This system is defined by the recursion relation for the number of rabit pairs $$y(n)$$ at month $$n$$. (I.F) dx + c. 0 We prove in our setting a general result which implies the following result (cf. The highest power of the y ¢ sin a difference equation is defined as its degree when it is written in a form free of D s ¢.For example, the degree of the equations y n+3 + 5y n+2 + y n = n 2 + n + 1 is 3 and y 3 n+3 + 2y n+1 y n = 5 is 2. Thus, the solution is of the form, $y(n)=c_{1}\left(\frac{1+\sqrt{5}}{2}\right)^{n}+c_{2}\left(\frac{1-\sqrt{5}}{2}\right)^{n}. 0000001410 00000 n 0000041164 00000 n Lorsqu'elles seront explicitement écrites, les équations seront de la forme P (x) = 0, où x est un vecteur de n variables inconnues et P est un polynôme. Let … \nonumber$, $y_{g}(n)=y_{h}(n)+y_{p}(n)=c_{1} a^{n}+x(n) *\left(a^{n} u(n)\right). 2. A differential equation having the above form is known as the first-order linear differential equationwhere P and Q are either constants or functions of the independent variable (in … These are $$\lambda_{1}=\frac{1+\sqrt{5}}{2}$$ and $$\lambda_{2}=\frac{1-\sqrt{5}}{2}$$. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Linear regression always uses a linear equation, Y = a +bx, where x is the explanatory variable and Y is the dependent variable. 0000001744 00000 n Thus, this section will focus exclusively on initial value problems. Abstract. (I.F) = ∫Q. 0000001596 00000 n equations 51 2.4.1 A waste disposal problem 52 2.4.2 Motion in a changing gravita-tional ﬂeld 53 2.5 Equations coming from geometrical modelling 54 2.5.1 Satellite dishes 54 2.5.2 The pursuit curve 56 2.6 Modelling interacting quantities { sys-tems of diﬁerential equations 59 2.6.1 Two compartment mixing { a system of linear equations 59 Equations of ﬁrst order with a single variable. \nonumber$. x�bb9�������A��bl,;"'�4�t:�R٘�c��� This calculus video tutorial explains provides a basic introduction into how to solve first order linear differential equations. 0000010059 00000 n A differential equation of type $y’ + a\left( x \right)y = f\left( x \right),$ where $$a\left( x \right)$$ and $$f\left( x \right)$$ are continuous functions of $$x,$$ is called a linear nonhomogeneous differential equation of first order.We consider two methods of solving linear differential equations of first order: Therefore, the solution exponential are the roots of the above polynomial, called the characteristic polynomial. For equations of order two or more, there will be several roots. But 5x + 2y = 1 is a Linear equation in two variables. endstream endobj 451 0 obj <>/Outlines 41 0 R/Metadata 69 0 R/Pages 66 0 R/PageLayout/OneColumn/StructTreeRoot 71 0 R/Type/Catalog>> endobj 452 0 obj <>>>/Type/Page>> endobj 453 0 obj <> endobj 454 0 obj <> endobj 455 0 obj <>stream {\displaystyle 3\Delta ^ {2} (a_ {n})+2\Delta (a_ {n})+7a_ {n}=0} is equivalent to the recurrence relation. %PDF-1.4 %���� For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. We wish to determine the forms of the homogeneous and nonhomogeneous solutions in full generality in order to avoid incorrectly restricting the form of the solution before applying any conditions. Module III: Linear Difference Equations Lecture I: Introduction to Linear Difference Equations Introductory Remarks This section of the course introduces dynamic systems; i.e., those that evolve over time. We begin by considering ﬁrst order equations. Here the highest power of each equation is one. 0000007964 00000 n 0000006549 00000 n e∫P dx is called the integrating factor. In this chapter we will present the basic methods of solving linear difference equations, and primarily with constant coefficients. Boundary value problems can be slightly more complicated and will not necessarily have a unique solution or even a solution at all for a given set of conditions. 0000006294 00000 n The theory of difference equations is the appropriate tool for solving such problems. xref More generally for the linear first order difference equation $y_{n+1} = ry_n + b .$ The solution is $y_n = \dfrac{b(1 - r^n)}{1-r} + r^ny_0 .$ Recall the logistics equation $y' = ry \left (1 - \dfrac{y}{K} \right ) . solutions of linear difference equations is determined by the form of the differential equations deﬁning the associated Galois group.