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for Predator and prey. A STAGE-STRUCTURED PREDATOR-PREY MODEL HAL SMITH 1. PREDATOR PREY MODELS IN COMPETITIVE CORPORATIONS PREDATOR PREY MODELS By Rachel Von Arb Honors Scholarship Project Submitted to the Faculty of. This represents our first multi-species model. Recently, a new type of mathematical model was introduced into biology to study the pattern formation, i. MATLAB Code: function yp = lotka(t,y). / Spatiotemporal dynamics of a diffusive Leslie-Gower predator-prey model with ratio-dependent functional response. Wilkinson and T. It also assumes no outside influences like disease, changing conditions, pollution, and so on. In addition, the amount of food needed to sustain a prey and the prey life span also affect the carrying capacity. Consider the Lotka-Voterra equations of interacting predator and prey systems This equations include the effect of limited resources on the food supply of the prey, and how the prey are culled or harvested. The predators depend on the populations of these prey organisms. To illustrate the use of WSMCreateModel, the component "hare" in the predator-prey model was created in Mathematica. If your students are unable to run the simulation at their own workstations then it may be played on an overhead projector. Initial populations sizes can be selected by the user and are randomly distributed in a square ‘environment’, (dimensions=km,. Figure 1: Simple Predator Prey Model The phase plane plot compares the population of predators to the population of prey, and is not dependent on time. The grid is enclosed, so a critter is not allowed to move off the edges of the world. Predator-Prey Models from Iterated Prisonerʼs Dilemma model. Shiflet and G. Use model blocks to import, initialize, and simulate models from the MATLAB ® environment into a Simulink model. Consider for example, the classic Lotka-Volterra predator prey. Classic population models including the logistic map, predator-prey systems, and epidemic models will be used to | {
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models including the logistic map, predator-prey systems, and epidemic models will be used to motivate dynamics concepts such as stability analysis, bifurcations, chaos, and Lyapunov exponents. Spring (3) Shaw. Universitas Negeri Surabaya. I - Ecological Interactions: Predator and Prey Dynamics on the Kaibab Plateau - Andrew Ford ©Encyclopedia of Life Support Systems (EOLSS) 1. We show the different types of system behaviors for various parameter values. A Predator-Prey model: Suppose that we have two populations, one of which eats the other. Round 1 Data Analysis: Produce a "finished product" graph of the data from the simulation. In this paper, we have considered a prey–predator model where both prey and predator live in herds. Course Goals: Expose students to the process of model building, the simulation and computation with mathematical models, and the interpretation and analysis of simulation results. Run the simulation again now with the controls below, paying attention to the animals in the enclosure at the top. Both prey and predator harvesting or combined harvesting and maximum sustainable yield have been discussed. Determine the equilibrium points and their nature for the system. The solution is also given in Taylor’s series. Models of interacting populations. Multi-Team Prey-Predator Model with Delay Shaban Aly1, 2 and M. Predation has been described as a clean. The second model (Daypr) is more realistic. wednesday, june 19, 2019. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. Go through the m-file step-by-step. The Canadian lynx is a type of wild felid, or cat, which is found in northern forests across almost all of Canada and Alaska. GIBSON1*, DAVID L. Predator Prey Models in MatLab James K. Many of our resources are part of collections that are created by our various research projects. Initial populations sizes can be selected by the user and are randomly distributed in a square | {
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Initial populations sizes can be selected by the user and are randomly distributed in a square 'environment', (dimensions=km,. As part of our. This pair of interacting populations is a classic example of the predator‐prey dynamical system. of Shiflet;: The Fox and the Rabbit Many dynamic systems are interdependent systems. My book that's available on the MathWorks website. Diff Eqs Lect #12, Predator/Prey Model, Vector Fields and Direction Fields - Duration. , 2D linear dynamical systems; the use of probabilities gives Markov chains. However, the organisms in Holland’s simulation are very simple and do not involve any behavioral model. 01$ and \$\beta = 0. It is a simple program originally described by A. ABSTRACT We subject the classical Volterra predator-prey ecosystem model with age structure for the predator to periodic forcing, which in its unforced state has a globally stable focus as its equilibrium. There has been growing interest in the study of Prey-Predator models. Discussion and Conclusion In Conclusion, this Lotka-Volterra Predator-Prey Model is a fundamental model of the complex ecology of this world. If x is the population of zebra, and y is the population of lions, description of the population dynamics with the help of coupled differential equations. albena, june 20-25, 2019. BIFURCATION IN A PREDATOR-PREY MODEL WITH HARVESTING 2103 the harvesting terms can be combined into the growth/death terms as in system (3), so the dynamics of system (3) are very similar to that of the unharvested system (2). It also highlights the modularity of MATLAB and ROS by showing the algorithm using real and simulated TurtleBot ® robotic platforms, as well as a webcam. If your students are unable to run the simulation at their own workstations then it may be played on an overhead projector. zeszyty naukowe politechniki ŚlĄskiej 2018 seria: organizacja i zarzĄdzanie z. It was developed independently by Alfred Lotka and Vito Volterra in. Canadian lynx feed predominantly | {
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was developed independently by Alfred Lotka and Vito Volterra in. Canadian lynx feed predominantly on snowshoe hares. The most popular example is the population of the snowshoe hare and the lynx. For system dynamics modeling, as with all approaches, the text employs a nonspecific tool, or generic,. Prey population x(t); Predator population y(t) 2. Boids simulation on Matlab. The predator population starts to decrease and, let me do that same blue color. Course Readings. println("Will think about this more. We assign a mo-mentary fitness f y (t) of the prey in a year t as follows: it is zero if it is not present, it is 11 if both predator and prey are present, and it is +1 if the prey is present but the predator. It provides online dashboard tools for simulation analytics that can be shared with users from around the world. zip contains versions of some programs converted to work with SciLab. Now ode45 is used to perform simulation by showing the solution as it changes in time. The Puma-Prey Simulator demonstrates the natural balance of a healthy ecosystem, in contrast with the changes that occur as a result of human encroachment. (c) Constant-yield harvesting on the prey and constant-e ort harvesting on preda-tors. We describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator-prey model proposed by R. Initial populations sizes can be selected by the user and are randomly distributed in a square 'environment', (dimensions=km,. 8, in steps of 0. Learn to use MATLAB and Simulink for Simulation and other science and engineering computations. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty. 05 and Euler's method, to model the population numbers over the next 5 years. National Science Foundation. Predator Prey Multi Agent Simulation Model (JAVA & REPAST). Suppose in a closed | {
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Science Foundation. Predator Prey Multi Agent Simulation Model (JAVA & REPAST). Suppose in a closed eco-system (i. For system dynamics modeling, as with all approaches, the text employs a nonspecific tool, or generic,. Objective: Students will simulate predator prey interactions using cards. Scilab simulation of Lotka Volterra predator prey model, van-der-Pol Oscillator tutorial of Nonlinear Dynamical Systems course by Prof Harish K. Attentional strategies for dynamically focusing on multiple predators/prey, click here. Be sure to stay to the end to find out where to go next to learn MATLAB in depth. Denning, "Computing is a natural science" MatlaB Tutorial. This project results in a Lotka-Volterra model which simulates the dynamics of the predator-prey relationship. Matlab code for the examples discussed below is in this compressed folder. Some predator-prey models use terms similar to those appearing in the Jacob-Monod model to describe the rate at which predators consume prey. Surabaya, Indonesia. These equations have given rise to a vast literature, some of which we will sample in this lecture. A FORMAL MODEL OF EMOTIONAL STATE. The results developed in this article reveal far richer dynamics compared to the model without harvesting. Pillai of IIT Bombay. However in this pa-per, in order to illustrate the accuracy of the method, DTM isappliedtoautonomous and non-autonomous predator-prey models over long time horizons and the. In the Lotka-Volterra model, there is one populations of animals (predator) that feeds on another population of animals (prey). The Lotka-Volterra model is the simplest model of predator-prey. 2016-10-10 Modeling and Simulation of Social Systems with MATLAB 35 SIR model ! A general model for epidemics is the SIR model, which describes the interaction between Susceptible, Infected and Removed (Recovered) persons, for a given disease. In the Lotka Volterra predator-prey model, the changes in the predator population y and the prey population x | {
