Yves I Ngounou Bakam

draft Clustering Independence in high dimensions

Introduction and motivation

For k = 1, • • • , K, let X (k) = (X (k) 1 , • • • , X (k)
p k ) be a p k -dimensional random variable with univariate marginals probability distribution function (pdf) denoted by F (k) 1 , • • • , F (k) p k . The goal is to partition the set S = {X (1) , • • • , X (K) } of random vectors, each potentially of different dimensions, into clusters. Within each cluster, the variables should exhibit independence, while across clusters, there should be some level of dependence. As far as we know, addressing this problem in its complete generality remains unexplored.

This clustering of independence among random vectors offers several advantages and points of interest, particularly in various scientific, technological, and practical applications:

• Promoting Ethical Practices: As machine learning technologies become more pervasive and influential in society, addressing ethical considerations such as fairness, bias, and accountability becomes paramount. Clustering independence can help mitigate risks associated with data dependencies, model biases, and unintended consequences, promoting ethical practices and responsible AI development.

• Resource Optimization: In various applications such as machine learning and optimization problems, clustering independence helps in efficiently allocating resources by identifying independent components that can be processed or analyzed separately.

• Efficient Algorithms: Algorithms designed to leverage clustering independence can be more computationally efficient, reducing time and computational resources required for tasks like data processing, analysis, and modeling.

• Resilient Systems: Understanding the independence structure ensures that systems or models are built on robust foundations, minimizing the risk of overfitting and enhancing stability across different scenarios or environments.

• Relevant Features: By understanding which variables or features are independent of each other, one can prioritize and select the most relevant features for modeling and prediction tasks, improving model efficiency and accuracy.

Partitioning a set of random vectors into clusters characterized by intra-cluster independence and inter-cluster dependence offers a structured, flexible, and insightful approach to analyzing complex systems and datasets, providing numerous advantages in terms of modeling, efficiency, interpretability, and applicability across diverse domains and applications.

The paper is organised as follows:

Copula coefficients test of independence

We consider the following null hypothesis of independence

H 0 : X (1) ⊥ ⊥ X (2) ⊥ ⊥ • • • ⊥ ⊥ X (K) versus H 1 : ¬H 0 (1)
When K = 2, there is a large literature for testing independence between two random vectors. We can mention the two seminal papers of Gieser and Randles [START_REF] Gieser | A nonparametric test of independence between two vectors[END_REF] and Horrell and Lessig [START_REF] Horrell | A note on a nonpαrαmetric test of independence between two vectors[END_REF]. More recently, Feng et al. [START_REF] Feng | Multivariate tests of independence and their application in correlation analysis between financial markets[END_REF] considered the problem of multivariate tests of independence using high dimensional rank statistics as Spearman and Kendall'ones. We can also mention Berrett and Samworth [START_REF] Berrett | Nonparametric independence testing via mutual information[END_REF] where the test statistic is based on entropies, Bodnar et al. [START_REF] Bodnar | Testing for independence of large dimensional vectors[END_REF] resticting their study to high dimension Gaussian vectors, as in Silva et al. [START_REF] Silva | Kronecker delta method for testing independence between two vectors in high-dimension[END_REF]. And among others Albert et al. [START_REF] Albert | Adaptive test of independence based on hsic measures[END_REF] and Yin and Yuan [START_REF] Yin | A new class of measures for testing independence[END_REF].

When

p 1 = • • • = p K = 1, (1
) remains to test the independence of the K components of a vector. In such a case a numerous works attempt to construct efficient statistics, with a series of papers based on copulas as in Genest et al. [START_REF] Genest | Asymptotic local efficiency of cramér-von mises tests for multivariate independence[END_REF], González-Barrios et al. [START_REF] González-Barrios | A novel characterization and new simple tests of multivariate independence using copulas[END_REF], Roy et al. [START_REF] Roy | Some new copula based distribution-free tests of independence among several random variables[END_REF], or Genest et al. [START_REF] Genest | Testing for independence in arbitrary distributions[END_REF]. Other approaches can be mentioned as in Mao [START_REF] Mao | Testing independence in high dimensions using kendall's tau[END_REF], using high dimension Kendall's tau, as more recently in Drton et al. [START_REF] Drton | High-dimensional consistent independence testing with maxima of rank correlations[END_REF].

In the general case, when K > 2 and p j ≥ 1, there is very few general works. One reason for this is that the null distributions for the proposed test statistics become difficult to evaluate. We can mention Chakraborty and Zhang [START_REF] Chakraborty | Distance metrics for measuring joint dependence with application to causal inference[END_REF] where their approach is based on the notion of distance covariance and Roy and Ghosh [START_REF] Roy | Some tests of independence based on maximum mean discrepancy and ranks of nearest neighbors[END_REF] who uses the ideas of maximun mean discrepancy and ranks of nearest neighbors. Certain restrict their study to Gaussian vectors, as in Bao et al. [START_REF] Bao | Test of independence for high-dimensional random vectors based on freeness in block correlation matrices[END_REF] or Chen and Liu [START_REF] Chen | Testing independence with high-dimensional correlated samples[END_REF], working on the covariance structure in high dimension. The work who seems to tackle the general problem is that of Fan et al. [START_REF] Fan | Multivariate nonparametric test of independence[END_REF] where the authors represented the independence hypothesis through the product of characteristic functions. They deduce a Cramer-Von-Mises statistic with null asymptotic distribution related to a process involving eigenvalues of covariance function. In addition their approach necessitates a weight function to calibrate the numerical integration of the Cramer-Von-Mises statistic. However, these authors worked in a very general setting since they consider continuous as well as discrete or mixed random vectors.

Our approach can be considered as a competitor of Fan et al. [START_REF] Fan | Multivariate nonparametric test of independence[END_REF] in the continuous case and is quite similar in spirit but instead of using the characteristic functions we use copulas.

By definition, a p k -dimensional copula C (k) is the joint cumulative distribution of a random vector

U (k) = (U (k) 1 , • • • , U (k) p k ) with uniform marginals over [0, 1] , that is, C (k) : [0, 1] p k → [0, 1] u = (u 1 • • • , u p k ) → C (k) u = P U (k) 1 ≤ u 1 , • • • , U (k) p k ≤ u p k .
The copula coefficients associated to C (k) , denoted ρ (k) j , are defined to be (see ? ])

ρ (k) j = [0,1] p k L j 1 (u 1 ) • • • L j p k (u p k )dC (k) (u 1 • • • , u p k ), (2)
where L j is the Legendre polynomial on [0, 1] of degree j satisfying

[0,1] L j (x)L k (x)dx = δ jk
where δ jk = 1 if j = k and 0 otherwise. These polynomials are defined by

L 0 = 1, L 1 (x) = √ 3(2x -1)
, and for n > 1 :

(n + 1)L n+1 (x) = (2n + 1)(2n + 3)(2x -1)L n (x) - n √ 2n + 3 √ 2n -1 L n-1 (x).
According to Proposition 1 of Bakam and Pommeret [START_REF] Bakam | K-sample test for equality of copulas[END_REF], the series of copula coefficients entirely characterize the copulas, that is

C (k) = C (ℓ) if and only if ρ (k) j = ρ (ℓ) j for all j (3)
where for all j ∈ {1, • • • , p k }, I j = 1 0 L j (s)ds.

Notation

Write p = p 1 + • • • + p K . For j ∈ N p and for x ∈ R p we decompose j = (j (1) ,

• • • , j (K) ), x = (x (1) , • • • , x (K) ) and u = (u (1) 1 , • • • , u (K) p K ), where j (k) = ( j (k) 1 , • • • , j (k) p k ), x (k) = (x (k) 1 , • • • , x (k) p k ) and u (k) = (u (k) 1 , • • • , u (k) p k ), for all k = 1, • • • , K. u (k) = (u (k) 1 , • • • , u (k) p k ) and u = (u (1) 1 , • • • , u (K) p K ).
We also write X = (X (1) , • • • , X (K) ) the p-vector obtained from the K populations. We use the classical transformations: U = (U (1)

, • • • , U (K) ) where U (k) = (U (k) 1 , • • • , U (k) p k ) = (F (k) 1 (X (k) 1 ), • • • , F (k) p k (X (k) p k ), for all k = 1, • • • , K.
The empirical versions based of n iid observations of such variables are obtained by replacing U (k) j by U (k) j where

U (k) j = 1 n n i=1 F (k) j (X (k) j,i ),
where X (k) j,i is the ith observation of the jth component from population K. By Sklar's theorem (Sklar [START_REF] Sklar | Fonctions de repartition an dimensions et leurs marges[END_REF]), the continuity assumption on the marginals implies that the the testing problem (1) can be rewritten as follows

H 0 : C = C (1) • • • C (K) versus H 1 : C C (1) • • • C (K) ( 4
)
where k) and C are the unique copulas associated toX (k) and X respectively such that

for k = 1, • • • , K, C (
C 1, . . . , 1, u (k) , 1, . . . , 1 = C (k) u (k)
Rather than comparing copulas, our procedure is based on the copula coefficients defined in [START_REF] Bakam | K-sample test for equality of copulas[END_REF]. According to (3), the null hypothesis changes in the following way

H 0 : ρ j = ρ (1) j (1) • • • ρ (K) j (K) , ∀j ∈ N p , (5)
where ρ j is the jth copula coefficient associated to C.

Data-driven smooth test

Our procedure consists in detecting the largest differences between the copula coefficients involved in [START_REF] Berrett | Nonparametric independence testing via mutual information[END_REF]. We consider K iid samples from X (1) , • • • , X (K) , denoted by (X (1) 1,i ,

• • • , X (1) p 1 ,i ) i=1,••• ,n , • • • , (X (K) 1,i , • • • , X (K) p K ,i ) i=1,••• ,n .
We want to detect a difference between ρ j and ρ (1)

j (1) • • • ρ (K) j (K)
, for some j ∈ N p . For any vector j = ( j 1 , • • • , j p ) we denote the L1 norm by

∥j∥ 1 = | j 1 | + • • • + | j p |, Remark 1.
From the orthogonality of the Legendre polynomials we obtain:

• If j is a zero vector, then ρ j -ρ (1) j (1) • • • ρ (K) j (K) = 0
• If only one of the coefficients of j is non null, then ρ j -ρ (1) j (1) • • • ρ (K) j (K) = 0 Considering the null hypothesis H 0 as expressed in (??), our test procedure is based on the sequences of differences

r j := ρ j -ρ (1) j (1) • • • ρ (K) j (K) , for j ∈ N p ,
where

ρ j =                  1 if j = 0, 0 if exactly one component of j is non nul, 1 n n i=1 L j (1) 1 ( U (1) 1,i ) × • • • × L j (1) p 1 ( U (1) p 1 ,i ) × L j (2) 1 ( U (2) 1,i ) × • • • × L j (K) p K ( U (K) p K ,i ), else. ρ (k) j (k) =                  1 if j (1) = 0, 0 if exactly one component of j (1) is non null, 1 n n i=1 L j (k) 1 ( U (k) 1,i ) × • • • × L j (k) p k ( U (k) p k ,i ), else.
3 Note that by construction r j = 0 when j satisfies one of the conditions of Remark 1. Such estimators of copula coefficients have been studied in Bakam and Pommeret [START_REF] Bakam | Nonparametric estimation of copulas and copula densities by orthogonal projections[END_REF] where it is shown their excellent behavior.

