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[EQUATION] where [EQUATION] As a result, the soft propagator in the Fourier-Laplace domain takes a reasonably simple form, however it still contains the function [MATH] which is given as an infinite sum. Fortunately, to compute moments of process and autocorrelation of velocity we need to know the corresponding derivat... |
The first and the second moment of the process in the Fourier-Laplace domain are [EQUATION] where [MATH] is the [MATH] -th moment of jump modules distribution [MATH] . The first moment of the directed process in the time space rises linearly in time [MATH] , exactly like for the process without memory. It is worth to n... |
Velocity Autocorrelation Function In the general case, the velocity autocorrelation function (VAF) in the time domain is given by |
[EQUATION] In the case of the directed CTRW process considered in this manuscript, it takes the form [EQUATION] where [MATH] is the inverse Laplace transform. To investigate the behavior of VAF in the limits [MATH] and [MATH] , we have to check the behavior in limits [MATH] and [MATH] of the expressions inside inverse ... |
To compare our model with empirical data we use two specific WTDs: exponential and double-exponential, with explicit results for both. First one is a simple distribution and its characteristics match stylized facts of financial time series. Second one can be fitted to empirical data with high accuracy TG_1 and still al... |
[EQUATION] and double-exponential WTD with partial mean waiting times equal to [MATH] and [MATH] and weighting parameter [MATH] is given as |
[EQUATION] Mean waiting time of double-exponential WTD is [MATH] For the exponential WTD, VAF is easily expressed as [EQUATION] where |
[EQUATION] Although exponential WTD does not describe properly the empirical WTD, one can easily interprete meaning of the parameters. Firstly, for [MATH] VAF is positive (unlike VAF in TG_1 ) and decreases exponentially. Relaxation time increases and the amplitude reduces with longer mean waiting time. Increasing para... |
For the double-exponential WTD, which satisfactorily fits empirical data, normalized VAF is [EQUATION] There are three exponents in this formula, except Dirac delta, all with positive amplitudes. The first one is worth mentioning, it does not depend on [MATH] . It implicates, that for directed processes VAF can be non-... |
Empirical results To compare our model with empirical data, we use tick-by-tick transaction data from Polish stock market (Warsaw Stock Exchange) from years 2011-2012. Presented results are calculated for KGHM - one of the most liquid stocks. We extract waiting times (periods between transactions) and jumps (price chan... |
Nonstationarity Varying mean inter-trade time during trading session is a stylized fact observed on stock prices on every market RCont2001 Hasbrouck2007 DacorognaGencayMullerOlsenPictet2001 This intra-day pattern, often called the ”lunch effect”, is characterized by low volatility and long inter-trade times in the midd... |
[EQUATION] where [MATH] can be interpreted as varying mean of the waiting times distribution during a trading session. Now, we can obtain an explicit expression for normalized VAF taking the seasonality into account: |
[EQUATION] where [MATH] is the length of a day, parameters [MATH] and [MATH] are fitted to data and come from the rational form of day seasonality and erf is the error function. As shown in Fig. , taking nonstationarity into account slightly improves the results. However, considering the day seasonality is not enough t... |
Conclusions We proposed and solved Directed CTRW model with one-step memory in jumps and obtained the analytical equation for propagator, first two moments and velocity autocorrelation function ( 20 ). Obtained VAF shows interesting properties: it is positive; it decays exponentially but slower than the VAF for the non... |
Presented simple model, despite being analytically solvable, turned out to be unable to describe empirical data. This result suggests that simple bid-ask bounce phenomenon is not sufficient to explain long memory in financial time series and volatility clustering. We suggest that taking into account at least one of the... |
# Source: arxiv 1807.01941 # Title: Statistics of Vector Manakov Rogue Waves # Sections: all # Downloaded: 2026-03-02T08:56:13.594023+00:00 |
Statistics of Vector Manakov Rogue Waves Abstract We present a statistical analysis based on the height and return time probabilities of high amplitude wave events in both focusing and defocusing Manakov systems. We find that analytical rational/semirational solutions, associated with extreme, rogue wave (RW) structure... |
pacs: 05.45.Yv, 02.30.Ik, 42.65.Tg Introduction The emergence, dynamics and prediction of rogue waves (RW), also referred to as freak waves or extreme events, has been in the focus of interest in diverse fields of science (oceanography, physics of fluids, optics, matter waves physics, sociology, bio-sciences,…) over th... |
Peregrine solitons c1 and Akhmediev breathers c2 are well-known RW candidates: they represent solutions of the scalar one-dimensional self-focusing nonlinear Schrodinger equation (NLSE); the Peregrine solitons with the property of being localized in both the transverse and evolution coordinates, the Akhmediev breathers... |
