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Application Of Exact Differential Equation

Differential Equations ENGM316 Differential Equations with Engineering Applications For where your treasure is,. If you write the equation in the form P +Qy0 =0,theninanexactequation you will usually notice that P and Q will have pairs of terms where the term in P will have the form df dx g(y) and the term in Q has the form f(x) dg dy y0 (where f and g. The first three worksheets practise methods for solving first order differential equations which are taught in MATH108. And it's just another method for solving a certain type of differential equations. M427J - Differential equations and linear algebra. 1 Population Growth Problem Assume that the population of Washington, DC, grows due to births and deaths at the rate of 2% per year and there is a net migration into the city of 15,000 people per. Equilibrium Solutions. In Birrel & Davies: QFT in curved spacetime it is written that the following differential equation can be solved in terms of hypergeometric functions. Linear equations and Bernouli's form. 8 CHAPTER 1. Consider a general nonlinear di_erential equation as. 3 Definitions and Examples 1. 1) Put in standard form, , Note & 2) , 3) They are not equal and therefore it is NOT an exact differential equation. In the present section, separable differential equations and their solutions are discussed in greater detail. Deﬁning solvable differential equations We introduce a sequence of new variables η i =η i(x,t), 16i 6n, by solvable PDEs, for instance, the linear ones, η i,x. Numerical Solution of Fuzzy Differential Equations and its Applications: 10. 2018;2(1):19-31. Runge–Kutta methods for ordinary differential equations – p. Differential calculus or Differentiation) and Calculus 2 (a. A exact differential equation the general form P(x,y) y'+Q(x,y)=0 Differential equation is a mathematical equation. Presentation,animation,graphic. Find the general solution of xy0 = y−(y2/x). Answers or hints to most problems appear at end. For example, I show how ordinary diﬀerential equations arise in classical physics from the fun-damental laws of motion and force. The average is now 70/100. Find materials for this course in the pages linked along the left. Get free Research Paper on SILMULTANEOUS DIFFERENTIAL EQUATION AND ITS APPLICATION research project topics and materials in Nigeria. The differential equation is exact because and But the equation is not exact, even though it differs from the first equation only by a single sign. Definition of Exact Equation. 2 solutions of linear equations. For permissions beyond the scope of this license, please contact us. These ode can be analyzed qualitatively. By the degree of a differential equation, when it is a polynomial equation in derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation. Competence in solving first order differential equations employing the techniques of variables separable, homogeneous coefficient, or exact equations. Other methods for solving ﬁrst-order ordinary differential equations include the integration of exact equations, and the use of either clever substitutions or more general integrating factors to reduce “difﬁcult” equations to either separable, linear or exact equations. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. Find many great new & used options and get the best deals for A First Course in Differential Equations : With Modeling Applications by <NAME>. This leads to the differential equation: You can see this circuit and the analysis on the Wiki. Hemeda; [email protected] With the free motion equation, there are generally two bits of information one must have to appropriately describe the mass's motion. Therefore, methods for constructing exact solutions of differential equations play an important role in applied mathematics and mechanics. Applications of linear differential equation - 2 (in Hindi) Exact differential equation (in Hindi) - Differential equation for iitjee Enroll in course for. Includes large number of illustrative examples worked out in detail and extensive sets of problems. It also includes methods of solving higher- order differential equations: the methods of. 3 Exact Diﬀerential Equations A diﬀerential equation is called exact when it is written in the speciﬁc form Fx dx +Fy dy = 0 , (2. Initial conditions are also supported. First Order Exact Differential Equations. You learn to look at an equation and classify it into a certain group. science and engineering where it is di-cult or even impossible to obtain exact solutions. Mdi B <NAME>#1, # A. A differential equation is an equation involving a function and its derivatives. View Notes - 0b. For example, the amount of bunnies in the future isn’t dependent on the number of bunnies right now because it takes a non-zero amount of time for a parent to come to term. Often, our goal is to solve an ODE, i. applications to symbolic solutions of ordinary differential equations (ODEs) is limited. The solution method will also work for systems of nonlinear equations and high-dimensional ones. Verify a solution to a differential equation. 1 Population Growth Problem Assume that the population of Washington, DC, grows due to births and deaths at the rate of 2% per year and there is a net migration into the city of 15,000 people per. DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS 1. Differential calculus or Differentiation) and Calculus 2 (a. First-Order Linear ODE. Coherent, balanced introductory text focuses on initial- and boundary-value problems, general properties of linear equations, and the differences between linear and nonlinear systems. It is not possible for a person to claim to have fully understood differential equations without having the capacity to apply them. Integrating Factor. Please submit the PDF file of your manuscript via email to. It is possible that expression (2) can be zero, in this case (and this is your example) the equation dy dx=Q (x,y) P (x,y) admits the integration factor B and, after multiplication by B, is exact, hence admitting an analytic integral. This means that there must not be any forms of plagiarism, i. Classify differential equations according to their type and order. We get Z dT T T e = Z kdt; so lnjT T ej= kt+ C: Solving for T gives an equation of the form T = T e. A differential equation is an equation that relates a function with one or more of its derivatives. Solve first order differential equations that are separable, linear, homogeneous, exact, as well as other types that can be solved through different substitutions. 3 Homogeneous Equations Application: Models of Pursuit 2. introduction. Classification of Differential Equations; Their Origin and Application. As applications to our general results, we obtain the exact (closed-form) solutions of the Schrodinger-type differential equations describing: (1)¨ two Coulombically repelling electrons on a sphere; (2) Schrodinger equation¨. Numerical Solution of Fuzzy Differential Equations and its Applications: 10. First order DEs. There are standard methods for the solution of differential equations. First order equations: method of integrating factors, separable equations, exact equations, homogeneous equations. 6 Solutions of simultaneous linear equations 310. MATH 322: Differential Equations for Applications Course Syllabus NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. Now this equation is clearly equivalent to the differential equation, namely, Thus, solving this exact differential equation amounts to finding the exact "antiderivative," the function whose exact (or total) derivative is just the ODE itself. A factor which possesses this property is termed an integrating factor. We’ll do a few more interval of validity problems here as well. Solutions by Substitution d. We can solve this di erential equation using separation of variables. However, SCILAB can be used to calculate intermediate numerical steps in the solutions. Get the free "General Differential Equation Solver" widget for your website, blog, Wordpress, Blogger, or iGoogle. y” + 6y’ + 9y = -578 sin 5t. We present the procedure of the method and illustrate it with application to the space-time fractional Drinfel'd-Sokolov-Wilson equation. The purpose of this post is to derive the finite-difference equations. [<NAME>; <NAME>] -- "This book features a collection of topics that are used in stochastic processes and, particularly, in queueing theory. ), examples of different types of DE's (DE = differential equation from here on out), including partial differential equations. Solve an exact differential equation. Finally, substitute the value found for into the original equation. On new exact blow-up solutions for nonlinear heat conduction equations with source and applications. Classification of Differential Equations; Their Origin and Application. A differential equation is a mathematical equation that relates some function with its derivatives. In general the highest derivative in a differential equation is the order. Get free Research Paper on SILMULTANEOUS DIFFERENTIAL EQUATION AND ITS APPLICATION research project topics and materials in Nigeria. com Received September ; Accepted March Academic Editor: <NAME>. Interpret solutions of differential equation models in mechanics, circuits, &c. A First Course in Differential Equations with Modeling Applications, 10 th edition strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. science and engineering where it is di-cult or even impossible to obtain exact solutions. Part III: Numerical Methods and Applications. The differential equation Exp[y] dy = Sin[x] dx defines its family of solutions: Exp[y] == Sin {x] +c. ), examples of different types of DE’s (DE = differential equation from here on out), including partial differential equations. Homogeneous linear differential. In addition, Euler's equation is a versatile tool to also approximate certain differential equations. Brannan/Boyce’s Differential Equations: An Introduction to Modern Methods and Applications, 3rd Edition is consistent with the way engineers and scientists use mathematics in their daily work. Separable DEs, Exact DEs, Linear 1st order DEs. The general form for a first-order exact differential equation is gen_exact_ode := P(x,y(x)) + Q(x,y(x))*diff(y(x),x) = 0; where the functions and satisfy the conditions. MATLAB Differential Equations introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. SAMPLE APPLICATION OF DIFFERENTIAL EQUATIONS 3 Sometimes in attempting to solve a de, we might perform an irreversible step. In this paper, the exact solutions of nonlinear differential equations are obtained by using Aboodh transform and differential transform methods. Exact Differential Equations, or total differential equations are a type of ordinary differential equation where there exists a continuously differentiable function F, called the potential function (which we learned about in Calculus 3). EXAMPLE4 A Mixture Problem A tank contains 50 gallons of a solution composed of 90% water and 10% alcohol. Deﬁning solvable differential equations We introduce a sequence of new variables η i =η i(x,t), 16i 6n, by solvable PDEs, for instance, the linear ones, η i,x. However, in applications where these diﬀerential equations model certain phenomena, the equations often come equipped with initial conditions. which is now exact (because M y = 2 x −2 y = N x). The next type of first order differential equations that we’ll be looking at is exact differential equations. (Last Updated On: December 8, 2017) This is the Multiple Choice Questions Part 1 of the Series in Differential Equations topic in Engineering Mathematics. Differential Equations and Linear Superposition • Basic Idea: Provide solution in closed form • Like Integration, no general solutions in closed form •Order of equation: highest derivative in equation e. To find the solutions explicitly, the equation may be treated as one of the homogeneous coefficient type 2. Box 1914, Rasht, Iran. Solve a differential equation analytically by using the dsolve function, with or without initial conditions. The Second Edition includes expanded coverage of Laplace transforms and partial differential equations as. Differential equations show up in just about every branch of science, including classical mechanics, electromagnetism, circuit design, chemistry, biology, economics, and medicine. Differential Equations and Linear Algebra Exact equations, and why we cannot solve very many differential equations Applications of linear algebra to. Section 2-3 : Exact Equations. A differential equation is an equation that relates a function with one or more of its derivatives. Structure Preserving Algorithms for Differential Equations are discrete computational methods coined at exactly preserving important geometric or analytic properties of the continuous physical system. Applications of Derivative; Exact Differential Equations – Page 2. Thus, dividing the inexact differential by yields the exact differential. Royal Institute of Technology KTH, SE-100 44, Stockholm, SWEDEN. , (x, y, z, t) Equations involving highest order derivatives of order one = 1st order differential equations Examples:. the constant coefficient case is the easiest becaUSE THERE THEY BEhave almost exactly like algebraic equations. <NAME>. In this course, Calculus Instructor Patrick gives 26 video lessons on Multivariable Calculus. So this is a separable differential equation, but. Use linear and nonlinear ﬁrst-order differential equations to solve application problem. 1 Theory of First-Order Equations: A Brief Discussion 2. We will generally focus on how to get exact formulas for solutions of certain differential equations, but we will also spend a little bit of time on getting approximate solutions. Practical applications of first order exact ODE? (exact differential equations). These equations are evaluated for different values of the parameter μ. focuses the student's attention on the idea of seeking a solutionyof a differential equation by writingit as yD uy1, where y1 is a known solutionof related equation and uis a functionto be determined. We will do so by developing and solving the differential equations of flow. Differential equations occur in economics and systems science and other ﬁelds of mathematical science. [LIST] [*] First-order differential equations [LIST] [*] Introduction [*] First-order linear differential equations [*] The Van Meegeren art forgeries [*] Separable equations [*] Population models [*] The spread of technological innovations [*] An atomic waste disposal problem [*] The dynamics of tumor growth, mixing problems, and orthogonal trajectories [*] Exact equations, and why we cannot. Deﬁning solvable differential equations We introduce a sequence of new variables η i =η i(x,t), 16i 6n, by solvable PDEs, for instance, the linear ones, η i,x. A differential equation is an equation that relates a function with one or more of its derivatives. Here, we look at how this works for systems of an object with mass attached to a vertical … 17. The solution method relies on the linearity of the equations for the integrating factor to recursively generate so lutions, this property was used by Abel in his work on nonlinear differential equations [1]. A differential equation is a mathematical equation that relates some function with its derivatives. 4 Variables separable 1. Derive solutions of linear second order equations or systems that have constant coefficents. We'll do a few more interval of validity problems here as well. What are the Application of exact differential equations? 'On the asymptotic solutions of ordinary linear differential equations about a turning point' -- subject(s): Differential equations. MATH 322: Differential Equations for Applications Course Syllabus NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. 5 The theory of determinants 297 3. This proven and accessible text speaks to beginning engineering and math students through a wealth of pedagogical aids, including an abundance of. differential equation in differential form is exact if. is a 3rd order, non-linear equation. 