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 genre did Margaret Way write in?

Read the following text and answer briefly.

Context 1: Margaret Way (b  Brisbane d  Cleveland, Queensland, Australia ) was an Australian writer of romance novels and women's fiction  A prolific author, Way wrote more than 120 novels since 1970, many through Mills & Boon, a romance imprint of British publisher Harlequin UK Ltd , owned by Harlequin Enterprises 

Biography
Before her marriage, she was a well-known pianist, teacher, vocal coach and accompanist  She began writing when her son, Laurence Way, was born, a friend took a pile of Mills & Boon books to her, she read all and decided that she also could write these types of novels  She began to write and promote her country with her stories set in Australia  She sold her first novels in 1970  Margaret Way lives with her family in her native Brisbane  Beginning in 2013, Margaret began to self-publish, releasing her first "e-book" mid-July 

Margaret died on the 10th of August 2022 in Cleveland, Queensland 

Bibliography

Single Novels
King Country (1970)
Blaze of Silk (1970)
The Time of the Jacaranda (1970)
Bauhinia Junction (1971)
Man from Bahl Bahla (1971)
Summer Magic (1971)
Return to Belle Amber (1971)
Ring of Jade (1972)
Copper Moon (1972)
Rainbow Bird (1972)
Man Like Daintree (1972)
Noonfire (1972)
Storm Over Mandargi (1973)
Wind River (1973)
Love Theme (1974)
McCabe's Kingdom (1974)
Sweet Sundown (1974)
Reeds of Honey (1975)
Storm Flower (1975)
Lesson in Loving (1975)
Flight into Yesterday (1976)
Red Cliffs of Malpara (1976)
Man on Half-moon (1976)
Swan's Reach (1976)
Mutiny in Paradise (1977)
One Way Ticket (1977)
Portrait of Jaime (1977)
Black Ingo (1977)
Awakening Flame (1978)
Wild Swan (1978)
Ring of Fire (1978)
Wake the Sleeping Tiger (1978)
Valley of the Moon (1979)
White Magnolia (1979)
Winds of Heaven (1979)
Blue Lotus (1979)
Butterfly and the Baron (1979)
Golden Puma (1980)
Temple of Fire (1980)
Lord of the High Valley (1980)
Flamingo Park (1980)
North of Capricorn (1981)
Season for Change (1981)
Shadow Dance (1981)
McIvor Affair (1981)
Home to Morning Star (1981)
Broken Rhapsody (1982)
The Silver Veil (1982)
Spellbound (1982)
Hunter's Moon (1982)
Girl at Cobalt Creek (1983)
No Alternative (1983)
House of Memories (1983)
Almost a Stranger (1984)
A place called Rambulara (1984)
Fallen Idol (1984)
Hunt the Sun (1985)
Eagle's Ridge (1985)
The Tiger's Cage (1986)
Innocent in Eden (1986)
Diamond Valley (1986)
Morning Glory (1988)
Devil Moon (1988)
Mowana Magic (1988)
Hungry Heart (1988)
Rise of an Eagle (1988)
One Fateful Summer (1993)
The Carradine Brand (1994)
Holding on to Alex (1997)
The Australian Heiress (1997)
Claiming His Child (1999)
The Cattleman's Bride (2000)
The Cattle Baron (2001)
The Husbands of the Outback (2001)
Secrets of the Outback (2002)
With This Ring (2003)
Innocent Mistress (2004)
Cattle Rancher, Convenient Wife (2007)
Outback Marriages (2007)
Promoted: Nanny to Wife (2007)
Cattle Rancher, Secret Son (2007)
Genni's Dilemma (2008)
Bride At Briar Ridge (2009)
Outback Heiress, Surprise Proposal (2009)
Cattle Baron, Nanny Needed (2009)

Legends of the Outback Series
Mail Order Marriage (1999)
The Bridesmaid's Wedding (2000)
The English Bride (2000)
A Wife at Kimbara (2000)

Koomera Crossing Series
Sarah's Baby (2003)
Runaway Wife (2003)
Outback Bridegroom (2003)
Outback Surrender (2003)
Home to Eden (2004)

McIvor Sisters Series
The Outback Engagement (2005)
Marriage at Murraree (2005)

Men Of The Outback Series
The Cattleman (2006)
The Cattle Baron's Bride (2006)
Her Outback Protector (2006)
The Horseman (2006)

Outback Marriages Series
Outback Man Seeks Wife (2007)
Cattle Rancher, Convenient Wife (2007)