Recently, progress has been made by extending the search for RW solutions to coupled-wave systems. Indeed, numerous physical phenomena require to model waves with two, or more components, in order to account for different modes, frequencies, or polarizations. In those cases, the focusing regime is not a prerequisite fo... |
, the three-wave resonant interaction equations 10 , the coupled Hirota equations 11 , and the long-wave-short-wave resonance 12 . It is crucial to add that new RW families can be created in the defocusing nonlinear regime too. This was shown theoretically and experimentally in |
13 b1 b2 b3 : its authors proved that, in the defocusing regime of the Manakov system, the range of existence of rational solutions of different types (bright-dark, dark-dark), which are the most serious candidates for RW, overlaps with the region of baseband modulation instability (MI). Moreover, it was demonstrated t... |
However, a basic question arises regarding the statistical description of high amplitude events in the course of nonlinear wave propagation. It should be considered that under realistic circumstances the propagation medium exhibits fluctuations of its parameters, hence of the background continuous wave (CW) solutions. ... |
Moreover, outside the context of discrete systems and numerical studies of supercontinuum generation new1 , the statistical analysis has not yet found a leading role in the studies of RWs, although RWs are statistically determined entities. In our research, we shall provide a new insight into the origin and dynamics of... |
In this paper we statistically investigate the behavior of high amplitude events in the integrable Manakov system. For this end, we numerically model the (light/matter) wave propagation in the nonlinear media (photonic/Bose-Einstein condensate), in the simplest case of a two component system. Physically this correspond... |
II The model equations The vector nonlinear Schrödinger equations, i.e., the Manakov system, can be written in dimensionless form as |
[EQUATION] where [MATH] and [MATH] represent the wave envelopes, [MATH] is the evolution variable, and [MATH] is a second independent variable. The meaning of variables depends on the particular applicative context (fluid dynamics, plasma physics, BEC, nonlinear optics, finance). The parameter [MATH] refers to the focu... |
13 In the focussing case, such rational solutions can be expressed in the form of different bright-dark breather composites : e.g., a boomeron-type soliton with a time-dependent velocity, a breather-like wave resulting from the interference between the dark and bright contributions, and more complex structures resultin... |
[EQUATION] with simultaneously added small periodic and random perturbations. The [MATH] parameters represent the initial amplitudes of component waves in the system, while [MATH] are the initial phases. The difference of phase factors [MATH] will be used to present our numerical results in the next sections. |
On the other hand, in the defocusing case the rational/semirational solutions were explicitly derived in 13 . They can be generated both analytically and numerically by starting from a plane wave solution ( ). It was analytically shown 13 that the region of rational wave existence, which is related to the domain of RW ... |
[EQUATION] In particular, the inequality ( ) implies that the background amplitudes have to be sufficiently large, for a fixed [MATH] , in order to allow for the rational wave formation, see Fig. . Here, we prefer not to use the term RW for high amplitude rational solutions, since an unique definition of RWs does not e... |
(a,b). By adding to the finite background small regular (periodic) and random perturbations in the parameter regimes associated with the presence of MI, we confirmed the analytical predictions and previous numerical results from the literature, 13 . The preparation of initial conditions included the presence of a super... |
The next step was to prepare initial conditions that can ensure the generation of a huge ensemble of localized, high amplitude events, which is necessary for the statistical analysis. We analyzed the results of numerical simulations with different initial conditions, namely, a plane wave (uniform background) with rando... |
Amplitude noise is numerically modeled as a uniform random process with zero mean. In order to have sufficient data for the statistical analysis, the long term evolution of the field was numerically simulated. The optimal width of the calculation window was estimated in each particular case by repeated numerical tests. |
III Statistics of the Manakov rogue waves The purpose of this study is the statistical analysis of the emergent peaks (dips) in the numerical solutions of the Manakov system. Such extreme amplitude wave events are usually referred to as RWs, whenever the significant height criterion is satisfied |
17 statistics . Here, the difference between the maximum value of the finite background elevation in between two zero-crossings and the minimum value of the background elevation in the adjacent (next or previous) zero crossing interval is called the wave height (Fig. ). In scalar models of water-wave propagation, the s... |
[MATH] is defined traditionally as the average height of one-third of the highest waves in the height distribution, and the RW threshold is estimated to be [MATH] (also, in the literature on ocean rogue waves, waves with height bigger than |