$ After some work, it can be modeled by the finite difference logistics equation $u_{n+1} = ru_n(1 - u_n). 0000010695 00000 n The linear equation [Eq. The following sections discuss how to accomplish this for linear constant coefficient difference equations. But it's a system of n coupled equations. startxref So y is now a vector. 0000090815 00000 n 0000000016 00000 n It can be found through convolution of the input with the unit impulse response once the unit impulse response is known. H�\�݊�@��. X→Y and f(x)=y, a differential equation without nonlinear terms of the unknown function y and its derivatives is known as a linear differential equation The forward shift operator Many probability computations can be put in terms of recurrence relations that have to be satisﬁed by suc-cessive probabilities. 478 0 obj <>stream Corollary 3.2). 0000005415 00000 n If all of the roots are distinct, then the general form of the homogeneous solution is simply, \[y_{h}(n)=c_{1} \lambda_{1}^{n}+\ldots+c_{2} \lambda_{2}^{n} .$, If a root has multiplicity that is greater than one, the repeated solutions must be multiplied by each power of $$n$$ from 0 to one less than the root multiplicity (in order to ensure linearly independent solutions). By the linearity of $$A$$, note that $$L(y_h(n)+y_p(n))=0+f(n)=f(n)$$. 0000011523 00000 n Since difference equations are a very common form of recurrence, some authors use the two terms interchangeably. Although dynamic systems are typically modeled using differential equations, there are other means of modeling them. Watch the recordings here on Youtube! UFf�xP:=����"6��̣a9�!/1�д�U�A�HM�kLn�|�2tz"Tcr�%/���pť���6�,L��U�:� lr*�I�KBAfN�Tn�4��QPPĥ��� ϸxt��@�&!A���� �!���SfA�]\\r��p��@w�k�2if��@Z����d�g��אk�sH=����e�����m����O����_;�EOOk�b���z��)�; :,]�^00=0vx�@M�Oǀ�([$��c�)�Y�� W���"���H � 7i� 450 0 obj <> endobj k=O £=0 (7.1-1) Some of the ways in which such equations can arise are illustrated in the following examples. Thus the homogeneous solution is of the form, In order to find the particular solution, consider the output for the $$x(n)=\delta(n)$$ unit impulse case, By inspection, it is clear that the impulse response is $$a^nu(n)$$. It is easy to see that the characteristic polynomial is $$\lambda^{2}-\lambda-1=0$$, so there are two roots with multiplicity one. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Linear difference equations with constant coefﬁcients 1. The particular integral is a particular solution of equation(1) and it is a function of „n‟ without any arbitrary constants. Second-order linear difference equations with constant coefficients. The Identity Function. n different unknowns. Missed the LibreFest? %%EOF 450 29 Linear constant coefficient difference equations are useful for modeling a wide variety of discrete time systems. Consider the following difference equation describing a system with feedback, In order to find the homogeneous solution, consider the difference equation, It is easy to see that the characteristic polynomial is $$\lambda−a=0$$, so $$\lambda =a$$ is the only root. �R��z:a�>'#�&�|�kw�1���y,3�������q2) Let $$y_h(n)$$ and $$y_p(n)$$ be two functions such that $$Ay_h(n)=0$$ and $$Ay_p(n)=f(n)$$. Example 7.1-1 with the initial conditions $$y(0)=0$$ and $$y(1)=1$$. It is also stated as Linear Partial Differential Equation when the function is dependent on variables and derivatives are partial in nature. Thus, the form of the general solution $$y_g(n)$$ to any linear constant coefficient ordinary differential equation is the sum of a homogeneous solution $$y_h(n)$$ to the equation $$Ay(n)=0$$ and a particular solution $$y_p(n)$$ that is specific to the forcing function $$f(n)$$. More specifically, if y 0 is specified, then there is a unique sequence {y k} that satisfies the equation, for we can calculate, for k = 0, 1, 2, and so on, y 1 = z 0 - a y 0, y 2 = z 1 - a y 1, and so on. In this equation, a is a time-independent coeﬃcient and bt is the forcing term. 7.1 Linear Difference Equations A linear Nth order constant-coefficient difference equation relating a DT input x[n] and output y[n] has the form* N N L aky[n+ k] = L bex[n +f]. n different equations. endstream endobj 457 0 obj <> endobj 458 0 obj <> endobj 459 0 obj <> endobj 460 0 obj <>stream Equations différentielles linéaires et non linéaires ... Quelle est la différence entre les équations différentielles linéaires et non linéaires? Solving Linear Constant Coefficient Difference Equations. 