Barons of the Outback Series Multi-Author
Wedding At Wangaree Valley (2008)
Bride At Briar's Ridge (2008)

Family Ties Multi-Author
Once Burned (1995)

Hitched! Multi-Author
A Faulkner Possession (1996)

Simply the Best Multi-Author
Georgia and the Tycoon (1997)

The Big Event Multi-Author
Beresford's Bride (1998)

Guardian Angels Multi-Author
Gabriel's Mission (1998)

Australians Series Multi-Author
7  Her Outback Man (1998)
17  Master of Maramba (2001)
19  Outback Fire (2001)
22  Mistaken Mistress (2002)
24  Outback Angel (2002)
33  The Australian Tycoon's Proposal (2004)
35  His Heiress Wife (2004)

Marrying the Boss Series Multi-Author
Boardroom Proposal (1999)

Contract Brides Series Multi-Author
Strategy for Marriage (2002)

Everlasting Love Series Multi-Author
Hidden Legacy (2008)

Diamond Brides Series Multi-Author
The Australian's Society Bride (2008)

Collections
Summer Magic / Ring of Jade / Noonfire (1981)
Wife at Kimbara / Bridesmaid's Wedding (2005)

Omnibus in Collaboration
Pretty Witch / Without Any Amazement / Storm Over Mandargi (1977) (with Lucy Gillen and Margaret Malcolm)
Dear Caliban / Heart of the Eagle / Swans' Reach (1978) (with Jane Donnelly and Elizabeth Graham)
The Bonds of Matrimony / Dragon Island / Reeds of Honey (1979) (with Elizabeth Hunter and Henrietta Reid)
The Man Outside / Castles in Spain / McCabe's Kingdom (1979) (with Jane Donnelly and Rebecca Stratton)
Winds From The Sea / Island of Darkness / Wind River (1979) (with Margaret Pargeter and Rebecca Stratton)
Moorland Magic / Tree of Idleness / Sweet Sundown (1980) (with Elizabeth Ashton and Elizabeth Hunter)
The Shifting Sands / Portrait of Jaime / Touched by Fire (1982) (with Jane Donnelly and Kay Thorpe)
Head of Chancery / Wild Heart / One-Way Ticket (1986) (with Betty Beaty and Doris Smith)
Heart of the Scorpion / The Winds of Heaven / Sweet Compulsion (1987) (with Janice Gray and Victoria Woolf)
One Brief Sweet Hour / Once More With Feeling / Blue Lotus (1990) (with Jane Arbor and Natalie Sparks)
Marry Me Cowboy (1995) (with Janet Dailey, Susan Fox and Anne McAllister)
Husbands on Horseback (1996) (with Diana Palmer)
Wedlocked (1999) (with Day Leclaire and Anne McAllister)
Mistletoe Magic (1999) (with Betty Neels and Rebecca Winters)
The Australians (2000) (with Helen Bianchin and Miranda Lee)
Weddings Down Under (2001) (with Helen Bianchin and Jessica Hart)
Outback Husbands (2002) (with Marion Lennox)
The Mother's Day Collection (2002) (with Helen Dickson and Kate Hoffmann)
Australian Nights (2003) (with Miranda Lee)
Outback Weddings (2003) (with Barbara Hannay)
Australian Playboys (2003) (with Helen Bianchin and Marion Lennox)
Australian Tycoons (2004) (with Emma Darcy and Marion Lennox)
A Mother's Day Gift (2004) (with Anne Ashley and Lucy Monroe)
White Wedding (2004) (with Judy Christenberry and Jessica Steele)
A Christmas Engagement (2004) (with Sara Craven and Jessica Matthews)
A Very Special Mother's Day (2005) (with Anne Herries)
All I Want for Christmas    (2005) (with Betty Neels and Jessica Steele)
The Mills and Boon Collection (2006) (with Caroline Anderson and Penny Jordan)
Outback Desire (2006) (with Emma Darcy and Carol Marinelli)
To Mum, with Love (2006) (with Rebecca Winters)
Australian Heroes (2007) (with Marion Lennox and Fiona McArthur)
Tall, Dark and Sexy (2008) (with Caroline Anderson and Helen Bianchin)
The Boss's Proposal (2008) (with Jessica Steele and Patricia Thayer)
Island Heat / Outback Man Seeks Wife / Prince's Forbidden Virgin / One Night Before Marriage / Their Lost-and-found Family / Single Dad's Marriage Wish (2008) (with Robyn Donald, Marion Lennox, Carol Marinelli, Sarah Mayberry and Anne Oliver)
Australian Billionaires (2009) (with Jennie Adams and Amy Andrews)
Cattle Baron : Nanny Needed / Bachelor Dad on Her Doorstep (2009) (with Michelle Douglas)

External links
Margaret Way at Harlequin Enterprises Ltd

Australian romantic fiction writers
Australian women novelists
Living people
Year of birth missing (living people)
Women romantic fiction writers



ground_truth: Romance novels and women's fiction.

