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Consider [MATH] . Take [MATH] [MATH] -independent elements in [MATH] . Suppose [MATH] and [MATH] . As [MATH] is an [MATH] -definable injection, we get [MATH] . Hence, [MATH] and [MATH] . Since [MATH] preserves [MATH] generically and [MATH] , we have |
[EQUATION] Hence, [MATH] . By Fact 22 [EQUATION] is an [MATH] -hyper-definable group, and [MATH] contains all [MATH] -generics in [MATH] |
Let [MATH] . Let [MATH] and [MATH] be [MATH] -independent tuples in [MATH] . Consider [EQUATION] As [MATH] in [MATH] we get [MATH] . Therefore, [MATH] . By Fact 22 [MATH] is a subgroup of [MATH] . Consider the projection [MATH] . It contains [MATH] , hence contains all [MATH] -generics of [MATH] . Thus [MATH] . Similar... |
Let [MATH] be an [MATH] -generic in [MATH] . Then [MATH] . Since [MATH] is also [MATH] -generic in [MATH] , we get that [MATH] satisfies [MATH] . As [MATH] is an [MATH] -definable embedding, we have |
[MATH] Since [MATH] , by Lemma 25 there is [MATH] realizing [MATH] such that [MATH] and [MATH] have the same [MATH] -type over [MATH] . Note that [MATH] . By Fact 20 (1), [MATH] . Hence, [MATH] realizes [MATH] , and [MATH] is [MATH] -generic in [MATH] . Therefore, the set of [MATH] -generics of [MATH] is contained in [... |
Now Lemma 26 implies that [MATH] is [MATH] -type-interpretable. Suppose [MATH] where [MATH] is an [MATH] -definable equivalence relation and [MATH] is [MATH] -type-definable. If [MATH] is definable, then [MATH] is the image of an [MATH] -definable function, hence [MATH] -definable. By compactness [MATH] is [MATH] -defi... |
Remark: Given an [MATH] -definable group [MATH] , without the assumption that [MATH] lives inside an [MATH] -definable group, we cannot generally have that [MATH] is [MATH] -definable. Here is an example. |
Example 30 Let [MATH] be a group without involutions of [MATH] -rank one in the language [MATH] . Let [MATH] be an [MATH] -structure. |
Define [MATH] as [MATH] if [MATH] ; and [MATH] if [MATH] . Let [MATH] be defined as [MATH] . Then the group [MATH] is [MATH] -isomorphic to [MATH] via [MATH] , but not [MATH] -definable. |
# Source: arxiv 1806.01500 # Title: Six-dimensional heavenly equation. Dressing scheme and the hierarchy # Sections: all # Downloaded: 2026-03-02T08:53:44.049339+00:00 |
Six-dimensional heavenly equation. Dressing scheme and the hierarchy. Abstract We consider six-dimensional heavenly equation as a reduction in the framework of general six-dimensional linearly degenerate dispersionless hierarchy. We characterise the reduction in terms of wave functions, introduce generating relation, L... |
Introduction Six-dimensional heavenly equation [EQUATION] where [MATH] [MATH] is the Poisson bracket [MATH] belongs to the class of quasiclassical self-dual Yang-Mills equations (SDYM equations for the Lie algebra of vector fields) and corresponds to the case of two-dimensional Hamiltonian vector fields. Four-dimension... |
It can be obtained from a standard SDYM type Lax pair (taken in a special gauge) [EQUATION] where [MATH] [MATH] (gauge field components) generally belong to some Lie algebra, |
[MATH] is a complex variable (spectral parameter). In our case [MATH] [MATH] are two-dimensional Hamiltonian vector fields. Commutativity condition for operators ( ) implies the existence of potential [MATH] (belonging to Lie algebra), [MATH] [MATH] satisfying the equation |
[EQUATION] Six-dimensional heavenly equation case corresponds to two-dimensional Hamiltonian vector fields [MATH] [EQUATION] and Lax pair ( ) takes the form |
[EQUATION] General properties of Lax pairs (more generally, involutive distributions) of this type were discussed in (see also and references therein, |
). According to Frobenius theorem for vector fields, linear equations [EQUATION] have four functionally independent solutions. Due to the special structure of vector fields, two of them are trivial, |
[EQUATION] and two others can be found in generic form of series in [MATH] [EQUATION] which for the case of Hamiltonian vector fields are canonically conjugate, |
[EQUATION] This is not a unique admissible form of series for solutions of linear equations ( ), they can be also constructed as series in nonnegative powers of [MATH] . Different basic sets of solutions of linear equations ( ) should be connected by diffeomorphism, and this observation leads to formulation of the dres... |
It is possible to introduce higher times and extend the Lax pair ( ) to the hierarchy of the form [EQUATION] where [MATH] are polynomials in [MATH] of the order |
[MATH] . The involutive distribution with the basis ), ( ) is of codimension four and the number of independent solutions of linear equations of the form ( ) defined by this distribution remains equal to four; functions ( ) retain their form, and ( ), ( ) are slightly modified to take into account the higher times, |