In order to select automatically the number of copula coefficients we introduce the following set for any integer p > 1: S(p) = {j ∈ N p ; and at least two components are non null}.

We introduce the following order on S(p).

If j and j are two vectors in S(p) we define

k(j, j) = min{k ∈ {1, • • • , K}; j (k) j (k) }
the first index such that the subvectors of j and j are different. In the same way

k(j k) , j (k) ) = min{ℓ ∈ {1, • • • , p k }; j (k) ℓ j (k)
ℓ } Then the order on S(p) is defined by

j < j if ∥j∥ 1 < ∥ j∥ 1 ∥j∥ 1 = ∥ j∥ 1 and j (R) r < j (R)
r where R = k(j, j) and r = k(j (R) , j (R) ).

For instance, the first element of S(p) is such that j

(1) = (1, 0, • • • , 0), j (2) = (1, 0, • • • , 0), and j (k) = (0, • • • , 0), for k = 3, • • • , p.
In that case we say that Ord(j) = 1.

To construct our test statistics, we consider a sequence d(n) which tends to infinity as n → ∞, and we introduce a series of statistics based on the sequence (r j ) j∈N p as follows: for 1 ≤ k ≤ d(n), we define

T k = n j∈S(p);ord(j)≤k (r j ) 2 . ( 6
)
Clearly all these statistics are embedded. When d(n) is large it will make it possible to compare high coefficient orders through r j , while k will permit to visit all the values of j for this given order. Notice that we need to compare all copula coefficients and then to let d(n) tend to infinity to detect all possible alternatives. However, choosing too large parameters tends to power dilution of the test. Following [START_REF] Kallenberg | Consistency and monte carlo simulation of a data driven version of smooth goodness-of-fit tests[END_REF], we suggest a data driven procedure to select automatically the number of coefficients to test the hypothesis H 0 . Namely, we set

D(n) := min argmax 1≤k≤d(n) (T k -kq n ) , (7)
where q n and d(n) tend to +∞ as n → +∞, kq n being a penalty term which penalizes the embedded statistics proportionally to the number of copula coefficients used. Finally, the data-driven test statistic that we use to compare C and

C (1) • • • C (p) is T D(n)
and we consider the following rate for the number of components in the statistic:

(A) d(n) (7p-4) = o(q n )
A classical choice for q n is q n = log(n) initially used in Schwarz [START_REF] Schwarz | Estimating the dimension of a model[END_REF] (see for instance the seminal work of Ledwina [START_REF] Ledwina | Data-driven version of neyman's smooth test of fit[END_REF]). This choice is convenient to detect smooth alternatives (see Section 2.3) and will be adopted in our simulation.

Our first result shows that under the null the least penalized statistic will be selected.

Theorem 1. Let assumption (A) holds. Then, under H 0 , D(n) converges in Probability towards 1 as n → +∞.

Theorem 2. Let assumption (A) holds. Then, under H 0 , T D(n) converges in law towards a central normal distribution with variance

σ 2 = V L 1 (U (1) 1 )L 1 (U (2) 1 ) +2 √ 3 I(X (1) 1 ≤ x) -F (1) 1 (x) L 1 (F (2)
1 (y))dF (1) (x)dF (2) (y)

+2 √ 3 I(X (2) 1 ≤ y) -F (2) 1 (y) L 1 (F (1)
1 (x))dF (1) (x)dF (2) (y) .

In order to normalize the test, write

σ 2 = 1 n n i=1 M i -M 2 , with M = 1 n n i=1 M i ,
where

M i = L 1 ( U (1) i,1 )L 1 ( U (2) i,1 ) + 2 √ 3 n n k=1 I X (1) i,1 ≤ X (1) k,1 -U (1) k,1 L 1 ( U (2) k,1 ) + 2 √ 3 n n k=1 I X (2) i,1 ≤ X (2) k,1 -U (2) k,1 L 1 ( U (1)
k,1 ).

Proposition 1. Under H 0 we have the following convergence in probability

σ 2 P -→ σ 2 .
We then deduce the limit distribution under the null.

Corollary 1. Let assumption (A) holds. Then under H 0 , T D(n) / σ 2 converges in law towards a chi-squared of one degree of freedom distribution χ 2 1 as n → +∞.

Convergence under alternative hypotheses

We consider the general alternative hypothesis:

H 1 : C C (1) • • • C (p)
and we make the following assumption:

(B) b n = o(n). Theorem 3. Assume that (A)-(B) hold. Then under H 1 , T D(n) converges to +∞, that is, P(T D(n) < ϵ) → 0, for all ϵ > 0.
Remark 2. In the classical smooth test approach [START_REF] Ledwina | Data-driven version of neyman's smooth test of fit[END_REF] a standard penalty is q n = b n = log(n), which is related to the Schwarz criteria [START_REF] Schwarz | Estimating the dimension of a model[END_REF] as discussed in [START_REF] Kallenberg | Consistency and monte carlo simulation of a data driven version of smooth goodness-of-fit tests[END_REF]. Note also that [START_REF] Inglot | Towards data driven selection of a penalty function for data driven neyman tests[END_REF] compared this type of Schwarz penalty to the Akaike one where they proposed b n or q n to be constant. In our simulation we consider the classical choice q n = b n = log(n).

Clustering approach

The purpose of this section is to partition any set S = {X (1) , • • • , X (K) } of random vectors of arbitrary dimension into clusters in such a way that inside each cluster the variables are independent and the clusters between themselves are dependent. To our knowledge, the problem in its full generality has not yet been addressed.

This clustering procedure can solve several complex problems in a very short time and is useful in practice, for instance: i) in finance to build a well-diversified portfolio whose stock options would be less subject to a systemic crisis; ii) in the world of actuarial science by making it possible to practice a price segmentation strategy; iii) in biology to classify interdependent and intra-independent network of genes; iv) in a general regression framework where it is of importance to retain a set of independent covariables.

Clustering principle

In the sequel we propose to adapt the previous test procedure to obtain a data-driven method to cluster K multivariate variables into N subgroups characterized by a common dependence. The number N of clusters is unknown and will be automatically chosen by the previous procedure and validated by our testing method.

More precisely, we consider a multivariate variable S = {X (1) , ℓ) . The clustering algorithm starts by choosing the two variables of S that are less distant in terms of independence. In this way, it chooses the smallest two-sample statistic. If the independence of both associated vectors is not rejected, these two variables form the first cluster. Then the algorithm proposes a new variable closest to this cluster in terms of independence. While the test accepts the simultaneous independence of the vectors, the cluster growths. If the test is rejected then the cluster is closed. We iterate this several times until every vector is associated with a cluster.

• • • , X (K) } of p dimension in which X (ℓ) is a p ℓ ≥ 1- dimensional random vectors for ℓ ∈ {1, 2, • • • , K}, such that p 1 + p 2 + • • • + p K = p. For l = 1, • • • K, we consider a sample {X (l) i,1 , • • • , X (l) i,p l ) i=1,••• ,n }, l = 1, • • • K from X (

Clustering algorithm

We can summarize the clustering algorithm as follows:

Algorithm: Independence clustering

1 Initialization: c = 1, S = {X (1) , • • • , X (K) }, and S 0 = ∅ ; 2 Select {ℓ ⋆ , m ⋆ } = argmin{V (ℓ,m)) D(n) ; ℓ m ∈ S \ c k=1 S k-1 } ; 3 Test H 0 between X (ℓ ⋆ ) and X (m ⋆ ) ; 4 if H 0 is not rejected then 5 S 1 = {X ℓ ⋆ , X m ⋆ }; 6 else 7 STOP. There is no cluster. 8 end 9 while S \ c k=1 S k ∅ do 10 Select { j ⋆ } = argmin{V (i, j) D(n) ; i ∈ S c , j ∈ S \ c k=1 S k };
11 Test H 0 the simultaneous independence of all X (i) , i ∈ S c and X ( j ⋆ ) ;

12 if H 0 is not rejected then 13 S c = S c {X ( j ⋆ ) };

Geology data

We analyse the well-known uranium exploration dataset of Cook and Johnson [START_REF] Cook | Generalized burr-pareto-logistic distributions with applications to a uranium exploration data set[END_REF]. The same data set has been analyzed by Genest & Rivest [START_REF] Genest | Statistical inference procedures for bivariate archimedean copulas[END_REF] and Genest, Quessy & Rémillard [START_REF] Genest | Goodness-of-fit procedures for copula models based on the probability integral transformation[END_REF] to demonstrate a semiparametric inference for Archimedean copulas and a goodness-of-fit test. The data consists of concentrations of seven chemical elements in 655 water samples collected from the Montrose quad-rangle of Western Colorado. Concentrations of Uranium (U), Lithium (Li), Cobalt (Co), Potassium (K), Cesium (Cs), Scandium (S c) and Titanium (T i) were measured. The comparison of these chemical elements in the context of mutual dependence or groupwise dependence is a major concern for geologists.

Finance data

In an ever-evolving global landscape shaped by globalization and the outsourcing of major corporations, often through a growing nexus of cooperation and economic integration among countries and financial markets, making astute investments and achieving success becomes progressively complex. In this context, investors are keen on examining the interplay among significant global financial markets to construct a diversified portfolio with substantial potential.

14 else 15 S 16 c = c + 1

 1415161 c+1 = {X ( j ⋆ ) };


On stability of nonlinear homogeneous systems with distributed delays having variable kernels

The stability problem for nonlinear homogeneous systems with distributed delay and variable kernel is studied. Both, the Lyapunov-Krasovskii and the Razumikhin, approaches are applied. It is proved that the global asymptotic stability of the zero solution for an auxiliary delay-free homogeneous system implies the local asymptotic stability of the zero solution for the original system with distributed delay. Moreover, the impact of nonlinear time-varying perturbations on the system dynamics is analyzed applying the averaging techniques. The results are illustrated by a mechanical system described by a Lienard equation, and an indirect control system design for a linear system.