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 does the paper aim to solve?

Read the following text and answer briefly.

Context 2: 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 3: Margaret Way (b  Brisbane d  Cleveland, Queensland, Australia ) was an Australian writer of romance novels and women's fiction  A prolific author, Way wrote more than 120 novels since 1970, many through Mills & Boon, a romance imprint of British publisher Harlequin UK Ltd , owned by Harlequin Enterprises 

Biography
Before her marriage, she was a well-known pianist, teacher, vocal coach and accompanist  She began writing when her son, Laurence Way, was born, a friend took a pile of Mills & Boon books to her, she read all and decided that she also could write these types of novels  She began to write and promote her country with her stories set in Australia  She sold her first novels in 1970  Margaret Way lives with her family in her native Brisbane  Beginning in 2013, Margaret began to self-publish, releasing her first "e-book" mid-July 

Margaret died on the 10th of August 2022 in Cleveland, Queensland 

Bibliography

Single Novels
King Country (1970)
Blaze of Silk (1970)
The Time of the Jacaranda (1970)
Bauhinia Junction (1971)
Man from Bahl Bahla (1971)
Summer Magic (1971)
Return to Belle Amber (1971)
Ring of Jade (1972)
Copper Moon (1972)
Rainbow Bird (1972)
Man Like Daintree (1972)
Noonfire (1972)
Storm Over Mandargi (1973)
Wind River (1973)
Love Theme (1974)
McCabe's Kingdom (1974)
Sweet Sundown (1974)
Reeds of Honey (1975)
Storm Flower (1975)
Lesson in Loving (1975)
Flight into Yesterday (1976)
Red Cliffs of Malpara (1976)
Man on Half-moon (1976)
Swan's Reach (1976)
Mutiny in Paradise (1977)
One Way Ticket (1977)
Portrait of Jaime (1977)
Black Ingo (1977)
Awakening Flame (1978)
Wild Swan (1978)
Ring of Fire (1978)
Wake the Sleeping Tiger (1978)
Valley of the Moon (1979)
White Magnolia (1979)
Winds of Heaven (1979)
Blue Lotus (1979)
Butterfly and the Baron (1979)
Golden Puma (1980)
Temple of Fire (1980)
Lord of the High Valley (1980)
Flamingo Park (1980)
North of Capricorn (1981)
Season for Change (1981)
Shadow Dance (1981)
McIvor Affair (1981)
Home to Morning Star (1981)
Broken Rhapsody (1982)
The Silver Veil (1982)
Spellbound (1982)
Hunter's Moon (1982)
Girl at Cobalt Creek (1983)
No Alternative (1983)
House of Memories (1983)
Almost a Stranger (1984)
A place called Rambulara (1984)
Fallen Idol (1984)
Hunt the Sun (1985)
Eagle's Ridge (1985)
The Tiger's Cage (1986)
Innocent in Eden (1986)
Diamond Valley (1986)
Morning Glory (1988)
Devil Moon (1988)
Mowana Magic (1988)
Hungry Heart (1988)
Rise of an Eagle (1988)
One Fateful Summer (1993)
The Carradine Brand (1994)
Holding on to Alex (1997)
The Australian Heiress (1997)
Claiming His Child (1999)
The Cattleman's Bride (2000)
The Cattle Baron (2001)
The Husbands of the Outback (2001)
Secrets of the Outback (2002)
With This Ring (2003)
Innocent Mistress (2004)
Cattle Rancher, Convenient Wife (2007)
Outback Marriages (2007)
Promoted: Nanny to Wife (2007)
Cattle Rancher, Secret Son (2007)
Genni's Dilemma (2008)
Bride At Briar Ridge (2009)
Outback Heiress, Surprise Proposal (2009)
Cattle Baron, Nanny Needed (2009)

Legends of the Outback Series
Mail Order Marriage (1999)
The Bridesmaid's Wedding (2000)
The English Bride (2000)
A Wife at Kimbara (2000)