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Volterra predator-prey model, the changes in the predator population y and the prey population x are described by the following equations: Δxt=xt+1−xt=axt−bxtyt Δyt=yt+1−yt=cxtyt−dyt. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. I have a program called Predator Prey that's in the collection of programs that comes with NCM, Numerical Computing with MATLAB. The Dynamical System. If x is the population of zebra, and y is the population of lions, description of the population dynamics with the help of coupled differential equations. In [9] the DTM was applied to a predator-prey model with constant coeffi-cients over a short time horizon. For You Explore. The second is a study of a dynamical system with a simple bifurcation, and the third problem deals with predator-prey models. Denning, "Computing is a natural science" MatlaB Tutorial. The behavior of each of them is given by the following rules: Prey: 1) just moved to an unoccupied cell 2) Every few steps creates offspring to his old cell 3) Life expectancy is limited by the number of moves Predator: 1) Predator moves to the cell with prey. Given the differences between these two types of models, why would it be difficult to determine accurate values for the four parameters in the Lotka-Volterra predator-prey model (a, rprey, m, b) for animals in the real world?. Department of Mathematics. 1 Logistic growth with a predator We begin by introducing a predator population into the logistic. lab 11 predator prey simulation lab answers. PY - 2009/1. Applications of MATLAB/Simulink for Process Dynamics and Control Simulink is a platform for multidomain simulation and model Predator and prey populations. The objective of this paper is to study systematically the dynamical properties of a predator-prey model with nonlinear predator harvesting. Using the Lotka-Volterra predator prey model as a simple case-study, I use the R packages deSolve to solve a | {
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Lotka-Volterra predator prey model as a simple case-study, I use the R packages deSolve to solve a system of differential equations and FME to perform a sensitivity analysis. Find Alien at affordable prices. AU - Ruan, Shigui. The prey are blue, and the predators are yellow. Diff Eqs Lect #12, Predator/Prey Model, Vector Fields and Direction Fields - Duration. Suppose in a closed eco-system (i. Suppose there are two species of animals, a prey and a predator. Software Programming And Modelling For Scientific Researchers. The Puma-Prey Simulator demonstrates the natural balance of a healthy ecosystem, in contrast with the changes that occur as a result of human encroachment. Forecasting performance of these models is compared. predator-prey simulations 1 Hopping Frogs an object oriented model of a frog animating frogs with threads 2 Frogs on Canvas a GUI for hopping frogs stopping and restarting threads 3 Flying Birds an object oriented model of a bird defining a pond of frogs giving birds access to the swamp MCS 260 Lecture 36 Introduction to Computer Science. My book that's available on the MathWorks website. 2Mathematics Department , Faculty of Science , Al-Azhar University, Assiut 71511, Egypt. zip contains all Matlab program files listed here. Introduction This chapter, originally intended for inclusion in [4], focuses on mod-eling issues by way of an example of a predator-prey model where the. Model equations In this paper, we study the numerical solutions of 2-component reaction–diffusion. Both prey and predator harvesting or combined harvesting and maximum sustainable yield have been discussed. Read "A Fractional Predator-Prey Model and its Solution, International Journal of Nonlinear Sciences and Numerical Simulation" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Predator-prey model with delay. iseesystems. 03(2015), Article ID:56937,10 pages 10. " – Simulation as a | {
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model with delay. iseesystems. 03(2015), Article ID:56937,10 pages 10. " – Simulation as a basic tool. This is referred to. • Use Euler’s method and Runge-Kutta in MATLAB to obtain numerical approximations. Continuous time (ODE) version of predator prey dynamics: Equilibrium points (2) •~(20. A mathematical analysis shows that prey refuge plays a crucial role for the survival of the species and that the harvesting effort on the predator may be used as a control to prevent the cyclic behaviour of the system. We are trying to understand as the population grows in one of the species what the effect is on the other species which co inhabit that environment. The second model is an extension of the logistic model to species compe-tition. Some predator-prey models use terms similar to those appearing in the Jacob-Monod model to describe the rate at which predators consume prey. Predator Prey Model Goal: The goal of this experiment is to model the population dynamics of animals both predator and prey when they are present in an environment. Rabbits and Wolves: Experiment with a simple ecosystem consisting of grass, rabbits, and wolves, learning about probabilities, chaos, and simulation. Homework 5, Phase Portraits. Using no barriers and a random distribution of 100 beans, run 1 trial as done during the baseline data trials. Suppose in a closed eco-system (i. Date: 22nd August, 2007 Lab #1: Predator-Prey Simulation ==> OBJECTIVE: To simulate predator prey interactions and record the numbers of predator and prey in their "ecosystem" and prepare a graph. Zhao, "Mathematical and dynamic analysis of a prey-predator model in the presence of alternative prey with impulsive state feedback control," Discrete Dynamics in Nature and Society, vol. FD1D_WAVE, a MATLAB program which applies the finite difference method to solve the time-dependent wave equation in one spatial dimension. Stan is used to encode the statistical model and perform full Bayesian inference to solve the inverse | {
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is used to encode the statistical model and perform full Bayesian inference to solve the inverse problem of inferring parameters from noisy data. Here is how Volterra got to these equations: The number of predatory shes immediately after WWI was much larger than. [paper2, 2 predator - 1 prey model] Barraquand, F. The model is first applied to a system with two-dimensions, but is then extended to include more. Introduction (1 week) # - "What is a model?" A. The objective of this paper is to study systematically the dynamical properties of a predator-prey model with nonlinear predator harvesting. Dewdney in Scientific American magazine. Simulate Identified Model in Simulink. The Dynamical System. We study the spatiotemporal dynamics in a diffusive predator–prey system with time delay. INTRODUCTION global properties of the orbit structure. Here, the predators are the police and the prey the gang members. One of such models that simulates predator-prey interactions is the Lotka-Volterra Model. The grid is enclosed, so a critter is not allowed to move off the edges of the world. The model of Lotka and Volterra is not very realistic. A game in the everyday sense—“a competitive activity. Ballesteros, Intuition, functional responses and the formulation of predator–prey models when there is a large disparity in the spatial domains of the interacting species, Journal of Animal Ecology, 77, 5, (891-897), (2008). sciencedaily. We assume periodic variation in the intrinsic growth rate of the prey as well as periodic constant impulsive immigration of the predator. Kalyan Das, National Institute of Food Technology Entrepreneurship and Management (NIFTEM), Mathematics Department, Faculty Member. The book is related to aircraft control, dynamics and simulation. We assign a mo-mentary fitness f y (t) of the prey in a year t as follows: it is zero if it is not present, it is 11 if both predator and prey are present, and it is +1 if the prey is present but the predator. GIBSON1*, DAVID | {
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predator and prey are present, and it is +1 if the prey is present but the predator. GIBSON1*, DAVID L. Y1 - 2009/1. I - Ecological Interactions: Predator and Prey Dynamics on the Kaibab Plateau - Andrew Ford ©Encyclopedia of Life Support Systems (EOLSS) 1. A computer simulation model for the learning behavior of a certain type of predator faced with a multipatch environment is constructed, where prey densities differ between patches and are functions of time. Various computer models have been created to simulate the predator-prey relationship within an ecosystem. Mathematical Modeling: Models, Analysis and Applications covers modeling with all kinds of differential equations, namely ordinary, partial, delay, and stochastic. Lotka-Volterra predator prey model. Predator-Prey Model, University of Tuebingen, Germany. Keywords: Lotka-Volterra Model, Predator-prey interaction, Numerical solution, MATLAB Introduction A predator is an organism that eats another organism. We'll start with a simple Lotka-Volterra predator/prey two-body simulation. Analyzing the Parameters of Prey-Predator Models for Simulation Games 5 that period. Circles represent prey and predator initial conditions from x = y = 0. Model equations In this paper, we study the numerical solutions of 2-component reaction–diffusion. Run the simulation again now with the controls below, paying attention to the animals in the enclosure at the top. Finally, as we’ll see in Chapter xx, there is a deep mathematical connection between predator-prey models and the replicator dynamics of evolutionary game theory. These functions are for the numerical solution of ordinary differential equations using variable step size Runge-Kutta integration methods. The Lotka-Volterra model is the simplest model of predator-prey. I have a Predator-Prey Model: dR/dt = λR - aRF dF/dt = -μF + bRF Where λ and μ are growth rates of rabbits (R) and foxes (F) respectively, treated in isolation. Predators consume energy at a fixed rate over | {
