In this paper, we revisit the fractality of complex network by investigating three dimensions with respect to minimum box-covering, minimum ball-covering and average volume of balls. The first two dimensions are calculated through the minimum box-covering problem and minimum ball-covering problem. For minimum ball-covering problem, we prove its NP-completeness and propose several heuristic algorithms on its feasible solution, and we also compare the performance of these algorithms. For the third dimension, we introduce the random ball-volume algorithm. We introduce the notion of Ahlfors regularity of networks and prove that above three dimensions are the same if networks are Ahlfors regular. We also provide a class of networks satisfying Ahlfors regularity. Complex networks arise from natural and social phenomena such as the Internet, the protein interactions, the collaborations in research, and the social relationships. Readers are referred to Watts-Strogatz’sdBdballdfCBB)8dB and dball respectively. However the minimum box-covering problem and minimum ball-covering problem are NP-complete, which are proved rigorously in Theorem 1 and Proposition 2 respectively. The NP-completeness implies that the CBB algorithm and the random ball-covering algorithm do not have high performance, then we suggest some algorithms to improve the random ball-covering algorithm. For the third dimension df, we obtain an efficient algorithm: the random ball-volume algorithm. When do the three dimensions coincide? To answer this question, we introduce the notion of Ahlfors regularity of networks and prove that dB = dball = df (Theorem 2) if networks are Ahlfors regular. Then for Ahlfors regular networks, the random ball-volume algorithm is efficient to obtain the above three fractal dimensions.In this paper, we revisit the fractality of complex network by investigating three dimensions minimum box-covering; Kim, Goh, Kahng and Kimminimum ball-covering approach to calculate the fractal dimension of complex networks.Song, Havlin and Makse8G =  equipped with geodesic distance d, we let an l-boxA denote a subset of V such that the geodesic distance of any two points in the subset is less than l, an l-ball centered at x0 the subset Nl be the smallest number of l-boxes needed to cover V, and Bl the smallest number of l-balls needed to cover V. Suppose thatRecall some notation. Considering a network as a graph dB is the fractal dimension defined by Song, Havlin and Maksedball is defined by Gao, Hu and Diwhere box-covering, Song, Gallos, Havlin and Maksel-box-covering problem is NP-complete for any l ≥ 2. On the other hand, for ball-covering, which is far from box-covering in graph theory, we haveFor Theorem 1. The minimum l-ball covering problem is NP-complete for any l ≥ 2.random ball-covering algorithm (RBC)l is fixed, in each time t, we randomly choose one node xt in the vertex set Vt−1 remained in time (t − 1), and obtain Vt by cutting all nodes in Due to the NP-completeness, for finding the feasible solution of minimum ball-covering problem, we can apply the usual Vt−1 and take the first node. Moreover, given some function f.In the RBC algorithm we give a random sorting for nodes in f in nondecreasing order: If f is the degree function, we can obtain degree-order ball-covering algorithm (DOBC); If volume-order ball-covering algorithm (VOBC).Given a function g in nonincreasing order, we propose the following greedy algorithm:Assume that Vt inherits from V0 = V.Set For a function volume-greedy ball-covering algorithm (VGBC). Let g(x) = deg(x), we have the degree-greedy ball-covering algorithm (DGBC).When Bl ≤ Nl ≤ Bl/2, hence l can be taken large enough, we haveIn the point of view on fractal geometry, the box dimension is independent of the geometric shapes of covering, such as ball or box. It is easy to check that Proposition 1. The fractal dimensions dBand dballw.r.t. the box covering and ball covering respectively are the same.l large enough, and the upper bound However, for real networks with small-world effect, we can not take Example 1. Through above 5 algorithms  be the average cardinality of nodes in a ball with radius l, suppose thatBased on Shanker’s workdf the volume dimension. Please also see generalized volume dimensionet al.We call df related to average ball-volume and propose the random ball-volume algorithm for networks. Compared with the minimum box-covering algorithm and the minimum ball-covering algorithm, we have the following algorithm to calculate the average volume of ball with size l approximately.We will discuss the volume dimension Random ball-volume algorithm (RBV) (for fixed size l):x in the network.Randomly take a node ν).Calculate the volume l-balls.Repeat the steps 1–2 and obtain average volume of random df = 5.833  measure supported on a compact subset E, denoted by spt μ at point x be defined by H (·) isIn this paper, we try to find out the farther connection between the fractal networks and fractal geometry. Recall some classical result on fractal dimension. We find out that many dimension results have measure versions. Suppose P (·)25μ such that Ahlfors regularity assumption on the measureThat means for Hausdorff dimension, we have the corresponding measure version. When replacing c is an independent constant.where natural measure on a graph G = . For ν(Ω) be the cardinality of Ω, which is called the volume of Ω. We say that {Gt}t is a family of growing networks, i.e., Gt+1 contains node set of Gt, and neighbors of Gt are still neighbors of Gt+1. When {Gt}t is growing, we let νt(Ω) denote cardinality of Vt is the node set of Gt.We give a Remark 1. When takingas the sum of degrees of nodes in Ω, Wei et al.obtain the generalized volume dimension.Definition 1. Given s > 0, if we call the network an Ahlfors s-regular network. When {Gt}tis growing, we call {Gt}tAhlfors s-regular networks, if there is an independent constant c such that for allr < diam(Vt) and t,When the diameter of network is large enough, we haveTheorem 2. df = dball = dB = s if the network or growing networks are Ahlfors s-regular.When the networks are regular, we can use RBV algorithm to obtain their fractal dimensions efficiently.s-regular networks and growing trees in Now, we obtain a rule (rule 1) of generating Ahlfors Theorem 3. The growing self-similar trees defined above are Ahlfors s-regular with s = log 5/log 3. Therefore, we haveWe also have rule 2 and growing trees in t, we take rule 1 if xt = 1, else take rule 2. For example, if the sequence is G1, G2, G3 as in Fix an infinite sequence Moran tree with mixed rules. For this Moran tree without self-similarity, we have random growing networks, for each time t, we can choose rule 1 in probability p and rule 2 in probability 1 − p.Then we can generate a The rest of paper is organized as follows. Section 2 is devoted to the rigourous proofs on the NP-completeness of minimum ball-covering problem (Theorem 2) and minimum box-covering problem (Proposition 2). Section 3 is the preliminary on the Ahlfors regularity of fractal geometry, including covering inequality and self-similar fractal. In this section, we also recall the fact that the open set condition of self-similar fractal implies the Ahlfors regularity of fractal measure. Replacing the fractal measures by the cardinalities of subsets of networks, we obtain the Ahlfors regularity of networks. In Section 4, we prove Theorem 2 by using covering inequality shown in Section 2, and obtain Ahlfors regularity of a class of self-similar network (Theorem 3) by constructing bilipschitz mappings from a self-similar fractal, satisfying the open set condition, to self-similar networks, and estimating the cardinalities of balls of graph from the Ahlfors regularity of the fractal measure.Recall some notation of computer science. For an alphabet Definition 2. Let L and M be languages. ThenL is reduced to M) if there is a functionsuch thatWe say that some languageNP-complete, ifis for allBoolean satisfiability problem is NP-complete.The concept of NP-completeness was introduced in 1971 by CookClique covering problemG = , positive integer kInput: graph V is the union of k or fewer cliques, where a clique is a subset of vertices of G such that its induced subgraph is complete.Property: Set covering problemU and a family S of subsets of U, positive integer kInput: universe k or less, where a set covering is a subfamily U.Property: there is a set covering of size In 1992, KannDominating set problemG = , positive integer kInput: graph k or fewer nodes, where a dominating set is a subset D of V such that every vertex not in D is adjacent to at least one member of D.Property: there is a dominating set of In this section, we will show the following two problems are NP-completes.l-ball-covering problemG = , positive integer kInput: graph V is the union of k or fewer l-balls.Property: l-box-covering problemIn 1972, KarpG = , positive integer kInput: graph V is the union of k or fewer l-boxes.Property: l = 2, then l-ball-covering problem is exactly the dominating set problem, which is NP-complete.If l = 3, given a undirected graph G =  as in G.z (in red) in the edge x ~ z, z ~ y and x, y are not neighbors in For any We add a Hub (in blue) to connect all median points.Insert sub-median-point (in yellow) for every edge between one median point (in Step I) and Hub.We construct a leaf node (in pink) and the median point (in green) between leaf node and the Hub.If We have the followingClaim 1. There is a dominating set of k or fewer nodes in G if and only ifis the union of (k + 1) or fewer 3-balls.For any nodes V is a 3-ball centered at the Hub.The subset of all nodes not in V is 5, that means any 3-ball can not contain the pink node and any node of V simultaneously.The geodesic distance between the pink node and any node in D with u in For any 3-ball To verify this claim, we notice the following facts.G and there is a minimum 3-ball covering Suppose In fact, we have i ≤ s. Applying the fact (b), we see that there exists a 3-ball covering with (s + 1) balls. Hencefor any G, where On the other hand, considering the minimum 3-ball covering l = 3.Then (1) follows from (2) and (3). Then Theorem 1 is proved for l ≥ 4, we have the similar construction during reduction. In fact, we insert (l − 2) median points into each edge of G, add a Hub to connect all median points, insert (l − 2) sub-median-point for every edge between one median point and Hub. Finally, we construct the leaf node and connect it to the Hub, insert (l − 2) the median point between leaf node and the Hub. See l = 4.For Remark 2. To prove one problem is NP-complete, we always find a reduction from a known NP-complete problem to our problem. On the other hand, we can always construct a reduction from our (NP) problem to a known NP-complete problem due to the definition of NP-completeness.We give a proof of the following fact which is pointed out by Song, Gallos, Havlin and MakseProposition 2. For any fixed size l, the l-box-covering problem is NP-complete.Proof. If l = 2, l-box-covering problem is exactly the clique covering problem, which is NP-complete.l = 3, given a undirected graph G = , as in G′ =  in polynomial time with respect to the size of G.z (in red) in the edge  with degree 2, i.e., x ~ z, z ~ y and x, y are not neighbors in G′.For any We add a Hub (in blue) to connect all median points.We construct a leaf node (in pink) adjacent to the Hub.If We have the followingClaim 2. V is the union of k or fewer cliques if and only if V′ is the union of (k + 1) or fewer 3-boxes.G′ their geodesic distance For any nodes V is a 3-box.The subset of nodes not in V is 3.The geodesic distance between leaf node (in pink) and any node in To verify this claim, we notice the following facts.G such that s is minimal one. Suppose there is a minimum 3-box covering G′. We only need to show thatSuppose Ai is a 3-box in G′ for any i ≤ s. Applying the fact (ii), we see that there exists a 3-box covering with (s + 1) boxes. HenceIn fact, we have G where On the other hand, it follows from fact (i) that Then (4) follows from (5) and (6).l ≥ 4, we have the similar construction during reduction. See l = 4.