[EQUATION] The coefficients of the polynomials [MATH] are connected with [MATH] by commutativity conditions of the higher flow with initial L,M operators. These conditions provide closed systems of equations, the systems [MATH] [MATH] are six-dimensional, and the systems [MATH] [MATH] |
are five-dimensional. This fact is connected with the degeneracy and special structure of wave functions ( ) for linearly degenerate quasiclassical YM type hierarchies |
, leading to simultaneous appearance of systems of different dimensionalities in the same framework. If we consider generic linearly degenerate six-dimensional dispesionless hierarchy |
, from which six-dimensional heavenly equation case can be obtained as a reduction (see below), this degeneracy disappears. A general feature of quasiclassical self-dual YM type hierarchies |
is that the basis of linear operators for the higher flows of the hierarchy ( ) can be represented in compact recursive form, [EQUATION] |
It is interesting to note that each of the operators [MATH] [MATH] is exactly of the same form as [MATH] [MATH] operators ( ) (for another set of variables), and a commutator of arbitrary pair of operators [MATH] [MATH] gives heavenly equation ) for the respective set of variables, |
[EQUATION] representing kind of intertwining equation for higher flows of the hierarchy. This phenomenon is known for Yang-Mills type hierarchies both in the standard |
and quasiclassical case. The hierarchy. General six-dimensional case We will describe six-dimensional heavenly equation hierarchy as a reduction of general six-dimensional linearly degenerate hierarchy and construct generating equations and Lax-Sato equations for 6D heavenly equation hierarchy. |
First we will briefly outline the picture of linearly degenerate hierarchy developed in (see also ). This picture starts from introducing the formal series for the wave functions, defining a Plücker form which is a dual object to the distribution of vector fields, and formulating a generating equation for the hierarchy... |
Let us consider the series [EQUATION] where [MATH] (for six-dimensional hierarchy case), depending on four infinite sequences of independent variables |
[MATH] [MATH] The hierarchy is [EQUATION] where for linearly degenerate case the differentials do not take into account [MATH] (considered as a parameter), and vector fields of respective distribution do not contain derivative over [MATH] Here [MATH] denotes the projection to the part of |
[MATH] with negative powers in [MATH] (respectively [MATH] projects to nonnegative powers) and [MATH] Relation ( 12 ) implies Lax-Sato equations defining the dynamics of wave functions over higher times, moreover, it is equivalent to the set of Lax-Sato equations |
Introducing the Jacobian matrix [EQUATION] it is possible to write the hierarchy in the Lax-Sato form, [EQUATION] where [MATH] [MATH] [MATH] |
First flows of the hierarchy (lowest level integrable distribution) read [EQUATION] where [MATH] To proceed to the case of six-dimensional heavenly equation hierarchy, we will need a reduction [MATH] corresponding to volume-preserving (divergence-free vector fields) case, generating relation in this case is |
[EQUATION] vector fields ( 14 ) are divergence-free, [MATH] The hierarchy. Description of reduction To obtain six-dimensional heavenly equation hierarchy, we consider a reduction of the hierarchy ( 15 characterised by the condition that two of the series [MATH] |
are equal to ‘vacuum’ functions [MATH] (for finite subsets of times they are polynomial) [EQUATION] Generating relation for six-dimensional heavenly equation hierarchy case is |
[EQUATION] We will restrict ourselves to the higher flows of six-dimensional heavenly equation type and drop higher times in [MATH] [MATH] also introducing new notations for functions [MATH] [MATH] and corresponding times to make the calculations more transparent. Thus we consider the wave functions of the form |
[EQUATION] where [MATH] [MATH] [MATH] [MATH] are independent variables and coefficients [MATH] [MATH] are considered as dependent variables, generating relation for six-dimensional heavenly equation hierarchy reads |
[EQUATION] Lax-Sato equations ( 13 ) have rather special structure in this case, taking into account that the only nonzero entries of last two lines of the Jacobian form the unity matrix: |
[EQUATION] here [MATH] [MATH] Vector fields are Hamiltonian due to the condition [MATH] Lax-Sato equations define the evolution of the series [MATH] |
[MATH] with the coefficients considered as functions of four variables [MATH] with respect to the higher times. The first two flows of the hierarchy read |
[EQUATION] and the first nontrivial order of expansion in [MATH] of the condition [MATH] gives [MATH] thus implying the existence of the potential [MATH] |
