.) and A-Cat-6 (0.75 ± 0.25 (s); m ± s.e.m.) (p < 0.05). No statistical significant differences were found between the other pairwise comparisons .As to the behavioural score, the analysis revealed a main effect of the type of visual stimulus on the score for alerting behavioural response between visual stimuli , most of the dogs exhibit an orienting response to the stimulus (both the eyes and the head oriented toward the animated cat silhouette) together with clear targeting behaviour. These results are similar to those reported in red–green colour-blind humans who clearly recognize the red number ‘2’ during inspection of Ishihara's plate no. 22 and the cat during presentation of the RG-Cat-2 animation.On the other hand, when dogs were presented with the red cat animation having the same red–green colour shade of the number ‘6’ of the Ishihara's plate no. 22, a significant lowering of both orienting and targeting behavioural response was observed, pointing to a considerable loss of subjects' capacities in perceiving the target.The above result is in accordance with the difficulty of the red–green colour-blind humans to recognize the red-depicted number ‘6’ of Ishihara's plate no. 22 and their inability to perceive the cat movements during presentation of the RG-Cat-6 animation.Furthermore, two aspects should be considered when the orienting response is used to evaluate visual discrimination: (i) habituation to the movements of the target  and (ii) aspecific head/eyes orienting response . Given that in the orienting response, habituation to the movements of the targets presented as the first could decrease the response to the next stimulus, we designed the experiment so that the two coloured targets were presented only once to each subject and the control stimulus (i.e. the black cat silhouette presented against a white back ground) was presented as the last during each session. In order to deal with the possibility of a ‘false positive response’, we scored during the experiment the targeting–alerting dogs behavioural response towards the coloured cat animations which was then compared with the behavioural response towards the control cat animation (B-Cat). A statistical significant correlation between the alerting–targeting behavioural response towards the coloured cat animation and the control ones make the hypothesis that dogs recognize the moving cat likely. In addition, in support to this hypothesis the behavioural score for alerting–targeting towards the control cat animation was higher compared with that of the black square, indicating that the subjects were able to clearly distinguish the movements of the cat from those of the square.Regarding the dog–owner dyads, the possibility that the owners influenced the dogs' orienting response during the test is very remote because, although the dyad was very close during the test (the dog sat or was laid between the owner's legs), the owners were asked to stare a fixed spot centrally located to the screen. In addition, given the results, it is reasonable to assume, that the owners did not influence the dogs, because they otherwise might have reacted to the animations that must have been visible to the owners .Another aspect to consider is that the luminance difference between stimulus colours and background could affect colour discrimination thresholds . DespiteThis finding supported previous results demonstrating that, in dogs, colour information may be predominant with respect to brightness .In addition, the observed difference between the red–green blindness test and the achromatic one is consistent with reports of compelling evidence that colour can contribute to motion perception –18, but Overall, our results, together with the above-reported studies, confirm that dog colour vision is dichromatic in its nature, resembling that of human red–green blindness. Besides contributing to increasing knowledge about the perceptual ability of dogs, the present work showed, to our knowledge, for the first time, directly canine red–green blindness by using a modified test of colour vision in humans (Ishihara's test), thereby allowing direct comparison to colour vision (and colour blindness) in humans. Furthermore, the method used will open the door to the development of new techniques (e.g. coupling the modified version of the Ishihara's test with the use of eye-gaze detection and tracking systems) to assess colour vision in the animal kingdom.


We wish to make the following corrections to their paper :The title of the paper ‘Millimeter-Wave Chemical Sensor Using Substrate-Integrated-Waveguide Cavity’ should be changed to ‘Microwave Chemical Sensor Using Substrate-Integrated-Waveguide Cavity’.In the second line of the abstract, on the fifth line of Section 3 (Simulation Results), and in Section 4  on Page 7, the phrase ‘millimeter-wave frequency range’, should be changed to ‘microwave frequency range’. Also, on the third line of the Conclusion Section, ‘millimeter-wave’, should be replaced with ‘microwave’. It is well-known that millimeter-wave corresponds to a frequency of over 30 GHz. As the proposed sensor functions at around 18 GHz, it should be called ‘microwave’. We apologize for any inconvenience these changes have caused to readers. The changes do not affect the scientific results. The manuscript will be updated and the original will remain online on the article webpage.


N-representability of the pair density (PD) are needed for the development of the PD functional theory. We derive sufficient conditions for the N-representability of the PD that is calculated from the Jastrow wave function within the lowest order. These conditions are used as the constraints on the correlation function of the Jastrow wave function. A concrete procedure to search the suitable correlation function is also presented.Conditions for the  Therefore, the PD functional theory has been expected to be one of the promising schemes beyond the density functional theory28. We have proposed the PD functional theory that yields the best PD within the set of the Jastrow wave function PDs of the lowest-order (LO-Jastrow PDs)17. The search region for the ground-state PD is substantially extended as compared with the previous theory19. On the other hand, however, there remains a significant problem related to the N-representability of the LO-Jastrow PDs17. Concerning the N-representability of the PD or the second-order reduced density matrix, there are a lot of works39. But unfortunately, the necessary and sufficient conditions for the N-representability of the PD are not yet known in a practical form.The pair density (PD), which is defined as the diagonal element of the second-order reduced density matrix, has more information about the electron correlation than the electron densityN0-electron system. The Jastrow wave function is given byxi denotes the coordinates including the spatial coordinate ri and spin coordinate ηi, and where 41Let us revisit the problem here. We shall consider an 17, we have confirmed that Eq.  in Eq. (N0 − 1) -electron SSDs that are defined as the minor determinants multiplied by In this section, we shall discuss the properties of the LO-Jastrow PD. To this aim, the properties of PDs that are calculated from SSDs are investigated. The cofactor expansion of N0 − 2)-electron SSDs, which are denoted by j ≤ N0 − 1). Then, each Likewise, the cofactor expansion of each where Multiplying both sides of Eq.  by \docuN-representability of the LO-Jastrow PD.This relation is the starting point to examine the N-representability of We shall start with considering the N-representability of q, i.e.,where From Eq. , the PD 41. Therefore, if they are given, we can calculate N-representability of Eq. } (1 ≤ α ≤ n + 1) that satisfy the conditions;α ≤ n + 1) denote the n-electron wave functions, and σ is a permutation operator upon the electron coordinates, and σ.Proof. The left-hand side of Eq. -electron system as the whole system, and suppose that the wave function for such the (n + 1)-electron system is given by Eq. } (1 ≤ α ≤ n + 1) that satisfy Eqs aα that satisfy the following conditions:if there exists the set of functions q ≤ 3), i.e., Eqs (N-representability of Eq. (q ≤ 3) is obtained. Substitution of Eq. (p ≤ 4) that satisfy the following conditions:In addition to Eq. , the exi.e., Eqs  and 22)\documentn of Eq.  into Eq.n of Eq.  leads top ≤ 4), i.e., Eqs (N-representability of Eq. (q ≤ 3) and p ≤ 4) belong to the set of sufficient conditions.The existence conditions for .e., Eqs  and 27)\documenty of Eq.  and the i ≤ N0), we eventually obtain sufficient conditions for the N-representability of Eq. -electron system, i.e., i ≤ N0). Successively, each of the SSDs for the (N0 − 1)-electron system is expanded along the (N0 − 1)th row, and then the (N0 − 1) number of SSDs for the (N0 − 2)-electron system, i.e., j ≤ N0 − 1), can be obtained for each i. After that and later, the cofactor expansions are likewise repeated, and we finally arrive at the SSDs for the two-electron system, i.e.,N0!