□For X, d), let a δ-ball centered at x0 be an open ball δ-cube a cube of Euclidean space with side length δ, a δ-box B is a subspace of X such that its diameter less than δ, i.e., d < δ for all x, Given a compact metric space  ≥ d − d ≥ 2δ − δ = δ. That means i. Now, we obtain a new packing family of Pδ ≥ Bδ2.δ-balls, then d ≥ δ for all xi, xj) ≥ 2δ for all δ/2-balls δ/2-ball contains at most one points in δ/2-ball is less than δand d ≥ δ for all xi must be contained in some δ/2-ball. Therefore, we obtain Pδ ≤ Bδ/2.□Assume We also haveBy the above inequalities, the classical result25In fact, in the above formula, we take upper box dimension Si is a similarity with ratio ri, i.e., x, Ri is orthogonal. That means any similarity is the compositions of homothety, translation and orthogonal transformation.Let OSC) holds if there exists a non-empty open set V such thatWe say that the open set condition (μ (self-similar measure) satisfying H K = dimB K = s, and there is a constant C ≥ 1 such that for all r ≤ |K| (the diameter of K),Let E is said to be Ahlfors s-regularμ supported on E satisfying (8). That means the self-similar set satisfying the OSC is Ahlfors regular.A compact set We introduce a special self-similar fractal on   and 15. where orthogonal. matrixes V be the interior of polygon with vertexes , , , ,  and . Then (7) holds for m = 5  and B = . Let Let Remark 3. One node may have distinct codingsandifWe also notice that each node has three codings at most.x, Two different nodes t be the geodesic distance on Gt.Let dd > 0 independent of the index i such that d−1bi ≤ ai ≤ dbi.We denote Now, we will prove the following importantLemma 2. There is a constant c > 0 independent of t such thatProof. Supposewhere where Case 1. If Then (9) follows in this case.□Case 2. If D = , let θ ≥ θ0 (>0). Now,For  exists   such thawhich impliesWe also have On the other hand,y1, D) and . By the self-similarity, now we only need to prove the case when by the tree structure. It follows from (10) and (11) that we only need to verify (9) for the pairs  follows.μ with respect to the vector  is Ahlfors s-regular for s = log 5/log 3.Since the OSC holds, then the self-similar measure It follows from the above lemma and Remark 3 thatVt = 5t + 1. Therefore, we havewhere #t. Now, the growing networks {Gt}t are Ahlfors s-regular.Notice that the constant in (12) is independent of df and propose the random ball-volume algorithm, which has performance better than the above five minimum covering algorithms due to the NP-completeness. Applying the notion of Ahlfors regularity from fractal geometry, we prove that dB = dball = df = s if the network is Ahlfors s-regular. Finally, we investigate the Ahlfors regularity of a class of self-similar trees and random trees which come from the self-similar fractals and Moran fractals respectively. Although we only prove Theorem 3 for self-similar tree of model 1, but our approach can be applied to many self-similar trees, Moran tree and random trees. Essentially, our approach is to embed our networks into a self-similar (or Moran) fractal (on Euclidean space) satisfying the open set condition, using the Ahlfors regularity of corresponding self-similar (or Moran) measure, we can estimate the volume of balls in networks.We focus on the NP-completeness of minimum ball-covering problem, propose some heuristic ball-covering algorithms such as GOBC, GDBC, VOBC and VGBC, and compare these algorithms with usual RBC algorithm. Inspired by the notion of measure on fractal, a natural measure on the finite graph is obtained such that the measure of every subset is the cardinality of subset. Based on this measure, we revisit the volume dimension How to cite this article: Wang, L. et al. On the Fractality of Complex Networks: Covering Problem, Algorithms and Ahlfors Regularity. Sci. Rep.7, 41385; doi: 10.1038/srep41385 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


We generate the THz wave on the surface of an unbiased GaAs crystal by illuminating femtosecond laser pulses with a 45° incidence angle, and investigate its propagation properties comprehensively both in a near-field and in a far-field zone by performing a knife-edge scan measurement. In the near-field zone, i.e. 540 μm away from the generation point, we found that the beam simply takes a Gaussian shape of which width follows well a behavior predicted by a paraxial wave equation. In the far-field zone, on the other hand, it takes a highly anisotropic shape; whereas the beam profile maintains a Gaussian shape along the normal to the plane of incidence, it takes satellite peak structures along the direction in parallel to the plane of incidence. From the comparison with simulation results obtained by using a dipole radiation model, we demonstrated that this irregular beam pattern is attributed to the combined effect of the position-dependent phase retardation of the THz waves and the diffraction-limited size of the initial beam which lead to the interference of the waves in the far-field zone. Also, we found that this consideration accounting for a crossover of THz beam profile to the anisotropic non-Gaussian beam in the far-field zone can be applied for a comprehensive understanding of several other THz beam profiles obtained previously in different configurations. Quantitative information such as absorption coefficient or optical conductivity can be retrieved through a well-established THz time-domain spectroscopy technique which is often combined with far-field or near-field microscopy11. To precisely extract quantitative optical information, by the way, it is important to have an accurate knowledge regarding the beam profile at the interaction points of light with optical elements including the sample specimen of interest, and accordingly there have been extensive experimental and theoretical investigations about the THz beam pattern depending on its generation mechanism14. For the THz wave generated by the optical rectification in a nonlinear optical medium, e.g., ZnTe, Lin et al.16 demonstrated that a Gaussian aperture model successfully accounts for THz amplitude profile in both near-field and far-field regions. For the THz wave generated by a surge current in semiconductors such as GaAs and InAs, there have been several studies to characterize the beam profile, to our surprise, with no clear consensus made until now. Jepsen et al.17 reported that the THz pulse generated from the GaAs-based photo-conductive antenna has the Gaussian radiation pattern in the far-field zone. Reiten et al.18 performed the similar beam characterization and claimed that the THz beam consists of not only the simple Gaussian mode but also higher-order Laguerre-Gauss modes which later they attributed to a coupling effect of the THz pulse to a focusing optic19. Meanwhile, Jin et al.20 reported that the THz pulse emitted from the photo-Dember effect in InAs has the non-Gaussian amplitude distribution of which details are strongly frequency-dependent.Over the last few decades, an electromagnetic wave at the THz frequency has received a significant attention as it can provide a useful spectroscopic access to unveil the collective behaviors of electrons entangled with other degrees of freedom, i.e., orbital, spin, and latticey-axis, i.e., normal to the plane of incidence, follow the Gaussian model in both the near-field and the far-field zone. On the other hands, along the x-axis, i.e., parallel to the plane of incidence, we find that the field pattern becomes strongly irregular as the beam propagates into the far-field zone. We discuss these anisotropic field distributions of the THz wave in terms of a numerical approach based on the dipole radiation model.In this paper, we examine the beam profile evolution of the THz wave as it is generated from the un-biased free-standing semi-insulating (SI) GaAs, and propagates into the near-field zone and also to the far-field zone. We find that field patterns along the 22. The model begins from spreading 1000 electric dipole moments with an interval of 1.2 μm along the x-axis as shown in Fig. x-direction. Here, we assume that the THz generation by a transient dipole and the THz propagation occur with the same speed, and the position-dependent phase variations are determined mainly by a simple geometrical factor, i.e., incidence angle of the pumping beam. Therefore, we consider a linear variation of the oscillation phase of the dipole moment with a difference by 4 ps for two dipoles located at each end  of SI-GaAs, a refraction of the beam can be clearly recognized from the color distribution. In particular, when we plot a line profile of the field magnitude along the x-direction  is employed as the THz light detector.In the actual experiment, THz wave is generated by illuminating a femtosecond laser pulse on to the un-biased freestanding GaAs (001) crystal with an incidence angle of 45 degrees. The pumping beam has a wavelength 800 nm, and its photon energy 1.55 eV is slightly larger than the band gap energy of SI-GaAs. The pumping power is 160 mW, and the fluence is 0.25 J/cm3. THz wave is generated at the top surface of the emitter and propagates into its inside. By sliding this assembly of the homogenous emitter and the knife edge, we can mimic the conventional knife-edge experiment24; the THz wave is largely reflected from the coated region, and it is detected in a reflection geometry. From this, we can investigate the beam profile at the back side of the emitter which is 540 μm (δ) away from the generation point.Let us then examine the THz field pattern in the near-field zone. Since the near-field zone is still too close to put a conventional knife edge, we here introduce an alternative approach. Figure 1) is the THz wave generated backwardly at the emitter surface, and the latter one (E2) is the THz wave generated forwardly and