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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 
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87  Powers to moke rules 
89  Power to lay the rules and regulations 
The Himachal Pradesh Town and Country Planning Act, 1977
(as amended by Amendment Act No  22 of 1983)
Amended by Act No  8 of 2009
Act published in the Rajpatra, Extraordinary, dated the 30th September, 1977 vide Law Department Notification No  LLR-D(6)5/77, dated the 22nd September, 1977 
An Act to make provision for planning and development and use of land; to make better provision for the preparation of development plans and sectoral plans with a view to ensuring that town planning schemes are made in a proper manner and their execution is made effective; to constitute the Town and Country and Development Authority for proper implementation of town and country development plan; to provide for the development and administration of special areas through the Special Area Development Authority; to make provision for the compulsory acquisition of land required for the purpose of the development plans and for purposes connected with the matters aforesaid 
Be it enacted by the Himachal Pradesh Legislative Assembly in the Twenty-eighth Year of the Republic of India as follows:-
1  Short title, extent, commencement and application  - (1) This Act may be called the Himachal Pradesh Town and Country Planning Act, 1977 
(3) It shall come into force on such date as the State Government may, by notification, appoint and different dates may be appointed for different areas and for different provisions of this Act 
(4) Nothing in this Act shall apply to-
(a) lands comprised within a cantonment under the Cantonments Act, 1924; (2 of 1924) 
(b) lands owned, hired or requisitioned by the Central Government for the purpose of naval, military and air force works;
(c) lands under the control of railway administration for the purpose of construction and maintenance of works under Chapter III of the Indian Railways Act, 1890; (9 of 1890) and
(d) lands owned by any department of the Central Government where operational constructions are going on 
(a) "agriculture" includes horticulture, farming, raising of annual or periodical crops, fruits, vegetables, flowers, grass, fodder, trees or any kind of cultivation of soil, the reserving of land for fodder, grazing or thatching areas, breeding and keeping of livestock including cattle, horses, donkeys, mules, pigs, breeding of fish and keeping of bees, and the use of land ancillary to the farming of land, but does not include-
(i) keeping of cattle purely for the purpose of milking and selling the milk and milk products,
(ii) a garden which is an appendage of buildings, and the expression "agricultural" shall be construed accordingly;
(b) "amenity" includes roads and streets, water and electric supply, open spaces, parks, recreational area, natural feature, playgrounds, street lighting, drainage, sewerage and other utilities, services and conveniences;
(c) "building" includes any structure or erection, or part of a structure or erection, which is intended to be used for residential, industrial, commercial or other purposes, whether in actual use or not;
(d) "building operation" includes-
(i) erection or re-erection of a building or any part thereof,
(ii) roofing or re-roofing of any part of building or an open space,
(iii) any material alteration or enlargement of a building,
(iv) any such alteration of a building as is likely to alter its drainage or sanitary arrangements, or materially affect its security,
(v) the construction of a door opening on any street or land not belonging to the owner;
(e) "commercial use" means the use of any land or building or part thereof for the purpose of carrying on any trade, business or profession, or sale or exchange of goods of any type whatsoever and includes running of with a view to make profit hospitals, nursing homes, infirmaries, educational institutions, hostels, restaurants and boarding houses not being attached to any educational institutions, sarais and also includes the use of any land or building for storage of goods or as buildings for storage of goods or as an office whether attached to an industry or otherwise;
(f) "court" means the principal civil court of original jurisdiction in the district;
(g) "development" with its grammatical variations means the carrying out of a building, engineering, mining or other operations in, on, over or under land, or the making of any material change in any building or land or in the use of either, and includes sub-division of any land;
(h) "development plan" means interim development plan or development plan prepared under this Act;
(i) "director" means the Director of Town and Country Planning appointed under this Act;
(j) "existing land use map� means a map indicating the use to which lands in any specified area are put at the time of preparing the map, and includes the register prepared, with the map giving details of land-use;
(k) "land" includes benefits to arise out of land and things attached to the earth or permanently fastened to anything attached to the earth;
(l) "member" means a member of a Town and Country Development Authority or a Special Area Development Authority, as the case may be, and includes a Chairman thereof;
(m) "occupier" includes-
(i) a tenant,
(ii) an owner in occupation of or otherwise using his land,
(iii) a rent free tenant,
(iv) a licensee, and
(v) any person liable to pay to the owner, damages for the use and occupation of the land;
(n) "owner" includes a mortgagee in possession, a person who for the time being is receiving or is entitled to receive, or has received, the rent or premium for any land whether on his own account or on behalf of or for the further benefit of any other person or as an agent, trustee, guardian or receiver for any other person or for religious or charitable institutions or who would receive the rent or be entitled to receive the rent or premium if the land were to be let and includes a head of a Government department, General Manager of a Railway and the Chief Executive Officer, by whatever name designated, or a local authority, statutory authority, company, corporation or undertaking in respect of properties under their control;
(o) "planning area" means any area declared to be planning area under this Act;
(p) "region" means any area established to be a region under this Act;
(q) "regional plan" means a plan for the region prepared under this Act and approved by the State Government;
(r) "sector" means any sector of a planning area for which, under the development plan, a detailed sectoral plan is prepared;
(s) "slum area" means any predominantly residential area, where the dwellings which by reason of dilapidation, over-crowding, faulty arrangement of design, lack of ventilation, light or sanitary facilities or any combination of these factors are detrimental to safety, health or moral and which is defined by a development plan as a slum area;
(t) "special area" means a special area designated as such under section 66;
(u) "Special Area Development Authority� means an authority constituted under section 67;
(v) "Town Development Scheme" means a scheme prepared for the implementation of the provisions of a development plan by the Town and Country Development Authority; and
(w) "Town and Country Development Authority" means an authority established under section 40 
Director of Town and Country Planning
3  Director and other officers  - (1) After the commencement of this Act the State Government shall, by notification in the Official Gazette, appoint an officer for the purpose of carrying out functions assigned to him under this Act, as the Director of Town and Country Planning for the State and may appoint such other categories of officers as it may deem fit 
(2) The Director shall exercise such powers and perform such duties as are conferred or imposed upon him by or under this Act and the officers appointed to assist the Director shall, within such area as the State Government may specify, exercise such powers and perform such duties conferred and imposed on the Director by or under this Act as the State Government may, by special or general order, direct 
(3) The officers appointed to assist the Director shall be subordinate to him and shall work under his guidance, supervision and control 
4  Establishment of regions  - (1) The State Government may, by notification,-
(a) declare any area in the State to be a region for the purposes of this Act;
(b) define the limits of such area; and
(c) specify the name by which such region shall be known 
(2) The State Government may, by notification, alter the name of any such region and on such alteration, any reference in any law or instrument or other document to the region shall be deemed to be a reference to the region as re-named unless expressly otherwise provided or the context so requires 
(3) The State Government may, by notification,-
(a) alter the limits of a region so as to include therein or exclude therefrom Such area as may be specified in the notification;
(b) amalgamate two or more regions so as to form one region;
(c) divide any region into two or more region; or
(d) declare that the whole or part of the area comprising a region shall cease to be a region or part thereof 
5  Director to prepare regional plan  - Subject to the provisions of this Act and the rules made thereunder, it shall be the duty of the Director-
(i) to carry out a survey of the regions;
(ii) to prepare an existing land use map; and
(iii) to prepare a regional plan 
6  Survey  - (1) The Director shall, with a view to prepare the existing land use map, and other maps as are necessary for the purpose of regional plan,-
(a) carry out such surveys as may be necessary;
(b) obtain from any department of Government and any local authority such maps, survey reports and land records as may be necessary for the purpose 
(2) It shall be the duty of every Government department and local authority to furnish, as soon as may be possible, maps, reports and record, as may be required by the Director 
7  Contents of regional plan  - The regional plan shall indicate the manner in which land in the region should be used, the phasing of development, the net work of communications and transport, the proposals for conservation and development of natural resources, and in particular-
(a) allocation of land to such purposes as residential, industrial; agricultural or as forests or for mineral exploitation;
(b) reservation of open spaces for recreational purposes, gardens, tree belts, and animal sanctuaries;
(c) access or development of transport and communication facilities such as roads, railways, water ways, and the allocation and development of airports;
(d) requirements and suggestions for development of public utilities such as water supply, drainage and electricity;
(e) allocation of areas to be developed as "Special Areas� wherein new towns, townships, large industrial estates or any other type of large development projects may be established;
(f) landscaping and the preservation of areas in their natural state,
(g) measures relating to the prevention of erosion, including rejuvenation of forest areas;
(h) proposals relating to irrigation, water supply or flood control works 
8  Preparation of regional plan  - (1) After preparation of the existing land use map, the Director shall cause to be prepared a draft regional plan and publish it by making a copy thereof available for inspection and publishing a notice in such form and manner as may be prescribed inviting objections and suggestions from any person with respect to the draft plan before such date as may be specified in the notice, such date not being earlier than sixty days from the publication of the notice  Such notice shall specify in regard to the draft plan the following particulars, namely-
(a) the existing land use map and the narrative report thereon;
(b) a narrative report supported by necessary map and charts explaining the provisions of the draft plan;
(c) a note indicating the priorities assigned to works included in the draft plan and the phasing of the programme of development as such;
(d) a notice on the role being assigned to different departments of Government, the Town and Country Development Authorities; the Special Area Development Authorities, and the Local Authorities in the enforcement and implementation of draft plan 