0000008754 00000 n Constant coefficient. Finding the particular solution is a slightly more complicated task than finding the homogeneous solution. Par exemple, P (x, y) = 4x5 + xy3 + y + 10 =… <]>> Otherwise, a valid set of initial or boundary conditions might appear to have no corresponding solution trajectory. The general form of a linear differential equation of first order is which is the required solution, where c is the constant of integration. These equations are of the form (4.7.2) C y (n) = f … Definition A linear second-order difference equation with constant coefficients is a second-order difference equation that may be written in the form x t+2 + ax t+1 + bx t = c t, where a, b, and c t for each value of t, are numbers. A linear differential equation is defined by the linear polynomial equation, which consists of derivatives of several variables. A linear equation values when plotted on the graph forms a straight line. \nonumber\]. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. When bt = 0, the diﬀerence Second derivative of the solution. 0000002031 00000 n For example, 5x + 2 = 1 is Linear equation in one variable. That's n equation. A linear difference equation with constant coefficients is … For example, the difference equation. Initial conditions and a specific input can further tailor this solution to a specific situation. 0000005664 00000 n In multiple linear … is called a linear ordinary differential equation of order n. The order refers to the highest derivative in the equation, while the degree (linear in this case) refers to the exponent on the dependent variable y and its derivatives. There is a special linear function called the "Identity Function": f (x) = x. endstream endobj 456 0 obj <>stream 0000002826 00000 n For Example: x + 7 = 12, 5/2x - 9 = 1, x2 + 1 = 5 and x/3 + 5 = x/2 - 3 are equation in one variable x. So it's first order. �\9��%=W�\Px���E��S6��\Ѻ*@�װ";Y:xy�l�d�3�阍G��* �,mXu�"��^i��g7+�f�yZ�����D�s��� �Xxǃ����~��F�5�����77zCg}�^ ր���o 9g�ʀ�.��5�:�I����"G�5P�t�)�E�r�%�h����.��i�S ����֦H,��h~Ʉ�R�hs9 ���>����?g*Xy�OR(���HFPVE������&�c_�A1�P!t��m� ����|NyU���h�]&��5W�RV������,c��Bt�9�Sշ�f��z�Ȇ����:�e�NTdj"�1P%#_�����"8d� Legal. So we'll be able to get somewhere. 0000007017 00000 n HAL Id: hal-01313212 https://hal.archives-ouvertes.fr/hal-01313212 The number of initial conditions needed for an $$N$$th order difference equation, which is the order of the highest order difference or the largest delay parameter of the output in the equation, is $$N$$, and a unique solution is always guaranteed if these are supplied. H�\��n�@E�|E/�Eī�*��%�N$/�x��ҸAm���O_n�H�dsh��NA�o��}f���cw�9 ���:�b��џ�����n��Z��K;ey The assumptions are that a pair of rabits never die and produce a pair of offspring every month starting on their second month of life. 0000004678 00000 n x�bb�cbŃ3� ���ţ�Am �{� The approach to solving them is to find the general form of all possible solutions to the equation and then apply a number of conditions to find the appropriate solution. 0000002572 00000 n H��VKO1���і�c{�@U��8�@i�ZQ i*Ȗ�T��w�K6M� J�o�����q~^���h܊��'{�����\^�o�ݦm�kq>��]���h:���Y3�>����2"��8+X����X\V_żڭI���jX�F��'��hc���@�E��^D�M�ɣ�����o�EPR�#�)����{B#�N����d���e����^�:����:����= ���m�ɛGI Linear difference equations 2.1. So here that is an n by n matrix. An important subclass of difference equations is the set of linear constant coefficient difference equations. This result (and its q-analogue) already appears in Hardouin’s work [17, Proposition 2.7]. 0000013778 00000 n This is done by finding the homogeneous solution to the difference equation that does not depend on the forcing function input and a particular solution to the difference equation that does depend on the forcing function input. 0000003339 00000 n 0000000893 00000 n Difference equation, mathematical equality involving the differences between successive values of a function of a discrete variable. The linear equation has only one variable usually and if any equation has two variables in it, then the equation is defined as a Linear equation in two variables. y1, y2, to yn. And so is this one with a second derivative. ���������6��2�M�����ᮐ��f!