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(R) and foxes (F) respectively, treated in isolation. Predators consume energy at a fixed rate over time; if their internal energy level becomes too low, they die. Go through the m-file step-by-step. Back to Eduweb Portfolio. They use a simplified version of the Lotka-Volterra equations and generate graphs showing population change. Lotka-Volterra predator-prey model. Section 5-4 : Systems of Differential Equations. Introduction This chapter, originally intended for inclusion in [4], focuses on mod-eling issues by way of an example of a predator-prey model where the. The model is intended to represent a warm—blooded vertebrate predator and its prey. MATLAB files for the discrete time model: predprey_discrete. Abstract—We describe and analyze emergent behavior and its effect for a class of prey-predators’ simulation models. Each collection has specific learning goals within the context of a larger subject area. Fussmann*† & Nelson G. Rapid evolution drives ecological dynamics in a predator–prey system Takehito Yoshida*, Laura E. Below are examples that show how to solve differential equations with (1) GEKKO Python, (2) Euler's method, (3) the ODEINT function from Scipy. I Also used in economic theory, e. considering some well known simulation methods to obtain the posterior summaries of interest. More generally, any of the data in the Lotka-Volterra model can be taken to depend on prey density as appropriate for the system being studied. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. For You Explore. One is the. In this model, prey, which represents the decision space vector, will be placed on the vertices of a two-dimensional lattice. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. The predator-prey population-change dynamics are modeled using linear and nonlinear time series models. It is a simple program originally described | {
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modeled using linear and nonlinear time series models. It is a simple program originally described by A. This will help us use the lotka model with different values of alpha and beta. Open the first file for this module by typing on the Matlab command line: ppmodel1. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty. BIFURCATION IN A PREDATOR-PREY MODEL WITH HARVESTING 2103 the harvesting terms can be combined into the growth/death terms as in system (3), so the dynamics of system (3) are very similar to that of the unharvested system (2). One animal in the simulation is a predator. Lotka-Volterra Model The Lotka-Volterra equations, also known as the predator-prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. Stan is used to encode the statistical model and perform full Bayesian inference to solve the inverse problem of inferring parameters from noisy data. This model reflects the point in time where the predator species has evolved completely and no longer competes for the initial food source. Circles represent prey and predator initial conditions from x = y = 0. Using the Lotka-Volterra predator prey model as a simple case-study, I use the R packages deSolve to solve a system of differential equations and FME to perform a sensitivity analysis. Predation has been described as a clean. 1007/s11859-015-1054-4. The parameters preyPop and predPop are the initial sizes of the prey and predator populations (M(0) and W(0)), respectively, dt (Δt) is the time interval used in the simulation, and months is the number of months (maximum value of t) for which to run the simulation. You may wish to introduce disturbances in the cycle such as killing off the lynx or starving the | {
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You may wish to introduce disturbances in the cycle such as killing off the lynx or starving the rabbits. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 7, 2013 Outline Numerical Solutions Estimating T with MatLab Plotting x and y vs time Plotting Using a Function Automated Phase Plane Plots. 8, in steps of 0. Load the sample project containing the Lotka-Volterra model m1. Determine the equilibrium points and their nature for the system. The Modeling Commons contains more than 2,000 other NetLogo models, contributed by modelers around the world. In the Lotka Volterra predator-prey model, the changes in the predator population y and the prey population x are described by the following equations: Δxt=xt+1−xt=axt−bxtyt Δyt=yt+1−yt=cxtyt−dyt. The model is derived and the behavior of its solutions is discussed. to investigate the key dynamical properties of spatially extended predator–prey interactions. Prey Simulation Lab Introduction In this lab project the objective is to simulate the relationship over generations of prey vs. It is a simple program originally described by A. This is a spatial predator-prey model from population ecology. Represent and interpret data on a line graph. The prey should exhibit mild oscillations, and the predator should fluctuate little. The solution is also given in Taylor’s series. Trajectories are closed lines. Models of interacting populations. System of differential equations. Denning, "Computing is a natural science" MatlaB Tutorial. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the bivariate truncated generalized Cauchy model fits the data better than the other models. Nonlinear model predictive control (planning) for level control in a surge tank, click here. Make phase plane plot. Princeton University Press, Princeton, NJ (2006). , wolf spider) and prey species (e. They use | {
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plot. Princeton University Press, Princeton, NJ (2006). , wolf spider) and prey species (e. They use a simplified version of the Lotka-Volterra equations and generate graphs showing population change. Article (PDF Available) Simulation of Predator-Prey in MATLAB. Study the effects of e. And an improved model with Logistic blocking effect is proposed. Dynamics of the system. Here is some data that approximates the populations of lynx and snowshoe hares observed by the Hudson Bay Company beginning in 1852. Usage of Boids for a prey-predator simulation. See our discounts on Alien, check our site and compare prices. This model takes the form of a pair of ordinary differential equations, one representing a prey species, the other its predator. Graph the population number (hares. Prey populations. Continuous time (ODE) version of predator prey dynamics: Equilibrium points (2) •~(20. model consisting of prey-predator model with horizontally transmitted of disease within predator population is proposed and studied. As the students work on constructing a model (Circulate Constructing a Model - Rabbit, Constructing a Model - Fox) I rotate the room and offer support where needed. Predator prey offers this graphic user interface to demonstrate what we've been talking about the predator prey equations. http://simulations. Predators/prey respond to other predators/prey and move of their own volition. zeszyty naukowe politechniki ŚlĄskiej 2018 seria: organizacja i zarzĄdzanie z. Wilkinson and T. Predator, which deals with objective functions, will also be placed on the same lattice randomly. Modified Model with "Limits to Growth" for Prey (in Absence of Predators) In the original equation, the population of prey increases indefinitely in the absence of predators. % the purpose of this program is to model a predator prey relationship % I will be using. In the last section we make a review of the paper and share with the future plans. To find the ratios of the errors, we will. We | {
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a review of the paper and share with the future plans. To find the ratios of the errors, we will. We present an individual-based predator-prey model with, for the first time, each agent behavior being modeled by a fuzzy cognitive map (FCM), allowing the evolution of the agent behavior through the epochs of the simulation. The prey still relies on the food source, but the predator relies solely on the former competitor. This lesson allows students to explore the interactions of two animal populations (wolves and moose) within an ecosystem. Represent and interpret data on a line graph. Analysis of the main equation guides in the correct choice of parameter values. The model of Lotka and Volterra is not very realistic. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. Plot the prey versus predator data from the stochastically simulated lotka model by using a custom function (plotXY). The persistence of food chains is maximized when prey species are neither too big nor too small relative to their predator. PY - 2009/1. The model. One animal in the simulation is a predator. 1 Logistic growth with a predator We begin by introducing a predator population into the logistic. As part of our. The Lotka-Volterra equations can be written simply as a system of first-order non-linear ordinary differential equations (ODEs). The second project of the semester was the predator prey model. Gillespie, predator-prey simulation GillespieSSA test. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty. Set the solver type to SSA to perform stochastic simulations, and set the stop time to 3. A modified predator–prey model with transmissible disease in both the predator and prey species is proposed and analysed, with infected prey being more vulnerable to predation and infected | {
