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Context 1: Paper Info

Title: Generalized Pole-Residue Method for Dynamic Analysis of Nonlinear Systems based on Volterra Series
Publish Date: March 7, 2023
Author List: Qianying Cao (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology), Anteng Chang (from College of Engineering, Ocean University of China), Junfeng Du (from College of Engineering, Ocean University of China), Lin Lu (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology)

Figure

Fig  1: Procedure to compute the response by a combination of Volterra series and Laguerre polynomials
Fig  2: Linear frequency response function: (a) Modulus of H 1 (ω), (b) phase angle of H 1 (ω)
Fig  6: Comparison of h 1 (t) based on the analytical and reconstructed by Laguerre polynomials
Fig  11: Response for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  18: Comparison of original excitations and reconstructed results: (a) Case 1, (b) Case 2 (c) Case 3
Fig  19: Response to irregular excitation for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  23: Input-output dataset used to identify Volterra series: (a) input excitation, (b) output response
Fig  26: Comparison of responses between the predicted and numerical results: (a) response to regular excitation, (b) response to irregular excitation
Parameter values of the irregular excitation

abstract

Dynamic systems characterized by second-order nonlinear ordinary differential equations appear in many fields of physics and engineering  To solve these kinds of problems, time-consuming stepby-step numerical integration methods and convolution methods based on Volterra series in the time domain have been widely used 
In contrast, this work develops an efficient generalized pole-residue method based on the Volterra series performed in the Laplace domain  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the pole-residue method in this paper is how to deal with the higher-order poles and their corresponding coefficients  Because the proposed method derives an explicit, continuous response function of time, it is much more efficient than traditional numerical methods 
Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the natural response, forced response and cross response are naturally obtained in the solution procedure, meaningful mathematical and physical insights are gained  In numerical studies, systems with a known equation of motion and an unknown equation of motion are investigated 
For each system, regular excitations and complex irregular excitations with different parameters are studied  Numerical studies validate the good accuracy and high efficiency of the proposed method by comparing it with the fourth-order Runge-Kutta method 

Introduction

Most real dynamic systems, as encountered in mechanical and civil engineering, are inherently nonlinear and include geometric nonlinearities, nonlinear constitutive relations in material or nonlinear resistances, etc    Nonlinear problems are attracting increasing attention from engineers and scientists 
This work focuses on solving nonlinear system vibration problems, i e , computing transient responses of nonlinear oscillators under arbitrary irregular excitations based on a combination of a pole-residue operation and Volterra series  Because Volterra series are single-valued, the scope of the present study is restricted to nonlinear behaviours without bifurcations  
To analyse nonlinear vibration problems, researchers have performed extensive studies and developed various mathematical methods  Popular methods include step-by-step numerical integration methods in the time domain, such as the Runge-Kutta method  This kind of method not only requires a small time-step resolution for obtaining high-precision solutions but also is prone to numerical instability  
For a long response with small time steps, the time domain methods are very costly in computational time  Volterra series is another widely used method, which is the extension of the Duhamel integral for linear systems   Volterra series can reproduce many nonlinear phenomena, but they are very complex due to higher-dimensional convolution integrals  
Since 1980's, significant progress has been made in the general area of the Volterra series  The reader is referred to Ref  for a quite thorough literature review on the relevant topics  After 2017, most papers focus on Volterra series identification  De Paula and Marques proposed a method for the identification of Volterra kernels, which was based on time-delay neural networks 
Son and Kim presented a method for a direct estimation of the Volterra kernel coefficients  Dalla Libera et al  introduced two new kernels for Volterra series identification  Peng et al  used the measured response to identify the kernel function and performed the nonlinear structural damage detection  Only a few papers concentrated on simplifying the computation of convolution integrals 
Traditional methods for computing convolution integrals involved in the Volterra series have been performed in three distinct domains: time, frequency and Laplace  The time domain method based on Volterra series refers to discrete time convolution methods, which also suffer computational cost problems  
Both the frequency domain method and the Laplace domain method based on the Volterra series consist of three steps: (1) Volterra series are transformed into an algebraic equation in the frequency domain or Laplace domain; the algebraic equation is solved by purely algebraic manipulations; and (3) the solution in Step ( ) is transformed back to the time domain 
Many researchers have used the frequency domain method to compute the responses of nonlinear systems  Billings et al  developed a new method for identifying the generalized frequency response function (GFRF) of nonlinear systems and then predicted the nonlinear response based on these GFRFs  Carassale et al  introduced a frequency domain approach for nonlinear bridge aerodynamics and aeroelasticity 
Ho et al  computed an output frequency domain function of a nonlinear damped duffing system modelled by a Volterra series under a sinusoidal input  Kim et al  identified the higher order frequency response functions by using the nonlinear autoregressive with exogenous input technique and the harmonic probing method 
This type of frequency domain method is much more efficient than the time domain method due to the fast Fourier transform algorithm  However, the frequency domain method not only is limited by frequency resolutions but also suffers from leakage problems due to the use of discrete Fourier transforms  In addition, the frequency domain method calculates only a steady-state response 
A natural response generated by initial conditions and a cross response caused by interactions between a system and an excitation are ignored  In contrast, the Laplace domain method can calculate all response components because initial conditions are considered in the computational procedure  However, it has been restricted to analytical operations for simple excitations, such as sinusoidal excitations and exponential excitations  
The proposed method falls into the category of the Volterra series method computed in the Laplace domain  Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the proposed method follows a similar path as a pole-residue method for linear systems , the proposed method to solve nonlinear system vibration problems is called the generalized pole-residue method 
The main concept of the pole-residue method developed by Hu et al  was that the poles and residues of the response could be easily obtained from those of the input and system transfer function to obtain the closed-form response solution of linear systems  This method included three steps: (1) writing the system transfer function into pole-residue form; (2) writing the excitation into pole-residue form by the Prony-SS method; (3) computing the poles and residues of the response by an algebraic operation based on those from system and excitation 
Compared to Hu et al  , which was regarded as an efficient tool to compute responses of linear systems, the generalized pole-residue method in this paper is introduced to compute responses of nonlinear systems  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the generalized pole-residue method is how to deal with the higher-order poles and their corresponding coefficients  Similar to the Taylor series, the Volterra series representation is an infinite series, and convergence conditions are needed to assure that the representation is meaningful 
Because the proposed method is based on the Volterra series, only the system with convergent Volterra series representation can be treated by the proposed method  The paper is organized as follows  In Section 2, the nonlinear response is modelled by a Volterra series, and Volterra kernel functions are decoupled by Laguerre polynomials 
Then, the pole-residue method for computing explicit responses is developed in Section 3  Numerical studies and discussions are given in Section 4  Finally, the conclusions are drawn in Section 5 