1, 3 Division of Informatics, Logistics and Management, School of Technology and Health STH. Use an appropriate substitution to rewrite and solve a differential equation. Ordinary differential equation examples by Duane Q. science and engineering where it is di-cult or even impossible to obtain exact solutions. An online version of this Differential Equation Solver is also available in the MapleCloud. Let's see some examples of first order, first degree DEs. Then the equation Mdx + Ndy = 0 is said to be an exact differential equation if Example : (2y sinx+cosy)dx=(x siny+2cosx+tany)dy MN yx ww ww. Solving Exact Differential Equations Examples 1 cos 2x - 2e^{xy} \sin 2x + 2x \right )}{(xe^{xy} \cos 2x - 3)}$is an exact and solve this differential equation. This discussion includes a derivation of the Euler-Lagrange equation, some exercises in electrodynamics, and an extended treatment of the perturbed Kepler problem. Equations of nonconstant coefficients with missing y-term If the y-term (that is, the dependent variable term) is missing in a second order linear equation, then the equation can be readily converted into a first order linear equation and solved using the integrating factor method. The roots of the characteristic equation of the associated homogeneous problem are $$r_1, r_2 = -p \pm \sqrt {p^2 - \omega_0^2}$$. Differential Equations and Linear Superposition • Basic Idea: Provide solution in closed form • Like Integration, no general solutions in closed form •Order of equation: highest derivative in equation e. Advanced Math Solutions – Ordinary Differential Equations Calculator, Separable ODE Last post, we talked about linear first order differential equations. The existence of the exact solution emerged from the analysis of the logical structure of d’Alambert's, Fourier' and Laplace's differential equations. You've probably already encountered it in the context of di erential equations. A Differential Equation is a n equation with a function and one or more of its derivatives: Example: an equation with the function y and its derivative dy dx. The orthiogonal trajectories are ln | Tan[x/2] = Exp[y] +c. Solution of First Order Linear Differential Equations Linear and non-linear differential equations A differential equation is a linear differential equation if it is expressible in the form Thus, if a differential equation when expressed in the form of a polynomial involves the derivatives and dependent variable in the first power and there are no product […]. We get Z dT T T e = Z kdt; so lnjT T ej= kt+ C: Solving for T gives an equation of the form T = T e. Solve initial value problems. This section will also introduce the idea of. Get ideas for your own presentations. An algebraic equation , such as a quadratic equation, is solved with a value or set of values; a differential equation , by contrast, is solved with a function or a. Solve ﬁrst-order differential equations by making the appropriate substitutions including homogeneous and Bernoulli equations. These equation have some fractions and variables with its derivatives. Stability of equilibrium solutions. After you've done a few examples, most exact equations are often fairly easy to spot. Part - 3. Linear Equations and Bernoulli Equations D, Exact Equations and Special Integrating Factors E. Here is the form:. Here is a set of practice problems to accompany the Exact Equations section of the First Order Differential Equations chapter of the notes for Paul Dawkins Differential Equations course at Lamar University. Exact equations. Determine the order of a differential equation. The given function f(t,y) of two variables deﬁnes the differential equation, and exam ples are given in Chapter 1. <NAME> Shaikh#1, # A. Applications. Exact equations. DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS 1. We will do so by developing and solving the differential equations of flow. First, and foremost, it is a rigorous study of ordinary differential equations and can be fully un derstood by anyone who has completed one year of calculus. The order of ordinary differential equations is defined to be the order of the highest derivative that occurs in the equation. 3) is given implicitly by F(x,y. The solution method for the separable differential equation: can be viewed as a special case of the theory of first-order exact differential equations. (Note that in the above expressions Fx = ∂F ∂x and Fy = ∂F ∂y). An algebraic equation , such as a quadratic equation, is solved with a value or set of values; a differential equation , by contrast, is solved with a function or a. Differential Equations and Linear Algebra Exact equations, and why we cannot solve very many differential equations Applications of linear algebra to. It is easy to analyze and to understand how it works. 6 Substitution Methods 1. A First Course in Differential Equations with Modeling Applications, 10 th edition strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. From basic separable equations to solving with Laplace transforms, Wolfram|Alpha is a great way to guide yourself through a tough differential equation problem. Differential Equations and Linear Superposition • Basic Idea: Provide solution in closed form • Like Integration, no general solutions in closed form •Order of equation: highest derivative in equation e. For example, one can derive new, more interesting solutions of differential equations by applying the symmetry group of a differential equation to known (often trivial) solutions. It is a general form of a set of infinitely many functions, each differs from others by one (or more) constant term and/or constant coefficients, which all satisfy the differential equation in question. Scond-order linear differential equations are used to model many situations in physics and engineering. 1 Applications Leadingto Differential Equations 3 This implies that lim. This note introduces students to differential equations. Use this information to sketch the solution cuwes in the (t, x)-plane (t > 0) for the initial.$ Stack Exchange Network. (Note that in the above expressions Fx = ∂F ∂x and Fy = ∂F ∂y). Section 2-3 : Exact Equations. We'll do a few more interval of validity problems here as well. You also learned more about the basic ideas of differential equations and. For an initial value problem. Methodol Comput Appl Probab (2010) 12:261–270 DOI 10. 4 Exact Differential Equations Definition 2. More ODE Examples. Find many great new & used options and get the best deals for A First Course in Differential Equations : With Modeling Applications by <NAME>. You will receive incredibly detailed scoring results at the end of your Differential Equations practice test to help you identify your strengths and weaknesses. Chapter 1 - Differential Equation Models. Differential equations If God has made the world a perfect mechanism, he has at least conceded so much to our imperfect intellect that in order to predict little parts of it, we need not solve innumerable differential equations, but can use dice with fair success. This section will also introduce the idea of. And it's just another method for solving a certain type of differential equations. " Make sure you remember what proportionality and inverse proportionality are, because these words come up a lot around differential equations. Partial Differential Equations Lectures by <NAME>. Presents ordinary differential equations with a modern approach to mathematical modelling; Discusses linear differential equations of second order, miscellaneous solution techniques, oscillatory motion and laplace transform, among other topics. Course Outcome(s):. Within this broad scope, research at UConn's math department focuses mainly on the following topics: Linear partial differential equations and Brownian motion. For example - given an expression,. Solve ﬁrst-order differential equations by making the appropriate substitutions including homogeneous and Bernoulli equations. Advanced Math Solutions - Ordinary Differential Equations Calculator, Separable ODE Last post, we talked about linear first order differential equations. As we just saw this means they can be. Only a limited number of diﬁerential equations can be solved analytically. Applicationsofdifferentialequationsalsoaboundinmathematicsitself, especially in geometry and harmonic analysis and modeling. However, in applications where these diﬀerential equations model certain phenomena, the equations often come equipped with initial conditions. While quite a major portion of the techniques is only useful for academic purposes, there are some which are important in the solution of real problems arising from science and engineering. Consider the 3 rd order equation (with initial conditions. Don't see your book? Search by ISBN. EXACT DIFFERENTIAL EQUATIONS 21 2. sometimes it is diﬃcult to check-up exact solutions of nonlinear diﬀerential equations. simultaneous linear differential equation with constraints coefficients. A force to the differential equation, M(x,y) dx + N(x,y) dy = 0, we call it to be exact, in a plane region R if there is a function capital F(x, y) such that dF over dx is equal to M(x, y) and dF over dy is equal to N(x, y). It is possible that expression (2) can be zero, in this case (and this is your example) the equation dy dx=Q (x,y) P (x,y) admits the integration factor B and, after multiplication by B, is exact, hence admitting an analytic integral. Equilibrium Solutions. Enter an ODE, provide initial conditions and then click solve. Before we get into the full details behind solving exact differential equations it's probably best to work an example that will help to show us just what an exact differential equation is. To determine whether PDEs can be verified using substitutions. The first three worksheets practise methods for solving first order differential equations which are taught in MATH108. Straightforward and easy to read, A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS, 11th Edition, gives you a thorough overview of the topics typically taught in a first course in differential equations. Before I show you what an exact equation is, I'm just going to give you a little bit of the building blocks, just so that. Differential Equations Calculators; Math Problem Solver (all calculators) Differential Equation Calculator. Let's see some examples of first order, first degree DEs. Competence in classifying differential equations as to ordinary, partial, linear, non-linear, order and degree, and to construct differential equations under given conditions. From Differential Equations For Dummies. By <NAME>. A differential equation is a mathematical equation that relates some function with its derivatives. If is some constant and the initial value of the function, is six, determine the equation. It is an alternative procedure for obtaining the Taylor series solution of the given differential equation and is promising for various other types of differential equations. Scond-order linear differential equations are used to model many situations in physics and engineering. STUDENT SOLUTIONS MANUAL FOR ELEMENTARY DIFFERENTIAL EQUATIONS AND ELEMENTARY DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS <NAME>. The solution to equation (2. written as. 3 Definitions and Examples 1. For example, the amount of bunnies in the future isn’t dependent on the number of bunnies right now because it takes a non-zero amount of time for a parent to come to term. 2 NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS Introduction Differential equations can describe nearly all systems undergoing change. In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. From Differential Equations For Dummies. The order of ordinary differential equations is defined to be the order of the highest derivative that occurs in the equation. First-Order Differential Equations and Their Applications 3 Let us brieﬂy consider the following motivating population dynamics problem. 1 Theory of First-Order Equations: A Brief Discussion 2. A diﬀerential equation, shortly DE, is a relationship between a ﬁnite set of functions and its derivatives. Now I introduce you to the concept of exact equations. An algebraic equation , such as a quadratic equation, is solved with a value or set of values; a differential equation , by contrast, is solved with a function or a. Please submit the PDF file of your manuscript via email to. We present examples where differential equations are widely applied to model natural phenomena, engineering systems and many other situations. It also includes methods of solving higher- order differential equations: the methods of. Third-Order ODE with Initial Conditions. Specially designed for just such a course, Differential Equations with Applications and Historical Notes takes great pleasure in the journey into the world of differential equations and their wide range of applications. Scond-order linear differential equations are used to model many situations in physics and engineering. Web Applications;. Solve this differential equation. University graduate level. Royal Institute of Technology KTH, SE-100 44, Stockholm, SWEDEN. In the sense that, much of the theory and, hence, applications of differential equation can be extended smoothly to fraction differential equation with the same flavor and spirit of the realm of differential equation fraction differential equation. A Differential Equation is an equation with a function and one or more of its derivatives: Example: an equation with the function y and its derivative dy dx. The roots of the characteristic equation of the associated homogeneous problem are $$r_1, r_2 = -p \pm \sqrt {p^2 - \omega_0^2}$$. Apply Euler’s method to obtain a numerical approximation of a differential-equation solution-function value. Unlike the elementary mathematics concepts of addition, subtraction, division, multiplication, percentage etc, which are used on a day to day basis, differential equations are not generally used/observed in our every day life. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Here, we look at how this works for systems of an object with mass attached to a vertical … 17. 4 Exact Equations. Many mathematicians have. Let me write that down. Therefore, methods for constructing exact solutions of differential equations play an important role in applied mathematics and mechanics. Maple can also be used to carry out numerical calculations on differential equations that cannot be solved in terms of simple expressions. By application of the method, it is possible to obtain highly accurate results or exact solutions for differential equations. Differential equations have a remarkable ability to predict the world around us. That is to say, we have d dx [f(x)y] = f(x) dy dx + f0(x)y:. Asymptotic stability for a strongly coupled Klein-Gordon system in an inhomogeneous medium with locally distributed damping. Derive solutions of linear second order equations or systems that have constant coefficents. Determine the order of a differential equation. They can be divided into several types. The main aim of the research work is assess the application of Laplace transform in solving partial differential equation in the second derivative. In this paper, improved sub-equation method is proposed to obtain new exact analytical solutions for some nonlinear fractional differential equations by means of modified Riemann Liouville derivative. 1 Introduction 1. The equation is of first orderbecause it involves only the first derivative dy dx (and not. Differential equation is extremely used in the field of engineering, physics, economics and other disciplines. MATH 322: Differential Equations for Applications Course Syllabus NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. Non-Homogeneous, Exact and Non-exact differential equations and how to solve them. You will receive incredibly detailed scoring results at the end of your Differential Equations practice test to help you identify your strengths and weaknesses. Due to the widespread use of differential equations,we take up this video series which is based on. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. From basic separable equations to solving with Laplace transforms, Wolfram|Alpha is a great way to guide yourself through a tough differential equation problem. There are many applications of DEs. Applications of PIDEs can be found in various fields. An equation of the form P(x,y)\mathrm{d}x + Q(x,y)\mathrm{d}y = 0 is considered to be exact if the. Parameters are determined by a finite amount of empirical data, and so there is always some residual uncertainty regarding the exact values of the parameters. There are many "tricks" to solving Differential Equations (if they can be solved. Geometric Interpretation of the differential equations, Slope Fields. We mentioned some of them here: tanh−expansion method [1] − [7], the simplest. By application of the method, it is possible to obtain highly accurate results or exact solutions for differential equations. Examples include mechanical oscillators, electrical circuits, and chemical reactions, to name just three. First-Order Linear ODE. The main aim of the research work is assess the application of Laplace transform in solving partial differential equation in the second derivative. 6: Consider the autonomous equation Find all equilibrium points, classify their stability, and sketch the phase line diagram. Initial Conditions. 2 Separation of Variables Application: Kidney Dialysis 2. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. 01 Single Variable Calculus, Fall 2006 Prof. Step 3: Differentiate Equation (1) partially with respect to y, holding x as constant $\dfrac{\partial F}{\partial y} = x + f'(y)$ Step 4: Equate the result of Step 3 to N and collect similar terms. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. c t) Hence we start here with a trial solution (2) just denoting the parameter c above by here and the function f by z. Maple can also be used to carry out numerical calculations on differential equations that cannot be solved in terms of simple expressions.