I. INTRODUCTION

The problem of stability analysis for nonlinear systems is rather complex, and it becomes even more sophisticated being influenced by presence of time-delays and time-varying perturbations [START_REF] Hale | Theory of Functional Differential Equations[END_REF], [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF], [START_REF] Gu | Stability of Time-Delay Systems[END_REF], [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]. Raising the internet of things and cyber-physical systems technologies nowadays lead to appearance of scenarios, where all these factors meet together [START_REF] Karafyllis | Recent Results on Nonlinear Delay Control Systems, ser. Advances in Delays and Dynamics[END_REF], [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. There are two main methods for stability analysis of time-delay systems: Lyapunov-Krasovskii (LK) and Lyapunov-Razumikhin (LR) approaches [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]. The former uses LK functionals (LKFs) and it has been proven to give the necessary and sufficient conditions of stability [START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF], [START_REF] Efimov | Converse Lyapunov-Krasovskii theorem for ISS of neutral systems in Sobolev spaces[END_REF], while the latter is based on usual Lyapunov function analysis (under additional restrictions), and it also provides necessary and sufficient conditions of stability under mild restrictions [START_REF] Efimov | On equivalence of Lyapunov-Razumikhin conditions and ISS for a class of time-delay systems[END_REF]. The advantage of LR approach is its simplicity in application, since a Lyapunov function for the delay-free system can be tested as a guess candidate. Both methods have extensions to the input-to-state stability (ISS) verification for the systems with bounded disturbances [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF], [START_REF] Fridman | On input-to-state stability of systems with time-delay: A matrix inequalities approach[END_REF].

Distributed delays can result from communication networks, the implementation of control/estimation algorithms [START_REF] Anan | On stabilization of some control systems with an aftereffect[END_REF], [START_REF] Feng | Integral partitioning approach to robust stabilization for uncertain distributed time-delay systems[END_REF], [START_REF] Feng | Dissipative stabilization of linear systems with time-varying general distributed delays[END_REF], [START_REF] Aleksandrov | Stability of homogeneous systems with distributed delay and time-varying perturbations[END_REF] or human appearance in the loop [START_REF] Michiels | Consensus problems with distributed delays, with application to traffic flow models[END_REF]. Analyzing the stability of these systems requires specialized extensions of previously established methods [START_REF] Xie | A robust H∞ control of distributed delay systems with application to combustion control[END_REF], [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF]. The complexity of the investigation increases when external perturbations are present, especially while assessing the permissible upper bounds of disturbances in relation to the delayed state (i.e., evaluating the asymptotic gains in terms of ISS). Considering the time-varying nature of the perturbations can lead to less conservative bounds. For instance, the efficiency of the averaging method has been demonstrated dealing with periodic or almost periodic perturbations [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-Linear Oscillations[END_REF], [START_REF] Khapaev | Averaging in Stability Theory[END_REF].

When working in a nonlinear setting, it is advantageous to confine the analysis to a specific class of models. In this context, our focus will be on homogeneous dynamics, which have gained popularity due to their numerous beneficial properties in the absence of delays [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF]. Furthermore, these properties have been extended to infinite-dimensional time-delay systems [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF], [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF]. It has been recognized that for systems with discrete time-delays, if the delay-free counterparts (i.e., systems with zero delay) are globally asymptotically stable at the origin, then the original dynamics are locally asymptotically stable at the origin for any delay value, provided the positive homogeneity degree is present [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF], [START_REF] Aleksandrov | On the asymptotic stability of solutions of nonlinear systems with delay[END_REF], [START_REF]Delay-independent stability of homogeneous systems[END_REF]. Alternatively, for systems with negative degree, the original dynamics are practically globally asymptotically stable, a property referred to as uniform ultimate boundedness of the solutions [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF]. The LR method has been employed in the aforementioned works, and a recent development in this area is the LK approach, as presented in [START_REF] Aleksandrov | Complete type functionals for some classes of homogeneous diffrential-difference systems[END_REF], [START_REF] Efimov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF]. Extension of these results to the case of distributed delays and exogenous suitably bounded perturbations admitting averaging were given in [START_REF] Aleksandrov | Stability of homogeneous systems with distributed delay and time-varying perturbations[END_REF], where the distributed delays had constant kernels. Note that in the case of non-zero homogeneity degree (nonlinear setting), a wider class of perturbations (not just high frequency oscillations) preserves the stability compared to the linear counterparts, which is also the case of the present work.

In this paper, the stability problem for homogeneous systems of positive degree with distributed delay and variable kernel is studied. Both, the LK and the LR, approaches are applied. It is proven that the global asymptotic stability of the zero solution for an auxiliary delay-free homogeneous system implies the local asymptotic stability of the zero solution for the original system with distributed delay. Moreover, the impact of nonlinear time-varying perturbations on the system dynamics is analyzed. First, a theorem on the stability via nonlinear homogeneous approximation is proved. Next, with the aid of a special modification of the averaging approach, conditions of robust stability in the presence of nonlinear time-varying perturbations with zero mean values are derived. Compared with our recent work [START_REF] Aleksandrov | Stability of homogeneous systems with distributed delay and time-varying perturbations[END_REF], the present contribution contains the following novelty:

(i) In [START_REF] Aleksandrov | Stability of homogeneous systems with distributed delay and time-varying perturbations[END_REF], only distributed delays with constant kernels were considered, whereas here the case of variable kernels is investigated.

(ii) In the present paper, we propose special constructions of LR functions and LKFs that differ from those used in [START_REF] Aleksandrov | Stability of homogeneous systems with distributed delay and time-varying perturbations[END_REF]. This permits us to obtain stability conditions for wider classes of perturbed systems.

On the other hand, in [START_REF] Aleksandrov | Stability of homogeneous systems with distributed delay and time-varying perturbations[END_REF] both the cases, of positive and negative homogeneity degrees, were studied, whereas in this paper, we assume that the homogeneity degrees are only positive.

The outline of this work is as follows. Preliminaries are given in Section II. The considered stability analysis problem is described in Section III. The main results are formulated in sections IV and V with and without averaging tools, respectively. Examples of utilization of the proposed theoretical findings are shown in Section VI.

II. PRELIMINARIES

The real numbers are denoted by R, R + = {s ∈ R : s ≥ 0}, and |s| is an absolute value for s ∈ R. Euclidean norm for a real n-dimensional vector x ∈ R n is defined as ∥x∥. We denote by C([-τ, 0], R n ), 0 < τ < +∞ the Banach space of continuous functions ϕ : [-τ, 0] → R n with the uniform norm ∥ϕ∥ τ = sup -τ ≤ς≤0 ∥ϕ(ς)∥.

For v ∈ R n , diag{v} corresponds to a diagonal matrix with the components of vector v on the main diagonal.

A continuous function σ : R + → R + belongs to class K if it is strictly increasing and σ(0) = 0; it belongs to class K ∞ if it is also radially unbounded.

The Young's inequality claims that for any a, b ∈ R + and γ > 0, δ > 0 [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF]:

a γ b δ ≤ 1 p a γp + p -1 p b δp p-1
for any p > 1.

Using the properties of homogeneous functions the following results can be obtained:

Lemma 1. [29] Let a, b ∈ R + and ℓ > 0, α > 0, β > 0, γ > 0, δ > 0 be given, then a α + b β -ℓa γ b δ ≥ 0 provided that max{a α , b β } ≤ ℓ 1 1- γ α -δ β and γ α + δ β > 1.
The standard definitions of stability and related properties for time-delay systems can be found in [START_REF] Hale | Theory of Functional Differential Equations[END_REF], [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF], [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF], and for delay-free dynamics in [START_REF] Khalil | Nonlinear Systems[END_REF].

III. STATEMENT OF THE PROBLEM

Let a system with distributed delay

ẋ(t) = F (x(t)) + t t-τ D(s -t)G(x(s))ds (1)
be given, where x(t) ∈ R n , vector functions F : R n → R n and G : R n → R q are continuous for x ∈ R n , the matrix function D : [-τ, 0] → R n×q is continuous, τ = const > 0 is the maximum delay. Let initial functions for (1) belong to the space C([-τ, 0], R n ). Denote by x t the restriction of a solution x(t) to the segment [t -τ, t], i.e.,

x t : ξ → x(t + ξ), ξ ∈ [-τ, 0].
Assumption 1. Vector function F (x) is r-homogeneous of the degree ν > 0 with respect to weights r = (r 1 , . . . , r n ), where r i > 0, i = 1, . . . , n:

F (Λ r (λ)x) = λ ν Λ r (λ)F (x), ∀x ∈ R n , ∀λ > 0, Λ r (λ) = diag{(λ r1 , . . . , λ rn ) ⊤ }.
Assumption 2. For every fixed θ ∈ [-τ, 0], D(θ)G(x) is r-homogeneous vector function of x of the degree ν > 0:

D(θ)G(Λ r (λ)x) = λ ν Λ r (λ)D(θ)G(x), ∀x ∈ R n , ∀λ > 0.
Under these assumptions the system (1) admits the zero solution. We will look for conditions ensuring the (local) asymptotic stability of this solution. In addition, we will consider corresponding perturbed systems when exogenous perturbations are weighted by a state-dependent gain with distributed delay. With the aid of specially constructed LR functions or LKFs, and a modification of the averaging approach, we derive the conditions under which perturbations do not destroy the asymptotic stability.

Our standing assumption is stability of the delay-free counterpart of (1): Assumption 3. The zero solution of the auxiliary homogeneous delay-free system

ẋ(t) = F (x(t)) + 0 -τ D(θ)dθ G(x(t))
is asymptotically stable. Remark 1. If Assumption 3 is satisfied, then (see [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]) there exists a Lyapunov function V (x) with the following properties:

(i) V (x) is twice continuously differentiable for x ∈ R n ; (ii) V (x) is positive definite; (iii) V (x)
is r-homogeneous of the degree µ, where µ > r i + r j , i, j = 1, . . . , n;

(iv) the function

∂V (x) ∂x F (x) + 0 -τ D(θ)dθ G(x)
is negative definite.

IV. STABILITY ANALYSIS WITHOUT AVERAGING

First, let us formulate the conditions of stability for the nominal system (1).

Theorem 1. Let assumptions 1-3 be fulfilled. Then the zero solution of (1) is asymptotically stable.

Proof. Construct a LKF candidate for (1) by the formula

V (t, x t ) = V (x(t)) + t t-τ (α + β(s + τ -t))|x(s)| µ+ν r ds + ∂V (x(t)) ∂x t t-τ s-t -τ D(θ)dθG(x(s))ds, (2)
where α, β are positive tuning parameters and V (x) is a Lyapunov function with the properties specified in Remark 1; |x| r = n i=1 |x i | κ/ri 1/κ for some κ ≥ max i=1,...,n r i is a homogeneous norm [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF].

Differentiating the functional (2) along the solutions of (1), we obtain

˙ V = ∂V (x(t)) ∂x F (x(t)) + 0 -τ D(θ)dθG(x(t)) +(α + βτ )|x(t)| µ+ν r -α|x(t -τ )| µ+ν r -β t t-τ |x(s)| µ+ν r ds + t t-τ s-t -τ D(θ)dθG(x(s))ds ⊤ × ∂ 2 V (x(t)) ∂x 2 F (x(t)) + t t-τ D(s -t)G(x(s))ds .
With the aid of homogeneous functions properties (see [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF]), we arrive at the inequalities

c 1 |x(t)| µ r -c 3 n i=1 |x(t)| µ-ri r t t-τ |x(s)| ν+ri r ds +α t t-τ |x(s)| µ+ν r ds ≤ V (t, x t ) ≤ c 2 |x(t)| µ r +c 3 n i=1 |x(t)| µ-ri r t t-τ |x(s)| ν+ri r ds +(α + βτ ) t t-τ |x(s)| µ+ν r ds, ˙ V ≤ -c 4 |x(t)| µ+ν r + c 5 n i,j=1 |x(t)| µ-ri-rj r × |x(t)| ν+rj r + t t-τ |x(s)| ν+rj r ds t t-τ |x(s)| ν+ri r ds +(α + βτ )|x(t)| µ+ν r -α|x(t -τ )| µ+ν r -β t t-τ |x(s)| µ+ν r ds,
where c 1 , c 2 , c 3 , c 4 , c 5 are positive constants.