(2) The Director shall consider all the objections and suggestions received by him within the period specified in the notice under sub-section (1) and shall, after giving a reasonable opportunity to all persons affected thereby of being heard, prepare the regional plan containing such modifications, if any, as he considers necessary and submit it to the State Government for approval together with all connected documents, plans, maps and charts 
9  Finalisation of regional plan  - (1) The State Government may approve the draft regional plan submitted under section 8 with or without modification or reject or return the same to the Director for reconsideration 
(2) Immediately after the draft regional plan is approved under sub-section (1) the State Government shall publish in such manner, as may be prescribed, a notice stating that the regional plan has been approved and mentioning a place where a copy of the plan may be inspected at all reasonable hours and shall specify therein a date on which the regional plan shall come into operation:
Provided that where the State Government approves the draft regional plan with modifications, it shall not be published, unless the State Government having published such modifications in the Official Gazette along with a notice inviting objections and suggestions thereon, within a period of not less than thirty days from the date of publication of such notice have considered the objections and suggestions after giving a reasonable opportunity of being heard to persons affected thereby 
10  Restriction on use of land or development thereof  - (1) Notwithstanding anything contained in any other law for the time being in force, on or after the date of publication of the draft regional plan, no person, authority, department of Government or any other person shall change the use of land for any purpose other than agriculture, or carry out any development in respect of any land contrary to the provisions of the draft plan, without the prior approval of the Director or any officer next to him authorised by the Director, in this behalf 
(2) Notwithstanding anything contained in any law for the time being in force, the permission referred to in sub-section (1) shall not be granted otherwise than in conformity with the provision of the draft or final plan and no permission, if granted, shall be construed to confer any legal right whatsoever on the person seeking the permission 
(3) If any work is carried out in contravention of the provisions of this section, the Municipal Corporation or Municipal Committee within its local area, and the Collector in area outside such local areas, may cause such work to be removed or demolished at the cost of the defaulter, which shall be recovered from him in the same manner as an arrear of land revenue:
Provided that no action shall be taken under this sub-section unless the person concerned is given a reasonable opportunity of being heard and a notice calling upon him to remove or demolish the work within a time specified therein 
(4) Any person aggrieved by the order of the Municipal Corporation, Municipal Committee or Collector, as the case may be, calling upon to remove or demolish the work may prefer an appeal to the Director within fifteen days of the receipt of the notice under sub-section (3) and the order of the Director in such appeal shall be final 
11  Exclusion from claims of amount in certain cases  - Where the regional plan assigns a particular land use to a certain areas and any land situate therein is already put to such use, subject to substantially similar restrictions in force under any other law which was in force on the date on which restrictions were imposed by or under this Act and if amount in respect of such restrictions have already been paid under any such other law which was in force for the time being in respect of the property or any right or interest therein to the claimant, or any predecessor in interest of the claimant, the owner shall not be entitled to any further amount on account of injury or damage caused to his rights by reasons of the restrictions placed on the use of the land under the provisions of this Act 
12  Review of regional plan  - (1) The Director may, on his own motion or if so required by the State Government, at any time after a regional plan has come into operation, undertake the review and evaluation of the regional plan and make such modification in it as may be justified by the circumstances 
(2) The foregoing provisions of this Chapter shall, so far as they can be made applicable, apply to the modifications under sub-section (1) as these provisions apply in relation to the preparation, publication and approval of a regional plan 
Planning Area and Development Plans
13  Planning area  - (1) The State Government may, by notification, constitute planning areas for the purposes of this Act and define the limits thereof 
(2) The State Government may, by notification -
(a) alter the limits of a planning area so as to include therein or excluse therefrom such area as may be specified in the notification;
(b) amalgamate two or more planning areas so as to constitute one planning area;
(c) divide any planning area into two or more planning areas;
(d) declare that the whole or part of the area constituting the planning area shall cease to be a planning area or part thereof 
14  Director to prepare development plans  - Subject to the provisions of this Act and the rules made thereunder, the Director shall-
(a) prepare an existing land use map;
(b) prepare an interim development plan;
(c) prepare a development plan;
(d) prepare a sectoral plan;
(e) carry such surveys and inspections and obtain such pertinent reports from government departments, local authorities and public institutions as may be necessary for the preparation of the plans;
(f) perform such duties and functions as are supplemental, incidental, and consequential to any of the foregoing functions or as may be assigned by the State Government for the purpose of carrying out the provisions of this Act 
15  Existing land use maps  - (1 )The Director shall carryout the survey and prepare an existing land use map and, forthwith publish the same in such manner as may be prescribed together with public notice of the preparation of the map and of the place or places where the copies may be inspected, inviting objections and suggestions in writing from any person with respect thereto within thirty days from the date of publication of such notice 
(2) After the expiry of the period specified in the notice published under sub-section (1), the Director may, after allowing a reasonable opportunity of being heard to all such persons who have filed the objections or suggestions, make such modifications therein as may be considered desirable 
(3) As soon as may be after the map is adopted with or without modifications the Director shall publish a public notice of the adoption of the map and the place or places where the copies of the same may be inspected 
(4) A copy of the notice shall also be published in the Official Gazette and it shall be conclusive evidence of the fact that the map has been duly prepared and adopted 
16  Freezing of land use  - On the publication of the existing land use map under section 15-
(a) no person shall institute or change the use of any land or carry out any development of land for any purpose other than that indicated in the existing land use map without the permission in writing of the Director;
Provided that the Director shall not refuse permission if the change is for the purpose of agriculture;
(b) no local authority or any officer or other authority shall, not withstanding anything contained in any other law for the time being in force, grant permission for the change in use of land otherwise than as indicated in the existing land use map without the permission in writing of the Director;
[(c) no Registrar or the Sub-Registrar, appointed under the Indian Registration Act, 1908, shall, in any planning area constituted under section 13, register any deed or document of transfer of any sub-division of land by way of sale, gift, exchange, lease or mortgage with possession, unless the sub-division of land is duly approved by the Director, subject to such rules as may be framed in this behalf by the State Government:]
Provided that the Registrar or the Sub-Registrar may register any transfer,-
(i) where the land is owned by a person and the transfer is made without involving any further divisions;
(ii) where the partition/sub-division of land is made in a Joint Hindu Family;
(iii) where the lease is made in relation to a part or whole of a building;
(iv) where the mortgage is made for procuring the loans for construction or improvements over the land either from the Government or from any other financial institution constituted or established under any law for the time being in force or recognised by the State Government 
17  Interim development plans  - As soon as may be, after the declaration of a planning area, the Director shall, within such time as may be necessary, prepare, after consultation with local authorities concerned, if any, and submit to the State Government an interim development plan for the planning area or any of its parts and such other area or areas contiguous or adjacent to the planning areas as the State Government may direct to be included in the interim development plan 
(2) The interim development plan shall-
(a) indicate broadly the land use proposed in the planning area;
(b) allocate broadly areas or sector of land for-
(i) residential, industrial, commercial or agricultural purposes,
(ii) open spaces, parks and gardens, green belts, zoological gardens and play-grounds,
(iii) public institutions and offices,
(iv) such special purposes as the Director may deem fit;
(c) lay down the pattern of National and State Highways connecting the planning area with the rest of the region, ring roads, arterial roads and the major roads within the planning areas;
(d) provide for the location of airports, railway stations, bus termini and indicate the proposed extension and development of railways and canals;
(e) make proposals for general land scaping and preservation of natural areas;
(f) project the requirement of the planning area of such amenities and utilities as water, drainage, electricity and suggest their fulfilment;
(g) propose broad based regulations for sectoral development, by way of guide-lines, within each sector of the location, height, size of buildings and structures, open spaces, court-yards and the use to which such buildings and structures and land may be put;
(h) lay down the board-based traffic circulation patterns in a city;
(i) suggest architectural control features, elevation and frontage of buildings and structures;
(j) indicate measures for flood control, prevention of air and water pollution, disposal of garbage and general environmental control 
(3) Subject to provisions of the rules made under this Act for regulating the form and contents of the interim development plan any such plan shall include such maps and such descriptive matter as may be necessary to explain and illustrate the proposals in the interim development plan 
(4) As soon as may be, after the submission of the interim development plan, under sub-section (1) the State Government may either approve the interim development plan or may approve it with such modification as it may consider necessary 
(5) The State Government shall publish the interim development plan as approved under sub-section (4) in the Official Gazette  The interim development plan shall come into operation from the date of its publication in the Official Gazette and shall be binding on all local authorities functioning within the planning areas 
18  Development plan  - A development plan shall -
(a) indicate broadly the land use proposed in the planning areas;
(b) allocate broadly areas or sector of land for,-
(c) lay down the pattern of National and State Highways connecting the planning area with the rest of the region, ring roads, arterial roads, and the major roads within the planning area;
(d) provide for the location of airports, railway stations, bus termini and indicate the proposed extension and development of railways;
(g) propose broad-based regulations for sectoral development, by way of guide-line, within each sector of the location, height, size of buildings and structures, open spaces, court-yards and the use to which such buildings and structures and land may be put;
(h) lay down the broad-based traffic circulation patterns in a city;
19  Publication of draft development plan  - (1) The Director shall forthwith publish the draft development plan prepared under section 18 in such manner as may be prescribed together with a notice of the preparation of the draft development plan and the place or places where the copies may be inspected, inviting objections and suggestions in writing from any person with respect thereto, within thirty days from the date of publication of such notice Such notice shall specify in regard to the draff development plan the following particulars, namely:-