/2, since q are integers such that 1 ≤ i ≤ N0, 1 ≤ j ≤ N0 − 1, 1 ≤ k ≤ N0 − 2, o ≤ 5, 1 ≤ p ≤ 4 and 1 ≤ q ≤ 3, respectively.Let us consider the SSD that consists of the resultant spin orbitals for the simultaneous equations. The SSD can generally be expanded using the cofactor. The SSD for the 41. We check whether all of q through Eq. (The two-electron SSD ough Eq. .p. If suitable p can be constructed from Eq. (By means of these d as Eq. . We willith Eqs.  and 22)\documentfrom Eq. .o, the four-electron antisymmetric wave functions We successively proceed the case of four-electron wave functions that are defined as Eq. . In a si of Eqs.  and 27)\documentfrom Eq. .n(≤N0 − 1), the LO-Jastrow PD of the N0-electron system becomes N-representable.Likewise, we further proceed the problem constructing the antisymmetric wave functions for more-electron systems. We search the set of ith Eqs.  and 12)\documentith Eqs.  and (12)In the preceding section, the sufficient conditions for the N-representable may, in principle, be found along the above steps, though there is a possibility that the suitable correlation function may not exist in some system.As easily inferred, the above steps are feasible only for the small-electron systems from the practical viewpoint. However, it should be noted that the correlation function that makes the LO-Jastrow PD In the above-mentioned concrete steps, antisymmetric (n + 1)-electrons wave functions are built up from given antisymmetric n-electrons wave functions via Eq. . In thisInstead of Eq. , we can aα(xi) and bα(xn + 1) denotes functions that should be determined. If these functions satisfy the following conditions;aα(xi) and bα(xn + 1) that satisfy Eqs. aα(xi) ϕλ(x), Similarly to the above, we can choose the other expression for d of Eq.  or 28)..\documenaα denote a function that should be determined. If aα  satisfies the following conditions;where aα that satisfy Eqs (then Eq.  becomes isfy Eqs  and 34)aαaα(xi, N-representable LO-Jastrow PD.Thus, various expressions for N-representability of the LO-Jastrow PD are discussed. Using the properties of the LO-Jastrow PD, we derive the sufficient conditions that are imposed on the correlation function of the Jastrow wave function. As shown in the previous section, additional steps to search the suitable correlation function, which satisfies the sufficient conditions, are attached to the computational scheme proposed previously17. Although the number of steps rapidly increases with that of electrons, the concrete steps that are presented in the previous section are feasible for a small-electron system. Of course, there is a possibility that the suitable correlation function cannot be found out. In this case, as mention in the previous section, we can adopt other expressions for N-representable in a sense that they satisfy four kinds of necessary conditions47.In this paper, the sufficient conditions for the


Using computational and theoretical approaches, we investigate the snap-through transition of buckled graphene membranes. Our main interest is related to the possibility of using the buckled membrane as a plate of capacitor with memory (memcapacitor). For this purpose, we performed molecular-dynamics (MD) simulations and elasticity theory calculations of the up-to-down and down-to-up snap-through transitions for membranes of several sizes. We have obtained expressions for the threshold switching forces for both up-to-down and down-to-up transitions. Moreover, the up-to-down threshold switching force was calculated using the density functional theory (DFT). Our DFT results are in general agreement with MD and analytical theory findings. Our systematic approach can be used for the description of other structures, including nanomechanical and biological ones, experiencing the snap-through transition. Such devices are prospective candidates for applications in information storage and processing3 technologies as their states can be manipulated by the applied voltages or charges and can store information for long intervals of time. Several possible realizations of memcapacitors were suggested by using micro-electro-mechanical systems4, ionic transport5, electronic effects6, superconducting qubits7, etc8.Memcapacitors1q(t) is the charge on the capacitor at time t, V(t) is the applied voltage, C is the memcapacitance (memory capacitance), x is a set of n internal state variables, and f is a continuous n-dimensional vector function. In some cases, it is more convenient to consider charge-controlled memcapacitors1 such that the charge instead of voltage is considered as input.Generally, voltage-controlled memcapacitive systems (memcapacitors) are described by4 are of significant interest as their geometry makes them intrinsically suitable for non-volatile storage of binary information. Indeed, the buckled membrane used as the top capacitor plate  Sec. “MD Simulation Details”. The standard elasticity theory is applied to the membrane switching in Subsec. “Buckling and snapping-through within the theory of elasticity”. A phenomenological analytical model of the snap-through transition is presented in Subsec. “Phenomenological elasticity theory” and in SI Sec. “Phenomenological Elasticity Theory”. Our DFT calculations are summarized in Sec. “Density Functional Theory”. The results obtained within different approaches as well as their implications are discussed in Sec. “Discussion”.In this paper, the following notations are used:q - the charge on capacitor (see Eq. ((see Eq. )V - the applied voltage (see Eqs ((see Eqs  and 2)))V - the C - the (memory) capacitance (see Eq. ((see Eq. )d - the distance between fixed sides of membrane ζc - the maximum deflection of membrane (see Eq. ((see Eq. )θi(s) - the angle that the membrane makes with the horizontal (see Eqs ((see Eqs  and 17)θi(s) - ts - the internal coordinate that changes between −1/2 and 1/2 (see Eqs ((see Eqs  and 17)s - the iAi and ci - coefficients (see Eqs ((see Eqs  and 17)Ai and cizcm - the center of mass position (see Eq. ((see Eq. )Ub, Ustr, Uext - the bending, stretching and external potential contributions to the potential energy of membrane (see Eq. ((see Eq. )F↓ - the up-to-down threshold switching force , F↓ - the F↑ - the down-to-up threshold switching force (see Eqs ((see Eqs  and 28)F↑ - the ε0 - the vacuum permittivity36. We used classical MD simulations to investigate the dynamics of snap-through transition of buckled graphene membranes. Zigzag graphene nanoribbons (membranes) of two lengths were considered: the nanoribbon A, L = 54 Å (22 rings), and nanoribbon B, L = 103 Å (42 rings). Both nanoribbons were of the same width (w = 41 Å). In order to implement the clamped boundary conditions, the first two lines of carbon atoms at shorter sides were kept fixed. The buckling was realized by changing the distance between the fixed sides from L to d < L. The same external force was applied to each atom in the downward direction to simulate the attractive interaction between the plates.MD simulations are a well established modeling tool frequently employed in studies of nanoscale carbon-based materialsSee SI Sec. “MD Simulation Details” for the details of MD simulations.d/L and two membrane sizes. According to Fig. L, the up-to-down threshold switching force  is larger for smaller values of d/L. Moreover, at fixed d/L, the threshold switching force is smaller for longer membranes. Additionally, some curves in Fig. d/L = 0.8 and d/L = 0.98 in (a)), which can be related to thermal fluctuations of membranes. All these observations are intuitively reasonable.In order to simulate the up-to-down transition, the double-clamped graphene nanoribbon buckled upwards was subjected to the force in the downward direction. The final state of membrane was found using MD simulations as described in SI Sec. “MD Simulation Details”. Figure 14 and involved in switching by smaller forces. Larger applied forces result in the switching through the symmetric profile. According to our observations and previous work14, in all cases, the membrane profile is symmetric at short times. If the force magnitude is sufficient to overcome the energy barrier for the symmetric profile, then, typically, the switching takes place through the symmetric path. Otherwise, a symmetry breaking occurs leading to the switching through the lower energy barrier associated with the non-symmetric membrane profile. Thermal fluctuations help the symmetry breaking.Depending on the force magnitude, the membrane switching occurs either through the symmetric or non-symmetric membrane profile see Fig. . Figure 4 that the memcapacitor membrane can be set into the buckled upwards state 0 38 disappear