[MATH] [MATH] , transforming equations ( 22 ) to the form corresponding to the Lax pair of six-dimensional heavenly equation ( ), |
[EQUATION] Higher flows ( 21 ) can be written in a simple recursive form, which can be also obtained directly from the generating relation ( 20 ) (see |
for more detail) [EQUATION] A comparison between Lax-Sato equations ( 21 and recursive relations ( 23 ) provides useful expressions of coefficients of expansion of Poisson brackets in [MATH] through the derivatives of potential |
[MATH] Remark 1 6D heavenly equation and the hierarchy can be easily generalized to the case of multidimensional Poisson bracket and Hamiltonian vector fields, in which the basic equation (connected to hyper-Kähler equations |
) reads [EQUATION] where [EQUATION] and the generating relation for the hierarchy is [EQUATION] the generalization of Lax-Sato equations is straightforward. |
Remark 2 We have already mentioned (see also that Yang-Mills type linearly degenerate dispersionless hierarchies are rather special, and, if we take into account different sets of times (submanifolds in the space of independent variables) in the generating relation, may contain equations of different dimensionalities. ... |
[EQUATION] where [MATH] is arbitrary (may be infinite) and [MATH] [MATH] are of general form ( 11 ), taking into higher times. Evidently, this relation contains several copies of 6DHE hierarchy considered in this work. It also contains copies of standard 4-dimensional heavenly equation hierarchy. Morover, any solution ... |
[EQUATION] where [MATH] is a polynomial of the order [MATH] and the set of independent variables consists of [MATH] [MATH] [MATH] [MATH] [MATH] |
[MATH] We believe that the hierarchy ( 25 ) containing subhierarchies of different dimensionalities may have an interesting geometric interpretation, probably in the language of exotic cohomologies developed in |
Dressing scheme and solutions Starting from the dressing scheme for general six-dimensional hierarchy ( 12 , by the reduction to the 6D heavenly equation hierarchy ( 20 ) (see also |
we obtain Riemann-Hilbert problem on the unit circle (or the boundary of some region [MATH] [EQUATION] where the diffeomorphism defined by [MATH] [MATH] |
for the case of Hamiltonian reduction should be area-preserving with respect to the variables [MATH] [MATH] or the [MATH] problem in the unit disk (or some region [MATH] |
[EQUATION] Here the Hamiltonian reduction is taken into account explicitly. The functional freedom of the dressing data consists of functions of 5 variables, that indicates that reduced equations are generically 6-dimensional, like equations of the unreduced linearly-degenerate dispersionless hierarchy. |
We search for the solutions of the form [EQUATION] where [MATH] [MATH] are analytic outside [MATH] and go to zero at infinity. The [MATH] problem can be obtained by the variation of the action |
[EQUATION] where one should consider independent variations of [MATH] [MATH] possessing required analytic properties, keeping times [MATH] fixed. Using the results of the work |
in our setting, we come to the following statement: Proposition 1 The function [EQUATION] i.e., the action ( 27 ) evaluated on the solution of the [MATH] problem ( 26 ), gives the potential for 6D heavenly equation hierarchy and satisfies 6D heavenly equation ). |
To prove this proposition, it is enough to check the relations [MATH] [MATH] A class of solutions Below we will consruct a class of solutions for the 6DHE hierarchy. We will go along the lines of similar calculation for general heavenly equation presented in |
A class of solutions for the 6D heavenly equation hierarchy in terms of implicit functions (similar to to solutions of hyper-Kähler hierarchy presented in |
can be constructed using the choice [EQUATION] where [MATH] [MATH] are two-dimensional delta functions in the complex plane, and [MATH] [MATH] are some (complex-analytic) functions of three variables. The [MATH] problem ( 26 ) in this case reads |
[EQUATION] The solutions of the [MATH] problem are then of the form [EQUATION] and from ( 26 ) the functions [MATH] [MATH] are defined as implicit functions, |
[EQUATION] The potential [MATH] solving the general heavenly equation hierarchy is then given by the formula ( 28 ), it depends on the set of arbitrary functions of three variables [MATH] [MATH] |
[EQUATION] where [MATH] [MATH] are given by 30 ), [MATH] [MATH] have the form 19 ), and the functions [MATH] [MATH] are defined as implicit functions by equations ( 31 ). Formula ( 32 ) corresponds to the special solution of hyper-Kähler hierarchies presented in |
, however, it is important to note that in our case the solution depends on the set of arbitrary functions of three variables, in contrast to the set of functions of one variable in |