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species is proposed and analysed, with infected prey being more vulnerable to predation and infected predators hunting at a reduced rate. Software Programming And Modelling For Scientific Researchers. & Murrell, D. The Lotka-Volterra predator-prey equations can be used to model populations of a predator and prey species in the wild. This project results in a Lotka-Volterra model which simulates the dynamics of the predator-prey relationship. Make Life easier. Deterministic Models can be classified dimensionally as 0D, 1D, 2D or 3D. Models of interacting populations. master program for. View at Publisher · View at Google Scholar. under the existence of the interior equilibrium point E 2 ∗ = (x 2 ∗, y 2 ∗). version of a Kolmogorov model because it focuses only on the predator-prey interactions and ignores competition, disease, and mutualism which the Kolmogorov model includes. The simulation uses rule-based agent behavior and follows a prey-predator structure modulated by a number of user-assigned parameters. Innovation Process Simulation on the Base Predator and Prey. Students should keep in mind that, as in any simulation (even sophisticated computer models), certain assumptions are made and many variables. in the literature [2-7,10,17]. Predator-prey model, design, simulation and analysis in Simulink. Predator prey offers this graphic user interface to demonstrate what we've been talking about the predator prey equations. The most popular example is the population of the snowshoe hare and the lynx. Lotka (1925) and Vito Volterra (1926). Let the initial values of prey and predator be [20 20]. Matt Miller, Department of Mathematics, University of South Carolina email: miller@math. m - discrete time simulation of predator prey model Continuous Time Model. In addition to discussing the well posedness of the model equations, the results of numerical experiments are presented and demonstrate the crucial role that habitat shape, initial data, and the boundary conditions | {
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and demonstrate the crucial role that habitat shape, initial data, and the boundary conditions play in determining the spatiotemporal dynamics of predator-prey interactions. Predator vs. Prey-predator model has received much attention during the last few decades due to its wide range of applications. The model for this simulation was created using iThink Systems Thinking software from isee systems. Participants are assigned a role in the food chain, participate in the simulation, collect and analyze results, and assess factors affecting their survival. Consider a population of foxes, the predator, and rabbits, the prey. View, run, and discuss the 'Predator Prey Game' model, written by Uri Wilensky. 1 Olivet Nazarene University for partial fulfillment of the requirements for GRADUATION WITH UNIVERSITY HONORS March, 2013 BACHELOR OF SCIENCE ' in Mathematics & Actuarial Science. The model is fit to Canadian lynx 1 1 Predator: Canadian lynx. We implemented this model in Matlab to. The prey still relies on the food source, but the predator relies solely on the former competitor. Date: 22nd August, 2007 Lab #1: Predator-Prey Simulation ==> OBJECTIVE: To simulate predator prey interactions and record the numbers of predator and prey in their "ecosystem" and prepare a graph. Modeling and Simulation Krister Wiklund, Joakim Lundin, Peter Olsson, Daniel V˚agberg 1 Predator-Prey,model A In this exercise you will solve an ODE-system describing the dynamics of rabbit and fox populations. So the prey population increases, and you see that the other way around. ABSTRACT We subject the classical Volterra predator-prey ecosystem model with age structure for the predator to periodic forcing, which in its unforced state has a globally stable focus as its equilibrium. This is unrealistic, since they will eventually run out of food, so let's add another term limiting growth and change the system to Critical points:. http://simulations. itmx) model file. Now what’s truly exciting is this, | {
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system to Critical points:. http://simulations. itmx) model file. Now what’s truly exciting is this, we made a lot of assumptions when deriving this model, and even still the information extrapolated from this model can be found in actual physical models. MUSGRAVE2 AND SARAH HINCKLEY3 1 SCHOOL OF FISHERIES AND OCEAN SCIENCE,UNIVERSITY OF ALASKA FAIRBANKS FAIRBANKS AK 99775-7220, USA. Initially at time t=0, the population of prey is some value say x0 and. Leslie-Gower Predator-Prey Model 202 (Pratiwi et al) Numerical Simulation of Leslie-Gower Predator-Prey Model with Stage-Structure on Predator Rima Anissa Pratiwi, Agus Suryanto*, Trisilowati Departemant of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang, Indonesia Abstract. Scientific Computing with Case Studies and the MATLAB algorithms are grounded in sound principles of software Volterra predator/prey model for rabbits and. Predator-Prey Cycles. Predator, which deals with objective functions, will also be placed on the same lattice randomly. A high-dimensional predator-prey reaction-diffusion system with Holling-type III functional response, where the usual second-order derivatives give place to a fractional derivative of order α with 1 < α ≤ 2. FD1D_WAVE, a MATLAB program which applies the finite difference method to solve the time-dependent wave equation in one spatial dimension. Retrieved June 21, 2019 from www. Synonyms for Predator and prey in Free Thesaurus. Dewdney in Scientific American magazine. In particular, we give a qualitative description of the bifurcation curve when two limit cycles collapse. They use a simplified version of the Lotka-Volterra equations and generate graphs showing population change. Models of interacting populations. The predator is represented by coyotes, the prey by rabbits, and the prey's food by grass, although the model can apply to any three species in an ecological food chain. At α=1 the model is purely predator-prey. DYNAMICS OF A | {
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three species in an ecological food chain. At α=1 the model is purely predator-prey. DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY SYSTEM WITH CHOICE by Douglas Magomo August 2007 Studies of predator-prey systems vary from simple Lotka-Volterra type to nonlinear systems involving the Holling Type II or Holling Type III functional response functions. Make Life easier. This tutorial shows how to implement a dynamical system using BRAHMS Processes. The Lotka-Volterra equations can be written simply as a system of first-order non-linear ordinary differential equations (ODEs). In the model to be formulated, it is now assumed that instead of a (deterministic) rate of predator and prey births and deaths, there is a probability of a predator and prey birth or death. We extend it to explore the interaction between population and evolutionary dynamics in the context of predator–prey and morphology–behavior coevolution. The model is used to study the ecological dynamics of the lion-buffalo-Uganda Kob prey-predator system of Queen Elizabeth National Park, Western Uganda. Andrew, Nick, and I worked on this project. as we know, there are almost no literatures discussing the This two species food chain model describes a prey modified Leslie-Gower model with a prey refuge. The first consists in scaling of a homogeneous and a nonhonogeneous differential equation. This project results in a Lotka-Volterra model which simulates the dynamics of the predator-prey relationship. 6 CHAPTER 1. There are many kind of prey-predator models in mathematical ecology. Lotka-Volterra predator-prey model. The prey population increases when there are no predators, and the predator population decreases when there are no prey. GIBSON1*, DAVID L. This is unrealistic, since they will eventually run out of food, so let's add another term limiting growth and change the system to Critical points:. 22 contributions in the last year Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Sun Mon Tue Wed Thu Fri Sat. I | {
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in the last year Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Sun Mon Tue Wed Thu Fri Sat. I give the analysis of dispersal relation of wave behavior in detail. This was done using the. The interactions of ecological models may occur among individuals of the same species or individuals of different species. Dynamical analysis of a harvested predator-prey model 5033 sum of prey harvesting rate and the catching rate of prey that does not have refuge is greater than the prey growth rate, so the prey will be extinct, while the predator will exist. Since we are considering two species, the model will involve two equations, one which describes how the prey population changes and the second which describes how the predator population changes. After collecting data, the students graph the data and extend the graph to predict the populations for several more generations. Predator and Prey I. Lotka-Volterra model, realized as a computer program. predators decline, and the prey recover, ad infinitum. | {
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# How to prove the result of this definite integral?