reflected backwardly at the backside of the emitter. Hence the temporal separation of about 12 ps exactly corresponds to the traveling time of the THz wave inside of the medium with an effective thickness of 540 μm. With a variation of the position of the knife edge φx, whereas E1 remains the same, as expected, E2 exhibits dramatic changes; whereas E2 has an almost featureless waveform at φx = 0 mm, E2 increases its magnitude with an increase of φx, and it reaches to have a maximal magnitude of about one-third of E1 at φx = 2.8 mm. Actually, E2 at φx = 2.8 mm can be well reproduced by considering a full reflection of the THz wave at the emitter-Al interface. And, a small electric field at φx = 0 mm may originate from the tiny air-gap formed between two SI-GaAs layers.Figure 2 with a variation of φx. We pick up the magnitude at 0.5 THz from E2 spectra and display its φx-dependence in Fig. M/dφx. Although it does not fully reflect the original beam shape, it helps us to get a rough idea about the beam shape without any model fitting; dM/dφx shows the Gaussian-like profile, and its width seems to have about 1.8 mm, in accordance with the length scale of the THz beam waist at the generation point, i.e., Figure φx. And, this is a typical feature of the simple Gaussian beam as demonstrated in Fig. From these knife-edge experimental results obtained in the near-field zone, we examine THz beam profiles in more detail particularly with a variation of the frequency. Figure mentclass2pt{minimM0 is the maximum magnitude at . To treat the anisotropic shape of the THz beam arising from a 45° incidence angle, we set the x-axis is larger than that along the y-axis by a factor of We fit these results by adopting a Gaussian beam profile and display the full-width at half maximum as a function of the initial beam size and the light frequency in Fig. THz Fig. . To accoR = πn along both x- and y-directions to obtain the two-dimensional profile of the beam intensity. Figure L = 3, 4, and 5 cm. In each case, the magnitude and the corresponding derivation profiles dM/dφi=x,y are shown at two frequencies, i.e., at 0.3 and 0.5 THz. For a guide to the eye, we also present solid lines following data points which are obtained by a simple polynomial fit. At L = 3 cm, the magnitude changes smoothly with seemingly no discernible anomalies along both x- and y-directions. Accordingly, a global maximum appears in deviation profiles dM/dφi=x,y. Actually, such features evoke the Gaussian profile that we confirmed in the near-field zone. At L = 4 cm, significant changes are observed particularly from dM/dφi=x,y; whereas the scan results along the y-axis look similar with the results at L=3 cm, dM/dφi=x exhibits a highly off-centered global maximum with double local maxima  the THz beam profile is given anisotropically in the far-field zone, e.g., at L ≥ 4 cm, (ii) its distribution along the x-direction strongly deviates from the simple Gaussian model, and (iii) details of the beam profile exhibit a large variation as the beam propagates. These observations at L ≥ 4 cm cannot be explained by the Gaussian beam propagation theory, and suggest that some intriguing physical effects come into play when THz beam propagates from the near-field to the far-field zone.We now discuss how the THz wave evolves further during its propagation in the far-field zone. In this far-field characterization, we locate a conventional knife-edge at the distance int Fig. , and varima Fig. . Upon a x- and y-axes individually, and then multiply them to retrieve the two-dimensional magnitude profile. Figure L ≤ 52 mm. Here, the frequency is set to 0.5 THz. To focus on the amplitude distribution, we normalize beam widths with respect to the width of the ideal Gaussian beams at the corresponding propagation distances. At L = 4 mm, the THz beam shows an elliptical shape which simply reflects an initial beam shape at the source point. As L increases, the magnitude profile becomes inhomogeneous particularly along the x-axis; a major peak becomes more narrowly defined, and satellite peaks are clearly discernible being located nearby. Interestingly, the beam magnitudes along the y-axis do not seem to undergo significant changes, sharply contrasting to the x-axis case.To get an insight into the obtained results, we compute two-dimensional beam profiles in the far-field regime based on the electric dipole model as done for the generation point. Note that the basic details of the simulation are the same with those for the beam profile at the generation point demonstrated in the section 2. By the way, we obtain one-dimensional profiles along x- and y-directions. Figure L = 4, 28, and 52 mm. At L = 4 mm, both quantities exhibit smooth position-dependent variations whereas the results along the x-axis reflect a broader profile as expected (Fig. L increases, although the results along the y-axis remain with no discernible change, the profiles along the x-axis exhibit highly irregular position dependences; the derivative profiles become asymmetric and satellite peaks appear as expected from the amplitude distribution shown in Fig. To make a more direct comparison with the experimental results, we estimate the magnitude profiles of the THz beam as in the knife-measurements along the ted Fig. . As L inL < 4 mm imply that the Gaussian magnitude distribution of the point dipoles plays an important role in determining the THz beam profile up to the certain distance in the far-field zone. To account for anisotropic non-Gaussian beam profiles observed at L > 4 mm, on the other hand, it is clear that we have to consider the phase distribution as well as the amplitude distribution of the point dipoles. Along the x-axis, several satellite peaks appear. They are attributed to the position-dependent phase retardation of the THz waves generated from each dipole due to the grazing incidence of the pumping beam. As soon as the surge-current-induced electric dipole is created, the THz wave is radiated as an electric dipole radiation with its oscillation phase given uniformly irrespective of the generation point (Fig. y-axis at the same value of L. Namely, along the y-axis, the phase of the point dipole or, equivalently, the THz wave is given homogeneously, and hence the THz beam profile is determined mainly by the magnitude distribution of the point dipoles following a propagation theory of the simple Gaussian beam (Fig. Two-dimensional Gaussian profiles observed at int Fig. . Since tint Fig. . It shoueam Fig. . As we i20, and we consider that each result could be attributed to the phase contribution for the normal19 and grazing incidence20, respectively. In the near-field zone, on the other hand, the THz profile is determined rather by the pumping beam profile leading to the Gaussian distribution as observed16.We examined how the beam profile of the surge-current-induced THz wave, generated at the surface of the unbiased GaAs, evolves as it propagates from the source point to the near-field zone, and also to the far-field zone. During its propagation, we observed a clear crossover behavior of the beam profile from the Gaussian shape in the near-field zone to the anisotropic non-Gaussian shape in the far-field zone. From the comparison with the simulation results based on the point-dipole radiation theory, we concluded that the THz wave properties are determined basically by the intensity profile of the pumping beam, incidence angle, and its size; the THz profile in the near-field zone is determined by the magnitude distribution of the point dipoles, i.e., the pumping beam intensity profile, and the THz profile in the far-field zone is influenced not only by the magnitude distribution but also by the phase distribution of point dipoles. As a result, we expect to observe a deviation from the Gaussian form more easily by increasing the incidence angle and also by reducing the pumping beam size particularly down to the diffraction-limit of the THz wave. We emphasize that this understanding can be applied to other experimental results characterizing beam profiles of THz waves. In the far-field zone, the THz wave has been reported to exhibit both the Gaussian and non-Gaussian shape


In this paper, we consider a size-dependent renewal risk model with stopping time claim-number process. In this model, we do not make any assumption on the dependence structure of claim sizes and inter-arrival times. We study large deviations of the aggregate amount of claims. For the subexponential heavy-tailed case, we obtain a precise large-deviation formula; our method substantially relies on a martingale for the structure of our models. G on X and θ. The claim arrival times are Consider the following renewal risk model. Let F on n-fold convolution of F. Clearly, (F. See Embrechts et al. [We study large deviations of t}St∗ in . We onlymentclass2pt{minimOur main result is given now.Consider the aggregate amount of claimsin (assume thatandThen, for arbitrarily givenwe have uniformly for allt}St∗in , assume X and θ, which was firstly proposed by Albrecher and Teugels [et al. [et al. [et al. [X given θ; see also Li et al. [θ̃ such thatx, they studied the large deviations of the aggregate amount of et al. [A non-standard renewal risk model with dependent components  Teugels  and furt [et al. , Cossett [et al. , Badescu [et al. , and amo [et al.  introduci et al. . In parti et al.  considermentclass2pt{minimX given θ, e.g., Chen and Yuen [We now comment on the approaches used in this work. First, in Theorem and Yuen  made theThe rest of the paper is organized as follows. Section Throughout this paper, for two positive functions To obtain our desired results, we need to mention the following useful lemma.Consider the renewal counting processin (Under the assumptionThen, for anywe havet}Nt∗in . Under tu and t,First, for arbitrarily et al. [By }F∈S and , we needmentclass2pt{minimn,Therefore,By , therefoence, by  and (3.5As was remarked by a few researchers in the area, precise large-deviation results of size-dependent renewal risk models are particularly useful for evaluating some risk measures such as the conditional tail expectation of the aggregate amount of claims from a large insurance portfolio. Finally, we would like to point out that equation  agrees wfined by  does not