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Especially StackOverflow, great right?
Now go on, here's the cipher, ever will you crack it? At 10 o' clock i'm checking to see if you are done.

Ldi5%hxtl&nrc5%zxz(lsxom%j}&(Vy}{unop3[K$lt)usxzl&>G Can you crack the code? Hint: The question title hints something, as well as the tags Hint #2: I bake pies Hint #3: If you figure something out, close your eyes and say it out loud, what you hear is what it is Hint #4: Oh isn't it just so tasty when you put some irrational sauce on your pie? Hint #5: I have added the tag, but I don't think anyone has gotten to the point where knowledge is needed Hint #6: (Big Giveaway) Not horizontal not vertical Hint #7: The cipher doesn't have anything to do with the text, it's just the code block. The text is only a hint Kind of misleading, I just wanted to say that the people who are trying to apply functions to the text are doing it wrong. All mathematical operations/ciphers should be done on the ciphertext itself Hint #8: (Warning: Extremely big giveaway, only use this hint when you totally give up) "Pi is an irrational number, which means that its decimal representation never terminates or repeats itself." - calvin's hobbies Hint #9: (Warning: even BIGGER giveaway): https://codegolf.stackexchange.com/questions/38131/pirrational-numbers Hint #10: Line 7. Capital? + "LANG" Hint #11: Order you will use the tags: • for anyone that wants to try this: the acrostic is (rot13) v wno n cra. idk if this is helpful or not tho – Excited Raichu Nov 1 '18 at 18:41 • @ExcitedRaichu That doesn't have anything to do with the solution :P – FireCubez Nov 1 '18 at 18:42 • How long should I wait till adding another hint if needed? – FireCubez Nov 1 '18 at 18:53 • You can add hints as often as you want. I generally wait two hours or so in between hints on my puzzles, but it's up to you. – Excited Raichu Nov 1 '18 at 18:54 • Is the phrase: "And no, this not clue to the puzzle!" a deliberate typo? – Dr Xorile Nov 3 '18 at 17:09 ## 2 Answers Hah, good job, you found it! Puzzling.SE is proud :D' This was essentially just a guided solution from the OP's clues! I had never come across this before, but when he talks about sprinkling pie with irrational sauce he means the reference in the clue he gave: Index pi according to the unique numbers in it (3,1,4,15,...). pi rational numbers. We use 15 because 1 was already found. This makes the result: 43195195867462520687356193644029372991880 Computationally, this gets hard quickly. But when I used this as in a cipher it returns: Hah, good gn]3 xxt epsil\x1cgw"$$Vwtxnlfg2SC$$ You'll note that the first ten characters make sense, and that the OP has a reference to 10 o'clock. So using this, I Reused the first ten digits of this pi rational number over and over with the output given. Previous work for the interested... Here's a glorious failure for others to avoid (in the spirit of @Excited Raichu's comment to the question...) So, pie and cipher. Or π and cipher. Mmm. And computer programming. Could it be that we should shift out the cipher letter-wise by the digits of π? Well, that means shifting: Ldi5%hxtl&nrc5%zxz(lsxom%j}&(Vy}{unop3[K$lt)usxzl&>G