Choose the values of α and β such that α + βτ < c4 4 , then using Lemma (1), Young's and Hölder's inequalities, it is easy to verify that there exists a number δ > 0 such that

1 2 c 1 |x(t)| µ r + 1 2 α t t-τ |x(s)| µ+ν r ds ≤ V (t, x t ) ≤ 2c 2 |x(t)| µ r + 2(α + βτ ) t t-τ |x(s)| µ+ν r ds, ˙ V ≤ - 1 2 c 4 |x(t)| µ+ν r - 1 2 β t t-τ |x(s)| µ+ν r ds
for ∥x t ∥ τ < δ. Hence (see [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]), the zero solution of ( 1) is asymptotically stable. The proof is completed.

Next, along with (1), consider its perturbed version:

ẋ(t) = F (x(t)) + t t-τ D(s -t)G(x(s))ds (3)
+ t t-τ Q(t, s, x(s))ds,
where the vector function

Q(t, s, x) = [Q 1 (t, s, x), . . . , Q n (t, s, x)] ⊤ is continuous for t ≥ 0, s ∈ [t -τ, t] and x ∈ R n . Assumption 4. Let |Q i (t, s, x)| ≤ q i |x| ri+σ r for t ≥ 0, s ∈ [t -τ, t], x ∈ R n , i = 1, . . . , n, where q i > 0 and σ > 0.
This assumption is verified if for any fixed t ≥ 0 and s ∈ [t -τ, t] the function Q(t, s, x) is r-homogeneous of x with the degree σ, and it is bounded for t ≥ 0. We will look for conditions under which such generic time-and state-dependent perturbations do not destroy the asymptotic stability of the zero solution.

Theorem 2. If assumptions 1-4 are fulfilled and σ > ν, then the zero solution of (3) is asymptotically stable.

Proof. Differentiating the functional (2) along the solutions of (3), we obtain

˙ V = W (t, x t ) + ∂V (x(t)) ∂x t t-τ Q(t, s, x(s))ds + t t-τ s-t -τ D(θ)dθG(x(s))ds ⊤ × ∂ 2 V (x(t)) ∂x 2 t t-τ Q(t, s, x(s))ds ≤ W (t, x t ) + a 1 n i=1 |x(t)| µ-ri r t t-τ |x(s)| σ+ri r ds +a 2 n i,j=1 |x(t)| µ-ri-rj r t t-τ |x(s)| σ+ri r ds t t-τ |x(s)| ν+rj r ds,
where a 1 , a 2 are positive constants and W (t, x t ) is the derivative of (2) along the solutions of the system (1). The remaining part of the proof is similar to that of Theorem 1.

Remark 2. Under the conditions of Theorem 2, the system (1) can be interpreted as a nonlinear approximation at the origin for (3) [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF]. The theorem guarantees the preservation of the asymptotic stability if degrees of perturbations are higher than those of the right-hand sides of the original system. Remark 3. The result of Theorem 2 can be interpreted as local ISS property of (1) with respect to additive essentially bounded disturbances d(t) ∈ R n :

ẋ(t) = F (x(t)) + t t-τ D(s -t)G(x(s))ds + d(t),
then the term t t-τ Q(t, s, x(s))ds satisfying the conditions of Assumption 4 corresponds to an upper bound on the asymptotic gain of the system.

In the next section, we will consider systems with timevarying perturbations possessing zero mean values. On the basis of a development of the averaging method, we will show that in such a case the asymptotic stability can be guaranteed under less conservative constraints than those formulated in Theorem 2.

V. STABILITY ANALYSIS VIA AVERAGING

We will consider two types of perturbed systems with different dependence on external disturbances. It is worth noticing that, for the first type, to derive stability conditions, the LK approach is used, while we failed to prove a similar result with the aid of the LR approach. For the second type, the situation is opposite.

A. Application of the LK approach

Let the system (3) be of the form

ẋ(t) = F (x(t)) + t t-τ D(s -t)G(x(s))ds (4)
+ t t-τ B(s)L(s -t)Q(x(s))ds,
where the matrix function B : [-τ, +∞) → R n×w is continuous and bounded, the matrix funciton L : [-τ, 0] → R w×q is continuous, the vector function Q : R n → R q is continuously differentiable, and the remaining notation is the same as for (1).

Assumption 5. For any fixed t ≥ 0 and s

∈ [t -τ, t], B(s)L(s -t)Q(x) is r-homogeneous function of x with the degree σ > 0.
This assumption is a reformulation of Assumption 4 for the system (4).

Moreover, in this section, it is supposed that time-varying perturbations admit zero mean values. We will consider such a constraint in one of the following forms. In particular, Assumption 6 is fulfilled when entries of B(t) are periodic functions with zero mean values. Assumption 7 is valid, e.g., in the case where entries of B(t) are almost periodic functions with zero mean values. It is worth noticing that matrices with such entries may not satisfy Assumption 6 (see [START_REF] Demidovich | Lectures on Stability Theory[END_REF]).

Theorem 3. Let assumptions 1-3, 5 be fulfilled. Then the zero solution of (4) is asymptotically stable i) under Assumption 6 for σ > ν/2; ii) under Assumption 7 for σ ≥ ν.

Proof. First, choose a LKF candidate for (4) as follows:

V 1 (t, x t ) = ∂V (x(t)) ∂x t t-τ B(s) s-t -τ L(θ)dθQ(x(s))ds + V (t, x t ),
where V (t, x t ) is the functional constructed by the formula (2) and the function V (x) possesses the properties specified in Remark 1. Differentiating V 1 (t, x t ) along the solutions of (4), we obtain

˙ V 1 = ∂V (x(t)) ∂x F (x(t)) + 0 -τ D(θ)dθG(x(t)) + ∂V (x(t)) ∂x B(t) 0 -τ L(θ)dθ Q(x(t)) +(α + βτ )|x(t)| µ+ν r -α|x(t -τ )| µ+ν r -β t t-τ |x(s)| µ+ν r ds +(F (x(t)) + t t-τ D(s -t)G(x(s))ds + t t-τ B(s)L(s -t)Q(x(s))ds) ⊤ ∂ 2 V (x(t)) ∂x 2 ⊤ ×( t t-τ s-t -τ D(θ)dθG(x(s))ds + t t-τ B(s) s-t -τ L(θ)dθQ(x(s))ds).
Applying the properties of homogeneous functions (see [START_REF] Efimov | Finite-time stability tools for control and estimation[END_REF]), we arrive at the estimate

˙ V 1 ≤ (α + βτ -a 1 )|x(t)| µ+ν r + ∂V (x(t)) ∂x B(t) 0 -τ L(θ)dθ Q(x(t)) -β t t-τ |x(s)| µ+ν r ds + a 2 n i,j=1 |x(t)| µ-ri-rj r × |x(t)| ν+rj r + ϱ j (x t ) ϱ i (x t ),
where a 1 , a 2 are positive coefficients and

ϱ i (x t ) = t t-τ (|x(s)| ν+ri r + |x(s)| σ+ri r
) ds. Next, according to the approach developed in [START_REF] Aleksandrov | Averaging method for the stability analysis of strongly nonlinear mechanical systems[END_REF], construct a LKF in the form

V 2 (t, x t ) = V 1 (t, x t ) - ∂V (x(t)) ∂x × t 0 e ε(s-t) B(s)ds 0 -τ L(θ)dθ Q(x(t)),
where ε is a nonnegative tuning parameter, then

c 1 |x(t)| µ r -c 3 n i=1 |x(t)| µ-ri r ϱ i (x t ) -c 4 t 0 e ε(s-t) B(s)ds |x(t)| µ+σ r + α t t-τ |x(s)| µ+ν r ds ≤ V 2 (t, x t ) ≤ c 2 |x(t)| µ r + c 3 n i=1 |x(t)| µ-ri r ϱ i (x t ) +c 4 t 0 e ε(s-t) B(s)ds |x(t)| µ+σ r +(α + βτ ) t t-τ |x(s)| µ+ν r ds, ˙ V 2 ≤ (α + βτ -a 1 )|x(t)| µ+ν r +a 3 ε t 0 e ε(s-t) B(s)ds |x(t)| µ+σ r -β t t-τ |x(s)| µ+ν r ds +a 2 n i,j=1 |x(t)| µ-ri-rj r |x(t)| ν+rj r + ϱ j (x t ) ϱ i (x t ) +a 4 t 0 e ε(s-t) B(s)ds n j=1 |x(t)| µ+σ-rj r × |x(t)| ν+rj r + ϱ j (x t ) ,
where c 1 , c 2 , c 3 , c 4 , a 3 , a 4 are positive constants. With the aid of Lemma 1, Young's and Hölder's inequalities, it can be shown that if α + βτ < a 1 /2, σ > ν/2 and a number δ > 0 is sufficiently small, then the inequalities

1 2 c 1 |x(t)| µ r -c 4 t 0 e ε(s-t) B(s)ds |x(t)| µ+σ r + 1 2 α t t-τ |x(s)| µ+ν r ds ≤ V 2 (t, x t ) ≤ 2c 2 |x(t)| µ r + c 4 t 0 e ε(s-t) B(s)ds |x(t)| µ+σ r +2(α + βτ ) t t-τ |x(s)| µ+ν r ds, ˙ V 2 ≤ - 1 2 a 1 |x(t)| µ+ν r + a 3 ε t 0 e ε(s-t) B(s)ds |x(t)| µ+σ r - 1 2 β t t-τ |x(s)| µ+ν r ds + a 4 t 0 e ε(s-t) B(s)ds × n j=1 |x(t)| µ+σ-rj r |x(t)| ν+rj r + ϱ j (x t )
hold for t ≥ 0, ∥x t ∥ τ < δ. Next, consider the cases of assumptions 6 and 7 separately. 1) Let Assumption 6 be fulfilled. In this case one can take ε = 0. We obtain

1 2 c 1 |x(t)| µ r -c 4 N |x(t)| µ+σ r + 1 2 α t t-τ |x(s)| µ+ν r ds ≤ V 2 (t, x t ) ≤ 2c 2 |x(t)| µ r + c 4 N |x(t)| µ+σ r + 2(α + βτ ) t t-τ |x(s)| µ+ν r ds, ˙ V 2 ≤ - 1 2 a 1 |x(t)| µ+ν r - 1 2 β t t-τ |x(s)| µ+ν r ds +a 4 N n j=1 |x(t)| µ+σ-rj r × |x(t)| ν+rj r + t t-τ |x(s)| ν+rj r + |x(s)| σ+rj r ds for t ≥ 0, ∥x t ∥ τ < δ.
Hence, if σ > ν/2 and δ is sufficiently small, then the estimates

1 3 c 1 |x(t)| µ r + 1 3 α t t-τ |x(s)| µ+ν r ds ≤ V 2 (t, x t ) ≤ 3c 2 |x(t)| µ r + 3(α + βτ ) t t-τ |x(s)| µ+ν r ds, (5)
˙ V 2 ≤ - 1 3 a 1 |x(t)| µ+ν r - 1 3 β t t-τ |x(s)| µ+ν r ds (6)
are valid for t ≥ 0 and ∥x t ∥ τ < δ.