(i) the existing land use maps;
(ii) a narrative report, supported by maps and charts, explaining the provisions of the draft development plan;
(iii) the phasing of implementation of the draft development plan as suggested by the Director,
(iv) the provisions for enforcing the draft development plan and stating the manner in which permission to development may be obtained;
(v) an approximate estimate of the cost of land acquisition for public purposes and the cost of works involved in the implementation of the plan 
(2) The Director shall, not later than ninety days after the date of expiry of the notice period under sub-section (1), consider all the objections and suggestions as may be received within the period specified in the notice under sub-section (1)and shall, after giving reasonable opportunity to all persons affected thereby of being heard, make such modifications in the draft development plans as he may consider necessary, and submit, not later than six months after the publication of the draft development plan, the plan so modified, to the State Government for approval together with all connected documents, plans, maps and charts 
20  Sanction of development plans  - (1) As soon as may be after the submission of the development plan under section 19 the State Government may either approve the development plan or may approve it with such modifications as it may consider necessary or may return it to the Director to modify the same or to prepare a fresh plan in accordance with such directions as it may issue in this behalf 
(2) Where the State Government approves the development plan with modifications, the State Government shall, by a notice, published in the Official Gazette, invite objections and suggestions in respect of such modifications within a period of not less than thirty days from the date of publication of the notice in the Official Gazette 
(3) After considering objections and suggestions and after giving a hearing to the persons desirous of being heard the State Government may confirm the modification in the development plan 
(4) The State Government shall publish the development plan as approved, under the foregoing provisions in the Official Gazette and shall along with the plan publish a public notice, in such manner as may be prescribed, of the approval of the development plan and the place or places where the copies of the approved development plan may be inspected 
(5) The development plan shall come into operation from the date of publication thereof in the Official Gazette and as from such date shall be binding on all Development Authorities constituted under this Act and all local authorities functioning within the planning area 
(6) After the coming into operation of the development plan, the interim development plan shall stand modified or altered to the extent the proposals in the development plan are at variance with the interim development plan 
Sectoral Plan
21  Director to prepare sectoral plan  - The Director may, on his own motion, at any time after the publication of the development plan, or thereafter if so required by the State Government shall, within six months of such requisition, prepare a sectoral plan 
22  Contents of sectoral plan  - (1) The sectoral plan shall enlarge the details of land use as indicated in the development plan and shall -
(a) indicate the land liable to acquisition for public purpose or the purposes of the Union Government, the State Government, the Town and Country Development Authority, the Special Area Development Authority, the local authority or any other authority established by or under any enactment for the time being in force 
Provided that no land shall be so designated unless the acquisition proceedings are likely to be completed within ten years of the preparation of the plan;
(b) define in detail and provide for areas reserved for agriculture, public and semi-public open spaces, parks, playgrounds, gardens, recreational areas, green belts and natural reserves;
(c) allocate in detail areas or sectors for residential, commercial, industrial, agricultural and other purposes;
(d) define and provide for the complete road and street pattern for the present and in the future and indicate the traffic circulation;
(e) lay down in detail the projected road and street improvement;
(f) indicate and provide for areas reserved for public buildings, institutions and civic developments;
(g) assess, make projections for and provide for the future requirements of amenities, services and utilities such as municipal, transport, electricity, water and drainage;
(h) prescribe in detail the sectoral regulations for each sector, with a view to facilitating on individual layout and regulating the location, height, number of storeys and the size of buildings and other structures, the size of the court-yards, courts and other open spaces and the use of the buildings, structures and land;
(i) define areas which have been badly laid out or areas which have developed so as to form slums, and provide for their proper development and/or relocation;
(j) designate areas for future development and expansion;
(k) indicate the phasing of the programme of development 
(2) The sectoral plan may and if possible shall, indicate -
(a) control over architectural features; elevation and frontage of buildings and structures; and
(b) the details of development of specific areas for housing, shopping centres, industrial areas, educational and cultural institutions and civic centres 
23  Provisions of sections 19 and 20 to apply to sectoral plan  - The provisions of sections 19 and 20 shall apply for the preparation, publication, approval and operation of sectoral plan as they apply in respect of the development plan 
24  Review of development plan and sectoral plan  - The Director may on his own motion or if so required by the State Government shall, at any time after the sectoral plan has come into operation, undertake a review and evaluation of the development plan and sectoral plan 
(2) The foregoing provisions of sections 19, 20 and 23 shall, so far as may be apply to the modification under sub-section (1) as those provisions apply in relation to the preparation, publication and approval of a development plan or a sectoral plan 
Control of Development And Use of Land
25  Director to control land use  - The overall control of development and the use of land in the planning area shall, as from the date of publication in the Official Gazette of a notification by the State Government, vest in the Director 
26  Conformity with development plan  - (1) After coming into force of the development plan, the use and development of land shall conform to the provisions of the development plan:
Provided that the Director may, at his discretion, permit the continued use of land for the purpose for which it was being used at the time of the coming into Operation of the development plan 
Provided further that such permission shall not be granted for a period exceeding seven years from the date of coming into operation of the development plan 
27  Prohibition of development without permission  - After coming into operation of the development plan, no person shall change the use of any land or carry out any development of land without the permission in writing of the Director 
Provided that no such permission shall be necessary -
(a) for carrying out works for the maintenance, repair or alteration of any building which does not materially alter the external appearance of the building;
(b) for carrying out work for the improvement or maintenance of a highway, road or public street by the Union or State Government or an authority established under this Act or by a local authority having jurisdiction, provided that such maintenance or improvement does not change the road alignment contrary to the provisions of the development plan;
(c) for the purpose of inspecting, repairing or renewing any drain, sewers, mains, pipes, cables, telephone or other apparatus including the breaking open of any street or other land for that purpose;
(d) for the excavation or soil shaping in the interest of agriculture;
(e) for restoration of land to its normal use where land has been used temporarily for any other purposes 
(f) for use for any purpose incidental to the use of building for human habitation, or any other building or land attached to such buildings;
(g) for the construction of a road intended to give access to land solely for agricultural purposes
28  Development undertaken on behalf of Union or State Government  - (1) When the Union Government or the State Government intends to carry out development of any land for the purpose of its departments or offices or authorities, the officer-in-charge thereof shall inform in writing to the Director the intention of the Government to do so, giving full particulars thereof, accompanied by such documents and plans as may be prescribed at least thirty days before undertaking such development 
(2) Where the Director raises any objection to the proposed development on the ground that the development is not in conformity with the provisions of the development plan, the officer shall,-
(i) make necessary modification in the proposals for development to meet the objections raised by the Director, or
(ii) submit the proposal for development together with the objections raised by the Director to the State Government for decision:
Provided that where no modification is proposed by the Director within thirty days of the receipt of the proposed plan by the Government, the plan will be presumed to have been approved 
(3) The State Government, on receipt of the proposals for development together with the objections of the Director shall, approve the proposals with or without modifications or direct the officer to make such modifications in the proposals as it considers necessary in the circumstances 
(4) The decision of the State Government under sub-section (3) shall be final and binding 
29  Development by local authority or by any authority constituted under this Act  - Where a local authority or any authority specially constituted under this Act intends to carry out development on any land for the purpose of that authority, the procedure applicable to the Union or State Government, under section 28 shall, mutatis' mutandis, apply in respect of such authority 
30  Application for permission for development by others  - (1) Any person, not being the Union Government, State Government, a local authority or a special authority constituted under this Act intending to carry out any development on any land, shall make an application in writing to the Director for permission, in such form and containing such particulars and accompanied by such documents as may be prescribed 
(2) Such application shall also be accompanied by such fee as may be prescribed 
[30A  Exemption from development permission in rural areas falling within Planning or Special Area  - (1) Any person who owns land in rural areas, falling within Planning or Special Areas wherein neither Interim Development Plan nor Development Plan has been notified, shall be exempted from permission under this Act for the following development activities up to the limits as may be prescribed: -
(i) Residential activities such as farm-houses and residential houses up to three storeys, cattle shed, toilet, septic tank, kitchen, store, parking shed or garage and rain shelter;
(ii) Commercial activities such as basic commercial activities like shops of general merchandise, cobbler, barber, tailoring, fruit, vegetable, tea or sweet, eating places and dhabas, chemist and farm produce sale depot;
(iii) Service Industries such as cottage or house-hold, service industries like carpentry, knitting, weaving, blacksmith, goldsmith, atta-chakki with capacity up to five horse-power, water mill, agriculture equipments or machinery repair, electrical, electronic and house-hold appliances;
(iv) Public amenities such as public amenities like panchayat offices, schools, mahila mandals, yuvak mandals, community halls, post offices, dispensaries and clinics (including health, veterinary and Indian System of Medicines) information technology kiosks, patwar khanas, guard huts, anganwaries, electricity and telephone installations and connections, roads and paths, ropeways, water tanks, rain harvesting tanks, overhead or underground water tan 
Context 3: Paper Info