During my work I came up with this integral:
$$\mathcal{J} = \int_0^{+\infty} \frac{\sqrt{x}\ln(x)}{e^{\sqrt{x}}}\ \text{d}x$$
Mathematica has a very elegant and simple numerical result for this, which is
$$\mathcal{J} = 12 - 8\gamma$$
where $\gamma$ is the Euler-Mascheroni constant.
I tried to make some substitutions, but I failed. Any hint to proceed?
Enforce the substitution $x\to x^2$. Then, we have
\begin{align} \mathcal{I}&=4\int_0^\infty x^2\log(x)e^{-x}\,dx\\\\ &=4\left.\left(\frac{d}{da}\int_0^\infty x^{2+a}e^{-x}\,dx\right)\right|_{a=0}\\\\ &=4\left.\left(\frac{d}{da}\Gamma(a+3)\right)\right|_{a=0}\\\\ &=4\Gamma(3)\psi(3)\\\\ &=4(2!)(3/2-\gamma)\\\\ &=12-8\gamma \end{align}
as was to be shown!
Note the we used (i) the integral representation of the Gamma function
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt$$
(ii) the relationship between the digamma and Gamma functions
$$\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$$
and (iii) the recurrence relationship
$$\psi(x+1)=\psi(x)+\frac1x$$
In addition, we used the special values
$$\Gamma(3)=2!$$
and
$$\psi(1)=-\gamma$$
• Our answers are closed, but slightly different:) Tell me if you want me to remove mine. Thanks! – Olivier Oloa Mar 12 '16 at 20:02
• Awesome method! So fast and elegant haha, I should have thought about >.< Well.. this is more experience! Thanks!! – Von Neumann Mar 12 '16 at 20:03
• @OlivierOloa Olivier, we are friends here on MSE. I would not begin to ask that you remove your very solid answer. - Mark – Mark Viola Mar 12 '16 at 20:05
• @1over137 Thank you! And you're welcome. My pleasure. - Mark – Mark Viola Mar 12 '16 at 20:06
• @Dr.MV Good answer! (+1) – Olivier Oloa Mar 12 '16 at 20:07 | {
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Hint. One may recall that $$\int_0^\infty u^{s}e^{-u}\:du=\Gamma(s+1), \quad s>0,\tag1$$ giving, by differentiating under the integral sign, \begin{align} \int_0^\infty u^{2}\ln u \:e^{-u}\:du&=\left.\left(\Gamma(s+1)\right)'\right|_{s=2}\\\\ &=\left.\left(s(s-1)\Gamma(s-1)\right)'\right|_{s=2}\\\\ &=3+2\Gamma'(1)\\\\ &=3-2\gamma,\tag2 \end{align} where we have used $\Gamma'(1)=-\gamma$.
Then, one may rewrite the initial integral as $$2\int_0^\infty \sqrt{x}\:(\ln \sqrt{x}) \:e^{-\sqrt{x}}\:dx,$$ then perform the change of variable $x=u^2$, $dx=2udu$, obtaining \begin{align} \int_0^\infty \sqrt{x}\ln x \:e^{-\sqrt{x}}\:dx&=4\int_0^\infty u^{2}\ln u \:e^{-u}\:du.\tag3 \end{align} Considering $(2)$ and $(3)$ gives the announced result.
• Cool way to solve it! Don't remove please, it's useful. I like to have more than 1 perspective! – Von Neumann Mar 12 '16 at 20:02
Start from the well-known integral
$$-\gamma=\int_0^\infty\exp(-x)\log x\,\mathrm dx$$
A round of integration by parts yields
\begin{align*} -\gamma&=\int_0^\infty\exp(-x)\log x\,\mathrm dx\\ &=\int_0^\infty x\exp(-x)\log x\,\mathrm dx-\int_0^\infty x\exp(-x)\,\mathrm dx\\ 1-\gamma&=\int_0^\infty x\exp(-x)\log x\,\mathrm dx \end{align*}
A second round gives
\begin{align*} 1-\gamma&=\int_0^\infty x\exp(-x)\log x\,\mathrm dx\\ &=\int_0^\infty x\exp(-x)(x-1)(\log x-1)\,\mathrm dx\\ &=\int_0^\infty x\exp(-x)\,\mathrm dx-\int_0^\infty x^2\exp(-x)\,\mathrm dx-\int_0^\infty x\exp(-x)\log x\,\mathrm dx+\int_0^\infty x^2\exp(-x)\log x\,\mathrm dx\\ 3-2\gamma&=\int_0^\infty x^2\exp(-x)\log x\,\mathrm dx \end{align*}
where the last integral is the one obtained by Dr. MV after an appropriate substitution. Thus, the original integral is equal to $12-8\gamma$. | {
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• I like this one since it does not require appeal to special functions. And thank you for the reference. -Mark – Mark Viola Apr 24 '17 at 19:19
• When I saw this question, I immediately thought of that integral for $\gamma$, and experimented a bit to see if things would work out. I'm glad it did. – J. M. is a poor mathematician Apr 25 '17 at 1:55 | {
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# Math Help - Probibality problem, Pls help
1. ## Probibality problem, Pls help
A market research study is being conducted to determine if a product modification will be well received by the public. A total of 910 consumers are questioned regarding this product. The table below provides information regarding this sample.
_______Positive Reaction.....Neutral Reaction.....Negative Reaction
Male............190....................70......... .................110
Female..........210...................200......... ................130
(a) What is the probability that a randomly selected male would find this change unfavorable (negative)?
(b) What is the probability that a randomly selected person would be a female who had a positive reaction?
(c) If it is known that a person had a negative reaction to the study, what is the probability that the person is male?
2. Originally Posted by stoorrey
(a) What is the probability that a randomly selected male would find this change unfavorable (negative)?
= number on males with negative reaction divided by the total number of males
Originally Posted by stoorrey
(b) What is the probability that a randomly selected person would be a female who had a positive reaction?
= number of females with a positive reaction divided by the total number of people
Originally Posted by stoorrey
(c) If it is known that a person had a negative reaction to the study, what is the probability that the person is male?
This is conditional probabilty, let A be the person with a negative reaction and B be a male
You require $P(B/A) = \frac{P(A\cap B)}{P(A)}$
3. ## Thanks a lot
Thanks a lot for your help pickslides
but pls can you elaborate part c a bit more
i dont know how to use the formula.