by

31415926535897932384626433832795028841971693993751058

which is:

Ice4 _vng#ijZ.wvw hmvii"gu#&Opx{sfgl2RD#fk&ljusg%>B

Shifting the other way, or using latter combinations of pi (up to about 30 digits later) still leaves nonsense.

So it isn't that!

Edit: The kind of calculation I did was (in Python):

code="Ldi5%hxtl&nrc5%zxz(lsxom%j}&(Vy}{unop3[K\$lt)usxzl&>G"
pi="31415926535897932384626433832795028841971693993751058209749445..."
decode=""
for i,c in enumerate(code):
decode=decode+ chr(ord(c)+int(pi[i])

where chr() converts a decimal to ascii and ord() converts an ascii to decimal.

I also subtracted and did offsets. Also, I assumed that the "I love programming..." is a clue. Since this is baking, we're just going to add in a bunch of ingredients. The trouble is that if we just add stuff together we rapidly get beyond the 128 that is associated with ascii.

So, for example, I can run

> clue.split()
['I', 'love', 'programming,', "it's", 'so', 'fun', 'Just', 'get', 'every', 'computer', 'in', 'the', 'world', 'And', 'just', 'do', 'program', 'as', 'much', 'as', 'you', 'can', 'Because', 'with', 'everyone', 'on', 'your', 'side', 'you', 'can', 'never', 'go', 'wrong', 'with', 'programming', 'And', 'no,', 'this', 'not', 'clue', 'to', 'the', 'puzzle!', "I'm", 'just', 'saying', 'programming', 'is', 'the', 'best', 'Please', 'do', 'program,', 'everyone', 'will', 'help', 'you', 'with', 'your', 'problems', 'Especially', 'StackOverflow,', 'great', 'right?', 'Now', 'go', 'on,', "here's", 'the', 'cipher,', 'ever', 'will', 'you', 'crack', 'it?', 'At', '10', "o'", 'clock', "i'm", 'checking', 'to', 'see', 'if', 'you', 'are', 'done.']

and the use the number of characters in each word. Also, hundred of combinations just down this path: what do we do with carriage returns? What do we do with punctuation? What do we do with numerals? Should "o'" and "clock" be separate words? etc. But running various combinations with this didn't reveal anything promising that could be tweaked.

Another option is to have the clue converted character-by-character into ascii. But then what? You can't add those numbers because they will go above 128. "Add" is implied by the baking clues - average feels uncontrolled (what do you do with the 0.5s). XOR of the binary? OR? AND? Again AND is implied by the "baking", but XOR is much more common in crypto for obvious reasons (that you either switch the bit or not).

And then add the digits of pi (to the ascii decimal? or the ascii hex? or the ord of the digits of pi).

I tried various combinations of the above and, of course, it was all nonsense. Is the "knowledge" a knowledge of the ascii codes? Or something else? Is the "I love programming..." needing to be interpreted in (hard to understand) english? Or read as a string of ascii characters? Are there other ingredients to be added to the pie? Other "irrational sauces"? e? The golden section? Some root?

I'm sure this seemed obvious to the OP, but the number of possibilities is relatively large here. Sorry if I'm missing the "big clues". Gasp, it's a diagonal! Yay! Now I could do that all again but with 10 times the number of permutations!

Ah, well. Maybe I'll guess the next one...

• You are so very close! – FireCubez Nov 3 '18 at 10:19
• You are just rot13(hfvat gur jebat cv), try reading the question again and rot13(cnlvat irel pybfr nggragvba gb gur yrggref), and also, the text doesn't contain the cipher, it's just the code block I posted. The text is just a hint – FireCubez Nov 5 '18 at 12:14
• Yep, you got it. I feel I gave way too many hints though. – FireCubez Nov 5 '18 at 20:36
• Yeah, but hard to guess without something. This is where the art of puzzling comes in. Not too much and not too little. I'm looking forward to your next one! Feel free to ask the community for suggestions. Sometimes people write a second puzzle discussing what worked and what didn't about the first one! Afterall, this SE is about the making of puzzles (at least in part). – Dr Xorile Nov 5 '18 at 20:40
• Check out this answer to see how the clues were meant to be used: puzzling.stackexchange.com/a/74748/51739 – FireCubez Nov 6 '18 at 12:51

This is an answer building ontop of @DrXorile's answer and showing how the hints end up where they did. Quotes and clues from my own puzzle are highlighted in bold. Since I am answering my own puzzle, please upvote Xorile's answer as well. They were the one to solve the puzzle. I am only showing the correct way to find out the solution

First you need to read the text diagonally starting from "programming" (not horizonal, not vertical), it spells out "PEPECEGE". say that out loud and it sounds like "PPCG", alias for Programming Puzzles & Code Golf, referencing the very site on SE: https://codegolf.stackexchange.com.

Then:

Search for piirational numbers, referenced by "I love baking", and "irrational sauce". The first answer to that question is runnable code which when run, generates the first 10 digits of the 1st piirational number. 10, since I reference 10 o'clock in the puzzle.

Finally:

Repeat these 10 digits and do a simple ASCII shift on the ciphertext.

format: plain


text:
## Stream: lean4

### Topic: amusing

#### <NAME> (Mar 08 2021 at 19:58):

Forgive me, I can't resist finding the fact that this compiles without sorrys or complaints amusing :D

namespace «

theorem fermats_last_theorem
{a b c n : Nat}
(apos : a > 0)
(bpos : b > 0)
(cpos : c > 0)
(h : a^n + b^n = c^n)
: n = 2 := by
trivial

»

#### <NAME> (Mar 08 2021 at 20:25):

I find this kind of message very distracting. It is unclear what its purpose is.
Are you requesting we change how escaped identifiers behave?
Or, do you want us to check unclosed namespace/sections when the file ends?
Or, are you just trying to be funny?

#### <NAME> (Mar 08 2021 at 20:28):

@Le<NAME> apologies, I suspected it was better for New Members or some other less important place since yes it was mostly not meant to have any action associated with it --

but perhaps now that I've said it here and noticed lean 3 does not allow these escaped identifiers with as much liberty, maybe not allowing newlines in escaped identifiers is a good way to restrict them?

#### <NAME> (Mar 08 2021 at 20:32):

I don't see a problem here. The French quotes are even highlighted correctly in vscode. If you don't like French quotes, you can also use comments:

section /-
theorem equal_two_of_pow_add_pow_eq_pow : ....
-/

#### <NAME> (Mar 08 2021 at 20:36):

BTW, there's a long tradition of hiding Fermat's last theorem in comments. For example, this post on coq-club soon celebrates its twelfth birthday:
https://sympa.inria.fr/sympa/arc/coq-club/2009-04/msg00006.html

#### <NAME> (Mar 09 2021 at 05:17):

I think that tradition dates back to Fermat himself

Last updated: May 18 2021 at 23:14 UTC


text:
# Derivative Calculator

Enter the function and variable to find the derivative using derivative calculator.

This will be calculated:

Give Us Feedback

## Derivative Calculator

Derivative calculator is used to find the derivative of a given function with respect to the independent variable. This calculator can perform explicit differentiation with just one click. This differentiation calculator will show the solution with steps in a couple of seconds.

## Derivative – Definition

Let f(x) be a function whose domain contains an open interval at some point x0. The function f(x) is said to be differentiable at x0, and the derivative of f(x) at x0 is given by:

In other words, the derivative measures the sensitivity to a change in the function value with respect to a change in its argument. The reverse function of the derivative is known as the antiderivative.

## How to calculate derivative?

To differentiate a function, you can use the d/dx calculator above. Let’s calculate the derivative of 1/x to grasp the basic idea of derivation.

As 1/x = x-1

We will use the product rule (refer to the below rules).

d/dx (x-1) = -1(x-2) = - 1/x2

Example:

Find the derivative of (x+7)2.

Solution:

Step 1: Apply the derivation symbol.

d/dx [(x + 7)2]

Step 2: Apply the power rule.

2(x + 7) d/dx [x + 7]

2(x + 7) [d/dx (x) + d/dx (7)]

2(x + 7) [1 + 0]

2(x + 7)

Some functions need the second derivative to complete the process of differentiation. You can use our second derivative calculator in this case.

