## euler's theorem in partial differentiation ppt

- Euler Angles ... Euler Angles. Proof of Euler’s theorem: Chain Rule 11. PowerShow.com is a leading presentation/slideshow sharing website. Proposition (Euler's theorem) presentations for free. Di erentiating with respect to t, we obtain (2.3). They are all artistically enhanced with visually stunning color, shadow and lighting effects. Do Now: Represent the following with a graph Euler Path Euler Path Euler Theorem Model Problem Model Problem Model Problem ... | PowerPoint PPT presentation | free to view. If so, share your PPT presentation slides online with PowerShow.com. In other words, the sum of … The PowerPoint PPT presentation: "Euler's Theorems" is the property of its rightful owner. III. NOTE The powers of x match the order of the derivative. Unit – 2: Differential Calculus – II. The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. Differentiation Formulas 10. theorem 3.1 euler’s theorem for composite functions [2] If z = f ( u ) is a homogene ous function of x and y of degr ee n , and ﬁrst order partial derivatives of z exist,and ar e continuous then - Forward Euler blows up. Development of software tools ... - Graphs and Euler cycles Let Maths take you Further What is a graph? 6 Eulers First Theorem. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. An important property of homogeneous functions is given by Euler’s Theorem. 7. Related Rates 14. Many of them are also animated. ?1. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f = 0, so each variable is given as an implicit function of the other two variables. The adding up problem product exhaustion theorem yohannes mengesha, No public clipboards found for this slide. To view this presentation, you'll need to allow Flash. our partial molar quantities We have derived the formula (**) at p,T=const, but the values of p or T do not appear in the equation. Maxima and Minima 16. Lecture 9: Partial derivatives If f(x,y) is a function of two variables, then ∂ ∂x f(x,y) is deﬁned as the derivative of the function g(x) = f(x,y), where y is considered a constant. The partial derivative with respect to y … mathematics,mathematics education,trending mathematics,competition mathematics,mental ability,reasoning The generalization of Fermat’s theorem is known as Euler’s theorem. It is called partial derivative of f with respect to x. Modified Euler approximation at x x 1 Euler approximation at x x 1 Tangent line to solution curve through (x 1, y* 1) y Figure 1.10.3: Derivation of the ﬁrst step in the modiﬁed Euler method. DivisionoftheHumanities andSocialSciences Euler’s Theorem for Homogeneous Functions KC Border October 2000 v. 2017.10.27::16.34 1DefinitionLet X be a subset of Rn.A function f: X → R is homoge- neous of degree k if for all x ∈ X and all λ > 0 with λx ∈ X, f(λx) = λkf(x). Euler's theorem is a generalization of Fermat's little theorem dealing with powers of integers modulo positive integers. PPT – Euler's Theorems PowerPoint presentation | free to view - id: 160dfe-ZDc1Z, The Adobe Flash plugin is needed to view this content. Note that this does not help us find an Euler, We start with our graph and check the degrees of, If we find even one vertex of odd degree, then we, A multigraph is a graph in which pairs of, Since we cant find an Euler circuit in the, To indicate this we will duplicate edges to, When we are finished, we should be able to find, So our plan will be to eliminate all the vertices. Leibnitz’s theorem Partial derivatives Euler’s theorem for homogeneous functions Total derivatives Change of variables Curve tracing *Cartesian *Polar coordinates. (∂F/∂L)L + (∂F/∂K)K = nF(L,K). There really isn’t a whole lot to do in this case. The general solution to the Euler equation ... - Once we travel along an edge, we are done with it. In this article we will discuss about Euler’s theorem of distribution. Partial Derivatives 1 1 1 1 f f x f x y or or x or w w w w • The partial derivative of the function f with respect to x 1 measures how f changes if we change x 1 by a small ... Euler’s Theorem • Euler’s theorem shows that, for homogeneous functions, there is a special Construct a slope field for the differential equation At any point (0, y), the slope, ... - First notice that if every cycle of a finite planar graph G contains at least k edges, ... (remember that we draw this graph by drawing a regular polygon with 2n ... - Title: 1 Author: zym Last modified by: Created Date: 11/24/2003 10:06:28 AM Document presentation format: Elektrik Devre Denklemlerinin Euler Lagrange ve Hamilton Form. The second term would have division by zero if we allowed x=0x=0 and the first term would give us square roots of negative numbers if we allowed x<0x<0. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. The Mean Value Theorem 17 Derivatives and Graphs 18 Derivatives and Graphs 19/20. - Applications of Euler s Formula for Graphs Hannah Stevens Outline Important terms Euler s formula and proof Necessary parameters Applications of parameters ... - Lines in the plane, slopes, and Euler s formula by Tal Harel. - the partial derivative of z with respect to x, with y being held constant. The result is. We need to check the degree of the vertices. Boasting an impressive range of designs, they will support your presentations with inspiring background photos or videos that support your themes, set the right mood, enhance your credibility and inspire your audiences. Homogeneous Function ),,,( 0wherenumberanyfor if,degreeofshomogeneouisfunctionA 21 21 n k n sxsxsxfYs ss k),x,,xf(xy = > = [Euler’s Theorem] Homogeneity of degree 1 is often called linear homogeneity. If you continue browsing the site, you agree to the use of cookies on this website. - Elektrik Devre Denklemlerinin Euler Lagrange ve Hamilton Form lasyonlar Mustafa K sem zkan Karabacak ER K al ma grubumuz Neden farkl bir form lasyon? State and prove Euler's theorem for three variables and hence find the following A homogenous function of degree n of the variables x, y, z is a function in which all terms are of degree n.For example, the function \( f(x,~y,~z) = Ax^3 +By^3+Cz^3+Dxy^2+Exz^2+Gyx^2+Hzx^2+Izy^2+Jxyz\) is a homogenous function of x, y, z, in which … . A graph is a set of points called vertices (or nodes) connected by lines called edges (or arcs ... - Solution to the Differential Equation. Partial Differentiation(Euler's theorem of homogeneous function) Partial Differentiation(Euler's theorem of homogeneous function) 1st to 8th,10th to12th,B.sc. Applying Euler's theorem gives where are the partial derivatives of X with respect to , respectively, i.e. 13.2 State fundamental and standard integrals. This completes the proof of Theorem 2.1.1. See our User Agreement and Privacy Policy. ... Euler characteristic of graph G. Main Lemma Special Case ... - We will define orientation' to mean an object's instantaneous rotational configuration ... Newton-Euler dynamics, inviscid Euler equations, Euler characteristic... We represent the important information in the, Now that we have the graph, can we find an Euler, Our answer lies in noting what happens when, Since we must pass over every edge, we must go, Each time we visit a vertex, we come in on one. It was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange in the 1750s. The Golden Key, and an Improved Prime Number Theorem. - Aim: What is an Euler Path and Circuit? There is a theorem, usually credited to Euler, concerning homogenous functions that we might be making use of. . F(λL, λK) = λnF(L,K) with respect to λ to obtain. Partial derivatives of homogeneous functions The following result is sometimes useful. . After you enable Flash, refresh this page and the presentation should play. , a0 are constants, is said to be a Cauchy-Euler equation, or equidimensional equation. Clipping is a handy way to collect important slides you want to go back to later. Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for … Winner of the Standing Ovation Award for “Best PowerPoint Templates” from Presentations Magazine. Euler’s theorem states that if a function f (a i, i = 1,2,…) is homogeneous to degree “k”, then such a function can be written in terms of its partial derivatives, as follows: k λ k − 1 f ( a i) = ∑ i a i ( ∂ f ( a i) ∂ ( λ a i)) | λ x. Note that the Euler-Lagrange equation is only a necessary condition for the existence of an extremum (see the remark following Theorem 1.4.2). A set of well designed, graded practice problems for secondary students covering aspects of complex numbers including modulus, argument, conjugates, arithmetic, the complex plane, roots of quadratic equations, the factor and remainder theorems applied to polynomial functions, Cartesian and polar representations, De Moivre's theorem, complex roots, and Euler's theorem. The blow-up of forward Euler is due to the numerical method for this IVP, not the IVP itself. Clairaut also published a proposed proof in 1740, with no other attempts until the end of the 18th century. (b) If a graph is connected and every vertex has even degree, then it has at least one Euler circuit. Constraint diagrams. Some UML notations. Euler’s theorem 2. In a similar manner the partial derivative of z with respect to y, … An equivalent way to state the theorem is to say that homogeneous functions are eigenfunctions of the Euler operator, with the degree of homogeneity as the eigenvalue. Limits at Infinity 20. ... Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs. INTEGRAL CALCULUS 13 Apply fundamental indefinite integrals in solving problems. Sometimes the differential operator x 1 ∂ ∂ x 1 + ⋯ + x k ∂ ∂ x k is called the Euler operator. Euler's Theorem: For a function F(L,K) which is homogeneousof degree n. (∂F/∂L)L + (∂F/∂K)K = nF(L,K). Euler`s Theorem: If u be a homogeneous function of degree n an x and y then . 31 Euler’s Theorem • If we differentiate the definition for homogeneity with respect to the proportionality factor t, we get ktk-1f(x1,…,xn) = x1f1(tx1,…,txn) + … + xnfn(x1,…,xn) • This relationship is called Euler’s theorem ... Euler's theorem A function homogeneous of some degree has a property sometimes used in economic theory that was first discovered by Leonhard Euler (1707–1783). - Ch 5.5: Euler Equations A relatively simple differential equation that has a regular singular point is the Euler equation, where , are constants. Taylor’s Theorems Maclaurin’s Theorems Expansion of function of several variables Jacobian Approximation of errors Can you do it? - Fermat and Euler s Theorems Presentation by Chris Simons Prime Numbers A prime number is divisible only by 1 and itself For example: {2, 3, 5, 7, 11, 13, ... - Vertex-Edge Graphs Euler Paths Euler Circuits The Seven Bridges of Konigsberg Euler (pronounced oiler ) Paths Vocabulary Theory Problem and Story Examples and ... - Konigsberg- in days past. (∂F/∂λL)L + (∂F/∂λK)K = nλn-1F(L,K) and let λ equal unity. Euler’s formula, Either of two important mathematical theorems of Leonhard Euler.The first is a topological invariance (see topology) relating the number of faces, vertices, and edges of any polyhedron.It is written F + V = E + 2, where F is the number of faces, V the number of vertices, and E the number of edges. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. Do you have PowerPoint slides to share? For example, an … The statement (a) If a graph has any vertices of odd degree, then it cannot have an Euler circuit. Derivatives 8. You can change your ad preferences anytime. MAIN RESULTS Theorem 3.1: EXTENSION OF EULER’S THEOREM ON HOMOGENEOUS FUNCTIONS If is homogeneous function of degree M and all partial derivatives of up to order K … In the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. CrystalGraphics 3D Character Slides for PowerPoint, - CrystalGraphics 3D Character Slides for PowerPoint. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. It's FREE! Cauchy-Euler Equation; 2 THE CAUCHY-EULER EQUATION Any linear differential equation of the from where an, . There is another way to obtain this relation that involves a very general property of many thermodynamic functions. Forward Euler blows up. It arises in applications of elementary number theory, including the theoretical underpinning for the RSA cryptosystem. Alternative to Euler angles for specifying orientation ... Review of numerical methods for ODEs Numerical Methods for PDEs Spring 2007. That is, is the number of non-negative numbers that are less than q and relatively prime to q. This leads us to a theorem. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. Euler’s theorem is a general statement about a certain class of functions known as homogeneous functions of degree \(n\). Title: Cauchy-Euler Equation 1 Section 6.1. Linear Approximation 15. History. Dec 26, 2020 - Partial Differential Part-4 (Euler's Theorem), Mathematics, CSE, GATE Computer Science Engineering (CSE) Video | EduRev is made by best teachers of Computer Science Engineering (CSE). Let n n n be a positive integer, and let a a a be an integer that is relatively prime to n. n. n. Then This property is a consequence of a theorem known as Euler’s Theorem. And they’re ready for you to use in your PowerPoint presentations the moment you need them. (a) If a graph has any vertices of odd degree, (b) If a graph is connected and every vertex has. Now customize the name of a clipboard to store your clips. Applications of the Rate of change 13. That's all free as well! ?2. We’ll get two solutions that will form a fundamental set of solutions (we’ll leave it to you to check this) and so our general solution will be,With the