Koomera Crossing Series
Sarah's Baby (2003)
Runaway Wife (2003)
Outback Bridegroom (2003)
Outback Surrender (2003)
Home to Eden (2004)

McIvor Sisters Series
The Outback Engagement (2005)
Marriage at Murraree (2005)

Men Of The Outback Series
The Cattleman (2006)
The Cattle Baron's Bride (2006)
Her Outback Protector (2006)
The Horseman (2006)

Outback Marriages Series
Outback Man Seeks Wife (2007)
Cattle Rancher, Convenient Wife (2007)

Barons of the Outback Series Multi-Author
Wedding At Wangaree Valley (2008)
Bride At Briar's Ridge (2008)

Family Ties Multi-Author
Once Burned (1995)

Hitched! Multi-Author
A Faulkner Possession (1996)

Simply the Best Multi-Author
Georgia and the Tycoon (1997)

The Big Event Multi-Author
Beresford's Bride (1998)

Guardian Angels Multi-Author
Gabriel's Mission (1998)

Australians Series Multi-Author
7  Her Outback Man (1998)
17  Master of Maramba (2001)
19  Outback Fire (2001)
22  Mistaken Mistress (2002)
24  Outback Angel (2002)
33  The Australian Tycoon's Proposal (2004)
35  His Heiress Wife (2004)

Marrying the Boss Series Multi-Author
Boardroom Proposal (1999)

Contract Brides Series Multi-Author
Strategy for Marriage (2002)

Everlasting Love Series Multi-Author
Hidden Legacy (2008)

Diamond Brides Series Multi-Author
The Australian's Society Bride (2008)

Collections
Summer Magic / Ring of Jade / Noonfire (1981)
Wife at Kimbara / Bridesmaid's Wedding (2005)

Omnibus in Collaboration
Pretty Witch / Without Any Amazement / Storm Over Mandargi (1977) (with Lucy Gillen and Margaret Malcolm)
Dear Caliban / Heart of the Eagle / Swans' Reach (1978) (with Jane Donnelly and Elizabeth Graham)
The Bonds of Matrimony / Dragon Island / Reeds of Honey (1979) (with Elizabeth Hunter and Henrietta Reid)
The Man Outside / Castles in Spain / McCabe's Kingdom (1979) (with Jane Donnelly and Rebecca Stratton)
Winds From The Sea / Island of Darkness / Wind River (1979) (with Margaret Pargeter and Rebecca Stratton)
Moorland Magic / Tree of Idleness / Sweet Sundown (1980) (with Elizabeth Ashton and Elizabeth Hunter)
The Shifting Sands / Portrait of Jaime / Touched by Fire (1982) (with Jane Donnelly and Kay Thorpe)
Head of Chancery / Wild Heart / One-Way Ticket (1986) (with Betty Beaty and Doris Smith)
Heart of the Scorpion / The Winds of Heaven / Sweet Compulsion (1987) (with Janice Gray and Victoria Woolf)
One Brief Sweet Hour / Once More With Feeling / Blue Lotus (1990) (with Jane Arbor and Natalie Sparks)
Marry Me Cowboy (1995) (with Janet Dailey, Susan Fox and Anne McAllister)
Husbands on Horseback (1996) (with Diana Palmer)
Wedlocked (1999) (with Day Leclaire and Anne McAllister)
Mistletoe Magic (1999) (with Betty Neels and Rebecca Winters)
The Australians (2000) (with Helen Bianchin and Miranda Lee)
Weddings Down Under (2001) (with Helen Bianchin and Jessica Hart)
Outback Husbands (2002) (with Marion Lennox)
The Mother's Day Collection (2002) (with Helen Dickson and Kate Hoffmann)
Australian Nights (2003) (with Miranda Lee)
Outback Weddings (2003) (with Barbara Hannay)
Australian Playboys (2003) (with Helen Bianchin and Marion Lennox)
Australian Tycoons (2004) (with Emma Darcy and Marion Lennox)
A Mother's Day Gift (2004) (with Anne Ashley and Lucy Monroe)
White Wedding (2004) (with Judy Christenberry and Jessica Steele)
A Christmas Engagement (2004) (with Sara Craven and Jessica Matthews)
A Very Special Mother's Day (2005) (with Anne Herries)
All I Want for Christmas    (2005) (with Betty Neels and Jessica Steele)
The Mills and Boon Collection (2006) (with Caroline Anderson and Penny Jordan)
Outback Desire (2006) (with Emma Darcy and Carol Marinelli)
To Mum, with Love (2006) (with Rebecca Winters)
Australian Heroes (2007) (with Marion Lennox and Fiona McArthur)
Tall, Dark and Sexy (2008) (with Caroline Anderson and Helen Bianchin)
The Boss's Proposal (2008) (with Jessica Steele and Patricia Thayer)
Island Heat / Outback Man Seeks Wife / Prince's Forbidden Virgin / One Night Before Marriage / Their Lost-and-found Family / Single Dad's Marriage Wish (2008) (with Robyn Donald, Marion Lennox, Carol Marinelli, Sarah Mayberry and Anne Oliver)
Australian Billionaires (2009) (with Jennie Adams and Amy Andrews)
Cattle Baron : Nanny Needed / Bachelor Dad on Her Doorstep (2009) (with Michelle Douglas)