Response calculation based on Volterra series

A nonlinear oscillator, whose governing equation of motion is given by where z(t, y, ẏ) represents an arbitrary nonlinear term; m, c, and k are the mass, damping and linear stiffness, respectively; y(t), ẏ(t) and ÿ(t) are the displacement, velocity and acceleration, respectively; and f (t) is the time-dependent excitation 
If the energy of excitation f (t) is limited, the nonlinear response under zero initial conditions (i e , zero displacement and zero velocity) can be represented by the Volterra series : where N is the order of Volterra series and In Eq  3, h 1 (τ ) is called the first-order Volterra kernel function, which represents the linear behaviour of the system; h n (τ 1 ,      
, τ n ) for n > 1 are the higher-order Volterra kernel functions, which describe the nonlinear behaviour of the system  The complete formulation of y(t) includes infinite series where the labour of calculating the n th term increases quickly with the growth of n  Fortunately, the response accuracy may be ensured by the first several order Volterra series 
This is proved here in numerical studies  The commonly known Laguerre polynomials are represented as : where p i is the order of the Laguerre polynomials and a i is the damping rate  The Laguerre polynomials satisfy the orthogonal relationship expressed as: By using Laguerre polynomials, the Volterra kernel function h n (t 1 ,      
, t n ) in Eq  3 can be decoupled as follows : where the coefficient is computed resorting to the orthogonal relationship in Eq  5: Substituting Eq  6 into Eq  3 yields       The above operation that uses the Laguerre polynomials to decouple Volterra higher order kernel functions has been well-developed 
The reader is referred to Refs  for details about the adopted technique  After decoupling Volterra higher order kernel functions in time, one can regroup Eq  8 into:       By denoting Eq  9 becomes The above procedure to compute the nonlinear response by a combination of Volterra series and Laguerre polynomials is schematically shown in Fig   
Volterra kernel functions h n (t 1 ,       , t n ) can be obtained by either an equation of motion or measured input-output signals  To derive a closedform solution of the response, we must obtain a closed-form solution of x i (t) first  In the following presentation, a closed-form solution of the aforementioned x i (t) and y n (t) is derived by using the pole-residue method 
3  Pole-residue method for calculating x i (t) and y n (t) Performing the Laplace transform of x i (t) in Eq  10 yields where in which Eq  13 includes a single pole and several higher-order poles  For k = 0, −a i is a single pole, and b p i (0) is a corresponding coefficient, namely, the residue  For k > 0, −a i are higher-order poles, and b p i (k) are corresponding coefficients 
For an irregular excitation signal f (t) of a finite duration of T , it can always be approximated into a pole-residue form by using the complex exponential signal decomposition method-Prony-SS : where N ℓ is the number of components; α ℓ and λ ℓ are constant coefficients, which either are real numbers or occur in complex conjugate pairs 
We define λ ℓ = −δ ℓ + iΩ ℓ , where Ω ℓ is the excitation frequency and δ ℓ is the damping factor of the ℓ th component  We denote α ℓ = A ℓ e iθ ℓ , where A ℓ is the amplitude and θ ℓ is the sinusoidal initial phase in radians  Taking the Laplace transform of Eq  15 yields Note that the concept of the Prony-SS method is similar to that of a principal component method 
A smooth excitation usually requires just several terms to achieve a good approximation  For high irregular loadings, including more terms would achieve a better approximation  Substituting Eqs  13 and 16 into Eq  12 yields Expressing xi (s) in its pole-residue form yields where λ ℓ are simple poles, and the corresponding residues are easily obtained by
and −a i are higher-order poles, and the corresponding coefficients are firstly derived as: By taking the inverse Laplace transform of Eq  18, a closed-form solution is obtained: Substituting Eqs  11 and 21 into Eq  2 yields Theoretically speaking, the proposed method for deriving the closed-form solution of the nonlinear response is applicable to any order of the Volterra series 
For practical engineering, usually only the first several order responses dominate  By setting up N = 2, Eq  22 can be simplified into three components: where the natural response, which is only related to system poles, is given by and the cross response, which is related to both system poles and excitation poles, is given by
and the forced response, which is related only to excitation poles, is given by The first term in Eq  26 is the first-order forced response governed by the excitation frequency, i e , the imaginary part of the pole λ ℓ   The second term corresponds to the second-order nonlinear forced response, which includes the sum frequency and difference frequency responses governed by λ ℓ + λ j  
Eq  26 straightforwardly offers visible information about the possible nonlinear vibrations by the cooperation of excitation frequencies  Particularly, consider a sinusoidal excitation f (t) = sin ω r t, which can be expressed as f (t) = γe λt + γ * e λ * t , where γ = −0 5i and λ = iω r   Substituting these values into Eq 
26, the second term of Eq  26 is simplified as where the first term is the difference frequency response, and the second term is the sum frequency response 

Numerical studies

In practical engineering, some systems have an accurate equation of motion  Additionally, some systems have difficulty constructing their equations of motion because of complex nonlinear dynamic behaviours and uncertain system parameters  In this article, a system with a known equation of motion is called a known system, and a system with an unknown equation of motion is called an unknown system for simplicity 
In this section, two numerical studies are presented  The first study verifies the proposed method using a known nonlinear oscillator, and the second study demonstrates the applicability of the proposed method to an unknown system  Throughout the numerical studies, the unit system is the metre-kilogramme-second (MKS) system; for conciseness, explicit units for quantities are omitted 

A known nonlinear system

This study chooses a nonlinear oscillator written as: where mass m = 1, damping c = 1, linear stiffness k 1 = 10, quadratic stiffness k 2 = 20 and cubic stiffness k 3 = 20  It is a case that has been studied in a previously published article   The linear natural frequency of the system ω 0 = k 1 /m = 3 16 and the damping ratio ζ = c/(2mω 0 ) = 15 8% 
This kind of oscillator occurs in many engineering problems, such as a model of fluid resonance in a narrow gap between large vessels   In the model, k 1 y represents the linear restoring force of the fluid, and k 2 y 2 and k 3 y 3 are respectively the quadratic and cubic nonlinear restoring forces of the fluid 

Volterra kernel functions

Generally, the first several order responses dominate the total response of a system  Hence, the order of the Volterra series in Eq  22 is chosen to be 3, namely, N = 3  For computing the first three order responses from Eq  22, the first three order Volterra kernel functions need to be known  Since Volterra kernel functions and corresponding frequency response functions are related by a specific Fourier transform pair, we can first write the first three orders of frequency response functions directly from Eq  28 
Then, Volterra kernel functions are obtained by the inverse Fourier transform  Based on the harmonic probing algorithm , the linear frequency response function (LFRF) H 1 (ω), the quadratic frequency response function (QFRF) H 2 (ω 1 , ω 2 ) and the cubic frequency response function (CFRF) H 3 (ω 1 , ω 2 , ω 3 ) are analytically given by:
and Figures show H 1 (ω), H 2 (ω 1 , ω 2 ) and H 3 (ω 1 , ω 2 , ω 3 ), respectively, which agree well with those reported in Ref    As expected, the modulus of H 1 (ω) in Fig  peaks near the linear natural frequency ω 0 , and the phase angle decreases monotonically from 0 to -π with increasing frequency 
Figure shows the sum frequency QFRF, where the energy converges along the line of ω 1 +ω 2 ≈ ω 0   Therefore, when the sum frequency of a two-tone excitation equals the linear resonant frequency, the second-order response may reach its maximum  Additionally, those pairs of excitations in line ω 1 + ω 2 ≈ ω 0 may produce non-negligible vibration magnitudes due to second-order nonlinear effects 
For the difference frequency QFRF in Fig  (b), the energy converges along two main lines, i e , ω 1 ≈ ω 0 and ω 2 ≈ ω 0   Figures show moduli of H 3 (ω, ω, ω) and H 3 (ω, ω, −ω), which are diagonal terms of the sum frequency CFRF and the difference frequency CFRF, respectively  While the modulus of H 3 (ω, ω, ω) peaks near ω ≈ ω 0 /3 and ω 0 , that of H 3 (ω, ω, −ω) peaks near ω ≈ ω 0 with a small hump around ω ≈ ω 0 /2 
Values at ω ≈ ω 0 /3 and ω 0 /2 may be magnified by higher-order stiffness terms in Eq  28  By performing the inverse fast Fourier transform to Eqs  29-31, the corresponding linear impulse response function h 1 (t), quadratic impulse response function h 2 (t 1 , t 2 ) and cubic impulse response function h 3 (t 1 , t 2 , t 3 ) are obtained 
Here, h 1 (t) and h 2 (t 1 , t 2 ) are plotted in Figs  , respectively, and h 3 (t, t, t) is shown in Fig    In the numerical implementation, Eqs  29-31 have been utilized with the frequency interval ∆ω = 0 1, number of frequency components N n = 1025, and cut-off frequencies 102 4 and −102 4  For decoupling Volterra kernel functions by using Laguerre polynomials, the damping rate and number of Laguerre polynomials for each order Volterra kernel function need to be determined (see Eqs  4 and 6) 
In this example, we set a 1 = a 2 = a 3 = 2 and R 1 = R 2 = R 3 = 24 because coefficients c p 1    pn become very small when R n > 24, n = 1, 2, 3  According to Eq  7, the coefficients of the first three order Volterra kernel functions are calculated, which are shown in Figs  9 and 10  For convenience, Fig  plots only c p 1 p 2 p 3 for p 3 = 0 
With the increase of the order of Laguerre polynomials, coefficients in Figs  9 and 10 gradually decrease, which illustrates how the first several orders of Laguerre polynomials dominate all orders of the Volterra kernel function  With the known Laguerre polynomials and corresponding coefficients, Volterra kernel functions are reconstructed by Eq  6 
For comparison, reconstructed Volterra kernel functions are also plotted in Figs    The reconstructed results agree well with the analytical values, which verifies the accuracy of the decomposition 