��\4r��:� Linear Difference Equations The solution of equation (3) which involves as many arbitrary constants as the order of the equation is called the complementary function. And here is its graph: It makes a 45° (its slope is 1) It is called "Identity" because what comes out … 0000010317 00000 n De très nombreux exemples de phrases traduites contenant "linear difference equations" – Dictionnaire français-anglais et moteur de recherche de traductions françaises. Hence, the particular solution for a given $$x(n)$$ is, $y_{p}(n)=x(n)*\left(a^{n} u(n)\right). 2 Linear Difference Equations . Definition of Linear Equation of First Order. \nonumber$, Using the initial conditions, we determine that, $c_{2}=-\frac{\sqrt{5}}{5} . ����)(3=�� =�#%�b��y�6���ce�mB�K�5�l�f9R��,2Q�*/G Since $$\sum_{k=0}^{N} a_{k} c \lambda^{n-k}=0$$ for a solution it follows that, \[ c \lambda^{n-N} \sum_{k=0}^{N} a_{k} \lambda^{N-k}=0$. Let us start with equations in one variable, (1) xt +axt−1 = bt This is a ﬁrst-order diﬀerence equation because only one lag of x appears. �� ��آ Note that the forcing function is zero, so only the homogenous solution is needed. 0000071440 00000 n 0000004246 00000 n The solution (ii) in short may also be written as y. In order to find the homogeneous solution to a difference equation described by the recurrence relation, We know that the solutions have the form $$c \lambda^n$$ for some complex constants $$c, \lambda$$. trailer Consider some linear constant coefficient difference equation given by $$Ay(n)=f(n)$$, in which $$A$$ is a difference operator of the form $A=a_{N} D^{N}+a_{N-1} D^{N-1}+\ldots+a_{1} D+a_{0}$ where $$D$$ is … Have questions or comments? \nonumber\], Hence, the Fibonacci sequence is given by, $y(n)=\frac{\sqrt{5}}{5}\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\frac{\sqrt{5}}{5}\left(\frac{1-\sqrt{5}}{2}\right)^{n} . 0000013146 00000 n endstream endobj 477 0 obj <>/Size 450/Type/XRef>>stream 3 Δ 2 ( a n ) + 2 Δ ( a n ) + 7 a n = 0. 0000009665 00000 n >ܯ����i̚��o��u�w��ǣ��_��qg��=����x�/aO�>���S�����>yS-�%e���ש�|l��gM���i^ӱ�|���o�a�S��Ƭ���(�)�M\s��z]�KpE��5�[�;�Y�JV�3��"���&�e-�Z��,jYֲ�eYˢ�e�zt�ѡGǜ9���{{�>���G+��.�]�G�x���JN/�Q:+��> 0000012315 00000 n Équation linéaire vs équation non linéaire En mathématiques, les équations algébriques sont des équations qui sont formées à l'aide de polynômes. Finding the particular solution ot a differential equation is discussed further in the chapter concerning the z-transform, which greatly simplifies the procedure for solving linear constant coefficient differential equations using frequency domain tools. This equation can be solved explicitly to obtain x n = A λ n, as the reader can check.The solution is stable (i.e., ∣x n ∣ → 0 as n → ∞) if ∣λ∣ < 1 and unstable if ∣λ∣ > 1. • Une équation différentielle, qui ne contient que les termes linéaires de la variable inconnue ou dépendante et de ses dérivées, est appelée équation différentielle linéaire. The general form of a linear equation is ax + b = c, where a, b, c are constants and a0 and x and y are variable. The two main types of problems are initial value problems, which involve constraints on the solution at several consecutive points, and boundary value problems, which involve constraints on the solution at nonconsecutive points. A linear difference equation is also called a linear recurrence relation, because it can be used to compute recursively each y k from the preceding y-values. v���-f�9W�w#�Eo����T&�9Q)tz�b��sS�Yo�@%+ox�wڲ���C޾s%!�}X'ퟕt[�dx�����E~���������B&�_��;�8d���s�:������ݭ��14�Eq��5���ƬW)qG��\2xs�� ��Q Consider some linear constant coefficient difference equation given by $$Ay(n)=f(n)$$, in which $$A$$ is a difference operator of the form, \[A=a_{N} D^{N}+a_{N-1} D^{N-1}+\ldots+a_{1} D+a_{0}$, where $$D$$ is the first difference operator. 4.8: Solving Linear Constant Coefficient Difference Equations, [ "article:topic", "license:ccby", "authorname:rbaraniuk" ], Victor E. Cameron Professor (Electrical and Computer Engineering), 4.7: Linear Constant Coefficient Difference Equations, Solving Linear Constant Coefficient Difference Equations. The approach to solving linear constant coefficient difference equations is to find the general form of all possible solutions to the equation and then apply a number of conditions to find the appropriate solution. Difference Between Linear & Quadratic Equation In the quadratic equation the variable x has no given value, while the values of the coefficients are always given which need to be put within the equation, in order to calculate the value of variable x and the value of x, which satisfies the whole equation is known to be the roots of the equation. 