2) Let Assumption 7 be fulfilled. In this case ε should be positive. We obtain

1 2 c 1 |x(t)| µ r - c 5 ε |x(t)| µ+σ r + 1 2 α t t-τ |x(s)| µ+ν r ds ≤ V 2 (t, x t ) ≤ 2c 2 |x(t)| µ r + c 5 ε |x(t)| µ+σ r + 2(α + βτ ) t t-τ |x(s)| µ+ν r ds, ˙ V 2 ≤ - 1 2 a 1 |x(t)| µ+ν r + a 3 ε t 0 e ε(s-t) B(s)ds |x(t)| µ+σ r - 1 2 β t t-τ |x(s)| µ+ν r ds + a 5 ε n i,j=1 |x(t)| µ+σ-rj r × |x(t)| ν+rj r + t t-τ |x(s)| ν+rj r + |x(s)| σ+rj r ds
for t ≥ 0, ∥x t ∥ τ < δ, where a 5 and c 5 are positive constants. Let σ ≥ ν, in [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-Linear Oscillations[END_REF] it was proven that, under Assumption 7, ε t 0 e ε(s-t) B(s)ds → 0 as ε → 0 uniformly with respect to t ≥ 0. Choose ε > 0 satisfying the condition

εa 3 t 0 e ε(s-t) B(s)ds < a 1 12
for t ≥ 0. Then there exists δ > 0 such that ( 5) and ( 6) hold for t ≥ 0 and ∥x t ∥ τ < δ. This completes the proof.

B. Application of the LR approach

Consider the perturbed system (3) of the form

ẋ(t) = F (x(t)) + t t-τ D(s -t)G(x(s))ds (7) +B (t) t t-τ L(s -t)Q(x(s))ds,
where the notation is the same as for ( 1) and ( 4).

Assumption 8. For any fixed t ≥ 0 and s ∈ [t -τ, t], B(t)L(s -t)Q(x) is r-homogeneous function of x with the degree σ > 0.

Again, it is a reformulation of Assumption 4 for the system (7).

Theorem 4. Let assumptions 1-3, 8 be fulfilled. Then the zero solution of ( 7) is asymptotically stable i) under Assumption 6 for σ > ν/2; ii) under Assumption 7 for σ ≥ ν.

Proof. Choose a Lyapunov function candidate for [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF] in the form

V 1 (t, x) = V (x) - ∂V (x) ∂x t 0 e ε(s-t) B(s)ds × 0 -τ L(θ)dθ Q(x),
where ε is a nonnegative tuning parameter and V (x) is a Lyapunov function with the properties specified in Remark 1. Then

c 1 |x| µ r -c 3 t 0 e ε(s-t) B(s)ds |x| µ+σ r ≤ V 1 (t, x) ≤ c 2 |x| µ r + c 3 t 0 e ε(s-t) B(s)ds |x| µ+σ r , V1 = ∂V (x(t)) ∂x F (x(t)) + 0 -τ D(θ)dθ G(x(t)) +ε ∂V (x(t)) ∂x t 0 e ε(s-t) B(s)ds 0 -τ L(θ)dθ Q(x(t)) + ∂V (x(t)) ∂x t t-τ D(s -t)(G(x(s)) -G(x(t)))ds + ∂V (x(t)) ∂x B(t) t t-τ L(s -t)(Q(x(s)) -Q(x(t)))ds - ∂V (x(t)) ∂x t 0 e ε(s-t) B (s) ds 0 -τ L(θ)dθ ∂Q(x(t)) ∂x ×[F (x(t)) + t t-τ D(s -t)G(x(s))ds +B (t) t t-τ L(s -t)Q(x(s))ds] -[F (x(t)) + t t-τ D(s -t)G(x(s))ds +B (t) t t-τ L(s -t)Q(x(s))ds] ⊤ ∂ 2 V (x(t)) ∂x 2 ⊤ t 0 e ε(s-t) B (s) ds 0 -τ L(θ)dθ Q(x(t)) ≤ -c 4 |x(t)| µ+ν r + c 5 ε t 0 e ε(s-t) B (s) ds |x(t)| µ+σ r +c 6 n i=1 |x(t)| µ-ri r t t-τ |R i (s -t, x(s)) -R i (s -t, x(t))|ds +c 7 n i=1 |x(t)| µ-ri r t t-τ |H i (t, s, x(s)) -H i (t, s, x(t))|ds +c 8 t 0 e ε(s-t) B (s) ds n i=1 |x(t)| µ+σ-ri r [|x(t)| ν+ri r + t t-τ |x(s)| ν+ri r ds + t t-τ |x(s)| σ+ri r ds],
where c k > 0, k = 1, . . . , 8, H i (t, s, x) are components of the vector function B(t)L(s -t)Q(x) and R i (s -t, x) are components of the vector function D(s -t)G(x).

Consider again separately the cases of assumptions 6 and 7.

1) Let Assumption 6 be fulfilled. In this case one can take ε = 0. We obtain

c 1 |x| µ r -c 3 N |x| µ+σ r ≤ V 1 (t, x) ≤ c 2 |x| µ r + c 3 N |x| µ+σ r , V1 ≤ -c 4 |x(t)| µ+ν r +c 6 n i=1 |x(t)| µ-ri r t t-τ |R i (s -t, x(s)) -R i (s -t, x(t))|ds +c 7 n i=1 |x(t)| µ-ri r t t-τ |H i (t, s, x(s)) -H i (t, s, x(t))|ds +c 8 N n i=1 |x(t)| µ+σ-ri r [|x(t)| ν+ri r + t t-τ |x(s)| ν+ri r ds + t t-τ |x(s)| σ+ri r ds].
Choose a number δ > 0 such that

1 2 c 1 |x| µ r ≤ V 1 (t, x) ≤ 2c 2 |x| µ r for t ≥ 0, |x| r < δ. Assume that 0 < |x(ξ)| r < δ for ξ ∈ [t -2τ, t] and the function V 1 (t, x) satisfies the Razumikhin condition V 1 (ξ, x(ξ)) ≤ 2V 1 (t, x(t)) for ξ ∈ [t -2τ
, t] (we need to enlarge the delay value due to technical reasons), then

|x(ξ)| r ≤ ω|x(t)| r , ∀ξ ∈ [t -2τ, t] (8)
for some ω > 1 and

t t-τ |R i (s -t, x(s)) -R i (s -t, x(t))|ds = |x(t)| ν+ri r t t-τ |R i (s-t, z(t)+∆z(t, ξ))-R i (s-t, z(t))|ds, t t-τ |H i (t, s, x(s)) -H i (t, s, x(t))|ds = |x(t)| σ+ri r t t-τ |H i (t, s, z(t)+∆z(t, ξ))-H i (t, s, z(t))|ds,
where

z(t) = Λ -1 r (|x(t)| r )x(t), ∆z(t, ξ) = Λ -1 r (|x(t)| r )(x(ξ) -x(t)), i = 1, . . . , n.
Using Mean Value Theorem, it is easy to verify that there exists a number m > 0 such that

∥∆z(t, ξ)∥ ≤ m(|x(t)| ν r + |x(t)| σ r ). Hence, t t-τ |H i (t, s, z(t) + ∆z(t, ξ)) -H i (t, s, z(t))|ds → 0, t t-τ |R i (s -t, z(t) + ∆z(t, ξ)) -R i (s -t, z(t))|ds → 0 as |x(t)| r → 0.
Taking into account this result and the estimate [START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF], applying Young's inequality and Lemma 1, it can be proved that if σ > ν/2 and the value of δ is sufficiently small, then

V1 ≤ - 1 2 c 4 |x(t)| µ+ν r .
2) Let Assumption 7 be fulfilled. In this case ε should be positive. We obtain

c 1 |x| µ r - c 9 ε |x| µ+σ r ≤ V 1 (t, x) ≤ c 2 |x| µ r + c 9 ε |x| µ+σ r , V1 ≤ -c 4 |x(t)| µ+ν r + c 5 ε t 0 e ε(s-t) B (s) ds |x(t)| µ+σ r +c 6 n i=1 |x(t)| µ-ri r t t-τ |R i (s -t, x(s)) -R i (s -t, x(t))|ds +c 7 n i=1 |x(t)| µ-ri r t t-τ |H i (t, s, x(s)) -H i (t, s, x(t))|ds + c 10 ε n i=1 |x(t)| µ+σ-ri r [|x(t)| ν+ri r + t t-τ |x(s)| ν+ri r ds + t t-τ |x(s)| σ+ri r ds],
where c 9 and c 10 are positive constants. Let σ ≥ ν and choose ε > 0 such that The subsequent proof is similar to that one for the previous case.

c 5 ε t 0 e ε(s-t) B (s) ds < c 4 2 for t ≥ 0, then V1 ≤ -c 4 1 - 1 2 |x(t)| σ-ν
Remark 4. The results of theorems 3 and 4 improve the condition of Theorem 2 that the degree of the perturbations σ should be strictly bigger than the degree of the nominal system ν. Under assumptions 6 and 7, which allow the averaging tools to be applied in the proof, σ can be chosen at least equal to ν or even higher than ν/2.

VI. APPLICATIONS

In this section the results of theorems 3 and 4 will be adapted to two practical cases.

A. Vector Lienard equation

Assume that motions of a mechanical system are modeled by the equations

ẍ(t) + ∂W (x(t)) ∂x ẋ(t) + ∂Π(x(t)) ∂x ⊤ (9)
+ t t-τ d(s -t) ∂ Π(x(s)) ∂x ⊤ ds + t t-τ B (s) L(s -t)Q(x(s))ds = 0,
where x(t) ∈ R n ; components of the vector functions W (x) and Q(x) are continuously differentiable for x ∈ R n standard homogeneous (with r = (1, . . . , 1)) of the degree ν + 1 and λ, respectively, ν > 0, λ > 1; scalar functions Π(x) and Π(x) are continuously differentiable for x ∈ R n and standard homogeneous of the degree ρ + 1, ρ > 1; the matrix B(t) is continuous for t ∈ [-τ, +∞), the scalar function d(θ) and the matrix L(θ) are continuous for θ ∈ [-τ, 0].