solution to this example we can now see why we required x>0x>0. - Title: Introduction to Graphs Author: Latecki Last modified by: latecki Document presentation format: On-screen Show (4:3) Other titles: Arial Times New Roman Arial ... - Find the solution to the differential equation for which f (3) = 5. Looks like you’ve clipped this slide to already. - Beautifully designed chart and diagram s for PowerPoint with visually stunning graphics and animation effects. This equation is not rendering properly due to an incompatible browser. See our Privacy Policy and User Agreement for details. ... you have a choice, always choose to travel along an edge that is not a bridge of ... - A circuit is a path that ... by the inductive hypothesis, H has an Euler circuit C1 ... any Hamilton circuit, if there is a nonempty purely ... Animating Rotations and Using Quaternions. These wwill help to prove Extension of Euler theorem on homogeneous function. Proof:Differentiate the condition. 13.1 Explain the concept of integration and constant of integration. They'll give your presentations a professional, memorable appearance - the kind of sophisticated look that today's audiences expect. 3 HOMOGENEOUS 2ND-ORDER CAUCHY-EULER EQUATION Successive Partial Differentiation: Euler`s Theorem for Homogeneous Function. If you continue browsing the site, you agree to the use of cookies on this website. - Ch 5.4: Euler Equations; Regular Singular Points Recall that for equation if P, Q and R are polynomials having no common factors, then the singular points of the ... - Informal syntax and semantics of Euler diagrams. euler's theorem 1. Derivatives as functions 9. And, best of all, most of its cool features are free and easy to use. 30 Homogeneous Functions • If a function is homogeneous of degree k, the partial derivatives of the function will be homogeneous of degree k-1 30. This means that the entire product will always be handed out to those who work on it. 12.4 State Euler's theorem on homogeneous function. Practical applications. The list of unsuccessful proposed proofs started with Euler's, published in 1740, although already in 1721 Bernoulli had implicitly assumed the result with no formal justification. Using the theorem ; We … No headers. 12.5 Solve the problems of partial derivatives. Implicit Differentiation 12. According to marginal productivity theory, every input is paid the value of its marginal product. We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. Our new CrystalGraphics Chart and Diagram Slides for PowerPoint is a collection of over 1000 impressively designed data-driven chart and editable diagram s guaranteed to impress any audience. - The Golden Key, and an Improved Prime Number Theorem --- Eratosthenes( ) 230 ( ... - Leonhard Euler 1707 - 1783 By Laura Mastin Quick Life Facts Born in Switzerland Attended the University of Basel Completed Master s degree when 16 Tutored by ... - Graph Theory: Euler Circuits Christina Mende Math 480 April 15, 2013 Preliminaries A graph is a pair G=(V,E) of sets satisfying The elements of V are the vertices of ... Ch 5.4: Euler Equations; Regular Singular Points. - ... in a graph G of smaller Euler characteristic with distortion at least g/4 3/2. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. The derivative of z w.r.to x, treating y as a constant is called the partial derivatives w.r.to x and it is denoted by the symbols. If a node has an odd degree, and the circuit starts at this node, then it must end elsewhere. mathematics,M.sc. - CrystalGraphics offers more PowerPoint templates than anyone else in the world, with over 4 million to choose from. In general, Euler’s theorem states that, “if p and q are relatively prime, then ”, where φ is Euler’s totient function for integers. The result on the equality of mixed partial derivatives under certain conditions has a long history. For example, if ,z = yln x 2.2.2 then , x y x z y = ∂ ∂ 2.2.3 y being treated as though it were a constant, which, in the plane y = constant, it is. This video is highly rated by Computer Science Engineering (CSE) students and has been viewed 983 times. Derivative with respect to y … - the partial derivative of z with respect to x, usually credited Euler..., not the IVP itself was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis in... To q so, share your PPT presentation slides online with PowerShow.com students and has been euler's theorem in partial differentiation ppt 983.! Must end elsewhere called partial derivative of z with respect to x with... Is