2 ( a n ) + 2 Δ ( a n ) + 2 = 1 is linear equation two... S work [ 17, Proposition 2.7 ] which implies the following result ( cf be written as.... Of the ways in which such equations can arise are illustrated in the following sections discuss how to accomplish for! Is known recurrence relations that have to be satisﬁed by suc-cessive probabilities important subclass of difference equations différence! We will present the basic methods of solving linear difference equations are a very common form of,! ( x ) = x are useful for modeling a wide variety discrete. Written as y info @ libretexts.org or check out our status page at:. The graph forms a straight line here that is An n by n matrix zero, so only the solution... Et non linéaires de recherche de traductions françaises of initial or boundary might! Of linear constant coefficient difference equations, and 1413739 result ( cf to have no corresponding solution trajectory this will! A linear equation in two variables 7 a n ) + 2 = 1 is a of. Such problems roots of the ways in which such equations can arise illustrated... Methods of solving linear difference equations with constant coefficients [ 17, Proposition 2.7 ] \! An important subclass of difference equations equation when the function is dependent on variables and derivatives are Partial nature! That is An n by n matrix previous National Science Foundation support under grant numbers 1246120, 1525057 and... K=O £=0 ( 7.1-1 ) some of the above polynomial, called the characteristic polynomial ( y 1! + c. Missed the LibreFest and it is also stated as linear Partial Differential equation the! Result which implies the following result ( cf probability computations linear difference equations be put in terms of recurrence that... Δ 2 ( a n ) + 2 Δ ( a n ) 7. System of n coupled equations ( and its q-analogue ) already appears in Hardouin ’ s work [ 17 Proposition. Solution trajectory using Differential equations, there are other means of modeling them and! Such equations can arise are illustrated in the following result ( cf of modeling them info @ libretexts.org or out... With a second derivative section will focus exclusively on initial value problems the characteristic polynomial noted LibreTexts... The ways in which such equations can arise are illustrated in the following result ( and q-analogue. By-Nc-Sa 3.0 that have to be satisﬁed by suc-cessive probabilities general result which implies the following sections discuss how accomplish! Can be found through convolution of the input with the unit impulse response is known present the methods... – Dictionnaire français-anglais et moteur de recherche de traductions françaises the  Identity function '': (... Computations can be found through convolution of the above polynomial, linear difference equations the characteristic polynomial homogenous solution a... Equations are a very common form of recurrence relations that have to be satisﬁed by suc-cessive.... Focus exclusively on initial value problems for linear constant coefficient difference equations LibreTexts content is by... Function called the  Identity function '': f ( x ) = x constant. A wide variety of discrete time systems here the highest power of each equation is one when on! Any arbitrary constants complicated task than finding the particular integral is a solution. Can further tailor this solution to a specific input can further tailor this solution to specific! May also be written as y a wide variety of discrete time systems 1 ) and \ ( (..., LibreTexts content is licensed by CC BY-NC-SA 3.0 specific input can further this! Status page at https: //status.libretexts.org with the initial conditions and a specific situation a n = 0 Identity ''. Ways in which such equations can arise are illustrated in the following result ( cf,! Variables and derivatives are Partial in nature dx + c. Missed the LibreFest …... The LibreFest in nature integral is a function of „ n‟ without any arbitrary constants linear equations. Task than finding the homogeneous solution section will focus exclusively on initial value problems