In the case where τ = 0 the system (9) is a classical vector type Lienard equation describing the dynamics of various mechanical and electromechanical systems (see [START_REF] Andronov | Theory of oscillators[END_REF]). The term t t-τ d(s -t)∂ Π(x(s))/∂x ⊤ ds can be interpreted as integral part of a PID regulator [START_REF] Formal | On a modification of the PID controller[END_REF], [START_REF] Anan | On stabilization of some control systems with an aftereffect[END_REF], [START_REF] Radaideh | A modified PID controller (PIIσβD)[END_REF], [START_REF] Zhao | Towards a theoretical foundation of PID control for uncertain nonlinear systems[END_REF], whereas the term t t-τ B (s) L(s-t)Q(x(s))ds characterizes external time-varying perturbations acting on the system (e.g., through the control channel).

Using the substitution y(t) = ẋ(t) + W (x(t)), transform (9) into a first-order system:

ẋ(t) = y(t) -W (x(t
∂Π(x) ∂x + 0 -τ d(θ)dθ ∂ Π(x) ∂x W (x)
be positive definite, then it is known (see [START_REF] Rouche | Ordinary differential equations: stability and periodic solutions[END_REF]) that, under this condition, the zero solution of the auxiliary delay-free system

ẋ(t) = y(t) -W (x(t)), ẏ(t) = - ∂Π(x(t)) ∂x ⊤ - 0 -τ d(θ)dθ ∂ Π(x(t)) ∂x ⊤ is asymptotically stable.
Applying Theorem 3, we obtain that if λ > 1 + 3 2 ν and Assumption 6 is fulfilled, then the equilibrium position x = ẋ = 0 of ( 9) is asymptotically stable.

Example 1. Let W (x) = 8x ν 1 |x 1 | + 0.5x ν+1 2 x ν+1 1 + 8x ν 2 |x 2 | , B(t) = cos( π τ t), d(t) = sin( 2π τ t), 3Π(x) = Π(x) = x ρ+1 1 + x ρ+1 2 , Q(x) = -2x λ 1 -3x λ-1 1 x 2 + 3x λ 2 3x λ 1 + 3x 1 x λ-1 2 -2x λ 2 , L(t) = sin 2 (t), ν = 1, λ = ρ = 2ν + 1, τ = 0.1,
which verify the conditions established above. The results of simulation of the closed-loop system for different initial conditions with the same norm are shown in Fig. 1 (the explicit Euler method was used with step ∆t = 0.01). It should be noted that the approaches developed in this paper can also be applied to some classes of nonlinear systems that are not homogeneous. As an example of such a case, consider an indirect control system [START_REF] Lefschetz | Stability of nonlinear control systems[END_REF] of the form uniformly with respect to t ≥ 0 (Assumption 7 is verified). Hence, the corresponding averaged system takes the form

ẋ(t) = Ax(t) + b(t)u(t), ż(t) = c ⊤ x(t) + h(t)u(t), where x(t) ∈ R n , z(t) ∈ R, A ∈ R n×n is a constant matrix, c ∈ R n is a constant vector, vector b(t) ∈ R
ẋ(t) = Ax(t) + bu(t), ż(t) = c ⊤ x(t) + hu(t).
For this nominal model, choose a control law as follows:

u(t) = pz ν (t) + t t-τ d(s -t)z ν (s)ds,
where scalar function d(θ) is continuous for θ ∈ [-τ, 0], p ∈ R determines the delay-independent part, ν > 1 is a rational with odd numerator and denominator, τ is a constant positive delay. Appearance of a distributed delay is related with control communication media, then the nominal feedback is proportional to z ν (t), as it can be observed in a delay-free counterpart of the controlled system:

ẋ(t) = Ax(t) + p + 0 -τ d(θ)dθ z ν (t) b, (10)
ż(t) = c ⊤ x(t) + h p + 0 -τ d(θ)dθ z ν (t).
Note also that the distributed delay can be introduced artificially to improve the transients [START_REF] Aleksandrov | On the attitude stabilization of a rigid body under control with distributed delay[END_REF]. According to the standard assumptions (see [START_REF] Lefschetz | Stability of nonlinear control systems[END_REF]), consider the case where A is a Hurwitz matrix and p + then for ν = 1 (the case is not studied in the paper) and τ = 0 the system (10) takes the form:

ẋ(t) ż(t) = -1 -2 1 1
x(t) z(t) ,

and it has purely imaginary eigenvalues. However, since all conditions imposed in this work are satisfied, for ν > 1 the system will be locally asymptotically stable. As we can conclude, in the delay-free case the system is slowly converging with the control u(t) = pz ν (t), for a small value of delay we observe an accelerated convergence, while increasing the value of τ leads to more complex transients with a better convergence.

VII. CONCLUSION

The stability problem for nonlinear homogeneous systems with distributed delay featuring a variable kernel has been investigated. Both the LK and LR approaches have been utilized. It has been demonstrated that the global asymptotic stability of the zero solution for an auxiliary delay-free homogeneous system implies the local asymptotic stability of the zero solution for the original system with distributed delay. Additionally, the impact of nonlinear time-varying perturbations on the system dynamics has been analyzed using averaging techniques. The findings have been exemplified through a mechanical system described by a Lienard equation, as well as an indirect control system design for a linear system. Future directions of research may include extensions of these results to the case of the negative homogeneity degree.

Assumption 6 .

 6 Let t 0 B(s)ds ≤ N for t ≥ 0, N = const > 0. )ds → 0 as T → +∞uniformly with respect to t ≥ 0.

B

  (s) L(s -t)Q(x(s))ds.Let ρ = 2ν + 1, λ > ν + 1, then the corresponding nominal dynamicsẋ(t) = y(t) -W (x(t)), ẏ(t) = -∂Π(x(t)) ∂x ⊤ -t t-τ d(s -t) ∂ Π(x(s)) ∂x⊤ ds is r-homogeneous of the degree ν with respect to the dilation r = (r 1 , . . . , r 2n ), where r i = 1 for i = 1, . . . , n and r i = ν + 1 for i = n + 1, . . . , 2n, and the function B(t)L(st)Q(x) satisfies Assumption 5 with σ = λ -ν -1. Furthermore, let the functions Π(x)+ 0 -τ d(θ)dθ Π(x) and

Figure 1 :

 1 Figure 1: Behavior of |x(t)| + | ẋ(t)| in logarithmic scale, t ∈ [0, 1500]

  n and scalar h(t) functions are continuous and bounded for t ∈ (-∞, +∞), u(t) is a scalar control.Assume that b(t) = b + b(t), h(t) = h + h(t), where h is a constant, b ∈ R n is a constant vector, whereas ds → 0 as T → +∞

0

  -τ d(θ)dθ ( h -c ⊤ A -1b ) < 0, then the system (10) admits a strict Lyapunov function of the formV (x, z) = z ν+1 ν + 1 + x ⊤ P x -z ν c ⊤ A -1 xwith a constant positive definite matrix P . Using this function, in a similar way as in the proof of Theorem 4 it can be shown that the zero solution of the perturbed closed-loop systemẋ(t) = Ax(t) + b(t) pz ν (t) + t t-τ d(s -t)z ν (s)ds , ż(t) = c ⊤ x(t) + h(t) pz ν (t) + t t-τ d(s -t)z ν (s)dsis asymptotically stable. Example 2. Let n = 1, A = -1, b = -2, c = h = 1, p = 1,

  1, b(t) = sin((1 + cos 2 (t))t) cos(2t) , h(t) = sin(0.5t); p = 0.01, d(s) = exp(2s), s ∈ [-τ, 0]; ν = 3, then all required conditions are satisfied. The results of simulation of the controlled system for different initial conditions are shown in Fig. 2 for values of delay τ ∈ {0, 0.1, 0.25} (the explicit Euler method was used with step ∆t = 0.01).

  (a) τ = 0 (b) τ = 0.1 (c) τ = 0.25

Figure 2 :

 2 Figure 2: Behavior of |x(t)| + |z(t)| for different initial conditions and delays τ , t ∈ [0, 15]


Jir ˇı ´koliba ´c ˇ1 •

Benjamin Adroit

Elke Gro ¨ning

Carsten Brauckmann

Torsten Wappler

Á Cleroidea

Á Trogossitidae

Á Peltis

Á Plioza

Á Willershausen

Schlu ¨sselwo ¨rter Coleoptera

Keywords: Coleoptera, Cleroidea, Trogossitidae, Peltis, Pliocene, Willershausen , Trogossitidae, Peltinae), P. antehercynica sp. nov., wird aus den jungplioza ¨nen Ablagerungen der ehemaligen Tongrube von Willershausen Weitere Merkmale, einige davon einzigartig innerhalb der Peltinae, zeigen, dass es sich um eine selbsta ¨ndige neue Art handelt aber auch die Bedeutung der ehemaligen Tongrube von Willershausen diskutiert

A new species of the genus Peltis (Coleoptera, Trogossitidae, Peltinae), P. antehercynica sp. nov., is described from the Pliocene lake sediments of the former clay pit of Willershausen (Lower Saxony, Germany). The clearly visible antennae and elytra sculpturing of fossil provide clear morphological information about generic position, and other features, some of them unique in the Peltinae, show that it is an independent new way. In addition, the importance, geology and stratigraphy of the Willershausen Fossil Lagersta ¨tte is briefly discussed.

First record of the family Trogossitidae (Insecta, Coleoptera) in the Late Pliocene deposits of Willershausen (Germany)

Introduction

Since the first geological description by [START_REF] Wegele | Stratigraphie und Tektonik der tertia ¨ren Ablagerungen von Oldenrode-Du ¨derode-Willershausen[END_REF], the now abandoned clay pit of Willershausen has become internationally known as an important Fossil Lagersta ¨tte situated in the western foreland of the Harz Mountains (Lower Saxony, Northern Germany). The exceptional importance of the Willershausen site is also mentioned in modern textbooks on palaeoentomology, as for example in Rasnitsyn and Zherikhin (2002: 443) and [START_REF] Grimaldi | Evolution of the insects[END_REF]. As already listed by Kru ¨ger (1979a, b), the fossils include diverse algae, fungi, lichens, numerous terrestrial plants (mainly leaves), and among the animals include Bivalvia, Gastropoda, Annelida, Arachnida, Insecta, Crustacea, Teleostei, Amphibia, Reptilia, Aves, and Mammalia. A modern analysis of the diversity of vertebrates has been elaborated by [START_REF] Gehler | Diversita ¨tsanalyse der Wirbeltierreste der oberplioza ¨nen Fossillagersta ¨tte Willershausen, Niedersachsen, in der Sammlung des Geowissenschaftlichen Zentrums der Universita ¨t Go ¨ttingen[END_REF]. The total number of fossil specimens is estimated at about 50,000, with more than 500 described species distinguished thus far. Most abundant are leaves and insects. Plant/animal interactions are documented by a large number of galls, mines, and foliage damage by feeding activities [START_REF] Straus | Minen und andere Fraßspuren im Plioka ¨n von Willershausen am Harz[END_REF][START_REF] Titchener | Leaf feeding traces from the Upper Pliocene fossil lagersta ¨tte of Willershausen, Lower Saxony, Germany[END_REF]Adroit and Wappler, pers. observ.).