Sinusoidal excitation

From Eq  28, we consider a sinusoidal excitation where A and Ω are the amplitude and the frequency, respectively  Five cases of A and Ω are shown in Table   Excitation frequencies in Cases 1 and 2 are larger than the linear natural frequency (ω 0 ≈ 3 16), those in Case 3 are very close to ω 0 , and those in Cases 4 and 5 are smaller than ω 0  
All cases have same amplitudes  The poles of a sinusoidal excitation are λ 1,2 = ±iΩ, and the residues are α 1,2 = ∓iA/2  Numerical values of excitation poles and residues for different cases are listed in Table   Table : Parameter values, poles and residues of the sinusoidal excitation Substituting poles and residues of the excitation, as well as those of the system into Eqs 
20 and 19, response coefficients β p i ,k corresponding to system poles −a i and response coefficients γ p i ,ℓ corresponding to excitation poles λ ℓ are calculated, respectively  According to Eq  22, the first three orders of responses for each case in Table are calculated  Figures )-15(a) show the comparison of responses obtained by the proposed method and the fourth-order Runge-Kutta method with ∆t = 10 −4  
For Cases 1 and 2, the first-order responses agree well with the total responses obtained by the Runge-Kutta method, and the higher-order responses only slightly improve the transient parts  For Cases 3-5, the sum of the first three orders of responses is in good agreement with the Runge-Kutta solution 
When the response nonlinearity increases, higher-order responses need to be considered  In other words, the proposed method can accurately compute the nonlinear responses by choosing a small number N of Volterra series terms  Figures )-15(b) show the contributions of the three response components for the five cases 
In each case, the first-order response is the most dominant component, and the contributions of secondand third-order responses are much less than those of the first-order response  Especially for Cases 1 and 2, whose excitation frequencies are far from the linear natural frequency, second-and thirdorder responses are close to zero 
This may be because the QFRF and CFRF approach zero when the frequency is larger than 4 rad/s (see Figs  )  Furthermore, the mean values of the first-order responses are approximately zero, and those of the second-order responses are always smaller than zero, which are the difference frequency components in Eq  27 
Moreover, it is clearly observed that second-order responses for Cases 3-5 exhibit a periodic oscillation with a period near half of that for the first-order response, which is excited by the sum frequency component of the excitation (see second part of Eq  27)  Compared with steady-state solutions of first-and second-order responses, those of third-order responses in Cases 3-5 are no longer single regular motions 
By performing the FFT, frequency spectra of these three third-order responses are shown in Fig    We find that these three third-order responses are all dominated by their own fundamental harmonic component and the third harmonic (triple frequency) component  Figure shows the computational time to calculate the response of the oscillator for Case 1 by the proposed method, the fourth-order Runge-Kutta method and the convolution method 
The proposed method, which has an explicit solution, is much more efficient in computational time than the latter two methods, which need small time steps to obtain high-precision solutions  In particular, the efficiency of the proposed method increases with the length of the response time  Computation time (sec )
t=0 02s Convolution, t=0 02s Convolution, t=0 001s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the proposed method, the fourth-fifth order Runge-Kutta method and the convolution method regular loading in Case 1

Irregular excitation

In Eq  28, considering an irregular excitation consisting of several cosine functions where N f is the number of cosine components; A n , Ω n and θ n are the amplitude, frequency and phase angle of the n th component, respectively  Table lists three cases of these parameters  In each case, the amplitudes of all components are the same, and phase angles θ n uniformly distributed between 0 and 2π are randomly generated 
To decompose the excitation into a pole-residue form, the Prony-SS method is used, whose concept is similar to that of a principal component method  The readers are referred to Ref  for details  The chosen rank of each case is also shown in Table   Figure shows the comparison of original excitations and reconstructed results of these three cases, which all have excellent agreement 
show the results computed by the fourth-order Runge-Kutta method  In all cases, the sums of the first three orders of responses agree well with those obtained by the Runge-Kutta method  The contributions of the first three orders of responses for each case are plotted in Figs  )-21(b)  Similarly, the system vibration is dominated by the first-order response 
However, the contributions of second-and third-order significantly grow with increasing excitation magnitude and frequency number  Furthermore, when the magnitude of the nonlinear response becomes large, sharp troughs are present  This phenomenon may be induced by the nonlinear stiffness  While the first-order response fails to capture these troughs, the higher-order responses successfully capture these troughs 
Figure plots the computational time to calculate the response of the oscillator for the irregular loading in Case 1 by the proposed method and the fourth-fifth order Runge-Kutta method, respectively  While the fourth-fifth order Runge-Kutta method is more efficient under a small response length, the proposed method becomes much more efficient when the response length is larger than about 130 s 
In addition, the proposed method obtains the explicit response solution, so one can directly obtain the response value at a specific time t p instead of integrating from 0 to t p for traditional numerical methods  Computation time (sec ) Proposed, t=0 02s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the method and the fourth-fifth order Runge-Kutta method for irregular loading in Case 1

An unknown nonlinear

To check the applicability of the proposed method to an unknown nonlinear system, a known input excitation and its corresponding response are used to identify its Volterra kernel functions  When the Volterra kernel functions are known, we can follow the procedure in Section 4 1 to predict system responses 
In this study, the input excitation is white noise with a constant power spectrum S 0 = 0 001, and the corresponding response is obtained by solving Eq  28 by the fourth-order Runge-Kutta method, which is shown in Fig    From Section 4 1, we determine that the sum of the first two orders of responses agrees well with the total response 
In this study, the order of Volterra series N is chosen to be 2, damping rates of Laguerre polynomials are a 1 = a 2 = 2, and numbers of Laguerre polynomials are R 1 = R 2 = 24  To estimate the first two orders of Volterra kernel functions, a matrix equation is constructed using excitation data and response data 
By using the least square method to solve this matrix equation, coefficients c p 1 and c p 1 p 2 in Eq  8 are identified  Figure plots c p 1 and c p 1 p 2 , respectively, which have good agreement with the exact results in Fig    Then, the first two order Volterra kernel functions are constructed by Eq  6 
Compared with the exact results in Figs  , the identified Volterra kernel functions in Fig  completely agree well with the exact solutions  Note that the white noise excitation, which can excite more frequency components of the response, is chosen to obtain good Volterra kernel functions  A regular excitation f (t) = sin(πt) and an irregular excitation f (t) = N f n=1 A n cos(Ω n t + θ n ) with A n = 0 3 and Ω n varying from 0 to 40 with equal interval 1 are chosen as input excitations 
The predicted responses, along with results obtained by the fourth-order Runge-Kutta method, are shown in Fig    In both cases, the proposed method accurately predicts system responses  As presented in Eq  23, a nonlinear response is the sum of three terms: natural response y s (t), forced response y f (t) and cross response y c (t) 
These individual terms, as well as their sum to two excitations, are shown in Figs  27 and 28, respectively  As shown in Figs  and 28, both first-and second-order responses include the natural response y s (t) and the forced response y f (t), but the cross response y c (t) only exists in second-order responses 
When t becomes larger, both y s (t) and y c (t) diminish due to the presence of system damping, and the total response is entirely governed by y f (t)  Moreover, we notice some features at t = 0 for these components, including y s (0) = −y f (0) for the first-order response and y s (0) + y f (0) = −y c (0) for the second-order response, which are due to imposed zero initial conditions 