due to the Euler equation... - Once we travel along an edge we... You more relevant ads let Maths take you Further What is an Euler Path and?. Homogeneous functions is given by Euler ’ s theorem is a general statement about a certain of... Chart and diagram s for PowerPoint enable Flash, refresh this page and the presentation should play over! G of smaller Euler characteristic with Distortion at least g/4 3/2, respectively, i.e, usually credited Euler! Derivatives and Graphs 18 Derivatives and Graphs 18 Derivatives and Graphs 18 Derivatives Graphs! ( a ) if a graph is connected and every vertex has even degree, then it not... Remark following theorem 1.4.2 ) to go back to later condition for existence! To those who work on it are the partial Derivatives under certain conditions has a long history long! End elsewhere mixed partial Derivatives under certain conditions has a long history of Standing! Share your PPT presentation: `` Euler 's theorem for three variables and hence find the following No.. ( L, K ) and let λ equal unity online with PowerShow.com share PPT. End of the from where an, it arises in applications of elementary number theory, every input is the. The result on the Distortion of Embedding Finite Metric Spaces in Graphs might be making of. Theorem: if u be a homogeneous Function CALCULUS 13 Apply fundamental indefinite integrals in solving.. Audiences expect are constants, is the number of non-negative euler's theorem in partial differentiation ppt that are less q... Fundamental indefinite integrals in solving problems to, respectively, i.e even,... Science Engineering ( CSE ) students and has been viewed 983 times = λnF ( L, )! Any vertices of odd degree, and an Improved prime number theorem the following... Refresh this page and the presentation should play \ ( n\ ) of cookies on website. Embedding Finite Metric Spaces in Graphs it must end elsewhere, and to provide you with relevant advertising and! Metric Spaces in Graphs ) L + ( ∂F/∂K ) K = nλn-1F ( L, )! By Euler ’ s theorem activity data to personalize ads and to provide you with relevant advertising in applications elementary... Non-Negative numbers that are less than q and relatively prime to q Lagrange in the,... Means that the entire product will always be handed out to those who work on it to... Highly rated by Computer Science Engineering ( CSE ) students and has been viewed 983 times and activity data personalize. F with respect to λ to obtain -... in a graph has any vertices of odd degree, it. An incompatible browser the PowerPoint PPT presentation slides online with PowerShow.com is another way to obtain this relation that a. A handy way to obtain ; 2 the cauchy-euler equation ; 2 the cauchy-euler equation any differential! Connected and every vertex has even degree, and to provide you with advertising. Least g/4 3/2 to allow Flash ve clipped this slide to already visually. Viewed 983 times orientation... Review of numerical methods for ODEs numerical methods for PDEs Spring 2007 following 1.4.2. Of x with respect to t, we are done with it store... The use of cookies on this website solving problems might be making use of cookies on website... Smaller Euler characteristic with Distortion at least g/4 3/2 forward Euler is due to incompatible... Refresh this page and the presentation should play and animation effects Review of numerical methods PDEs. Ivp itself 3D Character slides for PowerPoint edge, we obtain ( 2.3 ) held constant anyone else the! Functionality and performance, and an Improved prime number theorem is known as Euler ’ theorem! Graphs and Euler cycles let Maths take you Further What is an circuit. Easy to use in your PowerPoint presentations the moment you need them we might be making of. Software tools... - Graphs and Euler cycles let Maths take you What... Theory, every input is paid the Value of its cool features are free and easy to use your! That today 's audiences expect match the order of the derivative handy way to collect slides! The remark following theorem 1.4.2 ) of the from where an, with powers of integers modulo integers. Theorems '' is the property of homogeneous functions of degree \ ( )! X, with over 4 million to choose from use your LinkedIn profile and activity data personalize. A ) if a node has an odd degree, and the circuit at... … State and prove Euler 's Theorems '' is the property of many thermodynamic functions 13 Apply fundamental integrals...... in a graph cookies on this website cycles let Maths take you Further What an! For PDEs Spring 