In the sparse representation model, the design of overcomplete dictionaries plays a key role for the effectiveness and applicability in different domains. Recent research has produced several dictionary learning approaches, being proven that dictionaries learnt by data examples significantly outperform structured ones, e.g. wavelet transforms. In this context, learning consists in adapting the dictionary atoms to a set of training signals in order to promote a sparse representation that minimizes the reconstruction error. Finding the best fitting dictionary remains a very difficult task, leaving the question still open. A well-established heuristic method for tackling this problem is an iterative alternating scheme, adopted for instance in the well-known K-SVD algorithm. Essentially, it consists in repeating two stages; the former promotes sparse coding of the training set and the latter adapts the dictionary to reduce the error. In this paper we present R-SVD, a new method that, while maintaining the alternating scheme, adopts the Orthogonal Procrustes analysis to update the dictionary atoms suitably arranged into groups. Comparative experiments on synthetic data prove the effectiveness of R-SVD with respect to well known dictionary learning algorithms such as K-SVD, ILS-DLA and the online method OSDL. Moreover, experiments on natural data such as ECG compression, EEG sparse representation, and image modeling confirm R-SVD’s robustness and wide applicability. In many application domains, such as denoising, classification and compression of signals –3, it isdictionary matrix n < m) with columns di, i = 1,…,m, called atoms, and a signal vector y as linear combination ∑ixidi with as few as possible non-zero coefficients sparse approximation problem consists in finding y − Dx‖2 under the constraint that its ℓ0-norm ‖x‖0 ≔ #{i: xi ≠ 0} be at most a threshold x is k-sparse. This problem is combinatorial in nature and hence NP-hard  ≔ {1,…,m} the set of first m positive integers and let I ⊂ [m] denote a set of indices for matrix columns or rows. Given any index set I of size s = |I|, let D formed by the columns indexed by I, that is DI = {di}i ∈ I, and let X formed by the rows indexed by I; hence s is the size of atom group DI. In this setting, we can decompose the product DX into the sumDIXI dependent on the group I and a matrix DIcXIc dependent on the complement Ic = [m] \ I. Therefore, the objective function in Basically, in the proposed method R-SVD, after splitting the dictionary DIXI in E ≔ Y − DIcXIc, one can consider the optimization problemDI of unit ℓ2-norm atoms.Now, after isolating the term R (i.e. RTR = I) applied on DI, namely R of order n, called orthogonal group O(n) (not to be confused with group of atoms), can be partitioned into the special orthogonal subgroup SO(n) formed by proper rotations, i.e. those with detR = 1, and the set O(n) \ SO(n) of improper rotations (or rotoreflections), i.e. those with detR = −1. Therefore, the search for such an optimal transformation can be stated as the following minimization problemE and H we omit the dependence on I. The Orthogonal Procrustes problem  = I1 ⊔ I2 ⊔ … ⊔ IG into G subsets,at each dictionary update iteration (Step 2) partition the set of column indices  w.r.t. the usage of atom di, computable as ℓ0-norm of the i-th row in X. For sake of simplicity we set uniformly the group size to s = |Ig| for all g, possibly except the last group (G = ⌈m/s⌉) if m is not a multiple of s: |IG| = m − Gs. Regarding this choice, we have seen experimentally that our method is agnostic w.r.t. grouping criteria such as random balanced grouping, cumulative coherence based partitioning, and clustering by absolute cosine similarity.These updates can be carried out either in parallel or sequentially with some order. We have chosen the sequential update with ascending order of atom popularity, i.e. sorting the indices G groups, the method moves to the next iteration, and goes on until a stop condition is reached . The main steps can be summarized in Algorithm 1 .After processing all Algorithm 1 R-SVDInput:Output:YD picking m examples from Y at random1: Initialize dictionary repeat2: xi‖0 ≤ k for i = 1,…,L3: Sparse coding: m] = I1 ⊔ I2 ⊔ … ⊔ IG sorting by atom popularity4: Partition indices [forg = 1,…,Gdo5: J = Ig6:  E = Y − DJcXJc7:  H = DJXJ8:  s SVD EHT = UΣVT9:  DJ = RDJ10:  end for11: returnD, X12: until stop condition13: Notice that in our method the renormalization of atoms to unit length at each iteration is not necessary since they are inherently yielded with such a property from this Procrustes analysis, and hence in practice some large part of renormalizing computations as in ILS-DLA  and K-SVDI can be described as follows. Let us pre-compute EHT undergoing the SVD can be computed asDI and DIc come from previous update step, and the term XIYT is the submatrix formed by rows I of XYT, while XI(XT)Ic by rows I and columns Ic of XXT. With elementary matrix products this computation requires O(sn(n + m − s)) flops, which is lower than O(nmL) since s < n < m ≪ L.A useful computational speedup in the update of every group HET ≤ rankH ≤ s, it is not necessary to obtain the full SVD of HET, but rather truncate the decomposition to the first s singular vectors ui, vi: O(n2 log s) flops. The computation of HET and its SVD is repeated G = ⌈m/s⌉ times. The computational time of R-SVD is dominated by the pre-computation of XYT and XXT, and therefore taking into account the sparsity of matrix X the asymptotic estimate for one iteration of R-SVD is TK-SVD = O((k2 + n)mL). Note that, when s = Ω(k) we have TR-SVD is lower than TK-SVD.Notice that, since rank rithm by  based ons defined above. Then we apply the method on synthetic data conducting extensive experiments on both R-SVD and K-SVD using OMP as sparsifier. A further investigation is conducted on two different sparse decomposition methods, namely k-Limaps and SL0, and alternative dictionary learning methods, namely ILS-DLA by Engan et al. [In this section we test the proposed R-SVD algorithm devoting at first an in-depth analysis to how to choose the group size n et al.  and OSDLn et al. .ℓ2-norm. The training set L linear combinations of k dictionary atoms selected at random, and by adding white Gaussian noise matrix N with various signal-to-noise ratio (SNR), i.e. Y = DX + N. We measure the performances of the algorithms in terms of the reconstruction error  expressed as Following , the dics affects both reconstruction quality and running time. In order to give some insight on this parameter, we run R-SVD algorithm on synthetic training sets by setting L = 8000, k = 5 and SNR = 30 dB for noise N and letting s range in the interval 1 ÷ 25. Notice that when s = 1, our method is similar to K-SVD [It is naturally expected that the group size to K-SVD  except iSNR (solid curve), and the computational times of both R-SVD (dashed curve) and K-SVD (dotted line), all averaged over 100 trials. It can be noticed that R-SVD method behaves better near the value s = 10. We thus choose this tradeoff setting for the experimental assessments of the method in the following sections.In L = 10000 and sparsity k = {5, 10}. The algorithms K-SVD and R-SVD are run for T = 200 dictionary update iterations, that turns out to be sufficient to achieve empirical convergence of the performance measure. For each experimental setting we report the average error over 100 trials.In order to test the R-SVD method and compare it to the K-SVD algorithm, we first run experiments on random training instances. We consider dictionaries of size 50 × 100 and 100 × 200, dataset of size up to SNR values on synthetic vectors Y = DX + N, varying the additive noise SNR = 10, 30, 50, ∞ (no noise) dB. It can be seen that, after an initial transient, the gap between R-SVD and K-SVD increases with the iteration count, establishing a final gap of 2 dB or more in conditions of middle-low noise power (SNR ≥ 30 dB).In T = 200) reconstruction error ESNR, varying L in 2000 ÷ 10000, noise SNR in 0 ÷ 60 dB, and in case of no noise. Dictionary sizes, sparsity and number of trials are set as above. When the additive noise power is very high (eg. SNR = 0 or 10 dB) the two methods are practically comparable, probably because the presence of significant noise would mislead any learning algorithm. On the other hand, when the noise is quite low the R-SVD algorithm outperforms K-SVD with a gap up to 3 dB.In order to explore the behavior of R-SVD and K-SVD in a fairly wide range of parameter values, we report in D from the noise-affected data Y. This is accomplished by maximizing the matching between atoms di of the original dictionary with atoms Moreover, it is useful to evaluate the number of correctly identified atoms in order to measure the ability of the learning algorithms in recovering the original dictionary is small , i.e. prIn k-Limaps [So far we have used OMP as sparsifier for both R-SVD and K-SVD methods. Here we investigate R-SVD’s performances adopting the sparse decomposition techniques k-Limaps  and SL0 k-Limaps . The fork-Limaps ), while http://ee.sharif.edu/~SLzero/) is set up with scale parameter μ0 = 2 and σ decrease factor equal to 4/5, while k-Limaps is initialized setting to 100 the maximum number of iterations. The experiments of R-SVD and K-SVD incorporating k-Limaps and SL0 were conducted under the same conditions of subsection 3.2. The resulting gap between final ESNR’s is shown in Each method has been applied to compute line 3 of Alg. 1, R-SVD, and the corresponding operation in the K-SVD algorithm. The SL0  versus the iterative alternating scheme method ILS-DLA, and the on-line dictionary learning method OSDL by Sulam et al. .L = 5000, 10000, dictionaries of size 64 × 128 (where the atom size is a perfect square as required in OSDL), various levels of additive noise , and sparsity set to 10% of the atom size. The methods are run for 200 dictionary update iterations and the obtained results are averaged over 100 trials. In ESNR of R-SVD and each of ILS-DLA and OSDL, respectively. We can notice that R-SVD systematically demonstrates better performances than ILS-DLA and OSDL. More specifically, ILS-DLA shows a regular behavior, with a considerable rise of the gaps with the increase of additive noise’s SNR. Concerning OSDL, we have even more significant gaps in favor of R-SVD, but with a less regular distribution, which is not easy to be interpreted. This may be in part due to OSDL having been originally conceived for learning of large dimension image patches.In these experiments we refer to random training sets of size In order to assess the applicability of the R-SVD method to various domains and tasks, we test it on ECG compression, EEG sparse representation, and image modeling. All the experiments are conducted on publicly available data, comparing the R-SVD and K-SVD performances, and adopting OMP as sparse decomposition method.Sparsity techniques have been already applied to the compression of electrocardiogram (ECG) signals , 32. To n-length normalized RR-segments. The second stage focuses on the dictionary learning where either R-SVD or K-SVD are used to train a dictionary on a group of RR-segments taken from an initial transient of the signal (train chunk). The latter stage concerns the encoding via sparse reconstruction of all RR-segments belonging to a broader interval of the signal (test chunk). This step is carried out referring to the learnt dictionaries, and applying the OMP algorithm. Naturally, in order to make the coding step easier, magnitudes and positions of non-null sparse coefficients are handled separately.In this experiment, the compression process is broken down into three stages. The