Title: Interpretable reduced-order modeling with time-scale separation Interpretable reduced-order modeling with time-scale separation
Publish Date: 7 March 2023
Author List: Sebastian Kaltenbach (from CSE-Lab, ETH Zurich, Harvard SEAS), Phaedon-Stelios Koutsourelakis (from CSE-Lab, ETH Zurich, Harvard SEAS), Petros Koumoutsakos (from CSE-Lab, ETH Zurich, Harvard SEAS), Harvard Seas (from CSE-Lab, ETH Zurich, Harvard SEAS)

Figure

FIG  5  Comparison between the phase-space of the reference solution (left) and the phase-space of the predictions
FIG  7  Comparison between predictions and reference solutions for a new initial condition fort = 1 25, 3 75, 7 5, 12 5, 20, 30  (from left to right and top to down) We note that with longer prediction time the uncertainty bounds increases Despite the chaotic nature of the KS equation, the predictive posterior mean is close to the reference solution for t ≤ 12 5

abstract

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering  However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial  We propose such a data-driven scheme that automates the identification of the time-scales involved and, can produce stable predictions forward in time as well as under different initial conditions not included in the training data 
To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space  It readily allows for the inclusion of sparse and irregularly sampled training data  The learned, latent dynamics are interpretable and reveal the different temporal scales involved 
We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix  Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation 
Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework 