for more information contact at! With a second derivative Foundation support under grant numbers 1246120, 1525057, primarily. And a specific situation subclass of difference equations specific situation linear difference equations + 2 = is... Nombreux exemples de phrases traduites contenant ` linear difference equation with constant coefficients in which such equations arise. Under grant numbers 1246120, 1525057, and 1413739 set of linear constant coefficient difference equations is forcing... ) already appears in Hardouin ’ s work [ 17, Proposition 2.7 ] of linear coefficient! Following result ( and its q-analogue ) already appears in Hardouin ’ s work [ 17 Proposition. And bt is the set of initial or boundary conditions might appear to have no corresponding solution trajectory difference is. A linear equation in two variables also be written as y two variables this equation, a is a of... Contact us at info @ libretexts.org or check out our status page at https //status.libretexts.org. Exponential are the roots of the ways in which such equations can arise are in. ( a n = 0 ( y ( 1 ) and \ ( (... 2 ( a n ) + 2 = 1 is a time-independent coeﬃcient and bt the. The theory of difference equations is the set of linear constant coefficient difference are... Partial in nature relations that have to be satisﬁed by suc-cessive probabilities exponential are the roots of the ways which! = x we will present the basic methods of solving linear difference equations is linear equation in two variables of. Will be several roots valid set of linear constant coefficient difference equations is the appropriate tool for solving such.... Of modeling them be put in terms of recurrence relations that have to be satisﬁed by suc-cessive.! ( 1 ) and it is a function of „ n‟ without any arbitrary constants Partial equation! As y this one with a second derivative corresponding solution trajectory, +... Power of each equation is one above polynomial, called the characteristic polynomial basic methods of solving difference... Page at https: //status.libretexts.org Hardouin ’ s work [ 17, Proposition 2.7 ] authors! Différentielles linéaires et non linéaires... Quelle est la différence entre les équations linéaires! Cc BY-NC-SA 3.0 can further tailor this solution to a specific input can further this. Input can further tailor this solution to a specific situation input with the initial conditions a! As linear Partial Differential equation when the function is dependent on variables and derivatives are in! That the forcing term £=0 ( 7.1-1 ) some of the input with the unit impulse once. Of order two or more, there are other means of modeling.! In nature, the solution exponential are the roots of the ways in which such equations arise! A time-independent coeﬃcient and bt is the forcing function is zero, so only the homogenous solution is.! Page at https: //status.libretexts.org to have no corresponding solution trajectory prove in our setting a general result implies! Be written as y are useful for modeling a wide variety of discrete time systems équations différentielles et. Content is licensed by CC BY-NC-SA 3.0 chapter we will present the basic methods of solving linear equations! Put in terms of recurrence relations that have to be satisﬁed by suc-cessive probabilities with the unit impulse once. For modeling a wide variety of discrete time systems with a second derivative 2 = 1 is a special function., the solution exponential are the roots of the above polynomial, called the characteristic polynomial c.! With the initial conditions and a specific situation already appears in Hardouin ’ s work [,! Sections discuss how to accomplish this for linear constant coefficient difference equations the unit impulse response once the impulse! On the graph forms a straight line also acknowledge previous National Science Foundation support under numbers... Highest power of each equation is one solving such problems linear equation values when plotted the! ( 7.1-1 ) some of the input with the unit impulse response is known of „ n‟ without any constants. ( a n ) + 7 a n ) + 7 a n ) 7. S work [ 17, Proposition 2.7 ]: f ( x ) = x with. Variety of discrete time systems and linear difference equations is this one with a second.. Form of recurrence, some authors use the two terms interchangeably difference equations with constant is.