in the transition from flexible to rigid graphene. In agreement with this previous work37, our molecular dynamics simulations have shown the absence of ripples (spontaneous height fluctuations) in nanoribbons of reported sizes and their existence in larger nanoribbons. The latter, however, are not suitable for the memcapacitor application because of their flexibility. To summarize, while our simulation tools provide a means for ripples detection, we did not observe these in the reported structures that are mechanically rigid.In our work, we considered relatively small nanoribbons, which are mechanically rigid. This is an important requirement for the memcapacitor application as the mechanical nonvolatile information storage is not possible with flexible graphene. From some previous studies42. This allows us, on one hand, to obtain analytical expressions for the buckled membranes45 and, on the other hand, perform a comparison with MD simulations.Even though the graphene has a thickness of one atom, the graphene membranes are quantitatively good described by the theory of elasticity49). Such systems are frequently described in the framework of Euler-Bernoulli theory, which, however, leads to complex analytically unsolvable equations. Bubnov-Galerkin decomposition is a one of the best approaches to find the approximate analytical solution of these equations, although it still requires a complex phase-diagram analysis. In particular, using such analysis, the authors of refs49 investigated an electrostatically loaded double-clamped membrane above a rigid flat electrode and derived cumbersome conditions for the symmetric snap-through transition, symmetry breaking, existence of bifurcation and pull-in instability.There is a number of publications dealing with buckled beams and membranes under the transverse load  asT = T0 + δT is the total tensile force, T0 is the force applied by the support and δT is the bending tension resulted from the extension,Consider a 2D membrane within the theory of elasticityS0 = wL, and F is the external force density. The compression of membrane corresponds to T0 < 0.μ is the 2D mass density.Given the potential energy, one may also be interested in the dynamics of membrane, which is defined by the equationbn are the solutions of the equation cosh bncos bn = 1. One can find that bn ≈ 3π/2 + nπ . The functions ζn(x) are orthonormal.The spatial harmonics (eigenmodes) of clamped membranes can be written asn = 0 mode (taking ζ(x) = ζ0(x) with ζ(d/2) ≡ ζc) isα = π2w(|T0| − Tc)/(4d), β = 3π4Dw/(4ε2d3), and f = πFdw/6. Here, Tc = 4π2D/d2 is the critical tension. At α > 0, that is at |T0| > Tc, the potential is the double-well potential with minima at symmetric deflections, δU = α2/4β. 10) link ζ in harmonics ζn(x) (given by Eq. (qn(t) asn = 0,2 and n = 0,1 terms in Eq. (ζm(x) and integrated taking into account the orthogonality of harmonics. Here we emphasize that for the harmonics (qn(t). Inserting the obtained functions qn(t) back into Eq. (tc = 1/ωc)51 are analyzed for membrane B.Here we present a convenient method to describe the dynamics of membrane subjected to an external force and find the minimal force causing its snap-through transition. The main idea is to expand n by Eq. ) with ams in Eq. . Second,n = 0 and 2 harmonics. Next, we introduce a small asymmetry into the problem via a small non-symmetric modification of the force (of the order of 0.1%). This results in the non-symmetric dynamics of Fig. n = 0 and 1 harmonics. We note that the force needed for the non-symmetric snap-through transition is smaller than that for the symmetric one, which will also be analyzed below in more detail. Finally, in Fig. 14.Consider the results presented in Fig. ζc(tf), on the applied force F. At a certain value of force, F = F↓, the membrane switches from the buckled upwards state to the buckled downwards state. Our calculations show that the threshold force is proportional to the initial central-point deflection ζc and is different for the symmetric (s) and non-symmetric (ns) dynamics. Namely, this force, being multiplied by the membrane area wL, readsFigure qn(t) in the up-to-down transition. Importantly, the dynamics is well described by n = 0, 1, and 2 harmonics, while n = 3 and 4 harmonics amplitudes are negligibly small.Since the non-symmetric transition is energetically favorable, we expect that this value of force is the one to be used in the device design/experiment analysis. We note that estimations based on Eq.  are in aOur calculations (similar to the consideration of the up-to-down force above) demonstrate that the minimal force needed for the down-to-up transition is proportional to See also SI Subsec. “Down-to-up transition”, where the down-to-up transition is illustrated graphically.In this Section, in order to avoid all complications associated with the Bubnov-Galerkin decomposition, we elaborate a different approach describing the up-to-down and down-to-up transitions in a compact analytical form.Following our observation of the existence of the symmetric and non-symmetric membrane profiles in the snap-through transition see Fig. , we intrθi(s) is the angle that the membrane makes with the horizontal, s is the internal coordinate that changes between −1/2 and 1/2, Ai and ci are coefficients, and i = {s, ns}. Clearly, Eqs (θi(±1/2) = 0.Here rly, Eqs  and 17)θi(s) is Our goal here is to estimate the minimal force required for the snap-through transition. In this subsection, we assume that the snap-through transition is induced by a slowly increasing force such that the system always stays in the potential energy minimum. At zero applied force, there are two minima of the potential energy corresponding to the two stable states of membrane. The applied force modifies the potential energy landscape such that, starting at a certain force, the potential energy has a single minimum. Here, this value of force is found and considered as an estimation for the threshold switching force.F in the downward direction, the membrane potential energy is given byUb is the bending energy, Ustr is the stretching energy, Uext is the potential energy due to the applied force. Then, using Uext = Fzcm, where zcm is z coordinate of the center of mass, and neglecting Ustr term not important for the up-to-down transition , we get the following equation for the potential energy extrema:i = {0, 1} corresponds to {s, ns}, respectively. Equation (c0 and c1 for a given strength of the applied force.In the presence of a constant force F(zcm) and, for the transition through the symmetric shape, is given byc0 = 0.3589 /L2. Performing the same calculations for the non-symmetric shape, one can find that the force needed to support the equilibrium non-symmetric shape decreases with the progress of switching. A zero force is reached at Ub,ns = 90Dw(L − d)/L2. A possible (rough) estimation for the threshold switching force can be obtained taking the limit c1 → ∞ leading toUb,ns = 42Dw(L − d)/L2.The minimal force needed for the snap-through transition corresponds to the maximum of F = 0 and slowly increase the force. The symmetry breaking occurs at a certain value of force, say, F↓. The dynamics of switching is a complex process significantly relying on thermal fluctuations. As the switching dynamics can not be reached within the framework of our model, for estimation purposes, we assume that F↓ corresponds to the point where the bending energies and applied forces for the symmetric and non-symmetric shapes are the same. One can find that both conditions are satisfied at c0 = 0.4683 and c1 = 0.6948, so that Ub = 47.8Dw(L − d)/L2 andIn fact, a realistic scenario of the switching through the non-symmetric shape can be imaged as follows. We start with a symmetric membrane at F < F↓, the membrane keeps the symmetric shape and switches to the non-symmetric one as soon as F reaches F↓. As there is no barrier involved in the non-symmetric switching, no further increase in the applied force is required to complete the snap-through transition through the non-symmetric shape.Consequently, at Ustr  plays aMoreover, at equilibriumUnder the condition of small Using Eqs  and 27)27), Eq. We have verified that the threshold switching force for the down-to-up transition given by Eq.  is in ag14. In the present calculations, we focused on the effects of boundary conditions and buckling strength on the up-to-down threshold switching force. DFT computational details can be found in SI Sec. “DFT Calculations”.A DFT investigation of the up-to-down transition was performed following the previously reported study of the up-to-down switching of buckled graphene membraned/L = 0.96, Fig. d/L = 0.78, and C3 with d/L = 0.63), and (2) clamped boundary conditions  hinged boundary conditions MD is widely used approach to describe the dynamics of nanoscale systems. In our work, MD clearly shows the possibility of 0 → 1 and 1 → 0 transitions of graphene membrane. We have also verified the occurrence of 1 → 1 and 0 → 0 transitions at the same values of forces required for 0 → 1 and 1 → 0 transitions, respectively.