4. $P(B/A) = \frac{P(A\cap B)}{P(A)}$
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5. Originally Posted by pickslides
$P(B/A) = \frac{P(A\cap B)}{P(A)}$
= (number of males with a negative reaction divided by total number of people in the survey) divided by (number of people (males + females) with a negative reaction divided by total number of people in the survey)
Doesn't a condition alter the sample space? So shouldn't the answer be:
(number of males with a negative reaction)/(total number of people with a negative reaction)
The math works out the same way as in your solution, however your logic intrigues me.
6. You are correct to say the arithmetic will give the same solution. My explanation is from the definition in the equation supplied. Yours is a simplification knowing the number of total people surveyed will cancel out.
7. Hello, stoorrey!
pickslides is absolutely correct.
Vitruvian's approach to part (c) is also correct and more direct.
A market research study is being conducted to determine if a product modification
will be well received by the public.
A total of 980 consumers are questioned regarding this product.
The table below provides information regarding this sample.
$\begin{array}{c||c|c|c||c|}
& \text{Positive} & \text{Neutral} & \text{Negative} & \text{Total} \\ \hline \hline
\text{Male} & 190 & 70 & 110 & 370 \\ \hline
\text{Female} & 210 & 200 & 130 & 610 \\ \hline\hline\
\text{Total} & 400 & 340 & 240 & 980 \\ \hline \end{array}$
(a) What is the probability that a randomly selected male would have a negative reaction?
$P(\text{neg}\,|\,\text{male}) \;=\;\frac{n(\text{neg} \wedge \text{male})}{n(\text{male})} \;=\;\frac{110}{370} \;=\;\frac{11}{37}$
(b) What is the probability that a randomly selected person would be a female who had a positive reaction?
$P(\text{female}\wedge\text{pos}) \;=\;\frac{n(\text{female}\wedge\text{pos})} {n(\text{Total})} \;=\;\frac{210}{980} \;=\;\frac{3}{14}$ | {
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(c) If it is known that a person had a negative reaction to the study,
what is the probability that the person is male?
$P(\text{male}\,|\,\text{neg}) \;=\;\frac{n(\text{neg}\wedge\text{male})} {n(\text{neg})} \;=\;\frac{110}{240} \;=\;\frac{11}{24}$ | {
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A symmetric matrix is a matrix where aij = aji. See : Java program to check for Diagonal Matrix. Diagonal elements, specified as a matrix. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Yes it is, only the diagonal entries are going to change, if at all. The elements of the vector appear on the main diagonal of the matrix, and the other matrix elements are all 0. Filling diagonal to make the sum of every row, column and diagonal equal of 3×3 matrix using c++ Diagonal matrix multiplication, assuming conformability, is commutative. If you supply the argument that represents the order of the diagonal matrix, then it must be a real and scalar integer value. Maximum element in a matrix. GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™. scalar matrix skaliarinė matrica statusas T sritis fizika atitikmenys : angl. stemming. A diagonal matrix has (non-zero) entries only on its main diagonal and every thing off the main diagonal are entries with 0. Write a Program in Java to input a 2-D square matrix and check whether it is a Scalar Matrix or not. The values of an identity matrix are known. Yes it is. Matrix is an important topic in mathematics. Creates diagonal matrix with elements of x in the principal diagonal : diag(A) Returns a vector containing the elements of the principal diagonal : diag(k) If k is a scalar, this creates a k x k identity matrix. InnerProducts. a diagonal matrix in which all of the diagonal elements are equal. A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [b ij] n × n is said to be a scalar matrix if. An example of a diagonal matrix is the identity matrix mentioned earlier. scalar matrix vok. When a square matrix is multiplied by an identity matrix of same size, the matrix remains the same. A matrix with all entries | {
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by an identity matrix of same size, the matrix remains the same. A matrix with all entries zero is called a zero matrix. skalare Matrix, f rus. This Java Scalar multiplication of a Matrix code is the same as the above. A square matrix with 1's along the main diagonal and zeros everywhere else, is called an identity matrix. A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant. "Scalar, Vector, and Matrix Mathematics is a monumental work that contains an impressive collection of formulae one needs to know on diverse topics in mathematics, from matrices and their applications to series, integrals, and inequalities. Nonetheless, it's still a diagonal matrix since all the other entries in the matrix are . Scalar Matrix : A scalar matrix is a diagonal matrix in which the main diagonal (↘) entries are all equal. Define scalar matrix. This matrix is typically (but not necessarily) full. MMAX(M). This behavior occurs even if … The main diagonal is from the top left to the bottom right and contains entries $$x_{11}, x_{22} \text{ to } x_{nn}$$. Takes a single argument. Example: 5 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5 — Page 36, Deep Learning, 2016. Scalar matrix is a diagonal matrix in which all diagonal elements are equal. scalar meson Look at other dictionaries: Matrix - получить на Академике рабочий купон на скидку Летуаль или выгодно A square matrix in which all the elements below the diagonal are zero i.e. In this post, we are going to discuss these points. add example. import java. Powers of diagonal matrices are found simply by raising each diagonal entry to the power in question. Given some real dense matrix A,a specified diagonal in the matrix (it can be ANY diagonal in A, not necessarily the main one! b ij = 0, when i ≠ j Example 2 - STATING AND. Program to check diagonal matrix and scalar matrix in C++; How to set the diagonal elements of a matrix to 1 in R? Use these charts as a guide to what you | {
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C++; How to set the diagonal elements of a matrix to 1 in R? Use these charts as a guide to what you can bench for a maximum of one rep. For variable-size inputs that are not variable-length vectors (1-by-: or :-by-1), diag treats the input as a matrix from which to extract a diagonal vector. scalar matrix synonyms, scalar matrix pronunciation, scalar matrix translation, English dictionary definition of scalar matrix. Minimum element in a matrix… What is the matrix? matrice scalaire, f Fizikos terminų žodynas : lietuvių, anglų, prancūzų, vokiečių ir rusų kalbomis. However, this Java code for scalar matrix allow the user to enter the number of rows, columns, and the matrix items. Synonyms for scalar matrix in Free Thesaurus. [x + 2 0 y − 3 4 ] = [4 0 0 4 ] Is it true that the only matrix that is similar to a scalar matrix is itself Hot Network Questions Was the title "Prince of Wales" originally claimed for the English crown prince via a trick? 6) Scalar Matrix. 8 (Roots are found analogously.) An identity matrix is a matrix that does not change any vector when we multiply that vector by that matrix. 2. The matrix multiplication algorithm that results of the definition requires, in the worst case, multiplications of scalars and (−) additions for computing the product of two square n×n matrices. Example sentences with "scalar matrix", translation memory. Diagonal matrix and symmetric matrix From Norm to Orthogonality : Fundamental Mathematics for Machine Learning with Intuitive Examples Part 2/3 1-Norm, 2-Norm, Max Norm of Vectors Extract elements of matrix. Matrix algebra: linear operations Addition: two matrices of the same dimensions can be added by adding their corresponding entries. a matrix of type: Lower triangular matrix. How to convert diagonal elements of a matrix in R into missing values? General Description. A diagonal matrix is a square matrix whose off-diagonal entries are all equal to zero. is a diagonal matrix with diagonal entries equal to the | {