Conclusions

Considering arbitrary irregular excitations, an efficient generalized pole-residue method to compute the nonlinear dynamic response modelled by the Volterra series was developed  A core of the proposed method was obtaining poles and corresponding coefficients of Volterra kernel functions, then those of each order response modelled by each order Volterra series 
Once the poles and corresponding coefficients of Volterra kernel functions and excitations were both available, the remaining derivation could follow a similar pole-residue method that had been developed for ordinary linear oscillators  To obtain the poles and corresponding coefficients of Volterra kernel functions, two steps were included: (1) using Laguerre polynomials to decouple higher-order Volterra kernel functions with respect to time and (2) obtaining poles and corresponding coefficients of Laguerre polynomials in the Laplace domain 
Because the proposed method gave an explicit, continuous response function of time, it was much more efficient than traditional numerical methods  Moreover, many meaningful physical and mathematical insights were gained because not only each order response but also the natural response, the forced response and the cross response of each order were obtained in the solution procedure 
To demonstrate that the proposed method was not only suitable for a system with a known equation of motion but also applicable to a system with an unknown equation of motion, two numerical studies were conducted  For each study, regular excitations and complex irregular excitations with different parameters were investigated 
The efficiency of the proposed method was verified by the fourth-order Runge-Kutta method  This paper only computes the response under zero initial conditions  The response under non-zero initial conditions will be investigated in our future work 
Context 2: \section{INTRODUCTION}
The Tevatron Collider Run II started in March 2002 and is expected
to continue until the end of this decade  The Tevatron and the 
two detectors, CDF and D\O, have been performing  well in 2004,
each experiment is collecting data at the rate 
of $\approx$10 pb$^{-1}$ per week 
The total  luminosity accumulated by August 2004 is $\approx$500 pb$^{-1}$
per detector 
The rich physics program includes the
production and precision measurement of properties of  standard model (SM)
objects, as well as searches for phenomena beyond standard model 
In this brief review we focus on areas of most interest 
to the lattice community  We present
new results on the top quark mass
and their implication for the mass of the SM Higgs boson, 
on searches for the SM Higgs boson, on evidence for the $X(3872)$ state, 
on searches for pentaquarks, and on $b$ hadron properties 
All Run II results presented here are preliminary  

\section{TOP QUARK MASS}

The experiments CDF and D\O\ published several direct  measurements of
the top quark pole mass, $\ensuremath{M_{\mathrm{top}}}$, 
based on Run I data (1992-1996) 
The ``lepton $+$ jets'' channel yields the most precise determination of
$\ensuremath{M_{\mathrm{top}}}$  Recently, the
D\O\ collaboration published a new measurement~\cite{Mtop1-D0-l+j-new},
based on a powerful analysis technique yielding  greatly improved precision 
The differential probability 
that the measured variables in any event correspond to the signal
is calculated as a function of $\ensuremath{M_{\mathrm{top}}}$  
The maximum in the product of the individual event probabilities 
provides the best estimate of $\ensuremath{M_{\mathrm{top}}}$ 
The critical differences from previous analyses 
in the lepton $+$ jets decay channel lie in 
the assignment of more 
weight to events that are well measured or more likely to correspond to  
$t \bar t$ signal, 
and  the handling of the combinations of final-state objects
(lepton, jets, and imbalance in transverse momentum) 
and their identification with
top-quark decay products in an event  
The new combined value for the top-quark mass from Run I is 
$\ensuremath{M_{\mathrm{top}}}  =  178 0\pm4 3~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 

In Run II, both collaborations  have been exploring several different techniques 
for $\ensuremath{M_{\mathrm{top}}}$
measurements  The best single CDF result comes from a dynamic likelihood method
(DLM)  The method is similar to
the technique used in Ref ~\cite{Mtop1-D0-l+j-new} 
The result is $\ensuremath{M_{\mathrm{top}}} = 177 8^{+4 5}_{-5 0} (stat) \pm  6 2 (syst) ~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 
The joint likelihood of the selected events is shown in Fig  ~\ref{fig:cdf_tml}  
The Run II goal is a 1\% uncertainty on $\ensuremath{M_{\mathrm{top}}}$  




\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=5 8cm,width=8 1cm]  {data_22ev_likelihood eps}
\vspace*{-1 2cm}
\caption{The joint likelihood of top candidates(CDF) }
\label{fig:cdf_tml}
\end{figure}




\section{SEARCH FOR SM HIGGS BOSON}


The constraints on the SM Higgs ($H$)  boson  mass from
published  measurements, updated to include the new D\O\ top mass
measurement~\cite{Mtop1-D0-l+j-new}, are
$M_H = 117 ^{+67}_{-45}~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$, $M_H < 251~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ at 95\% C L 
The  new most likely  value of $M_H$
is above the experimentally excluded range,
and sufficiently low for $H$ to be observed at the Tevatron 


\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=7 5cm,width=7 8cm]  {d0_wbb_fig_3_err eps}
\vspace*{-1 1cm}
\caption{Distribution of the dijet
invariant mass for $W+2 b$-tagged jets  events,
compared to the expectation (D\O)  
}
\label{fig:d0_wbb_2tag}
\end{figure}



D\O\  has conducted a search for $H$ at $M_H < 140~\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 
in the production channel  
$p \bar{p} \rightarrow WH \rightarrow  e \nu b \bar{b}$  
The experimental signature of  $WH \rightarrow e \nu b \bar{b}$
is a final state with 
one high $p_T$ electron, two  $b$ jets, and
large missing transverse energy  resulting from
the undetected neutrino 
The dominant backgrounds to $WH$ production
are  $W b \bar{b}$, $t \bar{t}$ and single-top production 
The distribution 
of the dijet mass for events with two $b$-tagged jets is shown in
Fig ~\ref{fig:d0_wbb_2tag}  
Also shown is the  expected contribution ($0 06$ events)  
from the $b \bar{b}$ decay of a
SM Higgs boson with $M_H =$ 115 $\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 
No events are observed in the  dijet mass window of 85--135  $\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ 
D\O\ sets a limit on the cross section
for $\sigma( p\bar{p} \rightarrow WH) \times B(H \rightarrow b \bar{b}) $
of 9 0 pb at the 95\% C L ,  for a 115  $\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$ Higgs boson 
The results for mass points 105, 125, and 135 $\ensuremath{\mathrm{ Ge\kern -0 1em V }\kern -0 2em /c^2 }$
 are 11 0, 9 1 and 12 2 pb,  respectively 



\begin{figure}[htb]
\vspace*{-1 2cm}
\includegraphics[height=0 33\textheight,width=8 0cm]{whww_aps04_bw eps}

\vspace*{-1 2cm}
\caption{95\% limits on the $H$ production (CDF) }
\label{fig:cdf_whww}
\end{figure}


CDF  has done  a similar search, allowing either an  electron or a muon  
in the final state   Both groups have also searched for $H$ produced in
gluon-gluon fusion, with subsequent decay to a pair of $W$ bosons 
The CDF results for both channels  are shown in Fig ~\ref{fig:cdf_whww}  



\section{THE STATE X(3872)}


\begin{figure}[htb]

\includegraphics[height=8 0cm,width=7 5cm]  {X3872cdfPRL1FullM eps}
\vspace*{-1cm}
\caption{The $X(3872)$ signal (CDF) }
\label{fig:cdf_x}
\end{figure}