Acknowledgements MVP’s work was partially supported by the grant of Presidium of RAS ”Fundamental Problems of Nonlinear Dynamics” and by the RFBR grant 17-01-00366. |
# Source: arxiv 1806.01616 # Title: Power-law cross-correlations: Issues, solutions and future challenges # Sections: all # Downloaded: 2026-03-03T04:47:28.079851+00:00 |
Power-law cross-correlations: Issues, solutions and future challenges Abstract Analysis of long-range dependence in financial time series was one of the initial steps of econophysics into the domain of mainstream finance and financial economics in the 1990s. Since then, many different financial series have been analyze... |
keywords: long-range dependence , power-law cross-correlations , correlations , regression , power-law coherency , econophysics Introduction |
Analysis of long-range dependence properties of financial time series was at the very beginning of the econophysics field in the early 1990s (Beran, 1994 ; Mantegna and Stanley, 2000 ; Samorodnitsky, 2006 following the early works of the Mandelbrot research group (Mandelbrot, 1967 ; Mandelbrot and Wallis, 1968 ; Mandel... |
Long-range dependence of time series is characteristic by a slowly decaying auto-correlation function, contrary to the quickly vanishing exponentially decreasing auto-correlation function standardly seen in autoregressive (integrated) moving-average processes (ARMA/ARIMA) (Box et al., 1994 and (generalized) autoregress... |
In this work, we study and review the methodological steps that needed to be taken when coming from long-range correlations towards long-range cross-correlations. Importantly, we focus on problematic parts of the latter and cover two approaches how to treat them. Specifically, we argue (and review the relevant literatu... |
From long-range dependence to power-law cross-correlations Persistent series can be characterized through its dynamic properties in both time and frequency domains. In the former, the auto-correlation function is standardly represented by an asymptotic hyperbolic decay, specifically [MATH] for [MATH] where [MATH] is th... |
The hyperbolic decay of the auto-correlation function has some interesting implications which are covered in various textbooks (we refer here to the “classics” of Beran ( 1994 and Samorodnitsky ( 2006 ) but specifically its connection to the scaling of partial sums has crucial application. We define a partial sum of pr... |
The partial sums divergence is utilized in various estimators of the Hurst exponent, most notably by the detrended fluctuation analysis (DFA) (Peng et al., 1993 1994 ; Kantelhardt et al., 2002 . DFA is based on several steps mainly focused on further reducing the noise in the estimation procedure as well as filtering o... |
It took more than a decade to come from DFA to a parallel examination of dependence between two series. And again, it was DFA in the center. Podobnik and Stanley ( 2008 introduced the detrended cross-correlation analysis (DCCA/DXA) that is built on a parallel idea – scaling of covariances between partial sums. Even tho... |
The issues with power-law cross-correlations Most of the literature building on the DCCA procedure has been empirical and it has become quickly clear that the relationship between the bivariate Hurst exponent [MATH] and the Hurst exponents of the separate processes [MATH] and [MATH] might play a crucial role. From one ... |
(He and Chen, 2011 ; Wang et al., 2013 ; Oswiecimka et al., 2014 . From the other, numerical and theoretical studies suggested that either [MATH] or [MATH] |
(Sela and Hurvich, 2012 . The clash was apparent and a more detailed theoretical treatment was clearly needed. The primary issue of the literature (both theoretical and empirical) on power-law cross-correlations was non-existence of a process that would generate power-law cross-correlated series and allow to control th... |
Kristoufek ( 2013a introduced the mixed-correlated ARFIMA process (MC-ARFIMA), which allowed for controlling the [MATH] parameter. MC-ARFIMA processes are defined as |
[EQUATION] where [EQUATION] and error terms are characterized by [EQUATION] To put it in words, the two processes are each a linear combination of two power-law auto-correlated processes with possibly correlated error-terms. The separate long-term memory parameters [MATH] are unrestricted. The [MATH] -notation is kept ... |
[EQUATION] The MC-ARFIMA introduction has had two main results. First, there was finally a data generator that could be used for simulation studies that also has well-defined statistical properties (Kristoufek, 2015a 2016 . And second, the possibility of having [MATH] seemed to have vanished as the MC-ARFIMA processes ... |
As a follow-up, Kristoufek ( 2015c studies the issue of [MATH] on a theoretical basis in more detail. As it turns out, the answer is almost trivial. The issue is solved through the squared spectrum coherency and its scaling close to the origin. The squared spectrum coherency is defined for two stationary series [MATH] ... |