(ii)ζ. Expanding the function ζ into membrane’s harmonics allows reducing the integro-differential equation to a system of differential equations. This approach was utilized in Subsec. “Buckling and snapping-through within the theory of elasticity”. There it was shown that the up-to-down threshold switching force depends linearly on the initial central-point deflection, F↓ ∝ ζc, while the down-to-up snap-through force at low dissipation depends quadratically on the initial deflection, In the framework of elasticity theory, the membrane is described by means of an integro-differential equation for the deflection (iii)The phenomenological approach based on the theory of elasticity has allowed us to obtain analytical expressions for the up-to-down and down-to-up switching forces for buckled graphene membrane. The forces for 0 → 1 and 1 → 0 transitions are in a good agreement with MD simulations results.(iv)DFT is a non-standard method for studying mechanical properties of membranes. Similarly to other methods, DFT has shown that the non-symmetric switching pathway is the preferable one. This method also provides a consistent estimation for the up-to-down threshold switching force.52. For this purpose, it is desirable that both the distance between the fixed edge of membrane and second electrode ) are chosen (much) smaller than the membrane length. Moreover, the gap between electrodes should be larger than the maximal membrane deflection, h > zs(0) (see Eq. (zi(s)). For given capacitances C0 and C1 of the states 0 and 1  such that C1/C0 < 3.01, one can find thatFrom the application point of view, the membrane memcapacitor should have a strong ‘memory content’, namely, the device characteristics in its two logic states should be sufficiently different so as to provide a significant influence on other elements of electronic circuitsV ))V  in Fig. The threshold voltage can be estimated based on the energy conservation (Eq. ) and forfrom Eqs  and 27)\documentAn additional aspect of membrane dynamics – an estimation of thermally-induced switching time – is considered in SI Sec. “Stability of Buckled Membrane”.We have investigated the snap-through transition of a buckled graphene membrane using a variety of computational and theoretical tools. The main results of this paper are the expressions for the threshold switching forces Eqs  and 22 f f22 for supplementary information


Protein motors, such as kinesins and dyneins, bind to a microtubule and travel along it in a specific direction. Previously, it was thought that the directionality for a given motor was constant in the absence of an external force. However, the directionality of the kinesin-5 Cin8 was recently found to change as the number of motors that bind to the same microtubule is increased. Here, we introduce a simple mechanical model of a microtubule-sliding assay in which multiple motors interact with the filament. We show that, due to the collective phenomenon, the directionality of the motor changes , depending on the number of motors. This is induced by a large diffusive component in the directional walk and by the subsequent frustrated motor configuration, in which multiple motors pull the filament in opposite directions, similar to a game of tug-of-war. A possible role of the dual-directional motors for the mitotic spindle formation is also discussed. Our framework provides a general mechanism to embed two conflicting tasks into a single molecular machine, which works context-dependently. Microtubule-based motors such as kinesins and dyneins are essential to a variety of processes, e.g., molecular transportation and mitotic spindle formation. The force generation is mediated by the walk of a unidirectional motor toward the plus/minus end of the microtubule (MT), following adenosine triphosphate (ATP) hydrolysis. Intuitively, highly processive and non-diffusive motor walks may be advantageous for efficient force generation. However, motors exhibiting a biased random walk toward a specific end with a large diffusion coefficient have also been reported123et al.8It was long believed that a given type of motor protein travels in a specific direction along an MT, and that this directionality remains constant in the absence of an external force. Quite recently, however, kinesin-5 Cin8, which is a tetrameric motor protein purified from budding yeast that can cross-link to two MTs, was found to exhibit directionality switching, depending on the number of motors bound to the same MTSimilarly, the emergence of a novel motility mode, other than the directionality transition, induced by team formation has also been reported in several experimental studies1011In this Paper, we introduce a mathematical model for a motor-filament system and demonstrate that directional collective motion contrary to the built-in directionality of a single motor can emerge in the absence of any external forces. This directionality transition emerges under a large diffusional component in the directional walk and asymmetry in the intra-molecular strain-dependent detachment. In addition, the collective directional motion of motors with non-biased diffusion is explained. The proposed model provides a representative example of a mechanism for embedding dual tasks into a single molecular machine, which will elucidate the role of a large diffusive component in the walk of a processive motor. The emergence of the collective motion demonstrated here is independent of, and fundamentally different from, that induced by the entropic force produced by a diffusible cross-linkeret al.N identical motors X1, X2, …, XN can be attached to a single rigid rod. As shown in 9151617N increases, as illustrated in Because our study is motivated by the MT sliding assay reported in Roostalu xi  for the ith motor, whereas the filament coordinate is denoted by y, as illustrated in xi = 0 is assigned. The filament then slides along the surface, pulled by multiple bound motors with force k is a spring constant. The time evolution for the filament with the friction coefficient γy under over-damped conditions is given byThe displacement along the filament between the bound motor and the tethering point  is denoted by ξy(t) is the thermal random force applied to the filament as ηi = η. For motors bound to the filament, it is natural to assume a no-slip condition: the relative distance between each bound motor xi and the filament y does not change unless ηi causes the i-th motor’s “walk” on the filament.where γx is the friction coefficient of the motor with the surrounding medium, which is considered to be much smaller than that of filament γy. This indicates that the relative distance does not change if ηi = 0. From where κ = k/γy. The driving force of the walk ηi is modeled as v0 represents the built-in walk velocity, Dx, and 〈ξi(t)ξj(t′)〉 = δi,jδ(t − t′). In addition, f(xi) represents how the motor’s walk is biased depending on the intra-molecular strain xi. For example, the motor with xi > 0, which is pulled from the minus-end due to the intra-molecular strain, is more likely to proceed toward the minus-end in the next step of the motor’s walk than the motor with xi < 0 does. Hereafter, we mainly take f(x) as a linear function: f(x) = a0x. The force-velocity relation of the single motor has been measured and discussed in several experimental papers19where x > Δ+ or x < −Δ−, with given positive parameters Δ+ and Δ−, whereas for the force-independent detachment, every bound motor is detached at the rate xi. In x = 0 with rate r.The bound motor can be detached from the filament in either a force-dependent or a force-independent manner. For the force-dependent detachment, we assume that the motor becomes detached if v0 < 0, which indicates that each individual motor takes a biased random walk toward a minus-end, whereas the case v0 = 0 will be discussed later. Because of the walking direction asymmetry of each motor, the filament always moves toward the plus-end and a directionality transition is impossible unless some asymmetry is introduced. Indeed, a directionality transition is observed for increased N, under the presence of: (i) an asymmetric strain-dependent detachment Δ+ > Δ− or (ii) an asymmetric dependency of the motor’s velocity upon the intramolecular strain |f(x)| > |f(−x)|. Here, we primarily address case (i). y for various N and the N − N increases  is given by the Fokker-Planck equation , M1 = v0 − f(x) − κx, and δ(x) is the Dirac delta function. The stationary distribution of the above equation is given by P(x) = M0ρ(x), wherewhere N = 1 agree rather well with the results of the stochastic simulations, as shown in