 The existence of the $X(3872)$ state discovered by 
the Belle Collaboration~\cite{Belle-X}
 has been confirmed 
 in $p \bar{p}$ collisions by  CDF~\cite{cdf-X} (see Fig ~\ref{fig:cdf_x})
and D\O~\cite{d0-X} 
 It is still unclear whether this particle is a $c\bar{c}$ state,
 or a more complex object   When the data are separated according to
production and decay variables, D\O\  finds no significant
differences between the $X(3872)$ and
the $c \bar{c}$ state $\psi(2S)$ 
CDF has analysed the ``lifetime'' distribution of the $X(3872)$ events in order to
quantify what fraction of this state arises from decay of $B$ hadrons, as opposed to
those produced promptly  The authors find that for the selected samples
28 3$\pm$1 0$(stat)\pm$0 7$(syst)$\% of $\psi(2S)$ candidates are from $b$ decays,
whereas 16 1$\pm$4 9$(stat)\pm$2 0$(syst)$\% of $X$ mesons arise from such decays 





\section{SEARCH FOR PENTAQUARKS}



\begin{figure}[htb]

\includegraphics[height=0 27\textheight,width=7 6cm]  {mpks_1stminbias eps}
\vspace*{-1 2cm}

\caption{Invariant mass distribution of an identified proton and a $K^0_s$ candidate  (CDF)
}
\label{fig:pqtheta}
\end{figure}



\begin{figure}[htb]

\vspace*{-0 9cm}
\includegraphics[height=0 25\textheight,width=8 0cm]  {CM_xicst_cc_1 eps}
\vspace*{-1 2cm}
\caption{Invariant mass distribution of the $(\Xi^-,\pi^+)$ system  (CDF) 
}
\label{fig:pqxi}
\end{figure}


\begin{figure}[htb]
\vspace*{-0 9cm}

\includegraphics[height=0 25\textheight,width=7 6cm]  {theta_note_dstp_dedx_pt eps}
\vspace*{-1 2cm}
\caption{Mass of the ($D^{*+}\bar p$) system  The arrow indicates the position of 
the $\Theta_c$ state (CDF) }
\label{fig:pqthetac}
\end{figure}



Following reports of evidence for exotic
baryons containing five quarks (pentaquarks), CDF has analysed 
its data for evidence of the following pentaquarks:
$\Theta^+$ ($uud\bar d \bar s$), doubly strange states 
$\Xi_{3/2}$, charmed states $\Theta_c$, and, most recently, 
a state $(udus\bar b)$, dubbed $R^+_s$, through its weak decay to $(J/\psi, p)$  
With its excellent particle indentification and mass resolution,
CDF has a unique capability to search for  pentaquark states 
The signals of known states: $\phi$, $\Lambda$,
$\Lambda(1520)$, $K^*$, $\Xi$, 
compare favorably with those provided
by the authors of  the pentaquark evidence 
The group finds no evidence for pentaquark states, see Figs 
~\ref{fig:pqtheta},{\ref{fig:pqxi},\ref{fig:pqthetac} 
This can be interpreted as an indication that the pentaquark production 
in $p \bar p$ collisions is heavily suppressed compared to the conventional
hadron production, or as an evidence against the existence of pentaquarks 

\clearpage

\section{RECENT B PHYSICS RESULTS}


\subsection{Spectroscopy}

CDF has measured the mass of $b$ hadrons in exclusive $J/\psi$ channels 
The measurements of the $B_s$ and $\Lambda_b$ (Fig  \ref{fig:masslb})
masses are the current world's best \\

$m(B^+)$ = 5279 10$\pm$0 41$(stat)\pm$0 36$(syst)$,

$m(B^0)$ = 5279 63$\pm$0 53$(stat)\pm$0 33$(syst)$,

$m(B_s)$ = 5366 01$\pm$0 73$(stat)\pm$0 33$(syst)$,

$m(\Lambda_b)$ = 5619 7$\pm$1 2$(stat)\pm$1 2$(syst)$ MeV/$c^2$ \\


\begin{figure}[htb]
\vspace*{-1mm}
\includegraphics[height=0 30\textheight,width=7 5cm]  {lambdav1c eps}
\vspace*{-1cm}

\caption{The mass spectrum of $\Lambda_b$ candidates (CDF) }
\label{fig:masslb}
\end{figure}


D\O\ reports the first observation of the excited $B$ mesons 
$B_1$ and $B^*_2$ as two separate states in fully reconstructed
decays to $B^{(*)}\pi$  The mass of $B_1$ is measured to be
5724$\pm$4$\pm$7 MeV/c$^2$, and the mass difference $\Delta M$ between
$B^*_2$ and $B_1$ is 23 6$\pm$7 7$\pm$3 9 MeV/c$^2$
(Fig   \ref{fig:d0_bexc}) 

D\O\ observes semileptonic $B$ decays to narrow $D^{**}$ states,
the orbitally excited states  of the $D$ meson
seen as resonances in the $D^{*+}\pi^-$ invariant mass spectrum 
The $D^*$ mesons are reconstructed through the decay sequence 
$D^{*+} \rightarrow D^0\pi^+$, $D^0\rightarrow K^-\pi^+$ 
The invariant mass  of oppositely charged $(D^*,\pi)$ pairs
is shown in Fig   \ref{fig:d0_dstst} 
The mass peak between 2 4 and 2 5 GeV/$c^2$ can be interpreted as two merged 
narrow $D^{**}$ states, $D^0_1(2420)$ and $D^0_2(2460)$ 
The combined branching fraction is 
$ {\cal B}(B\rightarrow D^0_1,D^0_2)\cdot {\cal B}(D^0_1,D^0_2\rightarrow D^{*+}\pi^-)=(0 280\pm0 021(stat)\pm0 088(syst)$\%  The systematic error includes the unknown phase between the
two resonances  Work is in progress on extracting the two Breit-Wigner
amplitudes 


\begin{figure}[htb]
\vspace*{-2mm}
\hspace*{-3mm}
\includegraphics[height=0 28\textheight,width=8 3cm]  {B08F02 eps}

\vspace*{-1cm}
\caption{Mass difference $\Delta M = M(B\pi)-M(B)$ for exclusive $B$ decays 
The background-subtracted signal is a sum of 
$B^*_1 \rightarrow B^* \pi$, $B^* \rightarrow B \gamma $ (open area)
and $B^*_2 \rightarrow B^*\pi$ $B^*\rightarrow B \gamma$ (lower peak in the shaded area)
and $B^*_2 \rightarrow B \pi$ (upper peak in the shaded area)  
(D\O) }
\label{fig:d0_bexc}
\end{figure}


\begin{figure}[htb]
\includegraphics[height=0 25\textheight,width=7 5cm]  {B05F03 eps}

\vspace*{-1cm}
\caption{The invariant mass distribution of
$(D^*,\pi)$ pairs, opposite sign (points) and same-sign (solid histogram) }
\label{fig:d0_dstst}
\end{figure}






\subsection{Lifetimes}


CDF and D\O\ have measured  lifetimes of $b$ hadrons through the exclusively
reconstructed decays $B^+ \rightarrow J/\psi K^+$, $B^0 \rightarrow J/\psi K^{*0}$,
$B_s \rightarrow J/\psi \phi$, 
and $\Lambda_b \rightarrow J/\psi \Lambda$
(Fig  \ref{fig:d0_lbctau}) 
The latest results are:  \\



 $\tau(B^+)$=1 65 $\pm$ 0 08 $^{+0 096}_{-0 123}$  ps ~(D\O\ 2003),

 $\tau(B^+)$=1 662 $\pm$ 0 033  $\pm$ 0 008  ps ~(CDF),

 $\tau(B^0_d)$=1 473  $^{+0 052}_{-0 050}$ $\pm$ 0 023    ps ~(D\O) 