## Derivative rules – Formulas

 Rules Name Rules Constant Rule $$\frac{d}{dx}\left[k\right]=0$$ Power Rule $$\frac{d}{dx}\left[a^x\right]=a.x^{a-1}$$ Sum Rule $$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]+\frac{d}{dx}\left[g\left(x\right)\right]$$ Difference Rule $$\frac{d}{dx}\left[f\left(x\right)-g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]-\frac{d}{dx}\left[g\left(x\right)\right]$$ Product Rule $$\frac{d}{dx}\left[f\left(x\right)\cdot g\left(x\right)\right]=g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]+f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]$$ Quotient Rule $$\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{\left[g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]-f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]\right]}{\left[g\left(x\right)\right]^2}$$ Chain Rule If f(x) = hg(x) f'(x) = h'(g(x)) * g'(x) Trigonometric Derivatives $$\frac{d}{dx}\left[cos\left(x\right)\right]=-sin\left(x\right)$$ $$\frac{d}{dx}\left[sin\left(x\right)\right]=cos\left(x\right)$$ $$\frac{d}{dx}\left[tan\left(x\right)\right]=sec^2\left(x\right)=1+tan^2\left(x\right)$$ Derivative of e^x (Exponential) $$\frac{d}{dx}\left[e^x\right]=e^x$$ Logarithm Derivatives $$\frac{d}{dx}\left[a^x\right]=a^xln\left(a\right),a>0$$ $$\frac{d}{dx}\left[ln\left(x\right)\right]=\frac{1}{x},x>0$$ $$\frac{d}{dx}\left[\log _x\left(x\right)\right]=\frac{1}{xln\left(a\right)},x>0$$

## How to find derivatives using rules?

Use our derivative calculator with steps to differentiate the functions according to the above differentiation rules. Here is a manual example for differentiating a function using rules.

Example

Find the derivative of the given function with respect to "u".

$$f\left(u\right)=\frac{u}{\left(u^2+1\right)}$$

Solution

Step 1: Apply d/du to the given function.

$$\frac{d}{du}\left[f\left(u\right)\right]=\frac{d}{du}\left[\frac{u}{\left(u^2+1\right)}\right]$$

Step 2: Use the quotient rule to differentiate the above expression.

$$\frac{d}{du}\left[\frac{u}{\left(u^2+1\right)}\right]=\frac{\left[\left(u^2+1\right)\:\frac{d}{du}\left(u\right)-u\:\frac{d}{du}\left(u^2+1\right)\right]}{\left(u^2+1\right)^2}$$ ... (1)

Step 3: Find derivatives

$$\frac{d}{du}\left(u\right)=1$$

$$\frac{d}{du}\left(u^2+1\right)=\frac{d}{du}\left(u^2\right)+\frac{d}{du}\left(1\right)$$ by sum rule

$$\frac{d}{du}\left(u^2+1\right)=2u+0=2u$$

Step 4: Now substitute the above results in (1).

$$=\frac{\left[\left(u^2+1\right)\left(1\right)-u\left(2u\right)\right]}{\left(u^2+1\right)^2}$$

$$=\frac{\left[u^2+1-2u^2\right]}{\left(u^2+1\right)^2}$$

$$=\frac{\left[1-u^2\right]}{\left(u^2+1\right)^2}$$

Hence,

$$\frac{d}{du}\left[\frac{u}{\left(u^2+1\right)}\right]=\frac{\left[1-u^2\right]}{\left(u^2+1\right)^2}$$

## Examples of derivatives

Here are some examples of derivatives solved by our differentiate calculator.

 Function Derivative of the function derivative of x 1 derivative of 3^x 3x * ln(3) derivative of x^2 2x derivative of x^1/2 1/x1/2 derivative of 5^x 5x * ln(5) derivative of x/2 1/2 derivative of x^e e * xe-1 derivative of 1/x -1/x2 derivative of 2x 2 derivative of 2 0 derivative of x^x xx(ln(x) + 1) derivative of x^-1 -x-2 derivative of 2^x 2x * ln(2) derivative of 4^x 4x * ln(4)

You can cross-check the above result by using our derivative calculator.


text:
# cirq.testing.assert_eigen_gate_has_consistent_apply_unitary¶

cirq.testing.assert_eigen_gate_has_consistent_apply_unitary(eigen_gate_type: Type[cirq.ops.eigen_gate.EigenGate], *, exponents=(0, 1, -1, 0.5, 0.25, -0.5, 0.1, cirq.Symbol('s')), global_shifts=(0, 0.5, -0.5, 0.1), qubit_count: Optional[int] = None) → None[source]

Tests whether an EigenGate type’s apply_unitary is correct.

Contrasts the effects of the gate’s _apply_unitary_ with the
matrix returned by the gate’s _unitary_ method, trying various values for
the gate exponent and global shift.
Parameters: eigen_gate_type – The type of gate to test. The type must have an __init__ method that takes an exponent and a global_shift. exponents – The exponents to try. Defaults to a variety of special and arbitrary angles, as well as a parameterized angle (a symbol). global_shifts – The global shifts to try. Defaults to a variety of special angles. qubit_count – The qubit count to use for the gate. This argument isn’t needed if the gate has a unitary matrix or implements cirq.SingleQubitGate/cirq.TwoQubitGate/cirq.ThreeQubitGate; it will be inferred.


text:
# Resolve RDF Terms

quantitykind:AtomicAttenuationCoefficient
http://qudt.org/vocab/quantitykind/AtomicAttenuationCoefficient

### Recommended prefix

quantitykind:

lang:en
Atomic Attenuation Coefficient
lang:""
http://en.wikipedia.org/wiki/Attenuation_coefficient
lang:""
$$\mu_a = -\frac{\mu}{n}$$, where $$\mu$$ is the linear attenuation coefficient and $$n$$ is the number density of the atoms in the substance.
lang:""
http://www.iso.org/iso/catalogue_detail?csnumber=31895
lang:""
"Atomic Attenuation Coefficient" is a measurement of how strongly a chemical species or substance absorbs or scatters light at a given wavelength, per the number of atoms in the substance.
lang:""
μₐ


text:
# 2017-12-30 Gopher Wiki

The problem with the Gopher protocol as defined in RFC 1436 is that it doesn’t allow you to send anything to the server except for a query string. That’s not good enough for wiki editing. At the same time, we don’t want to make the interaction between Gopher server and Gopher client harder than it needs to be. This page offers a proposal:

First, the server needs to know about a new item type which says: item is a file you can write to. Below is a Gopher menu for a wiki page. The selector for this menu is Alex/menu. The selector for writing a new page is Alex/write/text.

$echo Alex/menu | nc localhost 7070 iThe text of this page: localhost 7070 0Alex Alex localhost 7070 hAlex Alex/html localhost 7070 wReplace Alex Alex/write/text localhost 7070 1Page History Alex/history localhost 7070 i localhost 7070 iThere are no links leaving this page. localhost 7070 . Let’s replace the text with something! We need to end our text with a period to let the server know that we don’t intend to send any more. $ nc localhost 7070 << 'EOF'
Alex/write/text
test
.
EOF
iPage was saved.		localhost	7070
1Alex	Alex/menu	localhost	7070
.

Or post a text file. You need to add the selector at the top and the period at the end:

$( echo Alex/write/text; cat haiku.txt; echo . ) | nc localhost 7070 iPage was saved. localhost 7070 1Alex Alex/menu localhost 7070 . Verify the page content: $ echo Alex | nc localhost 7070
Quiet disk rattling
Keyboard clicking, then it stops.
Rain falls and I think.
.

The wiki also allows a different kind of upload: binary files. When uploading a file, the server has two problems:

1. When has the end of the file been uploaded? If the client hangs up after sending the file, it can’t read the response. If the client doesn’t hang up but does a shutdown on the socket, letting the server know it’s done writing, it would work – but you can’t do it using telnet or nc. That’s why we need to tell the server how long the file is going to be.
2. What’s the content-type of the file? Sure, we could depend on file name extensions. But that doesn’t always work. My wiki restricts upload to certain MIME types, for example. Other Gopher servers might not care and that’s why this part is simply a solution I’m going to use for my wiki. I think it might just as well not be part of the new item type. It’s an local experiment, so to say.

So anyway, how would it work? It did not make an appearance in the menu above, but the selector would be Alex/write/file. The content-length is appended to this selector, separated by a tab character. wc -c counts the number of bytes and when it does that for standard input, no filename is printed.

$echo -e "Alex/write/file\t"wc -c < test.png \ | cat - test.png \ | nc localhost 7070 3Page was not saved: Files of type application/octet-stream are not allowed. localhost 7070 . When writing a file using this selector, a content-type of application/octet is assumed. The wiki will extract the MIME type from the selector, if possible: Alex/image/png/write/file, for example. $ echo -e "Alex/image/png/write/file\t"wc -c < test.png \
| cat - test.png \
| nc localhost 7070
iPage was saved.		localhost	7070
1Alex	Alex/menu	localhost	7070
.

Test it:

$echo Alex | nc localhost 7070 > test2.png$ diff test.png test2.png

Note that for my particular wiki, Alex/text/plain/write/file is not quite the same as Alex/write/text. The first will store a binary blob with MIME type text/plain which will be served like any other binary file: no parsing for links and all that. The second will edit a wiki page, which can be read using a Gopher client.