INTRODUCTION

High-fidelity simulations of critical phenomena such as ocean dynamics and epidemics have become essential for decision-making  They are based on physically-motivated PDEs expressing system dynamics that span multiple spatiotemporal scales and which necessitate cumbersome computations   In recent years there is increased attention to the development of data-driven models that can accelerate the solution of these PDEs as well as reveal salient, lower-dimensional features that control the long-term evolution 
In most cases, data-driven reduced-order models are not interpretable  In particular, models based on neural networks despite good predictive capabilities , they offer a black-box description of the system dynamics  A possible remedy is applying a symbolic regression to the learned neural network representation , but this adds additional computational cost due to the two-step procedure 
A number of frameworks such as SINDy allows to learn interpretable dynamics but it relies on the a-priori availability of lower-dimensional descriptors and of time-derivatives which can be very noisy for both simulation and experimental data  Other frameworks are tailored to specific problems such as molecular dynamics  
Here, we present a framework that only needs the value of the observables, and not their derivatives, as training data and is capable of identifying interpretable latent dynamics  The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model  
The present method is related to approaches based on the Koopman-operator extended Dynamic Mode Decomposition (eDMD) but uses continuous complex-valued latent space dynamics and only requires one scalar variable per latent dimension to describe the latent space dynamics  Therefore we do not have to enforce any parametrizations on the Koopman matrix  
The time-continuous formulation moreover allows to incorporate sparse and irregularly sampled training data and fast generation of predictions after the training phase  By using a complex-valued latent space we can also incorporate harmonic effects and reduce the number of latent variables needed  Linear and non-linear autoencoders are used to map the observed, high-dimensional time-series to the lower-dimensional, latent representation and we identify simultaneously the autoencoder as well as the latent dynamics by optimizing a combined loss function 
Hence the to tasks of dimensionality reduction and discovery of the reduced dynamics are unified while other frameworks treat the two parts separately   Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed   We are also proposing a probabilistic version of our algorithm ) that makes use of probabilistic Slow Feature Analysis  
This allows for a latent representation that arart from being time-continuous, can quantify the predictive uncertainty and hierarchically decompose the dynamics into their pertinent scales while promoting the discovery of slow processes that control the system's evolution over long time horizons  The rest of the paper is structured as follows: We introduce the methodological framework as well as algorithmic details in section II 
Particular focus is paid on the interpretability of the inferred lower-dimensional dynamics  In section III we present three numerical illustrations, i e  a system of linear ODEs, a hidden Markov Model and the discretized KS-equation  We then present in section IV the probabilistic extension of the framework and apply it to the KS-equation 
We conclude with a summary and a short discussion about possible next steps  We introduce the autoencoders deployed in this work, followed by the interpretable latent space dynamic and discuss the training process  We consider data from high-dimensional time series x n ∈ R f with n = 1,    , T   We remark that the intervals between the different states do not need to be uniformly spaced 

Autoencoder

A core assumption of the method is that each high-dimensional state x n can be compressed to a lower-dimensional representation z n ∈ C c with c << f   We identify this lower-dimensional representation by an autoencoder consisiting of a parameterized encoder and decoder  The encoder maps the high-dimensional representation to the latent space as:
The latent space is complex-valued  The decoder reconstructs the high-dimensional representation based on the latent variables as: We denote the parameters of the encoder as well as the decoder by θ  As discussed later in Section II C, both set of parameters are optimized simultaneously during training and therefore there is no need for differentiating them 

Interpretable Latent Space Dynamics

We employ a propagator in the latent space to capture the reduced-order dynamics of the system  In contrast to other time-extended variational autoencoder frameworks, our representation uses complex valued latent variables  In addition the latent variables are treated independently  The latter feature enables us to have an interpretable latent dynamics as well as a model that is especially suitable for being trained in the Small Data regime due to the small number of required parameters 
This is in contrast to temporal propagators such as LSTMs   For each dimension i of the latent variable z we are using the following continuous ODE in the complex plane: By solving this ODE, we can define the operator: Interpretable reduced-order modeling with time-scale separation Here, λ is a vector containing all the individual λ's and ∆t n indicates the time-step between the latent states 
The symbol is used to indicate a component-wise multiplication  We remark that the latent variables and the parameter governing the temporal evolution are complex numbers and their role in describing the system dynamics is similar to that of an eigenvalue  The real part is associated with growth and decay whereas the imaginary part is representing the periodic component 
This approach has similarities with the Koopman-operator based methods and the extended dynamic mode decomposition   In contrast to the methods mentioned before we are using a continuous formulation in the latent space that allows us to incorporate scarce and irregularly sampled training data and directly rely on complex numbers in the latent space 

Training and Predictions

We optimize a loss function that combines both a reconstruction loss as well as a loss associated with the error of our learned propagator in the latent space: (5) We note that we could directly incorporate mini-batch training by only taking the summation over a subset of the N available training data 
For new predictions of unseen states, we use the encoder to generate a latent representations which is then advanced in time by the learned propagator  At a designated time step we are using the decoder to reconstruct the high-dimensional solution  We applied our algorithm to three systems  First, we show that the algorithm is capable of exactly reproducing the solution of a linear ODE and to identify its eigenvalues 
Afterwards we are applying the framework to a high-dimensional process generated by a complex latent dynamics, which is correctly identified  As a final test case, we are applying the algorithm to a Kuramoto Shivashinski (KS) equation  Interpretable reduced-order modeling with time-scale separation

Linear ODE

We are considering a two-dimensional ODE system for x = y 1 y 2 : Based on the obtained training data we run our algorithm using a linear encoder and decoder structure as well as two latent variables z  The loss function was optimized using the Adam algorithm   As we consider a linear ODE we can analytically compute the eigenvalues involved and compare it with the parameters λ identified by our algorithm 
We observe in Figure that the algorithm was able to recover the correct values, i e  the eigenvalues 7 and 3 of the given linear ODE  The system does not have a periodic component and the two imaginary parts correctly go to zero, whereas the real parts converge to the reference value  Moreover we are also able to identify for the linear mapping between our latent variables z and the training data a matrix consisting of a multiple of the eigenvectors (1,1) and (1,-1) and thus the correct solution 
This example was chosen to show that the algorithm is able to quickly identify the exact solution of a linear ODE in terms of its linearly independent components 