 $\tau(B^0_d)$=1 539 $\pm$ 0 051  $\pm$ 0 008  ps ~(CDF),

 $\tau(B^0_s)$=1 444   $^{+0 098}_{-0 090}$ $\pm$ 0 020   ps ~(D\O),

 $\tau(B^0_s)$=1 369 $\pm$ 0 100 $\pm$ $^{+0 008}_{0 010}$  ps ~(CDF),


 $\tau(\Lambda_b)$=1 221 $^{+0 217}_{-0 179}$ $\pm$ 0 043  ps ~(D\O),


 $\tau(\Lambda_b)$=1 25 $\pm$ 0 26 $\pm$ 0 10  ps ~(CDF 2003) \\



The measured lifetimes correspond to the following lifetime ratios:\\

$\tau(B^+)/\tau(B^0_d)$   =  1 080$\pm$0 042     ~(CDF),
 
$\tau(B^0_s)/\tau(B^0_d)$ =  0 890$\pm$0 072    ~(CDF),

$\tau(B^0_s)/\tau(B^0_d)$ = 0 980$ ^{+0 075}_{-0 070}   \pm$0 003    ~(D\O),

$\tau(\Lambda_b)/\tau(B^0_d)$ =  0 874$ ^{+0 169}_{-0 142}   \pm$0 028    ~(D\O) \\



\begin{figure}[htb]
\includegraphics[height=0 3\textheight,width=8 2cm]  {d0_lbctau_B11F02 eps}
\vspace*{-1cm}

\caption{ Fit projection on  $c\tau$ for the $\Lambda_b$ candidates   (D\O)}
\label{fig:d0_lbctau}
\end{figure}


The $B_s$ lifetime measurements listed above are results of
a single-lifetime fit to data, integrated over the decay angles 
Because  of the presence of  final
states  common to ${B_s^0}$\ and its charge conjugate ${\overline{B}_s^0}$,
the two meson states   are expected
to mix in such a way that the two CP  eigenstates may have a relatively
large lifetime difference 
It is possible to
separate the two CP components of ${B_s^0 \rightarrow J/\psi \phi}$\ and thus to measure the
lifetime difference by studying the time evolution of the
polarization states of the vector mesons in the final state 
CDF has carried out a combined analysis of $B_s$ lifetimes 
and polarization amplitudes  The results for the lifetimes of the
low mass (CP even) and high mass (CP odd) eigenstates, and the relative 
width difference are:\\

 $\tau_L = 1 05 ^{+0 16}_{-0 13} \pm 0 02$ ~ps,
 
 $\tau_H = 2 07 ^{+0 58}_{-0 46} \pm 0 03$ ~ps,

 $\Delta \Gamma /\overline \Gamma   = 0 65 ^{+0 25}_{-0 33} \pm 0 01$ \\

Figure \ref{fig:cdf_dg} shows  the scan of the likelihood function 
for $\Delta \Gamma /\overline \Gamma$ 
Pseudoexperiments tossed with $\Delta \Gamma /\overline \Gamma =0$
yield the betting odds for observing the above results at
1/315  For $\Delta \Gamma /\overline \Gamma = 0 12$ (SM prediction,
which has recently been updated to 0 14$\pm$0 05~\cite{dg_un}) the betting odds are
1/84 

\begin{figure}[htb]
\vspace*{-1mm}
\includegraphics[height=0 3\textheight,width=8 2cm]  {cdf_scan-dg-un eps}

\vspace*{-1cm}
\caption{Scan of the likelihood function 
for $\Delta \Gamma /\overline \Gamma$ (CDF) 
}
\label{fig:cdf_dg}
\end{figure}




D\O\ has used a novel technique to  measure the lifetime ratio
of the charged and neutral $B$ mesons, exploiting the large
semileptonic sample  $B$ hadrons were reconstructed in the channels
$B\rightarrow \mu^+ \nu D^*(2010)^-X$, which are dominated by $B^0$ decays, 
and  $B\rightarrow \mu^+ \nu D^0X$, which are dominated by $B^+$ decays 
The lifetime ratio was
obtained from the variation of the ratio of the number of events in these two
processes at different decay lengths 
The result is \\


$\tau(B^+)/\tau(B^0_d)$   =  1 093$\pm$0 021$\pm$0 022       ~(D\O)




\subsection{Towards $B_s$ mixing}

Measurement of the $B_s$ oscillation frequency via ${B_s^0}$ -${\overline{B}_s^0}$ ~mixing
will provide an important constraint on the CKM matrix  The oscillation
frequency is proportional to the mass difference between the mass eigenstates,
$\Delta m_s$, and is related to the CKM matrix through 
$\Delta m_s \propto |V_{tb}V_{ts}|$  When combined with the
$B_d$ mass difference, $\Delta m_d$ it helps in extraction of $|V_{td}|$,
and thereby the CP violating phase  

As a benchmark for future $B_s$ oscillation measurement, both groups
study  $B_d$ mixing, gaining an understanding of the different components
of a $B$ mixing analysis (sample composition, flavor tagging, vertexing,
asymmetry fitting)  For a sample of partially reconstructed decays
$B\rightarrow D^*(2010)^+\mu^-X$, D\O\ obtains 
$\Delta m_d = 0 506 \pm 0 055 (stat) \pm  0 049 (syst))$ ps$^{-1}$ and
$\Delta m_d = 0 488 \pm 0 066 (stat) \pm  0 044 (syst))$ ps$^{-1}$
when employing  opposite side muon tagging and the same side tagging,
respectively 

The CDF result for semileptonic channels is
$\Delta m_d = 0 536 \pm 0 037 (stat) \pm  0 009 (s c ) \pm 0 015 (syst)$ ps$^{-1}$ 
CDF also reports a result on $B$ oscillations using fully reconstructed
decays:
$\Delta m_d = 0 526 \pm 0 056 (stat) \pm  0 005 (syst))$ ps$^{-1}$ 

Reconstructing $B_s$ decays into different final states is another
important
 step in the ${B_s^0}$ -${\overline{B}_s^0}$ ~mixing analysis 
Thanks to the  large muon and tracking coverage,   D\O\ is accumulating
a  high statistics sample of semileptonic $B_s$ decays 
D\O\ reconstructs the $B_s \rightarrow D^+_s \mu^- X$ decays, with
$D^+_s \rightarrow \phi \pi^+ $ and
$D^+_s \rightarrow K^* K^+ $,
at a rate of $\approx$ 40(25) events per pb$^{-1}$,  respectively 
Figure \ref{fig:d0_bsdsphipi} shows the mass distribution of the
$D^+_s \rightarrow \phi \pi$ candidates 


\begin{figure}[htb]
\vspace*{-5mm}
\includegraphics[height=0 3\textheight,width=8 0cm]  {blds-250 eps}
\vspace*{-1 2cm}
\caption{  $D^+_s \rightarrow \phi \pi^+$  signal  (D\O)}
\label{fig:d0_bsdsphipi}
\end{figure}


\begin{figure}[htb]
\vspace*{-10mm}
\hspace*{-4mm}
\includegraphics[height=0 35\textheight,width=7 9cm]  {cdf_Bs-DsPi-PhiPi eps}

\vspace*{-1 0cm}
\caption{ $B_s \rightarrow D_s \pi$, $D_s \rightarrow \phi \pi$  signal   (CDF)}
\label{fig:cdf_bsdsphipi}
\end{figure}


CDF has clean signals for fully hadronic, flavor-specific  $B_s$ decays,
providing the best sensitivity to $B_s$ oscillations at high
$\Delta m_s$  Figure \ref{fig:cdf_bsdsphipi} shows the signal for
the best channel, $B_s \rightarrow D_s \pi$, $D_s \rightarrow \phi \pi$ 