[MATH] qualify to be in this category Kharif In the preparatory phase of our study, we searched for proper RW classifiers. Recently, a two-dimensional (2D) equivalent of the significant wave height was defined as a classifier in vector models ( 18 ). In the framework of the complex RW patterns that are observed in our ... |
[EQUATION] In this expression, the abbreviation [MATH] indicates the transpose operation. Thus, the height threshold, ( [MATH] , is a vector quantity consisting of the height thresholds with respect to two spinor components, [MATH] ). Finally, if at least a height of one of the components reaches the corresponding thre... |
[MATH] . However, the proper definition of the height criterion for a RW in multi-component system remains still an open issue. For the sake of simplicity, the vector abbreviations for significant height and threshold height will be omitted in the following [MATH] |
We calculated different statistical measures which have been developed in the literature on extreme events, and considered their relevance for expressing the dynamical properties of high amplitude events in the Manakov system. It was shown that the most adequate statistical measure for our system is that based on the h... |
nash new , coupled with the probability distribution of the return time among to successive RW events, i.e., [MATH] 20 In the following, we will discuss the shape of the [MATH] curves (associated with the corresponding moments) as a function of |
[MATH] , along with the probability of RW occurrence [MATH] , which can be derived from [MATH] . The tails of the HPD are related to the presence of extreme events. The probability of RW occurrence is defined as [MATH] (with respect to both vector components), and it is obtained by integration of the normalized [MATH] ... |
For a deeper insight into the time statistics of RWs, the probability distribution of the return time (time is a synonym of propagation length/duration), [MATH] of these (vectorial) events was also calculated. The return time [MATH] is defined as the time interval between the appearance at given position of two success... |
IV Results and discussion The first step was to generate numerically rational solutions which can be classified as RWs. The existence of these solutions had been related, at least initially, with the development of MI |
13 , which is by itself threshold determined. Intensive numerical checking has shown that the rational solutions of the types presented in 13 , see also Figs. and |
, can be obtained from both coherently or noise driven MI 14 , and represent short-lived or transient wave structures. It should be noted that the exact choice of the initial excitation is crucial for the generation of rational solutions in the defocusing case. In this respect, the structures which were analytically de... |
Regardless of the initial perturbations, the long term dynamics of high amplitude events in the Manakov system, observed in the presence of MI, shows similar tendencies. This is the case for both types of nonlinearity, that is either focusing or defocusing. Therefore, statistical ensembles were obtained from long term ... |
IV.1 High amplitude events in the focusing case The evolution of wave amplitudes for two different initial conditions, corresponding to parameters above the MI threshold is presented in Fig. . Two different regimes can be distinguished on these plots: an initial, transient phase, and a long-term (long propagation lengt... |
On the basis of numerical simulations, we can distinguish between an initial, transient, and subsequent long-term dynamical regime for the ensemble of the high amplitude events. Inside the transient regime, MI is expected to be the governing mechanism for the creation of localized waves, including the rational solution... |
Now concerning the statistical measures, the HPD curves (i.e. [MATH] vs. wave component height) for the sets of parameters corresponding to Fig. are presented in Fig. |
((a) and (b)) in a linear scale, and in Fig. ((c) and (d)) in a log-linear scale. The statistical distributions are obtained for different intervals of the evolution coordinate |
[MATH] , as indicated in the legend of Fig. . As far as the overall behavior of these curves is concerned, we may observe that the [MATH] curves that characterize the statistics of extreme waves in the initial phase (black squares) differ from those obtained for the irregular phase (red triangles). Also, the [MATH] cur... |
In addition, we searched for the best fitting function for the HPDs, following the ideas already presented in the literature new new1 As expected, the observed HPD deviates from a Gaussian probability distribution (this is a known feature of RW statistics). Alternatively, it is possible to model the HPD by means of a g... |
[EQUATION] where [MATH] and [MATH] are shape parameters, and [MATH] is a scale parameter. In order to account for the normalization of our HPDs, the GGD distribution was multiplied with a parameter [MATH] [MATH] ). From Fig. (c) and Fig. (d), it is obvious that the HPDs associated with the long-term (blue circles) and ... |
[MATH] and [MATH] are listed in Table The corresponding values of the significant height, threshold height and [MATH] are listed in Table . All of these quantities were derived from the [MATH] distribution. The values of [MATH] and [MATH] are of the same order in both selected parameter cases and calculation windows. T... |