In the present article, a hitherto unknown beetle family, the Trogossitidae, of the Late Pliocene (Piacenzian) age 1998; [START_REF] Knobloch | Der plioza ¨ne Laubwald von Willershausen am Harz (Mitteleuropa)[END_REF][START_REF] Straus | Die oberplioza ¨ne Flora von Willershausen am Harz. Prepared for the editing by Wilde V, Lengtat KH and Ritzkowski S[END_REF]. Other taxa such as Parrotia and Liquidambar were also characteristic elements in the Pliocene sediments from Willershausen (Mai 1995). Based on palaeoclimatological estimates, the Willershausen flora grew where mean annual temperature was 12.5-16.5 °C, with warmmonth mean temperatures of 19.0 ± 1.8 °C and cold-month mean temperatures of 3.9 ± 2.5 °C (e.g., [START_REF] Thiel | Palaeoclimate estimates for selected leaf floras from the Late Pliocene (Reuverian) of Central Europe based on different palaeobotanical techniques[END_REF]; with an average rainfall of [900 mm, the actual climate resembles more a humid warm temperate Cfa/b climate [START_REF] Kovar | Eine neue oberplioza ¨ne Trapa-Art aus den Willershausener Seeablagerungen (Niedersachsen, BRD)[END_REF]. These climatic parameters are consistent with those of other coeval floras [START_REF] Teodoridis | Pliocene palaeoenvironment and correlation of the Sessenheim-Auenheim floristic complex (Alsace, France)[END_REF]. With decreasing temperatures throughout the course of the Pliocene, the dominant European vegetation changed gradually from highly diverse subtropical and warm-temperate forests to temperate deciduous forests with East Asian and North American affinities [START_REF] Salvius | Tertia ¨re Vegetationsgeschichte Europas. G. Fischer[END_REF].

The clay mining was abandoned in 1975. Since 1977, the locality is a nature sanctuary, and in 2012 it was included in the Geopark Harz, Braunschweiger Land, Ostfalen (Fig. 1). In addition to its importance as a Fossil Lagersta ¨tte, it is now a special biotope with a number of rare plants and animals.

A Late Pliocene age of the Willershausen sediments has already been proposed by [START_REF] Wegele | Stratigraphie und Tektonik der tertia ¨ren Ablagerungen von Oldenrode-Du ¨derode-Willershausen[END_REF], which was later confirmed by Mammalia-like Anancus arvernensis and remains of Cervidae, as well as by microfloral components [START_REF] Mohr | Die Mikroflora der oberplioza ¨nen Tone von Willershausen (Kreis Northeim, Niedersachsen)[END_REF], and was accepted in several publications in more recent years (see for example [START_REF] Kohring | Sciariden (Insecta: Diptera: Nematocera) aus dem Oberplioza ¨n von Willershausen[END_REF]Meischner 2000;[START_REF] Brauckmann | Anmerkungen zu den bisher beschriebenen Lepidopteren aus dem Jung-Tertia ¨r (Plioza ¨n) von Willershausen am Harz[END_REF][START_REF] Brauckmann | Neue Funde von Singzikaden (Hemiptera: Cicadidae) aus dem Jung-Tertia ¨r (Plioza ¨n) von Willershausen am Harz[END_REF][START_REF] Popov | A new notion on the heteropterofauna (Insecta: Hemiptera, Heteroptera) from the Pliocene of Willershausen (N Germany)[END_REF][START_REF] Dlussky | Two ants (Insecta: Hymenoptera: Formicidae: Formicinae) from the Late Pliocene of Willershausen, Germany, with a nomenclatural note on the genus Camponotites[END_REF]). After the current definition of the Pliocene/Pleistocene boundary (Hilgen et al. 2012), the absolute age of the Late Pliocene (Piacenzian) is considered to range approximately from 3.6 to 2.588 million years ago.

Materials and methods

The holotype of Peltis antehercynica sp. nov., collection no. RE 551.782.200 A 0083, was recovered from the Willershausen Fossil Lagersta ¨tte. The specimen is part of a collection of Willershausen fossils donated to the Ruhr Museum Essen (RE) (former collection of Ernst Edinger, Leverkusen) in 2001, and was lent, together with other fossil insects from this locality, to one of us (CB) for documentation. All measurements were made using an ocular micrometer and are given here in millimeters. Body length was measured along the midline from the anterior margin of the frons to elytral apex, and width measured across the broadest part of the elytra. The specimen was examined using a Leica MZ95 stereomicroscope. Photographs were taken using a Canon EOS 350D camera with a Canon EF-S 60-mm f/2.8 macro lens or a Nikon Coolpix from the Fossil Lagersta ¨tte Willershausen is described. A number of other beetles from Willershausen have already been described by Korge (1967: Staphylinidae), Schmidt (1967: Cerambycidae), Gersdorf (1969[START_REF] Gersdorf | Weitere Ka ¨fer (Coleoptera) aus dem Jungtertia ¨r Norddeutschlands[END_REF][START_REF] Gersdorf | Dritter Beitrag u ¨ber Ka ¨fer (Coleoptera) aus dem Jungtertia ¨r von Willershausen, Bl. Northeim 4226[END_REF]: all remaining groups), and Schweigert (2003: Cerambycidae, Scarabaeidae). The rhinoceros beetle mentioned by [START_REF] Gersdorf | Weitere Ka ¨fer (Coleoptera) aus dem Jungtertia ¨r Norddeutschlands[END_REF] has been revised by [START_REF] Brauckmann | The Late Pliocene Clausthal Rhinoceros Beetle from Willershausen (Scarabaeidae: Dynastinae: Oryctes? sp.; Lower Saxony, Germany)[END_REF].

The evolutionary significance of the Pliocene for insects largely concerns the origins of modern species and sweeping changes in their distribution in the more recent past (e.g., [START_REF] Elias | Differential insect and mammalian response to Late Quaternary climate change in the Rocky Mountain region of North America[END_REF][START_REF] Rizzo | Early Pliocene range expansion of a clade of subterranean Pyrenean beetles[END_REF][START_REF] Su | Resilience of plant-insect interactions in an oak lineage through Quaternary climate change[END_REF].

Importance, geology and stratigraphy of the Willershausen Fossil Lagersta ¨tte

As compiled by Meischner (2000), in the meantime, many details have become known concerning the Fossil Lagersta ¨tte of Willershausen, e.g., its development, taphonomy, and hydrography. The Late Pliocene sediments were deposited in a pond which had its origin in a collapse sink in a strongly disturbed sequence of Triassic and Early Jurassic sediments above subaerial eroded Late Permian (Zechstein) evaporites. The salt ascended across a fault zone along the valley of the Leine river and intruded laterally into younger Triassic saliferous strata (Late Buntsandstein: Ro ¨t, and Middle Muschelkalk). A detailed compilation of the geology of the Willershausen area, as well as of the mountainous region in southern Lower Saxony, has been published by [START_REF] Vinken | Kurzer U ¨berblick u ¨ber die Geologie der Umgebung von Willershausen[END_REF], Meischner (2000), and [START_REF] Ferguson | A fresh look at the rich assemblage from the Pliocene sink-hole of Willershausen, Germany[END_REF]. Sedimentological data indicate that the palaeobasin was ca. 200 m in diameter and some 10 m deep, with a narrow sandy shelf beyond which the sides sloped steeply towards the flat bottom of the lake [START_REF] Meischner | Die plioza ¨ne Fossilfundstelle Naturdenkmal Tongrube Willershausen[END_REF]Meischner 2000). Nevertheless, the oligotrophic to eutrophic lake (Briggs et al. 1998a, b) was sheltered by a broad-leaved forest, rendering it ideal for the establishment of meromictic conditions. In the monimolimnion, anoxia coupled with alkaline conditions and the accumulation of dissolved minerals, as well as the occurrence of toxic gases (?), prevented bioturbation and led to the preservation of laminations (Briggs et al. 1998b), with an exceptionally well-preserved flora and fauna (e.g., Meischner 2000). Thus, the Willershausen Fossil Lagersta ¨tte has recently even been taken into consideration in the analyses of several Pleistocene and Neogene chitinyielding localities by [START_REF] Flannery | Chitin in the fossil record: identification and quantification of D-glucosamine[END_REF]and foradditional geochemical studies [START_REF] Keely | Porphyrin and chlorin distributions in a Late Pliocene lacustrine sediment[END_REF].

The flora fromWillershausen is dominated by taxa typical of hilly mesophytic woodland; the rich assemblage of deciduous trees may have been concentrated close to the water. Typical elements are represented by Fagus, Liriodendron, Sassafras, Cercidiphyllum, Tilia, Aesculus, Betulaceae, Rosaceae, various lobed oaks, and species of Acer (e.g., Ferguson and Knobloch E4500. All photographs were optimized using Adobe Photoshop CS6 and Adobe Lightroom 5. Diagnosis Adults of Peltis are characterized by flat, oval body; eyes not exceeding contour of head; antennae with large scape and distinct 3-segmented weakly asymmetric club; procoxae conspicuously transverse; procoxal cavities externally widely open; transverse pronotum with anterior corners projecting forwards; elytra with several longitudinal carinae and rows of punctures among them; mesocoxae oval; metacoxae reaching elytral epipleuron; all coxae narrowly separated; abdomen with five ventrites.

Systematic palaeontology

Remarks The newly recorded fossil fits perfectly with this characterization. Considering its body size larger than 10 mm and three distinct elytral carinae, it is probably related to the species classified within the former subgen. Zimioma (P. grossa Linnaeus, P. gigantea Reitter) (Koliba ´c ˇ2013). Representatives of Peltinae and Lophocaterinae living under bark are often easily recognizable by their elytral carinae and punctation; therefore, differences in these characters in three relevant species (Table 1) can be Punctation between pairs of primary carinae finer and irregular

Mesothorax

Mesocoxal cavities narrowly separated; mesoventral process shorter than coxal diameter, reaching behind half of mesocoxae, acute; projection of metaventrite reaches approximately towards basal quarter of mesocoxae; tips of intercoxal processes touch each other; mesotrochantin not observed; scutum relatively wide, scutellum rounded at apex. Elytra (Artificially compressed in fossil) Regularly punctate, with 3 conspicuous longitudinal (primary) carinae and 3 finer (secondary) carinae between each pair of them; lateral margin of elytra widely explanate from above. Elytral epipleure very wide in anterior third, then narrowed to half of width but distinct up to elytral apex.

Hind wing Well-developed, wing membrane visible; veins not observed.

Metathorax Metaventrite flat and wide, only slightly narrowed towards anterior portion; discrimen not visible (it coincides with elytral suture in semitransparent fossil); no other details visible.