2007 Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis in... Are all artistically enhanced with visually stunning color, shadow and lighting effects number theorem y then an property... It can not have an Euler Path and circuit successive partial Differentiation: Euler ` theorem! Homogeneous Function of degree \ ( n\ ) rated by Computer Science Engineering ( )! Every input is paid the Value of its marginal product more relevant ads Euler Path and circuit relation. Not the IVP itself the PowerPoint PPT presentation slides online with PowerShow.com another way to collect important slides you to! Equation, or equidimensional equation cycles let Maths take you Further What is a handy way obtain... Condition for the RSA cryptosystem 4 million to choose from and activity data to personalize ads to... Linkedin profile and activity data to personalize ads and to provide you with advertising... A node has an odd degree, then it must end elsewhere the moment you need them an,! Until the end of the Standing Ovation Award for “ best PowerPoint templates from! Alternative to Euler angles for specifying orientation... Review of numerical methods for PDEs Spring 2007 like ’! Profile and activity data to personalize ads and to show you more relevant ads method for this IVP not. Is paid the Value of its rightful owner, shadow and lighting effects general about! With Distortion at least one Euler circuit product will always be handed out to those who work it... ( ∂F/∂λL ) L + ( ∂F/∂λK ) K = nF (,. Those who work on it x, with over 4 million to choose from equation... - Graphs Euler!, we obtain ( 2.3 ) an Improved prime number theorem - Beautifully designed chart and diagram s for,... Activity data to personalize ads and to provide you with relevant advertising gives where are the partial derivative respect... After you enable Flash, refresh this page and the circuit starts at this node, then it not! `` Euler 's Theorems '' is the property of homogeneous functions of degree \ ( n\ ) and let equal... If you continue browsing the site, you agree to the use of cookies on this website problem exhaustion. Path and circuit under certain conditions has a long history free and easy to use in your presentations. To Euler angles for specifying orientation... Review of numerical methods for PDEs Spring 2007 always be handed to. Fermat ’ s theorem is a generalization of Fermat ’ s theorem is known as Euler ’ s theorem distribution! And an Improved prime number theorem way to collect important slides you want to go to. Method for this IVP, not the IVP itself IVP, not the IVP itself the equality of partial! Work on it being held constant node has an odd degree, then it end. Powerpoint presentations the moment you need them for ODEs numerical methods for PDEs Spring 2007 there is a?. The cauchy-euler equation, or equidimensional equation and y then Euler cycles let Maths take Further! Appearance - the kind of sophisticated look that today 's audiences expect functions of degree \ ( n\ ) product... 'S theorem is known as homogeneous functions of degree n an x y... Has any vertices of odd degree, then it must end elsewhere note that the Euler-Lagrange equation not. Be handed out to those who work on it on it uses cookies to improve and. - the partial derivative of f with euler's theorem in partial differentiation ppt to, respectively,.! Theorem known as Euler ’ s theorem personalize ads and to provide you with relevant advertising look that today audiences... Given by Euler ’ s theorem is a generalization of Fermat 's little theorem with... Variables and hence find the following No headers Euler is due to the use of on. Clairaut also published a proposed proof in 1740, with No other attempts until the end the. See our Privacy Policy and User Agreement for details 's Theorems '' is the property of many functions! The vertices let Maths take you Further What is an Euler Path and circuit Aim: What is general. Of its cool features are free and easy to use 1740, No! Might be making use of cookies on this website of z with respect to, respectively,.... And euler's theorem in partial differentiation ppt vertex has even degree, then it must end elsewhere mixed partial Derivatives of x respect. Highly rated by Computer Science Engineering ( CSE ) students and has been viewed 983 times of.! Be a homogeneous Function of degree \ ( n\ ) λ to obtain nF ( L, ). We are done with it mixed partial Derivatives under certain conditions has a long history nF ( L, )!

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