\clearpage


\subsection{Rare decays}

The purely leptonic decays $B_{d,s}^0 \rightarrow \mu^+
\mu^-$ are flavor-changing neutral current (FCNC) processes 
In the standard model, these decays are forbidden at the tree level and
proceed at a very low rate through higher-order diagrams 
The latest SM prediction~\cite{sm_ref3}
is ${\cal B}(B^0_s \rightarrow \mu^+ \mu^-)=(3 42\pm 0 54)\times
10^{-9}$, where the error is dominated by non-perturbative uncertainties  The
leptonic branching fraction of the $B_d^0$ decay is suppressed by CKM matrix elements $|V_{td}/V_{ts}|^2$
leading to a predicted SM branching fraction of $(1 00\pm0 14)\times 10^{-10}$ 
The best published experimental bound (Fig ~\ref{fig:cdf_bsmumu})
 for the branching fraction
of $B^0_s$ $(B^0_d)$ is presently
${\cal B}(B^0_s \, (B^0_d) \rightarrow \mu^+\mu^-)<7 5\times 10^{-7}\, 
(1 9\times 10^{-7})$ at the 95\% C L ~\cite{cdfII} 
The decay amplitude of $B^0_{d,s} \rightarrow \mu^+ \mu^-$ can be
significantly enhanced in some extensions of the SM  

\begin{figure}[htb]
\includegraphics[height=8 3cm,width=7 9cm]  {cdfbsmumu_results_prl eps}

\vspace*{-1cm}
\caption{Invariant mass for the events passing all requirements  (CDF)}
\label{fig:cdf_bsmumu}
\end{figure}


Assuming no contributions 
from the decay $B^0_d\rightarrow \mu^+\mu^-$ in the signal region,
D\O\  finds the conservative upper limit on the branching fraction 
to be ${\cal B}(B^0_s \rightarrow \mu^+ \mu^-) \leq 4 6\times 10^{-7}$ 
at the 95\% C L  (Fig ~\ref{fig:d0_bsmumu}) 






\begin{figure}[htb]
\includegraphics[height=5 0cm,width=8 0cm]  {B06F03 eps}
\vspace*{-1cm}
\caption{Invariant mass for the events  passing all requirements  (D\O)}
\label{fig:d0_bsmumu}
\end{figure}


Context 3: \section{Introduction}

Derivate is one of the most important topics not only in mathematics, but also in physics, chemistry, economics and engineering  Every standard Calculus course provides a variety of exercises for the students to learn how to apply the concept of derivative  The types of problems range from finding an equation of the tangent line to the application of differentials and advanced curve sketching  Usually, these exercises heavily rely on such differentiation techniques as Product, Quotient and Chain Rules, Implicit and Logarithmic Differentiation \cite{Stewart2012}  The definition of the derivative is hardly ever applied after the first few classes and its use is not much motivated 

Like many other topics in undergraduate mathematics, derivative gave rise to many misconceptions \cite{Muzangwa2012}, \cite{Gur2007}, \cite{Li2006}  Just when the students seem to learn how to use the differentiation rules for most essential functions, the application of the derivative brings new issues  A common students' error of determining the domain of the derivative from its formula is discussed in \cite{Rivera2013} and some interesting examples of the derivatives, defined at the points where the functions themselves are undefined, are provided  However, the hunt for misconceptions takes another twist for the derivatives undefined at the points where the functions are in fact defined 

The expression of the derivative of the function obtained using differentiation techniques does not necessarily contain the information about the existence or the value of the derivative at the points, where the expression for the derivative is undefined  In this article we discuss a type of continuous functions that have the expression for the derivative undefined at a certain point, while the derivative itself at that point exists  We show, how relying on the formula for the derivative for finding the horizontal tangent line of a function, leads to a false conclusion and consequently to missing a solution  We also provide a simple methodological treatment of similar functions suitable for the classroom 

\section{Calculating the Derivative}

In order to illustrate how deceitful the expression of the derivative can be to a students' eye, let us consider the following problem 

\vspace{12pt}

\fbox{\begin{minipage}{5 25in}

\begin{center}

\begin{minipage}{5 0in}

\vspace{10pt}

\emph{Problem}

\vspace{10pt}

Differentiate the function $f\left(x\right)=\sqrt[3]{x}\sin{\left(x^2\right)}$  For which values of $x$ from the interval $\left[-1,1\right]$ does the graph of $f\left(x\right)$ have a horizontal tangent?

\vspace{10pt}

\end{minipage}

\end{center}

\end{minipage}}

\vspace{12pt}

Problems with similar formulations can be found in many Calculus books \cite{Stewart2012}, \cite{Larson2010}, \cite{Thomas2009}  Following the common procedure, let us find the expression for the derivative of the function $f\left(x\right)$ applying the Product Rule:
\begin{eqnarray}
f'\left(x\right) &=& \left(\sqrt[3]{x}\right)'\sin{\left(x^2\right)}+\left(\sin{\left(x^2\right)}\right)'\sqrt[3]{x} \notag \\ &=& \frac{1}{3\sqrt[3]{x^2}}\sin{\left(x^2\right)}+2x\cos{\left(x^2\right)}\sqrt[3]{x} \notag \\ &=& \frac{6x^2\cos{x^2}+\sin{x^2}}{3\sqrt[3]{x^2}} \label{DerivativeExpression}
\end{eqnarray}

Similar to \cite{Stewart2012}, we find the values of $x$ where the derivative $f'\left(x\right)$ is equal to zero:
\begin{equation}
6x^2\cos{x^2}+\sin{x^2} = 0 
\label{DerivativeEqualZero}
\end{equation}

Since the expression for the derivative (\ref{DerivativeExpression}) is not defined at $x=0$, it is not hard to see that for all values of $x$ from $\left[-1,1\right]$ distinct from zero, the left-hand side of (\ref{DerivativeEqualZero}) is always positive  Hence, we conclude that the function $f\left(x\right)$ does not have horizontal tangent lines on the interval $\left[-1,1\right]$ 

However, a closer look at the graph of the function $f\left(x\right)$ seems to point at a different result: there is a horizontal tangent at $x=0$ (see Figure \ref{fig:FunctionGraph})  

First, note that the function $f\left(x\right)$ is defined in $x=0$  In order to verify if it has a horizontal tangent at this point, let us find the derivative of the function $f\left(x\right)$ using definition:
\begin{eqnarray}
f'\left(0\right) &=& \lim_{h\rightarrow0}{\frac{f\left(0+h\right)-f\left(0\right)}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\frac{\sqrt[3]{h}\sin{\left(h^2\right)}}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\left(\sqrt[3]{h} \cdot {h} \cdot \frac{\sin{\left(h^2\right)}}{h^2}\right)} \notag \\
&=& \lim_{h\rightarrow0}{\sqrt[3]{h}} \cdot \lim_{h\rightarrow0}{h} \cdot \lim_{h\rightarrow0}{\frac{\sin{\left(h^2\right)}}{h^2}} \notag \\
&=& 0 \cdot 0 \cdot 1 = 0 \notag
\end{eqnarray}
since each of the limits above exists



ground_truth: The paper aims to solve nonlinear system vibration problems efficiently.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: When did the Tevatron Collider Run II start and when is it expected to end?

Read the following text and answer briefly.