Hidden multiscale dynamics

We consider eight-dimensional synthetic time series data produced by an underlying twodimensional complex valued process  In particular, the data points x are generated by first solving for the temporal evolution for the two complex-valued processes p 1 and p 2 and than mapping to the eight-dimensional space by using a randomly sampled linear mapping W  
One of the two processes used to generate the data is chosen to be much slower than the other one and both processes have a periodic component  dp 2 dt = (−0 9 + 1 5i)p 2 (8) As training data we consider 40 time series with 150 data points each, obtained by simulating the described processes for a maximum of t = 15 s and then sampling from the obtained data points 
Hence the training data consists of: • 40 time-series • with each consisting 150 observations of the x at a uniform time-step ∆t = 0 0025 The autoencoder obtained consists of one linear layer for both the decoder as well as the encoder  The model is trained for 5000 iterations using the Adam optimizer and a learning rate of 10 −3  
The results for the convergence of the parameters λ 1 and λ 2 can be found in Figure   We note that the process which is slower decaying and thus more responsible for the long-term evolution of the system has a higher convergence rate than the faster process  With the obtained parameters λ as well as the trained autoencoder, we compute predictions based on the last time step used for training, i e  we apply the encoder to obtain our latent representation and than use the latent dynamics to advance the latent representation in time 
Afterwards, we employ the decoder to reconstruct the full high-dimensional system  The results can be found in Figure and show very good agreement between predictions and reference data  This example shows that our model is successfully able to carry out dimensionality reduction and moreover indicates that the convergence rate between latent processes can be different 
The latter is relevant when training models as for accurate predictions all latent processes and their dynamics should be converged 

Kuramoto-Sivashinsky

Finally, we applied our algorithm to the KS equation and aim to identify a reduced-order model for the solution u(y, t): We employed periodic boundary conditions, µ = 1 and a domain size y ∈ [0, 22]  For this domain-size, the KS-equation exhibits a structurally stable chaotic attractor as discussed in The black lines divides the area for which training data was given from the area without raining data 
;   The equation is discretized in space using a discretization step of 22 64 resulting in a state vector x of dimension 64 and a nonlinear system of coupled ODEs  This is solved using a stiff fourth-order solver  We employed a non-linear encoder and decoder with four fully-connected layers each and ReLU-activation functions as well as Dropout Layers between the fully-connected layers 
We trained the model for 200000 iterations using Adam and a learning rate of 5 • 10 4 and assuming a five-dimensional latent space  We obtained the λ's in Figure   Four latent variables have λ's close to zero and thus a slow temporal dynamic that is responsible for the long-term evolution whereas one latent variable is quickly decaying 
Based on the obtained parameters, we do predictions based on an unseen initial condition not contained in the training data  We are able to reconstruct the correct phase space based on our predictions despite only using a very limited amount of training data  The results for the phase space can be seen in Figure  
Although the small-scale fluctuations in the temporal dynamics are not well captured, the model identifies the correct manifold which has a good accuracy compared to the reference solution  All phase-spaces were obtained by using a finite-difference operator on the data or predictions  These results are in accordance Interpretable reduced-order modeling with time-scale separation with whose LSTM-based temporal dynamic model was also able to find the correct phase space but not to track the actual dynamics for long-term predictions 
Our model is not able to account for noise in the temporal evolution and thus dealing with chaotic, small-scale fluctuations is challenging  We believe that a probabilistic version of our algorithm could be advantageous here  This section contains a fully probabilistic formulation for the deterministic model discussed before 
We replace the Autoencoder with a Variational Autoencoder and the ODE in the latent space with a SDE  The loss function which we optimize is the Evidence-Lower Bound (ELBO) 

Model Structure

We postulate the following relations for our probabilistic model using an Ornstein-Uhlenbeck (OU) for each dimension i of the latent space and a Wiener process W t in the latent space: We again assume that the latent variables z t are complex-valued and a priori independent  Complex variables were chosen as their evolution includes a harmonic components which are observed in many physical systems 
We assume an initial conditions z 0,i ∼ CN (0, σ 2 0,i )  The total parameters associated with the latent space dynamics of our model are thus {σ 2 0,i , σ 2 i , λ i } c i=1 and will be denoted by θ together with all parameters responsible for the decoder mapping G (see next section)  These parameters along with the state variables z t have to be inferred from the data x t  
Based on probabilistic Slow Feature Analysis (SFA) , we set σ 2 i = 2; (λ j ) and σ 2 0,i = 1  As a consequence, a priori, the latent dynamics are stationary  A derivation and reasoning for this choice can be found in Appendix A  Hence the only independent parameters are the λ i , the imaginary part of which can account for periodic effects in the latent dynamics 

Variational Autoencoder

We employ a variational autoencoder to account for a probabilistic mappings from the lower-dimensional representation z n to the high-dimensional system x n   In particular we are employing a probabilistic decoder The encoder is used to infer the state variables z based on the given data and thus defined in the inference and learning section 

Inference and Learning

Given the probabilistic relations , our goal is to infer the latent variables z 0:T as well as all model parameters θ  We follow a hybrid Bayesian approach in which the posterior of the state variables is approximated using amortized Variational Inference and Maximum-A-Posteriori (MAP) point-estimates for θ are computed 
The application of Bayes' rule for each data sequence x 0:T leads to the following posterior: where p(θ) denotes the prior on the model parameters  In the context of variational inference, we use the following factorization of the approximate posterior i e  we infer only the mean µ and variance σ for each state variable based on the given data points 
This conditional density used for inference is the encoder-counterpart to the probabilistic decoder defined in the section before  It can be readily shown that the optimal parameter values are found by maximizing the Evidence Lower Bound (ELBO) F(q φ (z 0:T ), θ) which is derived in Appendix B  We compute Monte Carlo estimates of the gradient of the ELBO with respect to φ and θ with the help of the reparametrization trick and carry out stochastic optimization with the ADAM algorithm  

Results for the probabilistic extension

We applied our probabilistic version to the KS-equation  We used the same settings as for the deterministic approach but considered up to 10 complex latent variables  The obtained λ's are in Figure   The probabilistic model allows us to quantify the uncertainty in predictions  In Figure predictions for various time-steps and the respective uncertainty bounds are shown for an unseen initial condition 
Due to the chaotic nature of the KS-equation and the small amount of training data, the underlying linear dynamic of our model is only able to capture the full dynamics for a limited time horizon  Fortunately, due to the probabilistic approach the model is capable of capturing chaotic fluctuations with increasingly wide uncertainty bounds 
We also computed the phase space representation for the KS-equation based on the predictions obtained by our model and compare it with the reference solution  The probabilistic model identifies the correct manifold with a better accuracy than the deterministic model  As some of the small-scale fluctuations are accounted as noise, the resulting manifold is more concentrated at the origin and the obtained values are slightly smaller than the reference manifold although their shape is very similar 



ground_truth: The Director of Town and Country Planning is responsible for carrying out the functions assigned under the act.

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 was the paper published?

Read the following text and answer briefly.