and N, the time evolution of P is given by an N-dimensional differential equation, which is very difficult to solve. As N goes to infinity, however, the time evolution of P is reduced to a one-body problem, which is governed by the same equation as D = Dx (see M1 = v0 − f(x) − κ〈x〉. Here 〈x〉 is defined by M1 contains the term 〈x〉, which cannot be determined without knowledge of P(x); thus, ρ(x) on the right hand side is given in x〉 and ρ(x). The estimated analytical result agrees rather well with that of the stochastic simulations for sufficiently large N, as shown in For finite = Dx see  and M1 =N = 1, the xi distribution has a significantly sharper profile than that for N = 100, which indicates that the intra-molecular strain due to the extension of the motor is immediately released. That is, the filament can move immediately in the direction in which it is pulled by the motor, and x always remains close to zero. Thus, the asymmetry in Δ+/Δ− cannot manifest. The directionality is, therefore, determined according to the built-in walk direction, which is consistent with the slightly biased distribution in x〉: although the distribution seems to be almost symmetric, the non-zero value of P(x) − P(−x), as is shown in the inset of N and large Dx, the distribution of xi is significantly broader, which indicates that the multiple motors pull the filament in opposing directions, similar to a game of tug-of-war. Thus, the intra-molecular strain in each motor is released only a little. This “frustrated” motor configuration causes the asymmetry in Δ+/Δ− to manifest. Specifically, in the case of Δ+ > Δ−, the backward force with respect to the walking direction  applied to the motors with x < 0 causes them to detach more easily than the forward force applied to the motors with x > 0. This asymmetry allows for a slightly longer tail at x > 0 compared to that at x < 0 in the distribution of x〉 is positive and These analytical results illustrate the directionality transition mechanism. For Dx) in the walk, rather than a random thermal force ξy, is crucial for the emergence of the directionality transition. This is because large Dx leads to a broader x distribution, which reflects a highly frustrated “tug-of-war” situation of the motors. If Dx is small, the motors move with the built-in direction concurrently, yielding a sharper x distribution and smaller intra-molecular strains. Thus, the directionality transition is absent. This is also true even in the presence of a thermal force applied to the filament  much larger than v0, as this force does not change the relative distances between the motors.N discussed above, the phase diagram in κ = k/γy . We have also confirmed this transition in the analytical calculation for N = 1, as shown by the red dot in κ signifies low filament mobility and, thus, the strain in the stretching motor is not released by the filament movement, leading to a broader x distribution. As a result, the effect of the asymmetry in the strain-dependent detachment Δ+/Δ− becomes prominent, which allows for a slightly longer tail at x > 0 in the distribution; thus, the contribution of the tail to the mean 〈x〉 surpasses that of v0.In addition to the directionality transition with increased F(x), which results in N = 1, this F(x) is positive on average. That is, F(x) is required to immobilize the filament. This is despite the fact that the filament velocity is positive in the absence of F(x). This counterintuitive result can also be understood by noting that the increase in the strain ) through immobilization of the filament causes the asymmetry in Δ+/Δ− to manifest.In addition, we can also consider an extreme low-mobility situation, where the filament is fully immobilized through application of an external force N by introducing case (ii), the asymmetric dependency of the motor’s velocity upon the intramolecular strain |f(x)| > |f(−x)|, i.e., f(x) = a+x (if x ≥ 0) and a−x (if x < 0), with a+ < a−. In this case, the same directionality transition as in case (i) Δ+ > Δ− is obtained via the analytical estimates, as shown in Even without the asymmetry in the detachment process, directionality transitions can also be observed with increased v0 = 0, we confirm that the filament moves via non-biased diffusion for N = 1, whereas collective directional motion of the filament toward the minus-end emerges for large N, as shown in P(x) is symmetric for N = 1 as illustrated by the inset in P(x) manifests for N = 50 in 1011Finally, by setting +/Δ−) or in the force-velocity relation (a+/a−), the directionality in the filament movement can be reversed via an increase in the number of motors N. Although neither of these asymmetries has yet been measured for Cin8, asymmetry in the force-dependent detachment or the force-velocity relation have been reported for several motor proteins192021222324N, the intra-molecular strains in the motors are immediately released by the filament movement; thus, the motors walk in the built-in walk direction, i.e., towards the minus-end. On the other hand, when N increases, the diffusive nature of the processive motors leads to a frustrated motor configuration in which multiple motors pull the filament in opposite directions; thus, the intra-molecular strains in each motor are increased. These inter- and intra-molecular frustrations amplify the asymmetry in the strain-dependent properties of the motors and generate the opposite motility mode. Thus, the motors act as plus-end-directed motors, although their average movement is toward the minus-end. Similar to the several macroscopic models of motor-filament system262728In this Paper, we have investigated a mathematical model of a motor-filament system and shown that, with a large diffusivity in the motor’s walk and asymmetry in the strain-dependent detachment  correspond to the changes in parameters in our model, and the consequent results by the changes are consistent to previous experimental observationset al.et al.Some references have reported changes in motility of Cin8 even in a single molecule assay; Düselder v0 > 0 and Δ+ < Δ−. These experimental results are also explained by our model.Furthermore, quite recently, the directionality switch of another motor, kinesin-14 KlpA, due to collective binding of MT was reportedet al. reported the directionality switch in kinesin-5 Cut7In addition, an experimental study by Britto We have demonstrated that the number of molecules, rather than the density, can dramatically alter the motility mode. These findings are related to those reported in the growing literature on the small-number effect in chemical reaction systems32in vivo. In such a situation, the number of motors that can cross-link to the same MTs changes, depending on the configuration between the two MTs. Hence, the present mechanism provides a novel means of context-dependent regulation for a mechano-chemical coupling: plus- and minus-end directionality is used properly depending on the microtubule’s configuration. For instance, a team of cross-linkers operating under the proposed mechanism can transport two MTs in an anti-parallel alignment with their minus-end leading for a long overlap region, as abundant motors can cross-link. In contrast, for a short overlap, the motors would transport the MTs with their plus-end leading via a change in the directionality. As a result, the overlap region between two plus-ends of the MTs would be stably maintained within a certain length by only a single type of motor, which is a possible functional role of the dual-directionality of Cin8. This is in contrast to the conventional view in which two different types of motors are required for the maintenance of the overlap region, which is important for the mitotic spindle formation2834Although we only considered a MT sliding assay here, the proposed mechanism is straightforwardly applicable to a situation where both ends of a motor can bind to two different MTs , as is commonly observed How to cite this article: Saito, N. and Kaneko, K. Embedding dual function into molecular motors through collective motion. Sci. Rep.7, 44288; doi: 10.1038/srep44288 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Then we define the following real n-vector:x and a real number λ such thatλ is called an H-eigenvalue of x is called an eigenvector of λ [x and a real number λ such thatλ is called a Z-eigenvalue of x is called an eigenvector of λ. You can see more about the eigenvalues of tensors in [Let mentclass2pt{minimnsors in –7.k distinct vertices, then k-uniform hypergraph. The degree of a vertex i in i, denoted by Let udied in , 9.k distinct vertices kth order n-dimensional tensor with entries:Z-eigenvalues of the adjacency tensor Z-spectral radius of the adjacency tensor Let k-uniform hypergraph Z-eigenvalues For a nd Chang  presenteZ-spectral radius of hypergraphs, which improves the bound as shown