[EQUATION] for a given frequency [MATH] . Using the definition of the power-law cross-correlations in the frequency domain, we can rewrite the coherency as |
[EQUATION] Now note that the squared coherency ranges between 0 and 1 everywhere (in fact even for non-stationary series with their pseudo-spectra). Therefore, it is so restricted for the long-range cross-correlations frequencies as well, i.e. [MATH] . This gives us two feasible and one infeasible possibilities: |
1. [MATH] 2. [MATH] 3. [MATH] This implies that [MATH] is impossible. Note that this holds for stationary as well as for non-stationary processes (and it can be easily shown for the DCCA fluctuations scaling as well). If the empirical literature reports otherwise, it is due to a bias. This bias might be due to various ... |
(Kristoufek, 2014c are not biased by the heavy tails. And third, there is a finite sample bias as showed in detail in Kristoufek ( 2015c . Unfortunately, this bias can be either positive, negative or none depending on the level of correlation between series for scales close to zero. This makes [MATH] or specifically it... |
What makes this finding even more alarming is the fact that in the financial econometrics and time series analysis literature, the impossibility of [MATH] is taken as an obvious property and it is pretty much a two-liner in Sela and Hurvich ( 2012 who quickly focus on the [MATH] case as the only relevant one for furthe... |
All in vain? One might then ask whether the whole research around power-law cross-correlations is in vain and futile. The short answer is “no” but it needs further work with more care about theoretical aspects of the topic. As it stands, most of the empirical literature reports either [MATH] or [MATH] . The former is i... |
4.1 Scale-specific correlations and regressions The DCCA procedure is built on scaling of the bivariate fluctuation function [MATH] , which eventually leads to a power-law scaling [MATH] , in the same way the DFA procedure is based on the fluctuation function [MATH] scaling. Asymptotically, these can be seen as covaria... |
[EQUATION] where [MATH] and [MATH] are scale-specific variances of processes [MATH] and [MATH] . This correlation coefficient has been shown to work well for non-stationary series as well and to outperform the standard Pearson correlation coefficient (Kristoufek, 2014a . In addition, its construction is so straightforw... |
When the scale-dependent correlations are defined, it is only a simple step towards regression. Kristoufek ( 2015b introduces a DCCA-based estimator of the scale-dependent [MATH] coefficient, defined as |
[EQUATION] Compared to [MATH] , which measures the strength of the relationship, [MATH] gives the specific effect, i.e. its level, which is much more useful for interpretation of economic and financial relationships where one is usually interested not only in whether the variables are strongly or weakly correlated but ... |
The work and insight of Zebende ( 2011 has thus given a very important alternative utility of the DCCA method (and other time domain [MATH] in general) and he has shown that the in-between steps of methods can sometimes lead to completely novel views on the topic. |
4.2 Power-law coherency As noted by Sela and Hurvich ( 2012 and Kristoufek ( 2015c 2017 , only the case of [MATH] is an interesting venue as it promises a new class of processes. Returning back to utilizing the squared spectrum coherency, if the two processes are power-law correlated so that [MATH] and [MATH] close to ... |
[EQUATION] close to the origin. The power-law coherency can be defined through parameter [MATH] as [MATH] Recall that the squared spectrum coherency [MATH] for all frequencies [MATH] which yields only two possible settings for the exponent – either [MATH] or [MATH] [MATH] gives us [MATH] and the coherency goes to a con... |
As in detailed shown by Kristoufek ( 2017 , the power-law coherency can be translated into the time domain easily. Eventually, one arrives at |
[EQUATION] so that the scaling exponent for both time ( [MATH] ) and frequency ( [MATH] ) domain power-law coherency is the same. Interestingly, the squared correlation [MATH] can be easily represented by the squared DCCA-based correlation coefficient [MATH] so that both approaches presented in this section and the pre... |
Discussion The whole issue and suggested solutions presented above point mainly to a general problem of interdisciplinary research (here specifically econophysics) – communities do not interact enough. The bivariate breakthrough in the sense of power-law cross-correlations came in 2008 (Podobnik and Stanley, 2008 but t... |
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