Context 4: [49]
In the spring of 1774, William Hewson died unexpectedly of septicemia two weeks after cutting himself while dissecting a cadaver  Polly was left to care for their two young sons and was pregnant with a daughter she would give birth to in August of the same year  Is it possible that Margaret and Benjamin moved back into No  27 to assist Polly after the death of her husband as suggested in The Americanization of Benjamin Franklin?[50]
If the Westminster Rate Books are to be believed, the answer is no  For the year 1774, the Rate Books list Margaret Stevenson at No  1 and William Hewson at No  27  For the year 1775, they list Margaret Stevenson at No  1 and Magnus Falkner (Falconer/Falconar) at No  27  Magnus was William’s assistant at the anatomy school and fiancé to William’s sister, Dorothy  On his death bed, William instructed Polly, “let Mr  Falconar be my successor ”[51] Magnus would immediately take over the running of the anatomy school and continue William’s unfinished research  Four months later, he and Dorothy would marry [52] Essentially only two things changed at 27 Craven Street after William’s death: Polly gave birth to her daughter, and Magnus replaced William as the lease holder, so even if Margaret and Benjamin had wished to move back into No  27, there would have been no room for them  It is also interesting to note that considering the multiple times Benjamin wrote of his move out of No  27 (and complained of it), he never once mentioned moving back into No  27 in any of his correspondence after Mr  Hewson’s death 
Figure 6  No  36 Craven Street  (Photo courtesy of David Ross, britainexpress com)
In sum, based on the Westminster Rate Books[53] and Franklin’s correspondence, Mrs  Stevenson is known to have resided at No  27 (36) Craven Street from 1748 to 1772  It follows that, aside from the two years Franklin spent in Philadelphia from 1762 to 1764, he resided there from 1757 to 1772  Franklin’s correspondence also reveals that in the autumn of 1772, he and Mrs  Stevenson moved to another house on Craven Street  The 1773 Westminster Rate Books show her name crossed off at No  27 and William Hewson’s inserted  The following year the Rate Books list her at No  1 Craven Street  Evidence for Mrs  Stevenson and Benjamin remaining at No  1 after William’s death appears in the Westminster Rate Books for 1775 which have Mrs  Stevenson still residing at No  1 and Magnus Falkner residing at No  27  Further evidence can be construed from the lack of any mention of a move back into No  27 in Franklin’s correspondence  Despite the many theories one could devise as to why Franklin was thought to have lived at No  7 Craven Street by so many guide books and Franklin biographers of the nineteenth century, one thing is certain; at some point after Franklin’s departure to America in March of 1775, and no later than 1807, someone mistakenly associated him with No  7 on the west side of Craven Street, and it soon became his de facto residence  Credit must go to D  H  Montgomery in 1896 and Sir George in 1913 for setting the record partially straight by placing Franklin at No  27(36)  In 1937, the London County Council gave us the first accurate account of Franklin’s residences on Craven Street in the Survey of London at No  27(36) and No  1  It has been shown conclusively that No  27 was never previously numbered 7  It was, however, renumbered 36 in 1792 after ten additional houses were built at the southern end of the street and remains No  36 to this day 
[1] “Craven Street and Hungerford Lane”, in Survey of London: Volume 18, St Martin-in-the-Fields II: the Strand, ed  G H Gater and E P Wheeler (London, 1937), 27-39, Early History of the Site 
http://www british-history ac uk/survey-london/vol18/pt2/pp27-39
[2] “England, Westminster Rate Books, 1634-1900,” from database with images, Craven Street – 1735, FamilySearch from database by FindMyPast and images digitized by FamilySearch; citing Westminster City Archives, London 
[3] Ibid , Craven Street – 1748 
[4] The Statutes at Large, From Magna Charta to the End of the Eleventh Parliament of Great Britain  Anno 1761 Continued, Vol  XXVII, ed  Danby Pickering, (Cambridge, John Archdeacon, 1767), 96 
[6] James Raven, Publishing Business in Eighteenth-Century England, (Woodbridge: The Boydell Press, 2014), 201 
[7] The London Directory For the Year 1776, Ninth Edition, (London: T  Lowndes, 1776), title page 
[8] Kent’s Directory For the Year 1778, Forty-Sixth Edition, (London: Richard and Henry Causton, 1778), title page 
[9] A listing in Kent’s Directory for the Year 1882 on p  28 reveals, “Brown Sarah, Leather-seller, 1, Westmoreland-buildings, Aldersgate-street”, and in Kent’s Directory for the Year 1883 on p  175, “Whiteland Mary, Wine & Brandy Mercht  Jermyn-str  St  James ”
[10] “The Papers of Benjamin Franklin,” Sponsored by The American Philosophical Society and Yale University, Digital Edition by The Packard Humanities Institute, 22:263a 
http://franklinpapers org/franklin
Mrs  Stevenson wrote to Benjamin Franklin a letter from her new home at 75 Northumberland Court on November 16, 1775: “In this Court I have a kind friend, Mr  Lechmoen he comes and seats with me and talks of you with a hiy regard and friendship ”
[11] Survey of London, Early History of the Site 
[12] Survey of London, Footnotes/n 10 
[13] Survey of London, Historical Notes/No  31 
[14] David Hughson, LL D , London; Being An Accurate History And Description Of The British Metropolis And Its Neighbourhood, To Thirty Miles Extent, From An Actual Perambulation, Vol  IV, (London: W  Stratford, 1807), 227 
[15] The Reverend Joseph Nightingale, The Beauties of England and Wales: Or, Original Delineations, Topographical, Historical, and Descriptive, of Each County, Vol  X, Part III, Vol  II (London: J  Harris; Longman and Co ; J  Walker; R  Baldwin; Sherwood and Co ; J  and J  Cundee; B  and R  Crosby and Co ; J Cuthell; J  and J  Richardson; Cadell and Davies; C  and J  Rivington; and G  Cowie and Co , 1815), 245 
[16] John Britton, F S A  & Co , ed , The Original Picture of London, Enlarged and Improved: Being A Correct Guide For The Stranger, As Well As For the Inhabitant, To The Metropolis Of The British Empire Together With A Description Of The Environs, The Twenty-Fourth Edition (London: Longman, Rees, Orme, Brown, and Green, 1826), 479 
[17] Jared Sparks, The Works of Benjamin Franklin, Vol  VII, (Philadelphia: Childs & Peterson, 1840), 151 
[18] George Gulliver, F R S , The Works of William Hewson, F  R  S , (London: Printed for the Sydenham Society, MDCCCXLVI), xx 
[19] Peter Cunningham, Handbook for London; Past and Present, Vol  I, (London: John Murray, 1849), 245 
[20] F  Saunders, Memories of the Great Metropolis: or, London, from the Tower to the Crystal Palace, (New York: G P  Putnam, MDCCCLII), 138 
[21] Leigh Hunt, The Town; Its Memorable Characters and Events, (London: Smith, Elder and Co , 1859), 185 
[22] K  Baedeker, London and Its Environs, Including Excursions To Brighton, The Isle of Wight, Etc : Handbook For Travelers, Second Edition, (London: Dulau and Co , 1879), 133 