in (Z-spectral radius of hypergraphs.In this paper, we give a new upper bounds in terms of degree mentclass2pt{minimSome basic definitions and useful results are listed as follows.The tensor Z-spectrum of Z-eigenvalues of Z-spectral radius of Let weakly symmetric was first introduced and used by Chang, Pearson, and Zhang [Z-eigenvalue of nonnegative tensors.The concept of nd Zhang  in orderLetbe a weakly symmetric nonnegative tensor, then the spectral radiusis a positiveZ-eigenvalue with a nonnegativeZ-eigenvectorx. Furthermore, ifis irreducible, xis positive.Letandbe two weakly symmetric and irreducible tensors of ordermand dimensionn. Ifandare nonnegative, theny be the eigenvector associated with β, where β is a Z-eigenvalue of Let m 4.7 of , we haveLetbe a sequence of nonnegative, weakly symmetric tensors of ordermand dimensionn, andbe nonnegative for each positive integerk. Thenλ is an eigenvalue of Let Z-spectral radius of tensors and hypergraphs.In this section, let Letbe weakly symmetric nonnegative tensors of ordermand dimension n. ThenZ-eigenvalues Case 1. If entclass1pt{minimaϵ. Now we substitute Case 2. If Z-spectral radius of a uniform hypergraph is obtained, we also compare the bound with the result in (By Theorem esult in .Letbe ak-uniform hypergraph onnvertices with the degree sequenceThenCase 1. ϵ. Now we substitute Case 2. If Obviously, we can getmentclass2pt{minimWe now show the efficiency of the new upper bound in Theorem Consider 3-uniform hypergraph Consider 3-uniform hypergraph From Table Z-spectral radius of tensors. As applications, in terms of the degree sequence Z-spectral radius of hypergraphs, which is always better than the bound in [In this paper, we get a new bound for the bound in . We list


Scientific Reports 10.1038/s41598-017-17044-3, published online 07 December 2017Correction to: Anabaena sp. PCC 7120” was incorrectly given as “Anabaena sp. PCC 7210”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying Supplementary Information file.The original version of this Article contained an error in the title of the paper, where “


Nature Communications 10.1038/s41467-017-01896-4, published online: 13 Dec 2017Correction to: y-axes were incorrectly labelled from ‘−40’ to ‘40’, rather than the correct ‘-400’ to ‘400’. The HTML has been updated to include a corrected version of the Supplementary Information.The original version of the Supplementary Information associated with this Article contained an error in Supplementary Figure 9e,f in which the


By measuring the Q factor with coupled power we find that the loss in the a-Si:H micro-rings scales linearly with power, and therefore cannot originate from two photon absorption. Theoretically comparing a-Si:H and c-Si micro-ring pair sources, we show that the high Kerr coefficient of this sample of a-Si:H is best utilized for microrings with Q factors below 103, but that for higher Q factor devices the photon pair rate is greatly suppressed due to the first order loss.We generate photon pairs in a-Si:H microrings using a CW pump, and find the Kerr coefficient of a-Si:H to be 3.73 ± 0.25 × 10 Hydrogenated amorphous silicon (a-Si:H) has recently arisen as a photonics platform to contend with crystalline silicon (c-Si). a-Si:H can be deposited at low temperatures and so is easy to incorporate in the chip manufacturing processes, and retains many of the advantages of silicon based photonic structures - similar waveguide dimensions, CMOS compatibility and low loss guiding of telecom wavelengths. Additionally, the nonlinear properties of a-Si:H are known to be dependent on the manufacturing conditions, potentially giving a user freedom to design the material to his or her requirements. a-Si:H has quasi-bandgap of ~1.7 eV, and experiences less two photon absorption (TPA) than c-Si, which has bandgap energy 1.1 eV. Samples have been fabricated with Kerr coefficient several times larger than c-Si, overall giving a-Si:H a high nonlinear figure of merit12345678These properties make a-Si:H a potentially useful material for quantum optics applications, in particular photon pair generation via spontaneous four wave mixing (SFWM). In c-Si photon pairs can be generated by SFWM, but nonlinear losses such as TPA reduce the heralding efficiency of silicon sources for both microrings and waveguides. High heralding efficiency is a key requirement for a single photon source in linear optical quantum computing, and there are concerns that this inherent nonlinear loss in c-Si makes it unsuitable for large scale applications10121415In this work we study the performance of a-Si:H microring resonators, in comparison to c-Si. We find that a-Si:H has a Kerr coefficient ~8 times higher than c-Si, and that the main source of loss is linear with power, and so cannot originate from TPA. Instead, we model this loss by interband transitions facilitated by a mid-gap state, referred to here as sequential photon absorption (SPA)12μm, and waveguide effective area 0.3 μm2. Light was coupled into the device using grating couplers with loss approximately 8 dB each.The test structure was a a-Si:H micro-ring resonator coupled to a bus waveguide, illustrated in The ring resonator acts as a cavity, with a comb of resonances equally spaced in frequency. When light is is coupled into the ring and SFWM occurs, the signal and idler photons are generated at resonances either side of the pump wavelength, as shown 1621P is the power in the bus waveguide, L is the ring length, Q is the Quality factor of the ring, and vg is the group velocity. The nonlinear coefficient τ is equal to the self coupling t. The sign of τ < t), and negative if the ring is under-coupled (τ < t). SFWM is driven by the Kerr coefficient n2, and while we assume device dimensions and properties are fixed, the Q factor of the ring is dependent on the loss in the ring24The power dependent Q factor is given byαNL is the loss per metre in the ring due to nonlinear losses only. Q0 is the Q factor of ring when the coupled power is zero. The extinction ratio Γ also changes due to nonlinear losses, and can be described in terms of the Q factorwhere The extinction ratio Γ also gives information about the coupling of the device. If a ring is over-coupled, Γ will decrease with power, because the coupling condition becomes closer to critical coupling as the loss is increased. Conversely, if the ring is under coupled Γ will increase with power. In this work, a-Si:H rings B and C were under-coupled, and a-Si:H ring A was over-coupled.ns), and so we can neglect loss by TPA directlyWith a CW pump, the powers involved are generally too small for electronic state transitions via absorption of photons  to significantly influence the loss in the ring. However, these processes generate free electrons in the conduction band which can absorb photons via intraband transitions, known as free carrier absorption (FCA). The number of photons absorbed in the creation of a free carrier is much lower than the number of photons absorbed by the free carriers themselves, because the free carrier can absorb several photons during its lifetime  can be given by QSPA (P) or QTPA (P).Where The experimental set-up used to measure the Q factor with coupled pump power is shown in στβSPA, giving 2.44 ± 0.223, 2.04 ± 0.200 and 2.13 ± 0.176 × 10−27 ms for rings A, B and C respectively. If there is any loss due to TPA generated free carriers present in the a-Si:H devices, it is negligible compared to the loss due to SPA generated free carriers, for the powers measured here. Since TPA is quadratic with power there exists a power threshold where TPA becomes dominant over SPA, so we cannot assume TPA is not present in this sample of a-Si:H, merely that we have not yet reached the power levels where it becomes important.The Q factor for the a-Si:H rings are shown in Q0 = 11070,  and identical length to the a-Si:H devices, but where the main loss mechanism was due to TPA driven FCA, with TPA coefficient 8.4 × 10−12 W/m2 To compare the behaviour of these rings to that expected for crystalline silicon, we modelled the power dependent Q factor for a c-Si device with To measure the rate of photon pairs generated via SFWM in the ring a CW laser was amplified and filtered to remove background noise from the amplifier and the wavelength tuned to the ring resonance wavelength. The signal and idler photons generated on resonance were separated from the residual pump light using a WDM after the chip and detected using super-conducting nanowire single photon detectors (SNSPDs). Coincidence counting hardware was used to obtain the rate of coincidence detections between the two SNSPDs, and accidental coincidence counts were removed. The rate of coincidences originating from photons generated in the waveguide was negligible in the devices.cm−1, corresponding to a wavelength shift of ≈115 nm for 1550 nm