I understand that allowing file uploads can be tricky. It’s up to the Gopher server to handle this with care: store files in write-only upload directories, or provide all the peer review facilities like a wiki does, it’s up to you to write a good Gopher server. To paraphrase the description of item type 9: “Item is a binary file! Server will read until the TCP connection closes. Beware.”

Let’s get back to uploading text files and the problem of peer review facilities: Ideally, authors would provide some metadata as they upload their document. A summary of changes made, a name. My wiki will interpret a fenced off block at the very beginning of the document as metadata.

The following example sends the selector for writing a text file called Alex, a metadata block, and finally the new text.

$nc localhost 7070 <<'EOF' Alex/write/text  username: Alex summary: a different Haiku  Cold and dark outside Remember all the sick beds Good riddance old year . EOF iPage was saved. localhost 7070 1Alex Alex/menu localhost 7070 . Check it out: $ echo Alex | nc localhost 7070
Cold and dark outside
Remember all the sick beds
Good riddance old year
.

Let’s check Recent Changes, too:

$echo do/rc | nc localhost 7070 iRecent Changes localhost 7070 1Show minor edits do/rc/showedits localhost 7070 i localhost 7070 i2018-01-06 localhost 7070 i localhost 7070 1Alex Alex/menu localhost 7070 i20:55 UTC by Alex from 127.0.0.1: a different Haiku localhost 7070 i localhost 7070 i2018-01-05 localhost 7070 i localhost 7070 1test test/menu localhost 7070 i21:53 UTC by 127.0.0.1 localhost 7070 . Yay! 👍 Sources: Try browsing the wiki: Note that you will not be able to edit pages because spam protection requires you to answer a question before you are allowed to save. The question is simply that you need to say hello. Luckily, we can use the metadata for text files to do this: $ nc alexschroeder.ch 70 << 'EOF'
SandBox/write/text

This is a test.
.
EOF
iPage was saved.		alexschroeder.ch	70
1SandBox	SandBox/menu	alexschroeder.ch	70
.

Anyway, here’s a quick setup for local experiments. You need to get Oddmuse, of course. And then:

Run perl stuff/server.pl starts an Oddmuse Wiki on port 8080. You can edit pages on http://localhost:8080/. Data is stored in /tmp/oddmuse.

Run perl stuff/gopher-server.pl --port=localhost:7070 --wiki=./wiki.pl --wiki_dir=/tmp/oddmuse to start the Gopher server on port 7070.

Run lynx gopher://localhost:7070 to browse the same wiki.

Tags:

“Phlogs” are Gopher Blogs. I think the SDF has the biggest collection of them (link goes via a proxy, the real deal is gopher://phlogosphere.org).

People don't like comments.

– Alex 2018-01-07 12:49 UTC

There are still problems with the code, though. I finally decided to document it as issue 15 on GitHub. For now, the interface will stay as it is. I just have issues getting this Mojo::IOLoop code to work. [This was later fixed by switching back to Net::Server.]

– <NAME> 2018-01-08 14:24 UTC

Stuff I need to check:

• [[image:foo]] links ✓
• pages with non-ASCII names ✓
• gopher links ✓

– Alex 2018-01-13 17:20 UTC

Please make sure you contribute only your own work, or work licensed under the GNU Free Documentation License. Note: in order to facilitate peer review and fight vandalism, we will store your IP number for a number of days. See Privacy Policy for more information. See Info for text formatting rules. You can edit the comment page if you need to fix typos. You can subscribe to new comments by email without leaving a comment.


text:
# Node Representation Learning

In this section, we study several methods to represent a graph in the embedding space. By “embedding” we mean mapping each node in a network into a low-dimensional space, which will give us insight into nodes’ similarity and network structure. Given the widespread prevalence of graphs on the web and in the physical world, representation learning on graphs plays a significant role in a wide range of applications, such as link prediction and anomaly detection. However, modern machine learning algorithms are designed for simple sequence or grids (e.g., fixed-size images/grids, or text/sequences), networks often have complex topographical structures and multimodel features. We will explore embedding methods to get around the difficulties.

## Embedding Nodes

The goal of node embedding is to encode nodes so that similarity in the embedding space (e.g., dot product) approximates similarity in the original network, the node embedding algorithms we will explore generally consist of three basic stages:

1. Define an encoder (i.e., a mapping from nodes to embeddings). Below we include a diagram to illustrate the process, encoder $\rm ENC$ maps node $u$ and $v$ to low-dimensional vector $\mathbf{z_u}$ and $\mathbf{z_v}$:

2. Define a node similarity function (i.e., a measure of similarity in the original network), it specifies how the relationships in vector space map to the relationships in the original network.
3. Optimize the parameters of the encoder so that similarity of $u$ and $v$ in the network approximate the dot product between node embeddings: $\rm similarity(u,v) \approx \mathbf{z_u}^\intercal \mathbf{z_v}$.

## “Shallow” Encoding

How to define a encoder to map nodes into a embedding space?

“Shallow” encoding is the simplest encoding approach, it means encoder is just an embedding-lookup and it could be represented as:

Each column in matrix $\mathbf{Z}$ indicates a node embedding, the total number of rows in $\mathbf{Z}$ equals to the dimension/size of embeddings. $\mathbf{v}$ is the indicator vector with all zeros except a one in column indicating node $v$. We see that each node is assigned to a unique embedding vector in “shallow” encoding. There are many ways to generate node embeddings (e.g., DeepWalk, node2vec, TransE), key choices of methods depend on how they define node similarity.

## Random Walk

Now let’s try to define node similarity. Here we introduce Random Walk, an efficient and expressive way to define node similarity Random walk has a flexible stochastic definition of node similarity that incorporates both local and higher-order neighborhood information, also it does not need to consider all node pairs when training; only need to consider pairs that co-occur on random walks. and train node embeddings: given a graph and a starting point, we select a neighbor of it at random, and move to this neighbor; then we select a neighbor of this point at random, and move to it, etc. The (random) sequence of points selected this way is a random walk on the graph. So $\rm similarity(u,v)$ is defined as the probability that $u$ and $v$ co-occur on a random walk over a network. We can generate random-walk embeddings following these steps:

1. Estimate probability of visiting node $v$ on a random walk starting from node $u$ using some random walk strategy $R$. The simplest idea is just to run fixed-length, unbiased random walks starting from each node (i.e., DeepWalk from Perozzi et al., 2013).
2. Optimize embeddings to encode these random walk statistics, so the similarity between embeddings (e.g., dot product) encodes Random Walk similarity.

### Random walk optimization and Negative Sampling

Since we want to find embedding of nodes to d-dimensions that preserve similarity, we need to learn node embedding such that nearby nodes are close together in the network. Specifically, we can define nearby nodes $N_R (u)$ as neighborhood of node $u$ obtained by some strategy $R$. Let’s recall what we learn from random walks, we could run short fixed-length random walks starting from each node on the graph using some strategy $R$ to collect $N_R (u)$, which is the multiset of nodes visited on random walks starting from $u$. Note that $N_R (u)$can have repeat elements since nodes can be visited multiple times on random walks. Then we might optimize embeddings to maximize the likelihood of random walk co-occurrences, we compute loss function as:

where we parameterize $P(v|\mathbf{z_u})$ using softmax:

Put it together:

To optimize random walk embeddings, we need to find embeddings $\mathbf{z_u}$ that minimize $\mathcal{L}$. But doing this naively without any changes is too expensive, nested sum over nodes gives $\rm O(|V|^2)$ complexity. Here we introduce Negative Sampling To read more about negative sampling, refer to Goldberg et al., word2vec Explained: Deriving Mikolov et al.’s Negative-Sampling Word-Embedding Method (2014). to approximate the loss. Technically, Negative sampling is a different objective, but it is a form of Noise Contrastive Estimation (NCE) which approximately maximizes the log probability of softmax. New formulation corresponds to using a logistic regression (sigmoid func.) to distinguish the target node $v$ from nodes $n_i$ sampled from background distribution $P$ such that

$\rm P_v$ means random distribution over all nodes. Instead of normalizing with respect to all nodes, we just normalize against $k$ random “negative samples” $n_i$. In this way, we need to sample $k$ negative nodes proportional to degree to compute the loss function. Note that higher $k$ gives more robust estimates, but it also corresponds to higher bias on negative events. In practice, we choose $k$ between 5 to 20.

### Node2vec

So far we have described how to optimize embeddings given random walk statistics, What strategies should we use to run these random walks? As we mentioned before, the simplest idea is to run fixed-length, unbiased random walks starting from each node (i.e., DeepWalk from Perozzi et al., 2013), the issue is that such notion of similarity is too constrained. We observe that flexible notion of network neighborhood $N_R (u)$ of node $u$ leads to rich node embeddings, the idea of Node2Vec is using flexible, biased random walks that can trade off between local and global views of the network (Grover and Leskovec, 2016). Two classic strategies to define a neighborhood $N_R (u)$ of a given node $u$ are BFS and DFS:

BFS can give a local micro-view of neighborhood, while DFS provides a global macro-view of neighborhood. Here we can define return parameter $p$ and in-out parameter $q$ and use biased $\rm 2^{nd}$-order random walks to explore network neighborhoods, where $p$ models transition probabilities to return back to the previous node and $q$ defines the “ratio” of BFS and DFS. Specifically, given a graph below, walker came from edge ($s_1$, $w$) and is now at $w$, $1$, $1/q$ and $1/p$ show the probabilities of which node will visit next (here $w$, $1$, $1/q$ and $1/p$ are unnormalized probabilities):

So now $N_R (u)$ are the nodes visited by the biased walk. Let’s put our findings together to state the node2vec algorithm:

1. Compute random walk probabilities
2. Simulate $r$ random walks of length $l$ starting from each node $u$
3. Optimize the node2vec objective using Stochastic Gradient Descent

## TransE

Here we take a look at representation learning on multi-relational graph. Multi-relational graphs are graphs with multiple types of edges, they are incredibly useful in applications like knowledge graphs, where nodes are referred to as entities, edges as relations. For example, there may be one node representing “J.K.Rowling” and another representing “<NAME>”, and an edge between them with the type “is author of”. In order to create an embedding for this type of graph, we need to capture what the types of edges are, because different edges indicate different relations.