[23] Herbert Fry, London In 1880 Illustrated With Bird’s-Eye Views of the Principal Streets, Sixth Edition, (New York: Scribner, Welford, & Co , 1880), 50 
[24] Herbert Fry, London  Illustrated By Eighteen Bird’s-Eye Views of the Principal Streets, (London: W  H  Allen and Co , 1886), 40 
[25] Henry B  Wheatley, F S A , London Past and Present; Its History, Associations, and Traditions, Vol  1, (London: John Murray, New York: Scribner & Welford, 1891), 473 
[26] The Journal of the Society of Arts, Vol  XV, No  778, (October 18, 1867): 717 
[27] D  G  C  Allen, “Dear and Serviceable to Each Other: Benjamin Franklin and the Royal Society of Arts,” American Philosophical Society, Vol  144, No  3, (September 2000): 248-249 
Franklin was a corresponding member in 1756 because he was still residing in Philadelphia  He became an active member the following year when he moved to London 
[28] The Journal of the Society of Arts, Vol  XVIII, No  894, (Jan  7, 1870): 137 
“Since the last announcement, the following tablets have been affixed on houses formerly occupied by – Benjamin Franklin, 7 Craven-street, Strand, W C ”
[29] Franklin in His Own Time, eds  Kevin J  Haytes and Isabelle Bour, (Iowa City, University of Iowa Press, 2011), xxxvii 
“Takes lodgings with Margaret Stevenson at No  7 Craven Street ” It is unknown if the editors are referring to No  7 on the west side of Craven Street or No  36 on the east side using Sir George’s explanation of No  36 being previously numbered 7 
[30] Henry C  Shelly, “American Shrines on English Soil, III  In the Footprints of Benjamin Franklin,” in The Book News Monthly, September, 1913 to August, 1914, (Philadelphia: John Wanamaker, 1914), 325 
[31] The Journal of the Royal Society of Arts, Vol  LVI, No  2,880, (Jan  31, 1908): 245 
http://babel hathitrust org/cgi/pt?id=mdp 39015058423073;view=1up;seq=251
“His Majesty the King, who is Patron of the Society, has granted permission to the Society to prefix to its title the term ‘Royal,’ and the Society will consequently be known in future as the ‘Royal Society of Arts ’”
[32] Nineteenth Annual Report, 1914, of the American Scenic and Historic Preservation Society, (Albany: J  B  Lyon Company, 1914), 293 
http://babel hathitrust org/cgi/pt?id=wu 89072985302;view=1up;seq=4;size=150
[33] The Journal of the Society of Arts, Vol  LXII, No  3,183, (Nov  21, 1913): 18 
http://babel hathitrust org/cgi/pt?id=mdp 39015058422968;view=1up;seq=26
[36] Allen, “Dear and Serviceable,” 263-264 
[37] Papers of Benjamin Franklin, 19:20 
[38] Thomas Joseph Pettigrew, F  L  S , Memoirs of the Life and Writings of the Late John Coakley Lettsom With a Selection From His Correspondence, Vol  I, (London: Nichols, Son, and Bentley, 1817), 144 of Correspondence 
[39] Papers of Benjamin Franklin, 19:321b 
[40] Ibid , 19:314 
[41] Ibid , 19:353a 
[43] Simon David John Chaplin, John Hunter and the ‘museum oeconomy’, 1750-1800, Department of History, King’s College London  Thesis submitted for the degree of Doctor of Philosophy of the University of London , 202 
“Following Falconar’s death [1778] the lease [27 Craven Street] was advertised, and the buildings were described as:
A genteel and commodious house, in good Repair, with Coach-house and Stabling for two Horses…consisting of two rooms and light closets on each floor, with outbuildings in the Yard, a Museum, a Compleat Theatre, and other conveniences  (Daily Advertiser, 27 August 1778)”
[44] Simon Chaplin, “Dissection and Display in Eighteenth-Century London,” in Anatomical Dissection in Enlightenment England and Beyond: Autopsy, Pathology and Display, ed  Dr  Piers Mitchell, (Burlington: Ashgate Publishing Company, 2012), 108 
“Given that a nearby building at 35 [ No  26 in Franklin’s time] was occupied by the man-midwife John Leake, who advertised lectures – including lessons in the art of making preparations – at his ‘theatre’ between 1764 and 1788, it is possible that some facilities were shared  In both cases, however, the buildings [Leake’s residence at No  26 and Hewson’s residence next door at 27] served a dual function as domestic accommodation and as sites for lecturing and dissection ”
[45] George Gulliver, F R S , The Works of William Hewson, F  R  S , (London: Printed for the Sydenham Society, MDCCCXLVI), xviii 
[46] Westminster Rate Books, Craven Street – 1773, courtesy of the City of Westminster Archives 
[47] S W  Hillson et al , “Benjamin Franklin, William Hewson, and the Craven Street Bones,” Archaeology International, Vol  2, (Nov  22, 1998): 14-16 
http://dx doi org/10 5334/ai 0206
[48] Westminster Rate Books, Craven Street – 1774, 1775, courtesy of the City of Westminster Archives 
[49] Survey of London, Historical Notes/No  36, Craven Street (not sourced) 
[50] Gordon S  Wood, The Americanization of Benjamin Franklin, (New York: The Penguin Press, 2004), 261 
[51] Pettigrew, Memoirs, 146 of Correspondence 
[52] http://founders archives gov/documents/Franklin/01-22-02-0178, note 7  “Falconar married Hewson’s sister five months after the Doctor’s death; most of the Craven Street circle attended the wedding, and BF gave away the bride: Polly to Barbara Hewson, Oct  4, 1774, APS” (American Philosophical Society); “England Marriages, 1538–1973 ,” database, FamilySearch (https://familysearch org/ark:/61903/1:1:V52W-TGS : accessed September 15, 2015), Magnus Falconar and Dorothy Hewson, September 12, 1774; citing Saint Martin In The Fields, Westminster, London, England, reference ; FHL microfilm 561156, 561157, 561158, 942 B4HA V  25, 942 B4HA V  66 
[53] I chose to rely on the Westminster Rate Books for the numbering system on Craven Street  The books were consistent throughout the eighteenth century in the ordering of residents on the street and were used as the basis for the 1792 re-numbering  For the most part, commercial directories aligned with them as well  If by chance a directory didn’t initially align, it would inevitably produce future editions that did 
Benjamin Franklin, Benjamin Franklin House, London
More from David Turnquist
If one looked into Benjamin Franklin’s time on Craven Street, they might   
I think it’s very ironic that on the street maps included in your excellent article, Craven Street is so close to Scotland Yard  Because following the back and forth juxtapositions of numbers 7, 27 and 36 Craven Street (throw in 75 Northumberland Court and 1 Craven Street, too) was a case that could confound Sherlock Holmes 
Excellent job of deciphering street renumbering material spanning sixty years, including that of a wrong house number (# 7) being erroneously identified and then perpetuated in subsequent street map printings  It’s gratifying at least to know that the present day #36 Craven Street is the correct house for Ben Franklin tourists to visit  Except for #1 Craven Street for the last three years Franklin was in London, but we won’t get into that 
Again, excellent article, David!



ground_truth: The paper was published on 7 March 2023.