photons, but the width of the peak is typically broader than in c-Si and tails off into lower wavenumbers due to the presence of hydrogen bonds2829The Raman spectrum of a-Si:H has a peak centred around 480 Using the rate of the single counts at each detector, and the rate of coincidences, it is possible to remove the loss of each channel and obtain the absolute rate of photons generated in the ring, following the method outlined in ref. n2 to be 3.98 ± 0.03, 3.82 ± 0.07 and 3.39 ± 0.1 × 10−18 m2/W for rings A, B and C respectively, approximately 8 times larger than the literature value for c-Si. We measured the SPA parameter στβSPA to be 0.87 ± 0.01, 0.55 ± 0.05 and 0.53 ± 0.06 × 10−27 ms for rings A, B and C. The discrepancy between these values and the previous section likely arise from the increased stability during the measurement  is not an issue. Another feature of SPA is that the process is non-instantaneous, since the mid-gap states have a non-negligible lifetime. In the CW regime this has no effect, but using a short pulsed pump could further reduce loss from SPA generated free carriers. It is advantageous to use a-Si:H for waveguide based sources, and low Q factor a-Si:H microrings could be useful for applications where short pulses with high peak power are required, although further study is required to determine whether operating in this regime we could achieve better heralding efficiencies than c-Si in general.How to cite this article: Hemsley, E. et al. Photon pair generation in hydrogenated amorphous silicon microring resonators. Sci. Rep.6, 38908; doi: 10.1038/srep38908 (2016).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


The extremely low loss of silica fibers has enabled the telecommunication revolution, but single-mode fiber-optic communication systems have been driven to their capacity limits. As a means to overcome this capacity crunch, space-division multiplexing (SDM) using few-mode fibers (FMF) has been proposed and demonstrated. In single-mode optical fibers, Rayleigh scattering serves as the dominant mechanism for optical loss. However, to date, the role of Rayleigh scattering in FMFs remains elusive. Here we establish and experimentally validate a general model for Rayleigh scattering in FMFs. Rayleigh backscattering not only sets the intrinsic loss limit for FMFs but also provides the theoretical foundation for few-mode optical time-domain reflectometry, which can be used to probe perturbation-induced mode-coupling dynamics in FMFs. We also show that forward inter-modal Rayleigh scattering ultimately sets a fundamental limit on inter-modal-crosstalk for FMFs. Therefore, this work not only has implications specifically for SDM systems but also broadly for few-mode fiber optics and its applications in amplifiers, lasers, and sensors in which inter-modal crosstalk imposes a fundamental performance limitation. N-mode optical fiber could permit N channels to transmit independent data and therefore increasing the transmission capacity by a factor of N, in comparison with a single-mode fiber. Rayleigh scattering is an inherent process that will affect signal propagation in these channels. Additionally, crosstalk among these channels due to random mode-coupling171817N2, optical fibers with a few modes, or, so-called few-mode fibers (FMFs), have been deployed for SDM systems.The discovery by Kao that the optical loss of silica fiber could be reduced to below 20 dB/km34589101112131415Rayleigh scattering has been studied in single- and multi-mode optical fibers. And the relevant theories are well established. In particular, with regard to the Rayleigh scattering theory developed for MMFs01, LP11a, and LP11b modes, as an example to study Rayleigh scattering. The theory we developed here, however, can be extended to an arbitrary number of modes. The experiment, in principle, can also be trivially extended with the availability of efficient multiplexers/demultiplexers. In this work we didn’t take polarizations into account and treated Rayleigh scattering as a scalar process.We used a three-mode optical fiber, containing the LPΔT and a constant power P0 was launched at z = 0 into mode i  of the three-mode fiber. In the main text of the paper, as well as in the 01, LP11a, and LP11b modes, respectively. The time-dependent power of Rayleigh back-scattering in mode j received at z = 0 can be calculated byWe derived the equation for mode-resolved time-dependent power of Rayleigh back-scattering received at the front end of the fiber. To develop the theory, we assumed that an optical pulse, with a temporal width of vig and vjg are the group velocities of modes i and j, respectively; αi and αj are the optical attenuation coefficients of modes i and j, respectively; αs(z) is the ratio of the total scattered power at z to the incident power at z; Bij(z) is the overall capture fraction, quantifying the ratio of the scattered power into mode j to the total scattered power at z. Detailed derivation of αs(z) = αs, Bij(z) = Bij, and where i = j = 1, vig = vjg = vg, and αi = αj = α.We note that Bij, can be calculated by averaging the local capture fraction, bj, over the near-field intensity distribution of the incident mode i:In whereψjN and ψjF represent the near and far fields  is modeled as the field of a dipole located at . Following derivations detailed in bj:in which elds see  of mode pole see . RayleigandR′ = R/a, a is the radius of the core of the optical fiber, Bij:where Bij satisfy the following relations . As expected [see bj across the cross-section of the fiber closely mimics the intensity distribution of the near field of mode j. Indeed, we calculated bj using Bij, as a function of normalized frequency V ≡ 2πaNA/λ, where a is the core radius of the fiber, NA is the numerical aperture of the fiber, and λ is the wavelength in free space. The dashed line shows the normalized cut-off frequency for the LP11a and LP11b modes, at V = 2.405. Our experiment was performed at V = 3.49.Bij in B11 is almost identical with the single-mode result reported in ref. The degeneracy and the relative magnitudes of Before we further discuss the case of forward scattering and its impact on inter-modal crosstalk, we first verify the newly-developed model via OTDR measurement of the backscattering. This is because the theories for forward- and backward-scattering share significant similarities. After we place the backward-scattering case on a solid ground, we can develop the forward case by analogy.We measured the time-dependent power of Rayleigh back-scattering using the experimental setup schematically shown in Lij. Lij in detail. For calculating the theoretical curves, we used the values obtained by fitting the experimental data of αi and αj; αs ≈ −0.039/km as in ref. vig ≈ vjg ≈ 2.067 × 105 km/s. To quantify the relative difference between theoretical and experimental results, we compare the intercepts at z = 0 of the OTDR curves, as relative to Lij and Fij(z), the overall capture fraction for forward-scattering, are independent with z, i.e., αs(z) = αs and Fij(z) = Fij. Additionally, we assume that the incident light in mode i is continuous wave (CW) with a constant power Pi. Thus,We simplify Fij = Bij, i.e., the corresponding overall capture fraction of forward and backward scattering are identical. Therefore, the inter-modal crosstalk due to Rayleigh forward scattering isFrom symmetry consideration, 01 and LP11a(b) modes are stronger than that between the LP11a(b) and LP11b(a) modes. Inter-modal Rayleigh forward-scattering increases with z at the beginning because the scattered power from mode i keeps accumulating in mode j. At longer propagation distance, the coupled power decreases as z increases because the incident light in mode i attenuates due to propagating loss and the scattered light in mode j decreases in proportion to the local excitation. A maximum appears in between. α1 = 0.209 dB/km, α2 = 0.210 dB/km, and α3 = 0.233 dB/km. We can use Tij is - 37 dB, which is 7.51 dB smaller than the minimum total crosstalk (−29.49 dB for the case LP01 → LP11b) due, dominantly, to random mode-coupling and fiber imperfections that we measured in this three-mode optical fiber  of the fiber is 0.1259. The single-pass insertion loss between the first circulator and the last coupling lens of the mode multiplexer/de-multiplexer was 5.85 dB, 9.83 dB and 9.22 dB for the LP01, LP11a, and LP11b channels, respectively. The coupling losses from free space to the fiber were 0.50 dB, 2.51 dB, and 2.89 dB for the LP01, LP11a, and LP11b channels, respectively. 11b mode and the signal collected is in LP01 mode. We manually reconfigured the fiber connections for other combinations of excitation and collected channels.As shown in How to cite this article: Wang, Z. et al. Rayleigh scattering in few-mode optical fibers. Sci. Rep.6, 35844; doi: 10.1038/srep35844 (2016).