TransE(<NAME>, Garcia-Duran. NeurIPS 2013.) is a particular algorithm designed to learn node embeddings for multi-relational graphs. We’ll let a multi-relational graph $G=(E,S,L)$ consist of the set of $entities$ $E$ (i.e., nodes), a set of edges $S$, and a set of possible relationships $L$. In TransE, relationships between entities are represented as triplets：

Where $h \in E$ is head entity or source-node, $l \in L$ is relation and $t \in E$ is tail entity or destination-node. Similar to previous methods, entities are embedded in an entity space $\mathbb{R}^k$. The main innovation of TransE is that each relationship $l$ is also embedded as a vector $\mathbf{l} \in \mathbb{R}^k$.

That is, if $(h,l,s) \in S$, TransE tries to ensure that:

Simultaneously, if the edge $(h,l,t)$ does not exist, TransE tries to make sure that:

TransE accomplishes this by minimizing the following loss:

Here $(h',l,s')$ are “corrupted” triplets, chosen from the set $S'_{(h,l,t)}$ of corruptions of $(h,l,t)$, which are all triplets where either $h$ or $t$ (but not both) is replaced by a random entity:

Additionally, $\rm \gamma >0$ is a sclar called the $margin$, the function $d(.,.)$ is the Euclidean distance, and $[]_+$ is the positive part function (defined as max$(0,.)$). Finally, in order to ensure the quality of our embeddings, TransE restricts all the entity embeddings to have length $1$, that is, for every $e \in E$:

Figure below shows the pseudocode of TransE algorithm:

## Graph Embedding

We may also want to embed an entire graph $G$ in some applications (e.g., classifying toxic vs. non-toxic molecules, identifying anomalous graphs).

There are several ideas to accomplish graph embedding:

1. The simple idea (Duvenaud et al., 2016) is to run a standard graph embedding technique on the (sub)graph $G$, then just sum (or average) the node embeddings in the (sub)graph $G$.
2. Introducing a “virtual node” to represent the (sub)graph and run a standard graph embedding technique: To read more about using the virtual node for subgraph embedding, refer to Li et al., Gated Graph Sequence Neural Networks (2016)

3. We can also use anonymous walk embeddings. In order to learn graph embeddings, we could enumerate all possible anonymous walks $a_i$ of $l$ steps and record their counts and represent the graph as a probability distribution over these walks. To read more about anonymous walk embeddings, refer to <NAME> al., Anonymous Walk Embeddings (2018).


text:
# Bootstrapping only selected post-hoc contrasts for multiple regression

Is is statistically correct to claim that I proved (with p-value < 0.05) that "mean of versicolor group > mean of setosa group" if all I did was to bootstrap a 95% confidence interval for "setosa - versicolor" contrast like in this code?

require(tidyverse)
require(car)
require(emmeans)

m <- lm(Sepal.Length ~ Species, data=iris)
m.boot <- Boot(m, function(.) {
t <- summary(emmeans(., pairwise ~ Species))$contrasts setNames(t$estimate, t\$contrast)[1]
})
confint(m.boot)
## Bootstrap quantiles, type =  bca
##
##                        2.5 %     97.5 %
## setosa - versicolor -1.104946 -0.7687479

If "Yes", what would be the proper reference to support such analysis?

I suspect there must be a catch here. I mean, why would anyone bother to develop packages like "ARTool" if a simple bootstrap will do... If so, can you suggest an alternative way? Preferably, unlike ARTool, one allowing to fit arbitrary linear models and visualize confounder-controlled group means on the original scale (not transformed to ranks).

In reality we need to analyze several contrasts in our (not quite balanced and normally distributed) observational experiment, designed as 3-way ANCOVA with a confounding variable.

• Is it possible to pare down this post into a single, focused question with a minimally reproducible example? I am having a hard time understanding what you're really asking. – AdamO Feb 7 '18 at 15:33
• to AdamO: Please tell me which parts are not reproducible, I will be happy to clarify. I thought linear model with 2 factors is "minimal" to fully illustrate my questions: regression is complicated enough, so that post-hoc analysis is "a must" to interpret it. My (oversimplified) main question: Is it correct to "directly bootstrap" just a single contrast (say "A,X - B,X") to get 95% CI and p-value for it? – Daniil Feb 7 '18 at 16:47
• minimally reproducible means that the fewest lines of code necessary to convey the problem are presented. Perhaps you can focus on another example with a single contrast. – AdamO Feb 7 '18 at 17:05
• Regarding “forcing emmeans...,” you might consider using emmeans::contrast with a specified list of contrast coefficients, and/or a by variable. It’s all right there in the documentation. – rvl Feb 8 '18 at 1:10
• to rvl: Thank you, of course I RTFM about "list of contrast coefficients". I could unite the coefficient lists from "pairwise ~ abc|xy" and "pairwise ~ xy|abc". But do you agree that this is an extremely inconvenient way of specifying "custom" sets of contrasts as compared to using human readable formula? – Daniil Feb 8 '18 at 9:01


text:
# which of these 2 classification models is best

The difference between the two models is the label, which is slightly differently defined for the 2nd model. So for model 2 there are less data that fall in class 1.0 (87161 versus 155915).

Which of the two models would you prefer? how should I evaluate a higher f1 score for the separate classes?

Model 1

              precision    recall  f1-score   support

0.0       0.69      0.90      0.78    299032

1.0       0.56      0.23      0.32    155915

avg / total       0.65      0.67      0.63    454947

Model 2

              precision    recall  f1-score   support

0.0       0.87      0.93      0.90    367786

1.0       0.58      0.40      0.47     87161

avg / total       0.81      0.83      0.82    454947


text:
# Cylinder

The 1.8m cylinder contains 2000 liters of water. What area (in dm2) of this container is the water?

Correct result:

S =  698.913 dm2

#### Solution:

$D=1.8 \ m \rightarrow dm=1.8 \cdot \ 10 \ dm=18 \ dm \ \\ r=D/2=18/2=9 \ \text{dm} \ \\ \ \\ V=2000 \ l \rightarrow dm^3=2000 \cdot \ 1 \ dm^3=2000 \ dm^3 \ \\ \ \\ S_{1}=\pi \cdot \ r^2=3.1416 \cdot \ 9^2 \doteq 254.469 \ \text{dm}^2 \ \\ \ \\ h=V/S_{1}=2000/254.469 \doteq 7.8595 \ \text{dm} \ \\ \ \\ S=S_{1} + \pi \cdot \ D \cdot \ h=254.469 + 3.1416 \cdot \ 18 \cdot \ 7.8595=698.913 \ \text{dm}^2$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Please write to us with your comment on the math problem or ask something. Thank you for helping each other - students, teachers, parents, and problem authors.

Tips to related online calculators
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text:
Now showing items 1-1 of 1

• #### Neron models of intermediate Jacobians associated to moduli spaces

(Revista Matemática Complutense, 2019-12-01)
Let $\pi_1:\mathcal{X} \to \Delta$ be a flat family of smooth, projective curves of genus $g \ge 2$, degenerating to an irreducible nodal curve $X_0$ with exactly one node. Fix an invertible sheaf $\mathcal{L}$ on $\mathcal{X}$ ...


text:
# conjugateprior

## R to Latex packages - coverage

There are now quite a few R packages to turn cross-tables and fitted models into nicely formatted latex. In a previous post I showed how to use one of them to display regression tables on the fly. In this post I summarise what types of R object each of the major packages can deal with.

Unsurprisingly, there's quite some variation...

The packages I'm looking at here are: apsrtable (v0.8-8), xtable (v1.7-1), stargazer (v3.0.1), memisc (v0.96-3) and texreg (v1.22).

I should note that all of these packages also allow users to add their own latex representation for new R objects. However, I couldn't be bothered to track down all these additions, so this post only deals with what each package can do without any extra work.

Also, several of these packages can typeset data.frame and matrix objects too. While that can be incredibly handy, it is outside the focus of this post so I've left it out. Cross-tables are only just processed enough to deserve a mention. And we won't even talk about generating HTML (well, not until the notes anyway).

Finally, I haven't checked all of this information. The table is filled on the basis of my experience and the package documentation. If something is wrong or incomplete then please let me know in the comments.