External links
Margaret Way at Harlequin Enterprises Ltd

Australian romantic fiction writers
Australian women novelists
Living people
Year of birth missing (living people)
Women romantic fiction writers



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 Margaret Way start self-publishing her books as e-books?

Read the following text and answer briefly.

Context 4: 338\,GHz with an angular resolution of $2 9''\times 1 4''$
(corresponding to an average linear resolution of 7700\,AU at the
given distance of 3 6\,kpc)  Figure \ref{18089} presents the
integrated C$_2$H emission with a contour overlay of the 860\,$\mu$m
continuum source outlining the position of the massive protostar  In
contrast to almost all other molecular lines that peak along with the
dust continuum \citep{beuther2005c}, the C$_2$H emission surrounds the
continuum peak in a shell-like fashion 

\section{Discussion and Conclusions}

To understand the observations, we conducted a simple chemical
modeling of massive star-forming regions  A 1D cloud model with a mass
of 1200\,M$_\sun$, an outer radius of 0 36\,pc and a power-law density
profile ($\rho\propto r^p$ with $p=-1 5$) is the initially assumed
configuration  Three cases are studied: (1) a cold isothermal cloud
with $T=10$\,K, (2) $T=50$\,K, and (3) a warm model with a temperature
profile $T\propto r^q$ with $q=-0 4$ and a temperature at the outer
radius of 44\,K  The cloud is illuminated by the interstellar UV
radiation field (IRSF, \citealt{draine1978}) and by cosmic ray
particles (CRP)  The ISRF attenuation by single-sized $0 1\mu$m
silicate grains at a given radius is calculated in a plane-parallel
geometry following \citet{vandishoeck1988}  The CRP ionization rate is
assumed to be $1 3\times 10^{-17}$~s$^{-1}$ \citep{spitzer1968}  The
gas-grain chemical model by \citet{vasyunin2008} with the desorption
energies and surface reactions from \citet{garrod2006} is used 
Gas-phase reaction rates are taken from RATE\,06 \citep{woodall2007},
initial abundances, were adopted from the ``low metal'' set of
\citet{lee1998} 

Figure \ref{model} presents the C$_2$H abundances for the three models
at two different time steps: (a) 100\,yr, and (b) in a more evolved
stage after $5\times10^4$\,yr  The C$_2$H abundance is high toward the
core center right from the beginning of the evolution, similar to
previous models (e g , \citealt{millar1985,herbst1986,turner1999}) 
During the evolution, the C$_2$H abundance stays approximately
constant at the outer core edges, whereas it decreases by more than
three orders of magnitude in the center, except for the cold $T=10$~K
model   The C$_2$H abundance profiles for all three models show
similar behavior 

The chemical evolution of ethynyl is determined by relative removal
rates of carbon and oxygen atoms or ions into molecules like CO, OH,
H$_2$O  Light ionized hydrocarbons CH$^+_{\rm n}$ (n=2  5) are quickly
formed by radiative association of C$^+$ with H$_2$ and hydrogen
addition reactions: C$^+$ $\rightarrow$ CH$_2^+$ $\rightarrow$
CH$_3^+$ $\rightarrow$ CH$_5^+$   The protonated methane reacts with
electrons, CO, C, OH, and more complex species at later stage and
forms methane   The CH$_4$ molecules undergo reactive collisions with
C$^+$, producing C$_2$H$_2^+$ and C$_2$H$_3^+$  An alternative way to
produce C$_2$H$_2^+$ is the dissociative recombination of CH$_5^+$
into CH$_3$ followed by reactions with C$^+$   Finally, C$_2$H$_2^+$
and C$_2$H$_3^+$ dissociatively recombine into CH, C$_2$H, and
C$_2$H$_2$  The major removal for C$_2$H is either the direct
neutral-neutral reaction with O that forms CO, or the same reaction
but with heavier carbon chain ions that are formed from C$_2$H by
subsequent insertion of carbon  At later times, depletion and
gas-phase reactions with more complex species may enter into this
cycle   At the cloud edge the interstellar UV radiation
instantaneously dissociates CO despite its self-shielding,
re-enriching the gas with elemental carbon 