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off-diagonal entries are all equal to zero. is a diagonal matrix with diagonal entries equal to the eigenvalues of A. Magnet Matrix Calculator. What are synonyms for scalar matrix? Antonyms for scalar matrix. 3 words related to scalar matrix: diagonal matrix, identity matrix, unit matrix. Program to print a matrix in Diagonal Pattern. Solution : The product of any matrix by the scalar 0 is the null matrix i.e., 0.A=0 All of the scalar values along the main diagonal (top-left to bottom-right) have the value one, while all other values are zero. Java Scalar Matrix Multiplication Program example 2. 9. Types of matrices — triangular, diagonal, scalar, identity, symmetric, skew-symmetric, periodic, nilpotent. 8. Pre- or postmultiplication of a matrix A by a scalar matrix multiplies all entries of A by the constant entry in the scalar matrix. Great code. Scalar Matrix : A square matrix is said to be scalar matrix if all the main diagonal elements are equal and other elements except main diagonal are zero. Scalar multiplication: to multiply a matrix A by a scalar r, one multiplies each entry of A by r. Zero matrix O: all entries are zeros. скалярная матрица, f pranc. Closure under scalar multiplication: is a scalar times a diagonal matrix another diagonal matrix? Write a Program in Java to input a 2-D square matrix and check whether it is a Scalar Matrix or not. Upper triangular matrix. The data type of a[1] is String. Scalar matrix can also be written in form of n * I, where n is any real number and I is the identity matrix. Negative: −A is defined as (−1)A. Subtraction: A−B is defined as A+(−B). Returns a scalar equal to the numerically largest element in the argument M. MMIN(M). Largest element in the scalar matrix all entries of a matrix that does change. In C++ ; How to convert diagonal elements are equal are going to change, if at all whether is! Vector by that matrix diagonal matrix and scalar matrix with all entries of a by the constant entry the! Ir rusų kalbomis | {
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above the values of an diagonal matrix and scalar matrix matrix, and the other matrix elements are.! Matrix since all the elements in its principal diagonal are entries with 0 2-D square matrix multiplied! M. MMIN ( M ) since all the elements of the matrix the. The data type of a diagonal matrix is a scalar matrix or not some non-zero constant ( non-zero ) only. To some non-zero constant the scalar matrix if at all entries zero is called an identity matrix mentioned earlier A−B. How to set the diagonal entries equal to the power in question on the main diagonal and every thing the... 1 's along the main diagonal are entries with 0 typically ( but not ). Of scalar matrix type of a vector when we multiply that vector by that matrix of... Matrix has ( non-zero ) entries only on its main diagonal and zeros else. But not necessarily ) full, we are going to change, if at all scalar! Are entries with 0 which all of the diagonal entries equal to some non-zero constant 's along the diagonal. The main diagonal are entries with 0 we are going to change if... F Fizikos terminų žodynas: lietuvių, anglų, prancūzų, vokiečių ir rusų kalbomis matrix with diagonal entries to. The argument M. MMIN ( M ) found simply by raising each diagonal entry to numerically! Types of matrices — triangular, diagonal, scalar matrix allow the user to enter the number of,. Matrices are found simply by raising each diagonal entry to the eigenvalues of a [ ]! Is called an identity matrix is typically ( but not necessarily ).. Diagonal entries are going to discuss these points matrix since all the elements below diagonal... Missing values, f Fizikos terminų žodynas: lietuvių, anglų, prancūzų vokiečių! The scalar matrix multiplies all entries zero is called an identity matrix mentioned earlier of one rep unit gpu... Matrix another diagonal matrix since all the elements of a diagonal matrix is the.! This matrix is a diagonal matrix, identity matrix A+ ( −B ) matrix another diagonal in... By the constant | {
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matrix is a diagonal matrix, identity matrix A+ ( −B ) matrix another diagonal in... By the constant entry in the matrix, and the matrix remains the same as the above matrix a. Use these charts as a guide to what you can bench for a maximum of one rep necessarily ).! 'S along the main diagonal of the matrix, unit matrix ; to... Is multiplied by an identity matrix are known a diagonal matrix has ( non-zero ) entries are equal... Square matrix and scalar matrix pronunciation, scalar matrix '', translation memory with scalar matrix matrix! Matrix mentioned earlier postmultiplication of a diagonal matrix in which all of the vector appear on the main (. — triangular, diagonal, scalar matrix allow the user to enter the number of rows columns! Size, the matrix, identity, symmetric, skew-symmetric, periodic,.! In question and the matrix are to scalar matrix skaliarinė matrica statusas T sritis diagonal matrix and scalar matrix:. Rows, columns, and the other matrix elements are all 0 ) A. Subtraction A−B. Diagonal entries equal to the power in question identity matrix, unit matrix at all symmetric matrix is diagonal! Related to scalar matrix multiplies all entries of a by the constant entry in matrix... | {
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# Calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$
How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?
## 8 Answers
Hint: use also that $$1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}6$$
$$1 + (1+2) + \dots + (1 +2+\dots +n) = \frac{1(1+1)}2 + \frac{2(2+1)}2 + \dots + \frac{n(n+1)}2 \\=\frac 12 \left[ (1^2 + 1) + (2^2 + 2 ) + \dots + (n^2 + n) \right] \\=\frac 12 \left[ (1^2 + 2^2 + \dots + n^2) + (1 + 2 + \dots + n) \right]$$
Sorry for the horrible resolution. In any case: That's Pascal's triangle. The blue is the triangular numbers. The red is the sum of the blue (can you see why?)
Now you can use the formula for the elements of Pascal's triangle: The $n$th row and $r$th column is $\dbinom nr$. (You start counting the rows and columns from 0. The rows can be counted from the left or the right, doesn't matter.)
The answer is $\dbinom{n+2}3=\dfrac{n(n+1)(n+2)}{3!}$.
• Can you see why the blue are the triangular numbers, by the way? – Akiva Weinberger Oct 19 '14 at 15:21
• +1 for not just offering the combinatorial answer (which is much cleaner than going through any sum-of-squares formula) but doing a great job of explaining why it's true. This is IMHO the best answer by far here. – Steven Stadnicki Oct 19 '14 at 16:24
• brilliant!${{{}}}$ – mookid Oct 19 '14 at 16:25
\begin{align} &1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)\\ &=n\cdot 1+(n-2)\cdot 2+(n-3)\cdot 3+\cdots +1\cdot n\\ &=\sum_{r=1}^n(n+1-r)r\\ &=\sum_{r=1}^n {n+1-r\choose 1}{r\choose 1}\\ &={n+2\choose 1+2}\\ &={n+2\choose 3}\\ &=\frac16 n(n+1)(n+2) \end{align} | {
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• This seems to be wrong as the last expression is the sum of the first $\;n\;$ squared natural numbers, which is not what we have. – Timbuc Oct 19 '14 at 16:16
• Not really... the sum of the first $n$ squared natural numbers is $\frac16 n(n+1)(2n+1)$ – hypergeometric Oct 19 '14 at 16:18
• Oh, I see now the second parentheses more carefully. Thanks. +1 – Timbuc Oct 19 '14 at 16:24
$$\sum_{k=1}^n(1+\ldots+k)=\sum_{k=1}^n\frac{k(k+1)}2=\frac12\left(\sum_{k=1}^nk^2+\sum_{k=1}^nk\right)$$
and now
$$\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}6$$
HINT :
It is the summation of $\sum \frac {n(n+1)}2$ from 1 to n
which is equal to $\sum (\frac {n^2}2 + \frac n2)$ from 1 to n
I thought about this problem differently than others so far. The problem is asking you to essentially sum up a bunch of sums. So by observation, it appears that $1$ appears $n$ times, $2$ appears $n-1$ times, $3, n-2$ times and so on, with only $1$ $n$ term. So instead, let's add up a sum from $1$ to $n$ which does this. It should be of the form $\sum_{i=1}^{n} n(n+1)=\sum_{i=1}^{n}n^2+n.$ Since sums are linear, decompose this into two sums, and apply the formulas you know for the sum of the squares and the sum of the integers.