[MATH] in the transient and long-term regimes are similar and very small, of the order of [MATH] , i.e. [MATH] . Depending on the values of the parameters (amplitudes and phases of initial plane-wave excitation), and therefore on the the position of the MI borderline, the value of [MATH] has a slight tendency to increa... |
On the other hand, in the presence of nonzero [MATH] , the transversely moving localized transient modes can be excited via the MI mechanism. In addition, for small heights, the growth rate of [MATH] with [MATH] is larger for simulations involving the long-term evolution, when compared with the corresponding growth rat... |
[MATH] in the early regime of evolution. Qualitative differences of the [MATH] curves corresponding to different calculation windows undoubtedly show that different types of high amplitude events govern the system behavior in the course of the vector wave propagation. On the other hand, the observed negligible quantita... |
An additional set of statistical measures for the RWs was derived from the statistics of the return time probability, [MATH] , as shown in Fig. . The [MATH] curves for two different initial conditions and with respect to (two) different thresholds, |
[MATH] , are comparatively presented in this figure. The shape of the [MATH] curves changes with the position of the calculation window and its width, as well as with the amplitude thresholds. For lower thresholds, the [MATH] curves corresponding to either transient or transient+long term evolution phases exhibit a sim... |
Additionally, for certain initial conditions, one can observe a turning point, i.e., a plateau, in the region of moderate values of the return time. By moving the calculation window from the early transient regime into the long-term limit, the slope of the |
[MATH] curves changes, and it becomes steeper. However, the tails of all these curves are power-law like. In the long-term regime, a plateau is no longer present on the [MATH] curves. All of this indicates the more frequent appearance of high amplitude events in the transient phase than in the long term situation. The ... |
IV.2 High amplitude events in the defocusing case The same approach of the previous subsection can also be applied to study RW statistics in the defocusing Manakov system. The particularity of this case is the strict dependence of the wave dynamics on the initial conditions, as already mentioned in Section II. The prep... |
[MATH] , which are outside and inside the baseband MI region, according to Eq. ( ), respectively. It should be noted that the properties and values of statistical measures can strongly depend on the system parameters, which are directly related to the position of the border of baseband MI, and the value of its growth r... |
The amplitude plots for both representative parameter sets are presented in Fig. . A clear distinction between two evolution phases, which was apparent in the focusing case, is absent in the defocusing regime. However, as we shall see below, the statistical study still shows that, in general, a competition exists among... |
In Fig. we present the wave height probability [MATH] curves together with their GGD fits, for a set of parameters that are either outside (i.e., [MATH] , see Fig. 7(a) and (c))) or inside (i.e., [MATH] , see Fig. 7(b) and (d)) the range of existence of baseband MI, respectively. We may note here the same qualitative b... |
On the other hand, Fig. shows that the [MATH] behavior for calculation windows in the long-term range is statistically the same in both selected parameter cases, namely, either outside or inside the region for baseband MI. Therefore, based on our results, one cannot claim that rational solutions, which have been report... |
follow the same scenario as they did in the focusing case. The [MATH] values are very small, of the order [MATH] , in all parameter regions which are related with the existence of high amplitude events (rational solitons). Modeling the HPD curves with the GGD gave similar results as in the focusing case (Fig. 7(c) and ... |
On the other hand, the return probability [MATH] behavior is illustrated in Fig. . For higher values of the threshold amplitude ( [MATH] ), the [MATH] curves show the same tendency with respect to the position of the calculation window for both sets of parameters. By moving the calculation windows towards the long-term... |
[MATH] curves does not change with further changes in the position or (width) of the calculation window. A similar tendency regarding the shape of the [MATH] curves can be recognized for higher threshold values. By comparing the return times of high amplitude events for the two selected thresholds, we can conclude that... |
. Note that this is the case for both sets of parameters, i.e., either outside or inside the baseband MI region. On the other hand, the differences in [MATH] and related quantities for calculations windows in the ’transient’ phase are obvious, and can be related to different types of RWs with respect to those in the la... |
Conclusions Let us summarize the results of our study of high amplitude events in the Manakov system, by pointing out the main findings. In both the focusing and defocusing nonlinearity regime, it was shown that the type of initial perturbation of the plane wave background did not have a significant influence on the lo... |
We decided to use the term ’high amplitude events’ instead of ’RWs’, on the basis of the unclear indications about the criteria for extracting RWs from a statistical analysis based on the height and return time probabilities. We have found that the statistics of heights of high amplitude events can be described very we... |