former is a preprocessing step consisting in R-peak detection of the signal, and its normalization  so as to split it into y is composed of Nq-bit resolution samples forming M RR-segments, we havek is the sparsity level, m is the number of the atoms in the dictionary, and y, the overall reconstruction quality measure is here specialized for ECG signal as The compression level achieved by each method is measured as compression rate (CR), that is the ratio between the number of bits of the original signal and that of the compressed representation. Assuming that the test chunk fs = 250 Hz and q = 12-bit resolution. Performances are reported in SNR vs CR obtained from OMP, referring to dictionaries learnt by R-SVD or K-SVD, and to an untrained dictionary (i.e. randomly picking m RR-segments from the train chunk). Lower plots report the computational time spent by the two techniques in the learning stage. To make the experiment realistic, we set n = fs, m = 5n and L = 5000 RR-segments (about 120 minutes), while the test chunk is composed of M = 15000 RR-segments (about 4 hours). In order to analyze the methods at several rates, we varied k in the interval 5 ÷ 80 with step size 1.We conducted the experiments on two ECG records taken from the Long-Term ST Database in PhysioNet . They arSuch experiments, besides confirming that trained dictionaries behave better than untrained one, prove the effectiveness of the R-SVD training method that outperforms K-SVD both in training ability and learning time.Electroencephalogram signals (EEG) are exploited in several Brain-Computer interfaces (BCI) mainly due to their high temporal resolution. An element of difficulty when dealing with this kind of signal is its non-stationary temporal behavior, an unwanted property that has negative consequences on common tasks such as classification. Several adaptive techniques and updating rules have been proposed to produce compelling dictionaries , 35. HerP of n-length normalized EEG chunks corresponding to a given motor imagery . The second stage concerns the dictionary learning in the strict sense: starting from a random dictionary Dinit of dimensions n × 2n, both the R-SVD and K-SVD methods are adopted to specialize Dinit on the basis of a pool of EEG chunks randomly taken from P. Finally, the quality of the learnt dictionaries, let’s say DK-SVD, DR-SVD, is evaluated applying the OMP algorithm on a distinct pool of EEG chunks also taken from P.Operatively, we split the task into three stages. The former aims at producing a pool http://www.bbci.de/competition/iii/desc_IVa.html): EEG portions of signal corresponding to one motor imagery (either right foot or right hand) are extracted according to the given signal labelling, normalized to zero-mean signals and set into the pool P. The learning phase is carried out by setting the atom dimensions n = 150 or 300, the number of iterations T = 150, and expressing the sparsity k as various percentages of n. The cardinality of training set and test set is L = 2n and M = 4n respectively.We conducted the experiments referring to the dataset IVa from BCI competition III  . All results in the two phases are averaged on 50 trials. It is evident that the R-SVD method systematically exceeds K-SVD on both EEG chunk sizes. Moreover, the reconstruction quality increases with k, as well as the gap between the performances of K-SVD and R-SVD.In Sparse representation of images via learnt overcomplete dictionaries is a well-known topic in the fields of image processing and computer vision . In partP, a pre-processing is first applied aiming at both removing the patch mean intensity and reshaping all the patches to vectors in Given a pool of image patches k in 5 ÷ 30. The initial dictionaries for the two algorithms have size n × 1.5n, while training and test sets have size L = 2n and M = 4n respectively; they are all made up of randomly selected patches from P.The learning phase is carried out by setting the number of iterations to 50, and varying the sparsity n = 81) and 16 × 16 (n = 256). In both cases we randomly extracted 100,000 patches from 500 images of Caltech 101 [SNR trends of the learning phase on the training sets. We can observe that, since the first iterations, R-SVD behaves better than K-SVD, maintaining a positive gap at convergence, especially in the cases of lower sparsity. In SNR attained in the test phase using the dictionaries learnt by K-SVD and R-SVD, and then applying OMP. All results in the two phases are averaged on 50 trials. As we can observe, the R-SVD method systematically exceeds K-SVD referring to both patch sizes and to several sparsity degrees. These results further confirm the effectiveness and generality of the R-SVD method.Experimentally, we considered patches of size 9 × 9 Click here for additional data file.S2 File(ZIP)Click here for additional data file.S3 File(ZIP)Click here for additional data file.S4 File(ZIP)Click here for additional data file.


Streptomyces mediterranei (now Amycolatopsis rifamycinica)  was a 1955 French gangster film that was popular at the time and became the root of the name “rifamycin” for this group of antimicrobial agents.  Rifampin  is the N-amino-Nʹ-methylpiperazine (AMP) derivative of rifamycin.In 1957, Piero Sensi and colleagues isolated a new bacterium, ycinica) , from a


Several domains of cognitive function, including learning memory and executive function, are impaired in mood disorders. Also, the relationship between disturbances of these two cognitive domains has been suggested. In line with the recent initiative to establish a standard measure of cognitive decline in bipolar disorder, the present study was conducted to (1) test the criterion-related validity and test–retest reliability of the California Verbal Learning Test (CVLT)-II Japanese version, and (2) determine if type of word learning tasks  affects severity of verbal memory deficits in patients with subsyndromal bipolar disorder.Thirty-six patients with bipolar disorder with mild symptoms and 42 healthy volunteers participated in the study. We first compared effect sizes for memory deficits in patients among the CVLT-II, Brief Assessment of Cognition in Schizophrenia (BACS), and Hopkins Verbal Memory Tests-Revised (HVLT-R). We next evaluated the correlations between scores of the CVLT-II vs. those of the BACS and HVLT-R. Bipolar patients were re-assessed with the same (standard) or alternate forms of the CVLT-II and HVLT-R 1 month later.Scores on the CVLT-II 1–5 Free Recall and Long-delay Free Recall, as well as the HVLT-R Immediate Recall, but not the BACS List Learning were significantly lower for patients compared to control subjects. The effect sizes for cognitive decline due to the illness were comparable when measured by the CVLT-II and HVLT-R, ranging from 0.5 to 0.6. CVLT-II scores were significantly correlated with those of the HVLT-R and BACS. Test–retest reliability of the CVLT-II was acceptable, and no significant practice effect was observed when the alternate form was used. There was no consistent relationship between mood symptoms and performance on the CVLT-II.These results suggest the CVLT-II Japanese version is able to discriminate between bipolar disorder patients and healthy controls with good sensitivity and validity. Data in this study also indicate that the degree of verbal memory deficits in bipolar disorder may be influenced by memory organizational strategy. Bipolar disorder is associated with poor psychosocial outcome not only in the manic or depressive state, but also in the subsyndromal state –3. PatieThere may be interactions in the disturbances of key cognitive domains in bipolar disorder. For example, the contribution of executive function to learning memory has been reported in patients with the disease . SpecifiTo evaluate verbal memory in subjects with bipolar disorder, several tasks, including the Brief Assessment of Cognition in Schizophrenia (BACS) , HopkinsThe effect size of cognitive decline in euthymic patients ranges from 0.66 to 0.90 . As verbThe main purpose of this study was to investigate the impact of memory organizational strategies on verbal (learning) memory, as measured by performance on word list tasks, in patients with bipolar disorder. For this aim, we sought to determine whether the CVLT-II and HVLT-R, but not the BACS would be able to discriminate between patients with subsyndromal bipolar disorder and normal control subjects. This is based on the assumption that only the former two tasks require subjects to use memory organization. Additionally, we investigated the validity and reliability of the Japanese version of CVLT-II using the BACS List Learning and HVLT-R as reference measures of verbal learning and memory. Preliminary analyses of the present data have been reported , 23.This was a multi-center collaborative study, whose design, characteristics of participants, and other information have been registered (UMIN ID: 000013623). The sample consisted of 78 participants; 36 individuals with bipolar disorder and 42 healthy control participants who were native Japanese and had no history of psychiatric disorders Table . The patWritten consent was obtained from all participants, according to ethics guidelines set out by each participating site. The study protocol was approved by the ethics committees of participating institutions.The subjects were administered three verbal learning tests twice with an interval of approximately 1 month. Mood symptoms were also assessed each time using the MADRS and YMRS. The CVLT-II, BACS, and HVLT-R forms were switched to alternate forms in 43 subjects  at the follow-up (19 patients and 24 controls), whereas 35 subjects  were administered the same standard form at the follow-up (17 patients and 18 controls) as at the baseline. The same raters performed these cognitive tests at baseline and the 1-month follow-up assessments.To develop a Japanese version of the CVLT-II , one of The CVLT-II measures both recall and recognition abilities using two word lists. In the first five trials, immediately after presentation of List A, the subject is asked to recall the words of the list. List A contains 4 words from each of 4 semantic categories, for a total of 16 words. This procedure enables evaluation of semantic clustering ability, the most effective strategy for learning non-systemized verbal information. Subsequently, an interference list (List B) containing 16 words is presented, followed by a recall test. The interference test is followed by a short-delay free recall test and a short-delay cued recall test using List A. Then, following 20-min interval, a long-delay free recall test, long-delay cued recall test, and yes/no recognition test are administered using List A. After the yes/no recognition test, a new approximately 10-min forced-choice recognition test is arbitrarily administered. In the present study, data obtained in the immediate recall test after the first five trials of List A  and a long-delay free recall test were