The transformation of C$_2$H into CO and other species proceeds
efficiently in dense regions, in particular in the ``warm'' model
where endothermic reactions result in rich molecular complexity of the
gas (see Fig ~\ref{model})   In contrast, in the ``cold'' 10\,K model
gas-grain interactions and surface reactions become important   As a
result, a large fraction of oxygen is locked in water ice that is hard
to desorb ($E_{\rm des} \sim 5500$~K), while half of the elemental
carbon goes to volatile methane ice ($E_{\rm des} \sim 1300$~K)  Upon
CRP heating of dust grains, this leads to much higher gas-phase
abundance of C$_2$H in the cloud core for the cold model compared to
the warm model  The effect is not that strong for less dense regions
at larger radii from the center 

Since the C$_2$H emission is anti-correlated with the dust continuum
emission in the case of IRAS\,18089-1732 (Fig \,\ref{18089}), we do
not have the H$_2$ column densities to quantitatively compare the
abundance profiles of IRAS\,18089-1732 with our model  However, data
and model allow a qualitative comparison of the spatial structures 
Estimating an exact evolutionary time for IRAS\,18089-1732 is hardly
possible, but based on the strong molecular line emission, its high
central gas temperatures and the observed outflow-disk system
\citep{beuther2004a,beuther2004b,beuther2005c}, an approximate age of
$5\times10^4$\,yr appears reasonable   Although dynamical and chemical
times are not necessarily exactly the same, in high-mass star
formation they should not differ to much: Following the models by
\citet{mckee2003} or \citet{krumholz2006b}, the luminosity rises
strongly right from the onset of collapse which can be considered as a
starting point for the chemical evolution  At the same time disks and
outflows evolve, which should hence have similar time-scales   The
diameter of the shell-like C$_2$H structure in IRAS\,18089-1732 is
$\sim 5''$ (Fig \,\ref{18089}), or $\sim$9000\,AU in radius at the
given distance of 3 6\,kpc   This value is well matched by the modeled
region with decreased C$_2$H abundance (Fig \,\ref{model})   Although
in principle optical depths and/or excitation effects could mimic the
C$_2$H morphology, we consider this as unlikely because the other
observed molecules with many different transitions all peak toward the
central submm continuum emission in IRAS\,18089-1732
\citep{beuther2005c}  Since C$_2$H is the only exception in that rich
dataset, chemical effects appear the more plausible explanation 

The fact that we see C$_2$H at the earliest and the later evolutionary
stages can be explained by the reactive nature of C$_2$H: it is
produced quickly early on and gets replenished at the core edges by
the UV photodissociation of CO  The inner ``chemical'' hole observed
toward IRAS\,18089-1732 can be explained by C$_2$H being consumed in
the chemical network forming CO and more complex molecules like larger
carbon-hydrogen complexes and/or depletion 

The data show that C$_2$H is not suited to investigate the central gas
cores in more evolved sources, however, our analysis indicates that
C$_2$H may be a suitable tracer of the earliest stages of (massive)
star formation, like N$_2$H$^+$ or NH$_3$ (e g ,
\citealt{bergin2002,tafalla2004,beuther2005a,pillai2006})  While a
spatial analysis of the line emission will give insights into the
kinematics of the gas and also the evolutionary stage from chemical
models, multiple C$_2$H lines will even allow a temperature
characterization  With its lowest $J=1-0$ transitions around 87\,GHz,
C$_2$H has easily accessible spectral lines in several bands between
the 3\,mm and 850\,$\mu$m   Furthermore, even the 349\,GHz lines
presented here have still relatively low upper level excitation
energies ($E_u/k\sim42$\,K), hence allowing to study cold cores even
at sub-millimeter wavelengths   This prediction can further be proved
via high spectral and spatial resolution observations of different
C$_2$H lines toward young IRDCs 

\acknowledgments{H B  acknowledges financial support
  by the Emmy-Noether-Programm of the Deutsche Forschungsgemeinschaft
  (DFG, grant BE2578)  }






ground_truth: Margaret Way started self-publishing her books as e-books in 2013.