Data derived from the return time probability mostly confirm previous statements, and show that the return time based quantities can be promising candidates for good classifiers of different types of RWs. The significance of the initial system preparation, width and position of the calculation window, on the values of ... |
Acknowledgements. A. M., Lj. H., and A. M. acknowledge support from Ministry of Education, Science, and Technological Development of Republic of Serbia [III 45010]. This project was partially supported from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreem... |
# Source: arxiv 1807.02075 # Title: Analysis of Nederlof's algorithm for subset sum # Sections: all # Downloaded: 2026-03-03T01:45:59.114477+00:00 |
Analysis of Nederlof’s Algorithm for Subset Sum Abstract We show that Nederlof’s algorithm [ Information Processing Letters , 118 (2017), 15-16] for constructing a proof that the number of subsets summing to a particular integer equals a claimed quantity is flawed because: 1) its consistence is not kept; 2) the propose... |
Keywords : subset sum problem, recurrence formula, dynamic programming. Introduction In computer science the subset sum problem is that: given a set (or multiset) of integers, is there a non-empty subset whose sum is equal to a given integer? In 1955, Gupta ever proved that |
Theorem 1 Let [MATH] denote the number of partitions of [MATH] into members of the set [MATH] [MATH] distinct positive integers), [MATH] . Then |
[EQUATION] In 1956, Bateman and Erdős proved that Theorem 2 If [MATH] is any non-empty set of positive integers, then the number of partitions of [MATH] into members of the set [MATH] [MATH] , is a non-decreasing function of [MATH] for large [MATH] , if and only if [MATH] either: (i) contains the element [MATH] or (ii)... |
In complexity theory, a proof system for subset sum problem is referred to as a Merlin-Arthur protocol. Babai and Moran ever discussed the Arthur-Merlin games and a hierarchy of complexity classes. In 2016, Williams |
pointed out the relation between strong ETH and Merlin-Arthur proof system. Austrin et al. pointed out that a special case of sub-set sum may be the hardest. In 2017, Nederlof |
proposed an algorithm for subset sum problem. In this note, we show that Nederlof algorithm is flawed. Review of Nederlof’s algorithm |
The subset sum problem discussed by Nederlof is that: given positive integers [MATH] along with a target integer [MATH] , the task is to determine whether there exists a subset [MATH] such that |
[EQUATION] Such an [MATH] is referred to as a solution. The Nederlof’s algorithm aims to construct a proof that the number of solutions of [MATH] is [MATH] To do so, the prover and the verifier execute the following algorithms, respectively. |
Algorithm P [MATH] . Prove that the number of solutions is [MATH] Output: Prime [MATH] for [MATH] 1: Initiate [MATH] and [MATH] for [MATH] |
2: for [MATH] do 3: for [MATH] do 4: if [MATH] then 5: [MATH] 6: else 7: [MATH] 8: Pick the smallest prime [MATH] such that [MATH] |
9: for [MATH] such that [MATH] do 10: [MATH] 11: return [MATH] Algorithm V [MATH] . Verify the proof for number of solutions. Output: [MATH] , if the proof is as output by P, NO with 1/2 probability otherwise. |
12: Pick a prime [MATH] satisfying [MATH] and a random [MATH] 13: Initiate [MATH] and [MATH] for [MATH] 14: for [MATH] do 15: for |
[MATH] do 16: [MATH] [MATH] denotes remainder of [MATH] divided by [MATH] 17: Compute [MATH] 18: if [MATH] then return [MATH] else return NO. |
Analysis of Nederlof’s algorithm 3.1 Inconsistency The correctness of Nederlof’s algorithm was not explained explicitly. For example, the choice of the prime [MATH] and the correctness of the recurrence formula |
[EQUATION] are not explained. Besides, the initial values [MATH] , are not specified. We find its consistency is not kept. To see this flaw, it suffices to investigate the following example. |
Example 1 . Suppose that [MATH] . Then [MATH] and [MATH] . By Nederlof’s algorithm, we have [EQUATION] for [MATH] . Hence, [EQUATION] |
The prover’s output is [MATH] Suppose that the verifier picks [MATH] . Then [MATH] By the recurrence formula [EQUATION] we have [EQUATION] |
To ensure [MATH] it has to specify [MATH] . This leads to a contradiction. That means the consistence of the algorithm is not kept. |
3.2 The correct recurrence formula Suppose that [MATH] , all members are positive integers and [MATH] for [MATH] . Let [MATH] [MATH] Denote the number of partitions of [MATH] into members of the set [MATH] by [MATH] Then the following recurrence formula |
[EQUATION] holds, where [MATH] is the number of solutions which do not contain [MATH] and [MATH] is the number of solutions which do contain [MATH] exactly once. Apparently, Nederlof |
confused Eq.(2) with Eq.(4). Conclusion We analyze the Nederlof’s algorithm for constructing a proof that the number of subsets summing to an integer is a claimed quantity. We also remark that it is somewhat difficult to theoretically compute the number of partitions of an positive integer into members of a finite set. |
# Source: arxiv 1807.02177 # Title: Higher Structures in Homotopy Type Theory # Sections: all # Downloaded: 2026-03-03T02:37:10.622538+00:00 |
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