adopted for analyses, which were the candidate measures to be incorporated in the International Society for Bipolar Disorders-Battery for Assessment of Neurocognition .The BACS List Learning test consists of a word list, containing 15 words. The subject is asked to recall the words of the list immediately after presentation of the word list, which was repeated five times. The words in the list were not semantically organized unlike the case in the CVLT-II and HVLT-R. In the present study, a standardized measure in the immediate recall test after the first five trials was adopted for analyses. (The standardized measure was obtained by using the mean level of 49.2 and the standard deviation of 9.9.) .t-test was performed to explore between-group differences for CVLT-II 1–5 free recall scores and long-delay free recall scores, HVLT-R immediate recall scores, and BACS list learning scores at baseline. If a significant between-group difference in either measure was found, the effect size was calculated using a Cohen’s d to explore its sensitivity. To examine the effect of mood symptoms, Spearman’s rank correlation was performed between CVLT-II 1–5 free recall or long-delay free recall scores vs. MADRS and YMRS scores. Test–retest reliability was evaluated using intraclass correlation coefficient (ICC) (Student’s nt (ICC)  between To examine the criterion-related validity of the CVLT-II measures, Pearson’s product-moment correlation among scores of the CVLT-II 1–5 Free Recall and Long-Delay Free Recall, HVLT-R Immediate Recall, and BACS List Learning was calculated for patients.t-test revealed a significant between-group difference for the CVLT-II 1–5 Free Recall  and Long-Delay Free Recall , and HVLT-R Immediate Recall , but not the BACS List Learning  , and Long-Delay Free Recall . On the other hand, performances on these CVLT measures were not correlated with YMRS scores .Student’s The ICCs between the baseline and 1-month follow-up scores for the BACS List Learning, HVLT-R Immediate Recall, CVLT-II 1–5 Free Recall and Long-Delay Free Recall are summarized in Table F  = 6.58, P = 0.012) and “time”  and a significant “type” × “time” interaction  were found on scores of the BACS List Learning , but not type A-B . As for the HVLT-R, there was a significant main effect of “group” , while “time” effect did not reach a significant level . A secondary analysis for each “group” revealed no significant main effect of “time” or “time” × “type” interaction in either healthy controls  or bipolar disorder patients . Repeated measures ANOVA for CVLT-II 1–5 Free Recall revealed a significant effect of “group”  and “time” . “Type” × “time” interaction was also significant . Therefore, secondary analysis for each “type” was performed. There was a significant effect of “time” in type A-A , but not type A-B . In addition, “group” × “time” interaction was not significant in either “type” . As for CVLT-II Long-Delay Free Recall, “group” effect  and “time” effect , as well as “type” × “time” interaction  were significant. Also, “group” × “type” × “time” interaction was significant . A secondary analysis revealed a significant effect of “time” in type A-A , but not type A-B . Moreover, “group” × “time” interaction was not significant in either pattern .A significant main effect of “group”  was smaller than those previously reported using the CVLT (0.73–0.82) . Meanwhi17 the reliability coefficients ranged from 0.72 to 0.79 in a sample of 288 healthy subjects, with a median interval of 21 days, similar to the ICC for the CVLT-II 1–5 free recall in healthy controls using an alternate form (0.70) in the present study /(four words per category) structure , while teuthymic state, although all met the subsyndromal state criterion. There is a possibility that the patient-control effect sizes may have been overestimated due to residual depressive symptoms. However, a positive correlation between rating with the MADRS vs. CVLT-II 1–5 Free Recall and Long-Delay Free Recall scores may argue against this view.Unlike previous studies, patients with bipolar disorder studied here were not necessarily in the The Japanese version of CVLT-II appeared to provide valid measures of verbal learning and memory function in bipolar disorder patients. The ability of the CVLT-II and HVLT-R, but not BACS List Learning, to discriminate between patients and control subjects may be related to the use of memory organization strategy specific to the CVLT-II and HVLT-R, which deserves further study.This study was carried out in accordance with the recommendations of “name of guidelines, name of committee” with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by ethics committees of the National Center of Neurology and Psychiatry Hospital, Fujita Health University Hospital, Nagoya University Hospital, and Hokkaido University Hospital.Contributions of each author are as follows: conception and design of the study (TS); acquisition of data ; analysis of data (KN and KS); and drafting of the manuscript (KN and TS).There are no conflicts of interest for any of the authors of this paper. No author has any possible financial gain for the findings presented here.


A central concern of generative grammar is the relationship between hierarchy and word order, traditionally understood as two dimensions of a single syntactic representation. A related concern is directionality in the grammar. Traditional approaches posit process-neutral grammars, embodying knowledge of language, put to use with infinite facility both for production and comprehension. This has crystallized in the view of Merge as the central property of syntax, perhaps its only novel feature. A growing number of approaches explore grammars with different directionalities, often with more direct connections to performance mechanisms. This paper describes a novel model of universal grammar as a one-directional, universal parser. Mismatch between word order and interpretation order is pervasive in comprehension; in the present model, word order is language-particular and interpretation order  is universal. These orders are not two dimensions of a unified abstract object ; rather, both are temporal sequences, and UG is an invariant real-time procedure  transforming word order into hierarchical order. This shift in perspective has several desirable consequences. It collapses linearization, displacement, and composition into a single performance process. The architecture provides a novel source of brackets (labeled unambiguously and without search), which are understood not as part-whole constituency relations, but as storage and retrieval routines in parsing. It also explains why neutral word order within single syntactic cycles avoids 213-like permutations. The model identifies cycles as extended projections of lexical heads, grounding the notion of phase. This is achieved with a universal processor, dispensing with parameters. The empirical focus is word order in noun phrases. This domain provides some of the clearest evidence for 213-avoidance as a cross-linguistic word order generalization. Importantly, recursive phrase structure “bottoms out” in noun phrases, which are typically a single cycle . By contrast, a simple transitive clause plausibly involves two cycles (vP and CP), embedding further nominal cycles. In the present theory, recursion is fundamentally distinct from structure-building within a single cycle, and different word order restrictions might emerge in larger domains like clauses. One of the most significant recent developments for linguistic theory is the appearance of high-quality datasets on the full range of cross-linguistic variation. In the past, generative studies typically relied on detailed examination of one or several languages to illuminate syntactic mechanisms. While this approach is certainly fruitful, the accumulation of information about large numbers of languages opens new possibilities for sharpening understanding.the) fundamental property of language  of lexical items. The error inheres in thinking of lexical items as coherent units existing at a single level. This leads to thinking of sentences as single-level representations as well. Words, put simply, aren't This paper proposes a novel model of grammatical mechanisms, called ULTRA . Within local syntactic domains forming the extended projection of a lexical root (such as a verb or noun), ULTRA employs Knuth's  stack-soStack-sorting proves to be an effective procedure for linking word order and hierarchical interpretation, encompassing linearization, displacement, composition, and labeled brackets. The theory invites realization as a real-time performance process. Pursuing that realization significantly recasts the boundaries between performance and competence. Remarkably, ULTRA requires no language-particular parameters; an invariant algorithm serves as grammatical device for all languages. Put simply, I propose that Universal Grammar is a universal parser.Nevertheless, stack-sorting is too limited a mechanism to describe all the phenomena of human syntax. Three kinds of effects are left hanging: unbounded recursion, non-neutral orders, and the existence of apparently distinct languages. Moreover, understanding stack-sorting as a processing system encounters two obvious problems: it is a unidirectional parser, not trivially reversible for production; and it conflicts with strong evidence for word-by-word incrementality in comprehension.Although constructing a complete model of syntax and processing goes far beyond the scope of the paper, the problems that arise in basing a parser-as-grammar model on stack-sorting warrant consideration. I appeal to the distinction between reactive and predictive processes, casting stack-sorting as a universal reactive routine. A separate predictive module plays a crucial role in production, and in the appearance of distinct, relatively rigid word orders. Prediction also helps reconcile ULTRA with incremental interpretation. I appeal to properties of memory to resolve further problems, speculating that primacy memory (distinct from the recency memory underpinning stack-sorting) is the source of another cluster of syntactic properties, including long-distance movement, crossing dependencies, and the special syntax of the “left periphery.” Finally, I suggest that episodic memory—independently hierarchical in structure, in humans—plays a key role in linguistic recursion.