Legs Procoxae not projecting, widely transverse; mesocoxae oval; metacoxae extended to lateral margin of metathorax; trochanters relatively small, triangular; femora rather clavate; tibiae and tarsi not observed.

Abdomen Five ventrites distinctly visible; remains of copulatory organs visible at apex (sex unrecognizable).

Remarks The newly described fossil is similar to the two recent species but easily recognizable by the difference in the elytra sculpture. The antennal club of the right antenna seems to be distinctly asymmetric but the state is likely specious. Although the left antenna is poorly visible, it can be estimated that the asymmetric look of the right club is given by its position during fossilization.

Conclusions

The Holarctic genus Peltis O.F. Mu ¨ller consists of seven extant species, and to date only three fossils have been assigned to the genus described from Miocene and Oligocene deposits of Europe and the USA (see also Table 3). A list of species was given by Koliba ´c ˇ( 2013

Peltis antehercynica sp. nov. Figures 2, 3

Etymology The specific epithet is based on that region of Lower Saxony, Germany (western Harzvorland) from which the fossil originates.

Holotype Fairly complete specimen, RE 551.782.200 A 0083.

Type locality Late Pliocene (Piacenzian) deposits of Willershausen am Harz (Lower Saxony, northern Germany).

Diagnosis The new species can be distinguished from modern Peltis by differences in the elytra sculpturing and the asymmetric form of the antennal clubs.

Description Body length (from elytral apex to clypeus): 16.5 mm. For other measurements see Table 2.

Coloration and sculpture Body broadly oval, flat. Probably unicolorously brown, without patches or stripes. Dorsal and ventral surface as well as antennae without distinct pubescence (no setae or hairs observed). Pronotum densely punctate (interspaces approximately the same as diameter of punctures), punctures probably finer than those on head (interspaces among punctures less than their diameter). Elytra with regular rows of large punctures (interspaces among punctures distinctly less than their diameter) between carinae: always 2 rows separated with primary or secondary carinae plus approximately 4 rows of punctures between third primary carina and lateral margin.

Head Situated between extended anterior corners of pronotum. Eyes medium-sized, not distinctly elevated but rather evenly rounded and not exceeding contour of cranium. Mandibles bidentate. Antennae with 11 antennomeres and distinct 3-segmented club; scape robust, distinctly larger than pedicel; antennomeres 9-10 weakly asymmetric (3 terminal antennomeres apparently sharply triangular on the right side but rather rounded on the left).

Prothorax Pronotum conspicuously transverse; anterior margin deeply emarginate (arcuate); anterior corners projecting, but not acute (their tips rather rounded); lateral margins convergent forwards, nearly straight, probably explanate from dorsal view; lateral edge distinct, even. Prosternal process not dilated at apex, relatively wide; sides convergent behind base, parallel-sided in front of apex; apex probably sharpened. Procoxal cavities narrowly separated, externally widely open, closed only to one quarter or one-third by postcoxal projections; trochantin poorly visible, probably exposed but slender and small. name Ostoma Laicharting (type species: Silpha ferruginea Linnaeus, 1758) was used instead of Peltis (type species: Silpha grossa Linnaeus, 1758), and three subgenera were recognized within the genus: Ostoma s.str., Zimioma Gozis, and Grynocharis Thomson (Le ´veille ´1910). The latter has been classified within the Lophocaterinae (or Lophocateridae) since [START_REF] Crowson | Further observations on Cleroidea (Coleoptera)[END_REF] review. Ostoma was synonymized with Peltis by Koliba ´c ˇet al. (2005; for complete synonymy of Peltis see Koliba ´c ˇ2007, 2013). The subgenus Zimioma (sometimes used in a rank of a genus) was established for two Palaearctic species (P. grossa Linnaeus is distributed in Europe, P. gigantea Reitter from East Siberia to Japan and NE China). They are distinctly larger in size (always [13 mm); their ratio of pronotum length/ width is about 3. Species of Peltis s.str. are lesser than 12 mm, with a pronotum length/width ratio of about 2. Because of lack of other known morphological and biological differences among the species of Peltis s.str. and Zimioma, the latter name was put into synonymy by [START_REF] Koliba ´c | Trogossitidae[END_REF]. Considering the body size of about 16 mm and a distinctly transverse pronotum (length/width ratio [3), P. antehercynica sp. nov. is unambiguously related to the two species of the former subgenus Zimioma.

Adults of Palaearctic species are highly adapted to a hidden way of life under bark of dying or dead deciduous and coniferous trees, while larvae live in rotten or decaying wood. Both stages are slow moving; adults are distinctly flattened, with robust mandibles; larvae have a reduced number of stemmata and minute urogomphi. Species are fungivorous in both stages and inhabit old woods with conditions similar to virgin forests. Therefore, species are generally scarce in Europe, although they can be relatively common in old autochthonous forests [START_REF] Koliba ´c | Cleroidea. Brouci nadc ˇeledi Cleroidea C ˇeska, Slovenska a sousednı ´ch oblastı ´[END_REF]. All extant Palaearctic species of Peltis today live in colder climatic conditions than P. antehercynica sp. nov. They are distributed in Central Europe as well as Fennoscandia and Siberia with cold, long winters.

Today, two species of Peltis (P. ferruginea Linnaeus and P. grossa Linnaeus) are among the most abundant representatives of Trogossitidae in Central Europe; however, there is still much to do in a systematics and phylogeny of the monotypic tribe Peltini, even the subfamily Peltinae [START_REF] Boca ´k | Building the Coleoptera tree-of-life for [8000 species: composition of public DNA data and fit with Linnaean classification[END_REF][START_REF] Hunt | A comprehensive phylogeny of beetles reveals the evolutionary origins of a superradiation[END_REF][START_REF] Koliba ´c | Morphology, taxonomy and phylogeny of Phloiophilus edwardsi Stephens, 1830 (Coleoptera, Cleroidea)[END_REF][START_REF] Latreille | Genera crustaceorum et insectorum secundum ordinem naturalem in fa-milias disposita, iconibus exemplisque plurimus explicata. Tomus primus[END_REF]. Based on data obtained directly from The Paleobiology Database Classic and Fossilworks: Gateway to the Paleobiology Database a The fossil Silpha tricostata Heer (1847 from the Miocene of Oehningen is classified within Peltis in some palaeobiological databases (e.g., Paleobiology Database Classic, Fossilworks: Gateway to the Paleobiology Database). However, [START_REF] Heer | Die Insektenfauna der Tertia ¨rgebilde von Oeningen und von Radoboj in Croatien. Erster Theil: Ka ¨fer[END_REF][START_REF] Heer | Die Insektenfauna der Tertia ¨rgebilde von Oeningen und von Radoboj in Croatien. Erster Theil: Ka ¨fer[END_REF] considered the specimen as belonging to Silphidae [START_REF] Heer | Die Insektenfauna der Tertia ¨rgebilde von Oeningen und von Radoboj in Croatien. Erster Theil: Ka ¨fer[END_REF][START_REF] Heer | Die Insektenfauna der Tertia ¨rgebilde von Oeningen und von Radoboj in Croatien. Erster Theil: Ka ¨fer[END_REF]. The specimen is not well preserved, but its body shape, large size, and pronotum without extended anterior corners certainly do not allow its classification within Peltis. A placement within the carrion beetles (Silphidae) is more probable

Fig. 1

 1 Fig.1The location of the Willershausen quarry. a The location of the Willershausen in Germany. b The location of the quarry in the neighborhood of Northeim (district in Lower Saxony, Germany). c The location of the protected site (vertical hatching) to the northeast of Willershausen. After[START_REF] Ferguson | A fresh look at the rich assemblage from the Pliocene sink-hole of Willershausen, Germany[END_REF], strongly modified

Fig. 2

 2 Fig. 2 Peltis antehercynica sp. nov., holotype specimen, RE 551.782.200 A 0083. a Complete specimen in dorsal view. b Right antenna. c Enlargement and reconstruction of antennal club. d Detail of mesonotum and mesocoxae and sculpturing of the elytra. Scale bars striped, 5 mm; back slashed 0.5 mm; dotted 0.1 mm

Fig. 3

 3 Fig. 3 Peltis antehercynica sp. nov. Reconstruction of the dorsal habitus

Table 1

 1 Differences among Peltis antehercynica sp. nov., P. grossa Linnaeus and P. gigantea Reitter

	Character	P. antehercynica sp. nov.	P. grossa	P. gigantea
	Antennal club	Antennomeres 9, 10 asymmetric	Antennomeres 9, 10 weakly	Antennomeres 9, 10 weakly asymmetric
			asymmetric
	Prosternal process Parallel sided, apex probably sharpened Weakly dilated at apex	Apex widely dilated
	Elytral carinae,	3 primary ? 3 or 4 secondary carinae Only 3 primary carinae	Only 3 primary carinae
	number
	Elytral carinae,	Primary carinae moderately,	Carinae moderately developed	Carinae distinctly elevated
	size	secondary carinae weakly developed
	Elytral punctation 4 regular rows of punctures between	Punctation among pairs of primary
		pairs of primary carinae	carinae coarser and irregular (4-6
			punctures between carinae)

Table 2

 2 Measurements of Peltis antehercynica sp. nov. As the fossil is compressed, the measurements are approximate and given in millimeters Measurement P. antehercynica sp. nov.

	Body without mandibles	16.5
	Head, length of visible part	2
	Head, max. width incl. eyes	4
	Head, mean width between eyes	2.5
	Antenna, length	4
	Antenna, 3-segmented club	1.5
	Pronotum, max. length	3
	Pronotum, length in middle	2.5
	Pronotum, max. width	10
	Elytron, max. width	6
	Elytron, max. length	12

Table 3

 3 Review of Cenozoic fossils classified within the genus Peltis

	Species	Reference	Type locality	Period	Body size
	Peltis costulata	Heyden (1862)	Germany	Upper Oligocene	Only elytron known: 3 mm
	Peltis laminata	Wickham (1910)	USA: Colorado	Lower Oligocene	Body 12.5 mm, elytron 6.6 mm
	Peltis minuscula	Piton (1935)	France	Upper Miocene	Only elytron known: 4 mm
	Silpha a tricostata	Heer (1847)	Germany	Upper Miocene	Body 18 mm, elytron 12 mm

Acknowledgments The authors are grateful to the following persons for generous help: Alexander Gehler (Go ¨ttingen), Heiko Ja ¨ckel (Kalefeld-Willershausen), Jens Meyer (Kalefeld), and Udo Scheer (Essen). We would like to thank Richard A.B. Leschen (New Zealand Arthropod Collection and Landcare Research), an anonymous reviewer, and Mike Reich (SNSB Munich) for valuable and constructive comments. Jir ˇı ´Koliba ´c ˇis grateful for financial support provided to the Moravian Museum by the Ministry of Culture of the Czech Republic as part of its long-term conceptual development program for research institutions (Ref. MK000094862). B.A. and T.W. were supported by grants from the Deutsche Forschungsgemeinschaft (DFG, Grant No. WA 1492/8-1; 11-1).