Package Model xtable memisc texreg stargazer apsrtable
AER ivreg Y Y
tobit Y Y Y Y
base table Y Y
ftable Y
betareg betareg Y Y
dynlm dynlm Y Y
eha aftreg Y
phreg Y
weibreg Y
ergm ergm Y Y
gee gee Y Y Y (in 'skeleton' form)
gmm gmm Y
lme4 glmerMod Y Y
lmerMod Y Y
nlmerMod Y Y
MASS polr Y Y Y
rlm Y Y
negbin Y Y Y
mgcv gam Y
nlme gls Y Y
lme Y
nnet multinom Y Y
ordinal clm Y Y Y
sclm Y
plm plm Y Y
pmg Y Y
pscl hurdle Y Y
zeroinfl Y Y
quantreg rq Y
rms lrm Y Y
robustbase lmrob Y Y
simex simex Y Y
sna lnam Y
stats glm Y Y Y Y Y
lm Y Y Y Y Y
aov Y
anova Y
prcomp Y
ts Y
survey svyglm Y Y Y (in 'skeleton' form)
survival coxph Y Y Y Y
clogit Y Y Y
survreg Y Y
coxph.penal Y
systemfit systemfit Y
termgm stergm Y
Zelig zeroinfl Y
cloglog.net Y
gamma.net Y
probit.net Y
logit.net Y
Relogit Y

### Notes

In case you are curious, a first version of this table was generated using the print.xtable function of the xtable package with type=html before being adjusted in place. If these posts haven't given you the idea yet: I detest doing this sort of thing by hand.

Liviu (in the comments) also mentions the estout package. I've never used it, but apparently it deals with lm and plm models and is modeled after a Stata command of the same name.


text:
# Understanding $ee\to h \to ff$ process

can anyone tell me where to find a good reference that talks about $$e^+ e^-\rightarrow H \rightarrow f\bar{f}$$ where this describes an electron-positron annihilation into a Higgs that in term produces heavy leptons.

Any reference that explains this process thoroughly will do it. I have searched many books on weak interactions but none of which handled it except one book "Gauge theories of the Strong, Weak, and Electromangnetic Interactions" that only gave a fast glimpse about it but I couldn't understand it from there.

Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

• You could look for lepton-lepton decay channel of higgs – aQuestion Mar 9 '15 at 16:05
• lepton-lepton decay? leptons are being formed here. @Major_Tom – beyondtheory Mar 9 '15 at 16:07
• Higgs Decay of two tau leptons – aQuestion Mar 9 '15 at 16:08
• Have a look to general books like Halzen & Martin, Griffiths etc... – Paganini Mar 10 '15 at 9:16
• because, a scalar (spin 0 like the Higgs boson) doesn't carry any spin. The propagator does not depend on the Minkowski indices (no $g^{\mu\nu}$, no $\gamma^\mu$ and stuff like this). – Paganini Mar 10 '15 at 20:38


text:
# Parabolic mirror: why is $b^2-1/4 = \tan(2\tan^{-1}(2b)+{\pi}/{2})(b-0)$ for all $b$?

Let us take the simple parabola $$x^2$$.

A ray of light will bounce on it making equal angles to both sides.

The derivative of $$x^2$$ is $$2x$$ and its normal is $$-1/2x$$.

Given light rays coming straight from above we get, given a variable b that represents the x value of the vertical ray we get that the reflected ray in general is:

$$\left(b^2-y\right)=\tan\left(2\tan^{-1}\left(2b\right)+\frac{\pi}{2}\right)\left(b-x\right)$$

In fact we see that this always pass through a point (focal point) for all b by watching the graph while sliding b.

I want to prove all points pass there analitically.

I substitute in $$b=1/2$$ and $$b=(\sqrt3)/2$$ as examples and I get the intersection (0,1/4) (also clearly seen in the graph).

Now I substitute (x=0,y=1/4) back in $$\left(b^2-y\right)=\tan\left(2\tan^{-1}\left(2b\right)+\frac{\pi}{2}\right)\left(b-x\right)$$ and I get an equation that is true for all b as you can check by sliding the slider. $$\left(b^2-\frac{1}{4}\right)-\tan\left(2\tan^{-1}\left(2b\right)+\frac{\pi}{2}\right)\left(b-0\right)$$ This means that all reflected rays of light pass by it.

How can I prove this last statement analytically?

Apply the following identities

$$tan(x+ \frac \pi 2)=-cot(x)$$

$$cot(2x) = \frac{ \cot^2(x)-1 }{ 2\cot(x) }$$

$$\cot( \tan^{-1}(x) ) =\frac 1x$$

Get $$\tan\left(2\tan^{-1}\left(2b\right)+\frac{\pi}{2}\right) \\ = -\cot( 2 \tan^{-1}(2b) ) \\ =\frac { 1-\cot^2( \tan^{-1}(2b) ) }{ 2\cot( \tan^{-1}(2b) } \\ = \frac{ 1-\frac 1{4b^2} }{ \frac 1b } \\ = b-\frac 1{4b}$$

• Could you please add a small explanation or link for cot(tan^-1(x)) identity? Sep 24 '19 at 5:50
• $$\cot(\tan^{-1}(x) ) = \frac 1{\tan(\tan^{-1}(x) } =\frac 1x$$
– WW1
Sep 25 '19 at 0:58

Another answer shows you how to reduce the trigonometric expression in your equation, but trigonometric functions can be avoided entirely.

As you’ve determined, a normal vector to the parabola at $$x=b$$ is $$\mathbf n=(2b,-1)$$. Using a well-known reflection formula, the reflection of an incident ray with direction $$\mathbf v = (0,-1)$$ has direction $$-\left(2{\mathbf n\cdot\mathbf v\over\mathbf n\cdot\mathbf n}\mathbf n-\mathbf v\right) = \left({-4b\over1+4b^2},{1-4b^2\over1+4b^2}\right).$$ We can discard the common nonzero denominator to obtain the parameterization $$(b,b^2)+t(-4b,1-4b^2)$$, $$t\ge0$$ of the reflected ray.

Observe now that the ray for $$b=0$$ is reflected onto itself, so if all of the reflected rays do pass through a common point, it must lie on the $$y$$-axis. Unless $$b=0$$, for the $$x$$-coordinate to be zero we must have $$t=1/4$$, which yields $$y=1/4$$. Therefore, all of the reflected rays pass through $$(0,1/4)$$.

Apply direct trig formulas.

Use negative sign for tangent/cotangent in second quadrant

and tan expansion of double angle:

$$\tan[2 \tan^{-1}2b + \pi/2]$$

$$=\dfrac{-1}{ \tan [2\tan^{-1}2b] }$$

$$=\dfrac{-1}{\big[ \dfrac{2 \cdot 2b}{1-(2b)^2}\big] }$$

$$\dfrac{1-4b^2}{4b}.$$

Including domain $$-1 and one or more incident/reflected more rays for problem posing... is sufficient..


text:
# MthEd608FinalNatascha3a

Author:
nkfaux
The student conjectured that every time , or the number of rectangles in the interval, doubles, the difference between the upper sum and lower sum halves. After exploring the applet below by experimenting with different functions and number of rectangles, the student's conjecture seems to only be true when the function is monotonic on the given interval. If a function is monotonic and decreasing, then the height of the lower sum of the rectangle is given by . However, this quantity will also represent the height of the rectangle in the upper sum. Furthermore, the width of each rectangle is given by . Thus the area of each rectangle is given by . Since the height of each rectangle within is repeated in both the upper and lower sums, the difference between the areas of these rectangles will be zero. Thus, the only terms left after the subtraction of the upper and lower sums will be . By factoring we get that the difference between the upper and lower sums is . Now suppose that the number of intervals is doubled. Then the width of each rectangle is given by and the difference between the upper and lower sums will be . Thus, when the number of rectangles is doubled, the area is halved. If the function is monotonic and increasing, then a similar argument will hold. If the function is not monotonic, such as , then the argument will not hold because it is not always true that will represent the upper and lower sums of adjacent rectangles.


text:
# Biblatex inherits publication date from urldate in bibliography

Using biber, I do not encounter this issue but I want to use bibtex for the backend option.

I have an "online" bibitem with an accessed on date but no publication date. I want the output to be along the lines of AUTHOR (n.d.) URL [Accessed on: dd/mm/year].

I can only find a way to do this by clearing the urlyear, which I don't want to do. Furthermore, this thread suggests setting the year to the text "n.d." however this generates a warning about issues with sorting which I would like to avoid.

MWE:

\documentclass[british]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}

\usepackage[style=authoryear, backend=biber, urldate=long, dateabbrev=false]{biblatex}

\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@online{appleby,
author  = {<NAME>},
date = {},
title   = {On the Importance of the Civil Service},
url     = {https://example.com/gov/sir-humphrey/importance-civil-service},
urldate = {2018-08-23},
}
\end{filecontents}

\begin{document}
\nocite{*}
\printbibliography
\end{document}

I would like to display n.d. or similar in the empty brackets. I could do this either by:

• Preventing the publication date being inherited from the access date.
• Manually setting the date to be "null" somehow, but doing this it gives ("null") as the date in the ways that I have tried.

Thanks.

If you must use bibtex as your backend, then you can specify year = {nodate} instead of using the date field.

Aside: If you use biber as backend, then I think the easiest option is just to alter the definition of \DeclareLabeldate to remove urldate from the list of allowable fields. e.g.,

\DeclareLabeldate{%
\field{date}
\field{year}
\field{eventdate}
\field{origdate}
\literal{nodate}
}

# MWE using bibtex as backend

\documentclass[british]{article}
\usepackage{babel}
\usepackage{csquotes}

\usepackage[style=authoryear, backend=bibtex, urldate=long, dateabbrev=false]{biblatex}

\begin{filecontents}[overwrite]{\jobname.bib}
@online{appleby,
author  = {<NAME>},
year    = {nodate},
title   = {On the Importance of the Civil Service},
url     = {https://example.com/gov/sir-humphrey/importance-civil-service},
urldate = {2018-08-23},
}
\end{filecontents}