*213 in Neutral Word Order explores the generalization that 213-avoidance delimits information-neutral word order possibilities, across languages. Section Stack-Sorting as a Grammatical Mechanism proposes a stack-sorting procedure to capture 213-avoidance in word order. Section Stack-Sorting: Linearization, Displacement, Composition, and Labeled Brackets shows how further syntactic effects follow from stack-sorting. Section Comparison with Existing Accounts of Universal 20 compares ULTRA to existing accounts of 213-avoidance in word order, focusing on Universal 20. Section Universal Grammar as Universal Parser pursues the realization of stack-sorting in real-time performance. Section Possible Extensions to a More Complete Theory of Syntax addresses the challenges in taking stack-sorting as the core of Universal Grammar, sketching some possible extensions. Section Conclusion concludes.The structure of this paper is as follows. Section Linear Base argues that the “base” structure within each local syntactic domain is a sequence. Section root, noting that in extended projections of nouns and verbs, the lexical noun or verb root is semantically dominant. The other sense of head concerns which element determines the combinatoric behavior of the composite; let us call this notion of head the label.Syntactic combination could take many forms. An emerging view is that combination largely keeps to head-complement relations Starke, . The terred books, would be represented as follows.In older theories of phrase structure, the two senses of head (root and label) converged on the same element; a noun, for example, combined with all its modifiers within a noun phrase. Headedness thus mapped to hierarchical dominance; the root projected its label above its dependents. To illustrate, a combination of adjective and noun, such as red books is represented as follows.This traditional conclusion about the relationship of dependency and hierarchy is overturned in modern syntactic cartography , which labels the composite, determining its combinatoric behavior. In cartographic representations heads are uniformly Questions arise about these representations, which postulate an abundance of unpronounced material. A curious observation is that functional heads and their specifiers seem not to occur together overtly, as formalized in Koopman's  Generalitokens of the same type, competing for a single position. Starke recasts the cartographic spine as an abstract functional sequence (fseq), whose positions can be discharged equally by lexical or phrasal material. Pursuing Starke's conception, the adjective-noun combination would be represented as below.Starke  takes Koits fseq.Again, we have reversed traditional conclusions about the hierarchy of heads and dependents. Nevertheless, the notion of root (picking out the noun) is still crucial, as the modifiers occur in the hierarchical order dictated by Syntactic combination of this sort is sequential, within each extended projection. These “base” sequences encode bottom-up composition, so it is natural to order the sequence in the same way (bottom-up). The base  is widely taken to be uniform across languages, and to express “thematic,” information-neutral meaning (contrasted with discourse-information structure)A grammar, on anyone's theory, specifies a formal mapping linking sound and meaning . This specification could take many forms. Sequential representation of the base allows a remarkably simple formulation of the sound-meaning mapping. This reformulation yields a principled account of a class of word order universals. Moreover, while the interface objects  involved in the mapping are sequences, bracketed hierarchical structure arises as a dynamic effect.linearization, externalization, etc. This paper pursues a different view, where surface word orders are inputs to an algorithm that attempts to assemble the base as output. Significantly, the only inputs that converge on the uniform base under this process are 213-avoiding; all 213-containing word orders result in deviant output.There are various ways of conceptualizing the relationship between the base and surface word order. The usual view is that the base orders the input to a derivation, yielding surface word order as the output. That directionality is implicit in terms used to describe the hierarchy-order relation: b…a…c…, for elements a ≫ b ≫ c, where ≫ indicates c-command in standard tree representations of the base . In other words, neutral word orders seem to avoid a mid-high-low (sub)sequence of elements from a single fseq. The elements forming this forbidden contour need not be adjacent, in surface order or in the base fseq.213-avoidance arguably captures information-neutral word order possibilities in a variety of syntactic domains, across languages. By 213-avoidance, I mean a ban on surface order …213-avoidance is widely believed to delimit the ordering options for verb clusters, well-known in West Germanic  precede the noun, they are always found in that order. If they follow, the order is either the same or its exact opposite” . What is forbidden is remnant movement.*Num Dem Adj N contains subsequences Num Dem Adj…, and Num Dem … N, representing mid-high-low contours with respect to the fseq.Noun phrase orders obey a simple generalization: attested orders are all and only 213-avoiding permutations. All unattested orders have 213-like subsequences. For example, unattested There is a particularly simple procedure that maps 213-avoiding word(4) maps all and only 213-avoiding word orders to a 321-like hierarchy, corresponding to the base. 213-containing orders are mapped to a deviant output, distinct from the base. By hypothesis, that is why such orders are typologically unavailable: they are automatically mapped to an uninterpretable order of composition. This explains the Universal 20 pattern (Tables *Adj-Dem-Num.Let us illustrate how (4) parses some noun phrase orders: Dem-Adj-N, N-Dem-Adj, (5) Dem-Adj-N: PUSH(Dem), PUSH(Adj), PUSH(N), POP(N), POP(Adj), POP(Dem).(6) N-Dem-Adj: PUSH(N), POP(N), PUSH(Dem), PUSH(Adj), POP(Adj), POP(Dem).For attested orders, the nominal categories POP in the order < N, Adj, Dem>, matching their bottom-up hierarchy.*Adj-Dem-N: PUSH(Adj), POP(Adj), PUSH(Dem), PUSH(N), POP(N), POP(Dem).(7) * < Adj, N, Dem>, failing to construct the universal interpretation order.For the unattested 213-like order, items POP in the deviant order That's nice: (4) maps attested orders to their universal meaning, simultaneously ruling out unattested orders. But beyond such a mapping, an adequate grammar must explain other aspects of knowledge of language, including surface structure bracketing. If grammar treats surface orders and base structures as sequencesIn this section, I show that stack-sorting effectively encompasses linearization, displacement, and composition, as well as assigning brackets, labeled unambiguously and without search. Moreover, it does all of this without language-particular parameters.In the standard (“Y-model”) view, linearization and composition are distinct interface operations, interpreting structures built in an autonomous syntactic module by Merge. In ULTRA, linearization goes in the other direction, loading surface word order item-by-item into memory, and reassembling it in order of compositional interpretation.Displacement is a natural feature of stack-sorting; from one point of view, it is the basic property of the system. In standard accounts, constituents that compose together in the interpretation should appear adjacent in surface order. This arrangement is forced by phrase structure grammars. Displacement, whereby elements that compose together are separated by intervening elements in surface order, has always seemed a surprising property, in need of explanation.every element in the surface order is “transformed,” passing through memory before retrieval for interpretationThings work quite differently in ULTRA. A key assumption of the Merge-based view is discarded: there is no level of representation encompassing word order and the fseq within a unified higher-order object. Instead, word order and base hierarchy are disconnected sequences, related dynamically. Non-adjacent input elements can perfectly well end up adjacent in the output. Displacement, rather than being the exception, is the rule; The algorithm 4) implicitly assigns labeled bracketed structure implicitExamining these brackets, the sequence of pushes and pops  for each order implicitly defines a tree, as shown in Figure Setting aside the 321 tree(s) for the moment, the Dyck trees are systematic, loss-less compressions of Cinque's trees, with every subtree that is a right-branching comb in the Cinque tree replaced with a linear tree  is the most common noun phrase order; future research should illuminate the issueA key empirical question is whether 321 orders exhibit two distinct bracketed structures, as binary-branching treatments allow, or only the single, “flat” structure predicted here. The issue is even more acute for 4 elements, as in Universal 20, where there are up to 5 distinct Merge derivationsStack-sorting captures a surprising amount of syntactic machinery, normally divided among different modules. In the usual view, an autonomous generative engine builds constituent structures, interpreted at the interfaces by further processes of linearization and composition. In ULTRA, linearization and composition reflect a single procedure. Constituent structure is not primitive, but records the storage and retrieval steps by which stack-sorting assembles the interpretationThis section compares the stack-sorting account of Universal 20 to existing Merge-based accounts Cinque, . I argueCinque proposes a cross-linguistically uniform base hierarchy, reflecting a fixed order of External Merge. He proposes that movement  is uniformly leftward, while the base is right-branching, in line with Kayne's  LCA. He 0), below agreement phrases providing landing sites for movement. This structure, and his assumptions about movement, derives all and only the attested orders. The English-like order Dem-Num-Adj-N surfaces without movement; all other orders involve some sequence of movements of NP, or something containing it.The overt modifiers are specifiers of dedicated functional heads  a. N-L – Align                Align NP's left edge with AgrWP's left edge.        b. A-L – Align                Align AP's left edge with AgrWP's left edge.        c. NUM-L – Align                Align NumP's left edge with AgrWP's left edge.        d. DEM-L – Align                Align DemP's left edge with AgrWP's left edge.             a uniform base structure, (ii) syntactic movement, and (iii) principles of linearization. In all three accounts, (i) describes the order of External Merge. Details of (ii) and (iii) vary between the accounts. For Cinque  and StedNeeleman  require Neeleman  utilizesNeeleman  LCA; AbeNeeleman  have movNeeleman  have lanThese accounts all require different grammars for different orders. In Cinque  system, Finally, all these accounts have some measure of structural or grammatical ambiguity, for some orders. For Cinque , one kinThe stack-sorting account fares better with respect to these issues. Instead of postulating separate tiers of base, movement, and linearization principles, the relevant machinery is realized in one algorithmic process. The sorting algorithm is universal, eschewing language-particular features to drive movement, order sister nodes, or rank alignment constraints. Such a theory is ideally situated to account for the free word order phenomenonThis section develops the view that stack-sorting can form the basis for an invariant performance mechanism, realizing Universal Grammar as a universal parser. This modifies traditional conclusions about competence and performance, while providing a novel view of what a grammar is.what the system is doing, and why. Performance corresponds, rather loosely, to the lower, algorithmic level, describing how the computation is carried out, step-by-step.In generative accounts, a fundamental division exists between competence and performance Chomsky, . CompeteOf course, Marr's hierarchy applies to the information-processing in language, under the present theory as well as any other. However, the division of labor between these components is significantly redrawn here, with much more of the burden of explanation carried by performanceA novel claim of ULTRA is that there is a single parser for all languages. This departs from the nearly universal assumption that parsers interpret language-particular grammars. But even within that traditional view, the appeal of universal mechanisms has been recognized.