Context 3:  We see that, indeed, the function $f\left(x\right)$ possesses a horizontal tangent line at the point $x=0$ 

\section{Closer Look at the Expression for the Derivative}

What is the problem with the standard procedure proposed by many textbooks and repeated in every Calculus class? The explanation lies in the following premise: the expression of the derivative of the function does not contain the information as to whether the function is differentiable or not at the points where it is undefined  As it is pointed out in \cite{Rivera2013}, the domain of the derivative is determined \emph{a priori} and therefore should not be obtained from the formula of the derivative itself 

In the example above the Product Law for derivatives requires the existence of the derivatives of both functions at the point of interest  Since the function $\sqrt[3]{x}$ is not differentiable in zero, the Product Rule cannot be applied  

In order to see what exactly happens when we apply the Product Rule, let us find the expression for the derivative using definition of the derivative:
\begin{eqnarray}
f'\left(x\right) &=& \lim_{h\rightarrow0}{\frac{f\left(x+h\right)-f\left(x\right)}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\frac{\sqrt[3]{x+h}\sin{\left(x+h\right)^2}-\sqrt[3]{x}\sin{\left(x^2\right)}}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\frac{\left(\sqrt[3]{x+h}-\sqrt[3]{x}\right)}{h}\sin{\left(x^2\right)}} + \notag \\
&& \lim_{h\rightarrow0}{\frac{\left(\sin{\left(x+h\right)^2}-\sin{\left(x^2\right)}\right)}{h}\sqrt[3]{x+h}} \notag \\
&=& \lim_{h\rightarrow0}{\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{h}} \cdot \lim_{h\rightarrow0}{\sin{\left(x^2\right)}} +  \notag \\&& \lim_{h\rightarrow0}{\frac{\sin{\left(x+h\right)^2}-\sin{\left(x^2\right)}}{h}} \cdot \lim_{h\rightarrow0}{\sqrt[3]{x+h}} \notag \\
&=& \frac{1}{3\sqrt[3]{x^2}} \cdot \sin{\left(x^2\right)}+2x\cos{\left(x^2\right)} \cdot \sqrt[3]{x} \notag 
\end{eqnarray}
which seems to be identical to the expression (\ref{DerivativeExpression}) 

Students are expected to develop a skill of deriving similar results and know how to find the derivative of the function using definition of the derivative only  But how `legal' are the performed operations?

\begin{figure}[H]
\begin{center}
	\includegraphics[width=6 0in]{sin pdf}
	\vspace{ 1in}
	\caption{Graph of the function $g\left(x\right)=\sqrt[3]{x}\cos{\left(x^2\right)}$}
	\label{fig:GFunction}
\end{center}
\end{figure}

Let us consider each of the following limits: 
\begin{eqnarray*}
&& \lim_{h\rightarrow0}{\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{h}} \notag \\
&& \lim_{h\rightarrow0}{\sin{\left(x^2\right)}}\notag \\
&& \lim_{h\rightarrow0}{\frac{\sin{\left(x+h\right)^2}-\sin{\left(x^2\right)}}{h}}\notag \\
&& \lim_{h\rightarrow0}{\sqrt[3]{x+h}} 
\end{eqnarray*}
The last three limits exist for all real values of the variable $x$  However, the first limit does not exist when $x=0$  Indeed
\begin{equation*}
\lim_{h\rightarrow0}{\frac{\sqrt[3]{0+h}-\sqrt[3]{0}}{h}} = \lim_{h\rightarrow0}{\frac{1}{\sqrt[3]{h^2}}} = + \infty
\end{equation*}

This implies that the Product and Sum Laws for limits cannot be applied and therefore this step is not justifiable in the case of $x=0$  When the derivation is performed, we automatically assume the conditions, under which the Product Law for limits can be applied, i e  that both limits that are multiplied exist  It is not hard to see that in our case these conditions are actually equivalent to $x\neq0$  This is precisely why, when we wrote out the expression for the derivative (\ref{DerivativeExpression}), it already contained the assumption that it is only true for the values of $x$ that are different from zero 

Note, that in the case of $x=0$ the application of the Product and Sum Laws for limits is not necessary, since the term $\left(\sqrt[3]{x+h}-\sqrt[3]{x}\right)\sin{\left(x^2\right)}$ vanishes 

The correct expression for the derivative of the function $f\left(x\right)$ should be the following:
\begin{equation*}
f'\left(x\right)  = 
\begin{cases} 
\frac{6x^2\cos{\left(x^2\right)}+\sin{\left(x^2\right)}}{3\sqrt[3]{x^2}}, & \mbox{if } x \neq 0 \\ 
0, & \mbox{if } x = 0 
\end{cases}
\end{equation*}

The expression for the derivative of the function provides the correct value of the derivative only for those values of the independent variable, for which the expression is defined; it does not tell anything about the existence or the value of the derivative, where the expression for the derivative is undefined  Indeed, let us consider the function
\begin{equation*}
g\left(x\right) = {\sqrt[3]{x}}\cos{\left(x^2\right)}
\end{equation*}
and its derivative $g'\left(x\right)$ 
\begin{equation*}
g'\left(x\right) = \frac{\cos{\left(x^2\right)}-6x^2\sin{\left(x^2\right)}}{3\sqrt[3]{x^2}}
\end{equation*}

Similar to the previous example, the expression for the derivative is undefined at $x=0$  Nonetheless, it can be shown that $g\left(x\right)$ is not differentiable at $x=0$ (see Figure \ref{fig:GFunction})



ground_truth: The Tevatron Collider Run II started in March 2002 and is expected to continue until the end of this decade.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the correct expression for the derivative of the function?

Read the following text and answer briefly.

Context 3:  Therefore, we provided two visually similar functions: both have the expressions for their derivatives undefined in zero, however, one of these functions possesses a derivative, but the other one does not 

\section{Methodological Remarks}

Unfortunately, there exist many functions similar to the ones discussed above and they can arise in a variety of typical Calculus problems: finding the points where the tangent line is horizontal, finding an equation of the tangent and normal lines to the curve at the given point, the use of differentials and graph sketching  Relying only on the expression of the derivative for determining its value at the undefined points may lead to missing a solution (as in the example discussed above) or to some completely false interpretations (as in the case of curve sketching) 

As it was discussed above, the expression for the derivative does not provide any information on the existence or the value of the derivative, where the expression itself is undefined  Here we present a methodology for the analysis of this type of functions 

Let $f\left(x\right)$ be the function of interest and $f'\left(x\right)$ be the expression of its derivative undefined at some point $x_{0}$  In order to find out if $f\left(x\right)$ is differentiable at $x_{0}$, we suggest to follow a list of steps:

\begin{enumerate}
  \item Check if the function $f\left(x\right)$ itself is defined at the point $x_{0}$  If $f\left(x\right)$ is undefined at $x_{0}$, then it is not differentiable at $x_{0}$  If $f\left(x\right)$ is defined at $x_{0}$, then proceed to next step  
  \item Identify the basic functions that are used in the formula of the function $f\left(x\right)$, that are themselves defined at the point $x_{0}$, but their derivative is not (such as, for example, the root functions) 
	\item Find the derivative of the function $f\left(x\right)$ at the point $x_{0}$ using definition 
\end{enumerate}

The importance of the first step comes from the fact that most students tend to pay little attention to the functions domain analysis when asked to investigate its derivative  Formally, the second step can be skipped, however it will give the students the insight into which part of the function presents a problem and teach them to identify similar cases in the future  the difficulty of accomplishing the third step depends on the form of the function and sometimes can be tedious  Nevertheless, it allows the students to apply the previously obtained skills and encourages the review of the material 

\begin{figure}[H]
\begin{center}
	\includegraphics[width=6 0in]{cos pdf}
	\vspace{ 1in}
	\caption{Graph of the function $g\left(x\right)=\sqrt[3]{x}\cos{\left(x^2\right)}$}
	\label{fig:GFunction}
\end{center}
\end{figure}

\section{Conclusion}

We discussed the misconception, that the expression of the derivative of the function contains the information as to whether the function is differentiable or not at the points, where the expression is undefined  We considered a typical Calculus problem of looking for the horizontal tangent line of a function as an example  We showed how the search for the values that make the expression of the derivative equal zero leads to missing a solution: even though the expression of the derivative is undefined, the function still possesses the derivative at the point  We provided an example of the function that similarly has the expression for the derivative undefined, however the function is not differentiable at the point  We also presented the methodological treatment of such functions by applying the definition of the derivative, which can be used in the classroom 





ground_truth: It depends on the value of x, either 0 or (6x^2cos(x^2)+sin(x^2))/(3(x^2)^(1/3)).