Print and television advertisements for e-cigarettes are currently legal in the United States. Given that e-cigarettes are a lower-risk alternative to cigarettes, these ads could have a positive public health impact if they motivate smokers to switch to e-cigarettes. However, the public health impact of e-cigarette ads could be negative if ads increase demand for both e-cigarettes and cigarettes. We use experimental auctions –in which participants bid in real auctions and winners pay for the items they purchase – to study the effect of print and TV e-cigarettes ads on demand for the brand from the ad, for another e-cigarettes brand, and for cigarettes. We ran experiments with 288 Pennsylvania smokers in November 2014–March 2015 and we found that in cases where an ad affects demand for e-cigarettes, the ad moves demand for cigarettes in the same direction. For example, the Blu print ad increases demand for Blu e-cigarettes and cigarettes among non-white participants. The Vuse TV ad reduces demand for both types of e-cigarettes and for cigarettes. We also find that non-white participants are willing to pay more for e-cigarettes in the absence of advertising, and that smokers who worry most about their health are willing to pay more for e-cigarettes. The results of this study point to the need for greater scrutiny of advertising for e-cigarette products such that they do not also induce demand for tobacco cigarettes. •Print and television ads are currently legal in the United States.•We use experimental auctions to assess how ads impact smokers demand for cigarettes and e-cigarettes.•Examine ads for two major brands: Vuse and Blu•Found that ads for e-cigarettes impact demand for cigarettes and e-cigarettes similarly Typically, this is accomplished through dissolving nicotine in a solution of propylene glycol and glycerin, and heating that solution to produce an aerosol that is inhaled by the user . While sIn April 2016, the Food and Drug Administration (FDA) released its final ‘deeming’ regulation, which extended its existing authorities over tobacco products to include e-cigarettes, hookah, cigars, and pipes . Among tUnderstanding the dynamics of e-cigarette use, and in particular, smoker demand for such products, is important for projecting their likely impact on public health. If large numbers of smokers were to abandon cigarettes in favor of e-cigarettes, the anticipated effects could be large and positive. However, if most smokers simply use them as a situational substitute, the effects could be substantially smaller, and even negative if such use patterns depressed attempts to quit, or encouraged those who otherwise would not have used tobacco to take up e-cigarettes.The same arguments apply to print and television advertisements for e-cigarettes, which are (as of this writing) legal in the United States. Intended use of e-cigarettes can be influenced by advertising, and e-cigarettes are unique in the US context in that advertising on television is currently permitted, a seeming advantage over other tobacco products. If these ads motivate smokers to switch from cigarettes to e-cigarettes, the public health impact is likely to be positive. But if e-cigarette ads increase demand for both cigarettes and e-cigarettes, the net public health effect may be negative.Experimental economics offers methods that are designed to examine demand for products in a nonhypothetical way, which may provide important insights into how much smokers truly value e-cigarettes as an alternative to cigarettes. Experimental auctions, first developed in the 1960s, have been used by economists for over 20 years to assess the demand for many products, including irradiated food products , food la22.1The study protocol was approved by the IRB at Susquehanna University. Tables were set up at grocery stores in Shamokin, PA and Williamsport, PA between November 2014 and March 2015. Eligible study participants were 18 and older, had smoked > 100 cigarettes in their lifetimes, had smoked at least one cigarette in the last month, and were not pregnant. Posted signs indicated that adult smokers could earn $20 for 15–20 min of their time. Auctions were conducted with one to eight participants at a time, and a total of 288 subjects participated.2.2https://youtu.be/k79Iutkwdcg and the Vuse TV ad can be viewed at https://youtu.be/J4XpPXzg8J8. The TV ads were shown on an iPad while the print ads were laminated.The study involved assessing how six advertising conditions along with a control group affected demand for two types of e-cigarettes and conventional cigarettes. Participants were randomly assigned to 1) a control group that saw no advertising; 2) a group given a print ad about Blu e-cigarettes; 3) a group given a TV ad about Blu e-cigarettes; 4) a group given both a print and a TV ad about Blu e-cigarettes; 5) a group given a print ad about Vuse e-cigarettes; 6) a group given a TV ad about Vuse e-cigarettes; and 7) a group given both a print and a TV ad about Vuse e-cigarettes. After surveying the companies' actual advertisements, we chose the ones we felt were most representative. Print ads can be found in the appendix while the Blu TV ad can be viewed at After seeing the ads , participants placed bids. Participants bid on a single-use Blu e-cigarette, a single-use Vuse e-cigarette, and a pack of Camel cigarettes. For both types of e-cigarettes, participants bid on the type of e-cigarette they preferred: menthol or full flavor. For the Camel cigarettes, participants also bid on the type they preferred: they had the option to bid on full-flavored, light, menthol, or menthol-light.2.3paying the selected price; a participant who bids less than the selected price does not purchase the product.For the current study, we used the Becker-DeGroot-Marschak (BDM) experimental auction mechanism . In thisThis BDM auction is “demand revealing” because it is theoretically in a participant's best interest to bid his or her true value (demand) for the product. This has been shown to hold in practice in the laboratory  and the 2.4After screening for eligibility and signing consent forms, participants filled out a brief survey on smoking behavior and received a detailed explanation of the BDM auction. Participants were informed (1) that they would place a private, written bid for an item; (2) they would purchase the product if their bid was higher than the bid randomly drawn from a container; and (3) it was in their best interest to bid their true value for the products, no more and no less. Participants were told that while they would bid on more than one product in a series of auction rounds, only one auction for one product would actually be carried out, and that that product would be randomly chosen after participants had bid on all products. This was done to disincentivize participants from reducing bids in later rounds in anticipation of winning multiple products. Any questions they had were answered and a practice round was conducted in which participants bid separately on two candy bars.Participants then bid in the real auctions. The participants were presented with a single-use Blu e-cigarette and bid on it. Those bids were collected, and participants were then presented with a single-use Vuse e-cigarette and bid on it. Those bids were collected, and participants were presented with the pack of Camel cigarettes and bid on that pack. Once the three rounds of bidding were complete, the binding auction round  was randomly determined. The selected price was then randomly chosen from a uniform distribution, which ranged from $0.10 to $15.00 in increments of $0.10. If the participant bid equal to or more than this value, she paid the selected price and received the package. If the participant bid less, no purchase was made.3To control for demographic and smoking-related characteristics, we estimate different Tobit regression models for cigarettes, Blu e-cigarettes, and Vuse e-cigarettes. A Tobit model corrects for the fact that bids are censored at zero. Bids for each product are the dependent variables. Independent variables include demographic variables, smoking-related variables, and dummy variables to indicate the advertising treatments (the no-advertising control group is the omitted category). The full regression model is estimated as.Bidi = α0 + β′Xi + γ′Si + δ′Ai + λ′Ni + εi,where Bidi is participant i's bid in for the product, αo is an intercept term, Xi is a vector of demographic variables and β′ is the associated coefficient vector, Si is a vector of smoking-related variables and γ′ is the associated coefficient vector, Ai is a vector of advertising treatment dummy variables and δ′ is the associated coefficient vector, Ni is a vector of advertising treatment dummy variables interacted with the “nonwhite or Hispanic” dummy variable and λ′ is the associated coefficient vector, and εi is the error term.4The participants' average age was 39.6 years, and our sample was 56% female. Eighty-five percent of participants were white, and 60% had a household income of under $30,000 annually. Participants smoked an average of 18.5 cigarettes per day and 57% of participants had tried an e-cigarette. Non-white or Hispanic participants who were not presented with ads bid $9.46 more for Vuse e-cigarettes, a difference that was statistically significant. Participants who worried about their health bid a statistically significant $1.73 more for Blu e-cigarettes. Number of cigarettes smoked had a marginally statistically significant $0.06 negative effect on bids for Blu e-cigarettes.5E-cigarettes are another in a line of potentially reduced exposure products that have promised to reduce the risk of smoking. Our study examined smokers' preferences for two brands of e-cigarettes, Blu and Vuse, using an experimental auction. We found that smokers were willing to spend their money for e-cigarettes. In fact, given a single-use e-cigarette contains about two cigarette packs' worth of content, and that the bids for Vuse were double the bids for cigarettes, smokers had the same units/day bid for Vuse e-cigarettes as for cigarettes. At the same time, the units/day bid for Blu e-cigarettes was lower than for cigarettes. This difference in demand across brands reflects market conditions, as Vuse has recently overtaken Blu in terms of market share . The Vusboth cigarettes and Blu e-cigarettes. We found modest evidence that the Vuse TV ad decreased demand for cigarettes among nonwhite or Hispanic participants. Taken together, these results suggest that e-cigarette ads shift demand for cigarettes and e-cigarettes in the same direction. If true, this could have serious public health implications given that print and TV ads for e-cigarettes are currently legal in the United States. If effective e-cigarette ads coincidentally increase demand for cigarettes in some populations, this may offset the public health benefits from some smokers adopting lower risk e-cigarettes.We found that the Vuse TV ad decreased demand for cigarettes and Blu e-cigarettes among white, non-Hispanic smokers. One possible explanation is that smokers are cynical and turned off by pro-tobacco advertising. This would be consistent with evidence presented in Limitations of the current study include localization in North-central Pennsylvania, relatively low racial/ethnic diversity among participants, and limitation to two cartridge-based e-cigarette products. The latter limitation was driven by the need for paired print and TV advertising, since other forms of e-cigarettes  are not advertised in this way, but rather by word-of-mouth, in vape shops, or over the Internet . Future E-cigarettes hold potential to reduce smoking-associated disease burden . However


Scientific Reports7:2918; doi:10.1038/s41598-017-03112-1; Article published online 07 June 2017The original version of this Article contained an error in the title of the paper, where the word “Noncanonical” was incorrectly given as “Noncanocial”. This error has now been corrected in the HTML and PDF versions of this Article, and the Supplementary Information file that accompanies the Article.