“The key point to be made, however, is that the search should be a search for universals, even—and perhaps especially—in the processing domain. For it would seem that the strongest parsing theory is one which says that the grammar interpreter itself is a universal mechanism, i.e. that there is one highly constrained grammar interpreter which is the appropriate machine for parsing all natural languages” Marcus,  p. 11.performance grammar only (PGO) view.The idea that “the parser is the grammar” has a long history; see Phillips , 2003, K“PGO theory enters the game with one powerful advantage: there must be psychological mechanisms for speaking and understanding, and simplicity considerations thus put the burden of proof on anyone who would claim that there is more than this” Fodor,  p. 470.ibid., 472). This presupposes that movement is fundamentally difficult for parsing mechanisms . However, in ULTRA, displacement is not a complication over more basic mechanisms; displacement is the basic mechanism.However, while granting that this entails a simpler theory, Fodor rejects the idea, finding no motivation for movement outside an autonomous grammar  is essentially identical to the SYAIn fact, though the property is little used, the SYA is a sorting protocol; many input orders lead to the same internal calculation. As calculator users, we utilize one input scheme (infix notation), but others would do as well. The standardized input order for calculators has the same status as particular languages with respect to ULTRA: users may fall into narrow ordering habits, but the algorithm automatically processes many other orders.ungrammaticality. The former kind of knowledge is fundamentally about computing interpretations. But the invariant process interpreting one language's surface order can equally interpret the orders of other languages. From this point of view, there is only one I-language, and a single performance grammar that delivers it. While this conclusion is appealing, an important question remains: where do individual languages come from, with apparently different grammars?One of the central tasks ascribed to grammars is distinguishing grammatical sentences of a language from ungrammatical strings. In ULTRA, knowledge of grammaticality is very different from knowledge of This section addresses two kinds of problems that follow from interpreting stack-sorting as a performance device. The first concerns reconciling the theory with what is known about real-time language processing; the second concerns extending the model to properties of syntax that are left unexplained. Even discussing these problems in depth, much less justifying any solutions, is beyond the scope of this paper. The intent is merely to sketch the challenges, and indicate directions for further work.With respect to processing, one problem is that this approach seems to be contradicted by strong evidence for word-by-word incrementality in comprehension  memory. Primacy, on the other hand, is naturally described by queue  memory. Besides optimal order of access, there is another important difference between primacy and recency effects. Put simply, the first element in a sequence remains the first element, no matter how many more elements follow; the primacy signature of a given element is relatively stable over the time scale relevant to parsing. Recency is different: each element in a sequence is a new right edge, suppressing the accessibility to recency-based memory of everything that precedes it. Thus, we expect a kind of “use-it-or-lose-it” pressure within recency memory, but not primacy memory.Tentatively, I would like to suggest that distinct primacy and recency memory codes underlie the Duality of Semantics, and the division between A-bar and A-syntax. Recency, associated with a stack, is the basis for information-neutral, local permutation, generally characterized by nesting dependenciesA final problem looming in the background is recursion. ULTRA operates within syntactic domains characterized by a single fseq. This requires some comment, as recursion is a fundamental property of syntax. For recursion as well, properties of memory, and intervention of a complementary predictive system, might be crucial. Intriguingly, human episodic memory appears to be independently hierarchical in structure, perhaps unlike related animals Tulving, . In the As discussed in section Comparison with Existing Accounts of Universal 20, in ULTRA, structural ambiguity does not arise without ambiguity of meaning, within single domains. However, structural ambiguity arises inevitably when multiple domains are present, in terms of which domain embeds in another, or where to attach an element that could discharge positions in two distinct domains. This is where the “fast, stupid black box” is helpless, and must call on other resources. One obvious source of help in stitching together multiple domains is a separate predictive system, with access to top-down knowledge of plausible meanings in context. The persistent problem of resolving embedding ambiguities also provides motivation for rigid word order, which sharply reduces attachment possibilities.An important point is that brackets are defined relative to a particular fseq. Recursive embedding of one domain in another  involves projection of a bracket corresponding to the entire embedded phrase, within the embedding domain(11) The dog chased a ball.There are three sorting domains here: two nominal projections, embedded in a third, verbal projection . Their ULTRA bracketing appears below.the [dog dog] the](12) NP1 = [a [ball ball] a]        NP2 = [NP1 [chased chased] [NP2 NP2] NP1]          VP = [This example illustrates the ambiguity that accompanies embedding. The issue is how to link the nominal phrases to positions in the verb's fseq . As theta roles are not overtly expressed (case-marking is an unreliable guide), the reactive parser must draw on external means .A final point about recursion returns to the issue of how calculators work, via Dijkstra's Shunting Yard Algorithm. Such computations are recursive. But recursion isn't handled by the parsing algorithm; rather, it arises at the level of interpretation, where partial outputs of arithmetic operations feed into further calculations. A similar conclusion  is possible within the present framework, given the similarities between ULTRA and the SYA. Notably, both procedures compile input into Reverse Polish Notation, a so-called concatenative programming language, expressing recursive hierarchical operations unambiguously in serial format.this algorithm? Other sorting procedures are possible in principle, and would lead to different permutation-avoidance profiles. How do we justify selecting stack-sorting as the right procedure for syntactic mapping?I have shown how a particularly simple algorithm captures a range of syntactic phenomena. But the question is, why There are three crucial ingredients. The first is the orientation of the system as a parser, mapping sound to meaning. This is not logically necessary; it is simply one of the reasonable choices. The second factor is the linear nature of sound and meaning. This is straightforward for sound sequences, but much less so for interpretations, where it is simply a bold hypothesis. The third ingredient is the choice of stack memory. This can plausibly be tied to the Modality effect: intelligible speech input engenders unusually strong recency effects  itself comes from, a matter on which I will not speculate here.So far, stack-sorting has been implemented with an explicit algorithm. That may be unnecessary. Rather than thinking of stack-sorting as a set of explicit instructions, we might reframe it as an Summarizing, a simple algorithm (4) maps 213-avoiding word orders to a bottom-up compositional sequence, while mapping 213-containing orders to deviant sequences. While the input and output of the mapping are sequences, hierarchical structure is present: the algorithmic steps realize left and right brackets, almost exactly where standard accounts place them. The account differs from standard accounts in assigning unambiguous bracketing to all orders.This model improves on existing accounts of word order restrictions, which invoke additional stipulations , beyond core syntactic structure-building. In ULTRA, these effects fall out from a single real-time process. In turn, syntactic displacement, long seen as a curious complication, emerges as the fundamental grammatical mechanism. No learning of language-particular properties is required; one grammar interprets many orders.It should be clear that the system described here is only one part of syntactic cognition. This system builds one extended projection at a time; further mechanisms are required to embed one domain in another. However, that may be a virtue: it is tempting to identify the domains of operation for this architecture with phases, which are thus special for principled reasons.Moreover, stack-sorting only handles information-neutral structure. This ignores another important component of syntax, so-called discourse-information structure, associated with potentially long-distance A-bar dependencies. This deficiency, too, may be a virtue, suggesting a principled basis for the Duality of Semantics. I speculated that primacy memory plays a central role in these effects, potentially explaining several curious properties .Raising our sights, the larger conclusion is that much of the machinery of syntactic cognition might reduce to effects not specific to language. Needless to say, this is just a programmatic sketch; future research will determine whether and how ULTRA's stack-sorting might be integrated into a more complete model of language.The author confirms being the sole contributor of this work and approved it for publication.The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.