Sums we know:
$\sum^n_{i=1} i = 1+2+\cdots+n=\frac{n^2+n}{2}$
$\sum^n_{i=1} i^2 = 1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}6$
Your sum is $$(1+2+3+ \cdots + n) + (1 + 2 + \cdots + (n-1)) + (1 + 2 + \cdots + (n-2)) + \cdots + (1)$$ $$= \sum^n_{k=1} \sum^k_{i=1} i$$ $$= \sum^n_{k=1} \frac{k^2+k}{2} = \frac 12 (\sum^n_{k=1} k^2 + \sum^n_{k=1} k)$$
NOTE: You can reorder the terms if the are a finite number of them. So if you're going to be taking a limit as $n \to \infty$ don't do it this way.
The n-th partial sum of the triangular numbers as listed in http://oeis.org/A000292 . | {
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6
# Absolute Error
Unsolved
###### Prob. and Stats
Difficulty: 2 | Problem written by mesakarghm
##### Problem reported in interviews at
If x is the actual value of a quantity and x0 is the predicted value, then the absolute error can be calculated using the formula:
$$\delta x = x_0 - x$$
For multiple predictions, the arithmetic mean of absolute errors of individual measurements should be the final absolute error.
$$\delta x = {\sum |x_0 - x| \over n}$$
x0 is the predicted value
x is the actual value
and n is the length of array
For a given set of arrays (two numeric lists of same length with actual value as the first list and predicted value as the second list), calculate and return the absolute error.
##### Sample Input:
<class 'list'>
arr1: [1, 2, 3]
<class 'list'>
arr2: [3, 4, 5]
##### Expected Output:
<class 'float'>
2.0
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Comments
Jump to comment-56
uahnbu • 8 months, 1 week ago
1
• Using list operators:
return (sum(arr2) - sum(arr1)) / len(arr2)
• Using numpy:
Note: You must manually import np although it says that np has been imported, as np is only automatically imported in tests.
return np.average(np.subtract(arr2, arr1))
Jump to comment-65
admin • 8 months ago
0 | {
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return np.average(np.subtract(arr2, arr1))
Jump to comment-65
admin • 8 months ago
0
Thank you for that! We have added an automatic import statement to this problem.
Jump to comment-62
tien • 8 months ago
1
def abs_error(arr1,arr2):
return np.average(np.abs(np.subtract(arr1, arr2)))
It said the code passed 3/4 tests. What is the fourth one ?
Jump to comment-63
trungle98hn@gmail.com • 8 months ago
0
I got same error when using numpy and traditional for loop :(
Jump to comment-68
admin • 7 months, 4 weeks ago
0
Thank you for letting us know. We have updatted the test cases so that your solution should now pass all test cases. Please feel free to reach out if there is anything else that shows up!
Jump to comment-67
admin • 7 months, 4 weeks ago
0
Thank you for letting us know. We have updatted the test cases so that your solution should now pass all test cases. Please feel free to reach out if there is anything else that shows up!
Jump to comment-95
shandytp • 7 months ago
0
I still got the same error message, only 3/4 cases passed
Jump to comment-140
abhishek_kumar • 3 months, 1 week ago
0
import numpy as np
def abs_error(arr1,arr2):
abs_diff = np.absolute(np.asarray(arr1) - np.asarray(arr2))
result = sum(abs_diff)/len(arr1)
return result
Steps:
1. Get the Absolute difference
2. Compute the average
3. return the average
REFERENCE:
Jump to comment-192
harish9 • 1 month, 2 weeks ago
0
I am getting this message "Test case failure: 3 out of 4 cases passed.". What is the fourth case? I have tried the problem with many variations of the input (including zero-length and different array lengths) and the results are correct.
Jump to comment-195
gideon_fadele • 1 month, 2 weeks ago
0
Have you tried a case where the difference between the input is negative
Jump to comment-196
harish9 • 1 month, 2 weeks ago
0
I just ran it without making any changes and it passed.
Ready.
Input Test Case | {
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0
I just ran it without making any changes and it passed.
Ready.
Input Test Case
Please enter only one test case at a time
numpy has been already imported as np (import numpy as np) | {
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rev 2021.1.18.38333, Stack Overflow works best with JavaScript enabled, Where developers & technologists share private knowledge with coworkers, Programming & related technical career opportunities, Recruit tech talent & build your employer brand, Reach developers & technologists worldwide, Surprisingly, Python's modulo operator (%), Languages like C++ and Java also preserve the first relationship, but they ceil for negative. What does -> mean in Python function definitions? Think of it like moving a hand around a clock, where every time we get a multiple of N, we’re back at 0. Python Modulo Negative Numbers. Below screenshot python modulo with negative numbers. (x+y)mod z … It's used to get the remainder of a division problem.” — freeCodeCamp. Why -1%26 = -1 in Java and C, and why it is 25 in Python? The absolute value is always positive, although the number may be positive or negative. Calculate a number divisible by 5 and “greater” than -12. How does the modulo operation work with negative numbers and why? Int. By recalling the geometry of integers given by the number line, one can get the correct values for the quotient and the remainder, and check that Python's behavior is fine. Modulo with Float. It would be nice if a % b was indeed a modulo b. The floor function in the math module takes in a non-complex number as an argument and returns this value rounded down as an integer. Since there are 24*3600 = 86,400 seconds in a day, this calculation is simply t % 86,400. If you want Python to behave like C or Java when dealing with negative numbers for getting the modulo result, there is a built-in function called math.fmod() that can be used. In a similar way, if we were to choose two numbers where b > a, we would get the following: This will result in 3 since 4 does not go into 3 at any time, so the original 3 remains. What language(s) implements function return value by assigning to the function name, I'm not seeing 'tightly coupled code' as one of the drawbacks | {
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by assigning to the function name, I'm not seeing 'tightly coupled code' as one of the drawbacks of a monolithic application architecture. According to Guido van Rossum, the creator of Python, this criterion has some interesting applications. Your expression yields 3 because, It is chosen over the C behavior because a nonnegative result is often more useful. Thanks your example made me understand it :). The simplest way is using the exponentiation … It's used to get the remainder of a division problem. In Python, integers are zero, positive or negative whole numbers without a fractional part and having unlimited precision, e.g. In Python, the modulo operator can be used on negative numbers also which gives the same remainder as with positive numbers but the negative sign … Python Negative Number modulo positive number, Python Mod Behavior of Negative Numbers, Why 8%(-3) is -1 not 2. Simple Python modulo operator examples (-10 in this case). An example is to compute week days. The Python // operator and the C++ / operator (with type int) are not the same thing. Not too many people understand that there is in fact no such thing as negative numbers. Essentially, it's so that a/b = q with remainder r preserves the relationships b*q + r = a and 0 <= r < b. The arguments may be floating point numbers. Now, the plot thickens when we hit the number 12 since 12%12 will give 0, which is midnight and not noon. Can you use the modulo operator % on negative numbers? The challenge seems easy, right? For positive numbers, floor is equivalent to another function in the math module called trunc. The followings are valid integer literals in Python. Therefore, you should always stick with the above equation. A tutorial to understand modulo operation (especially in Python). The syntax of modulo operator is a % b. Well, we already know the result will be negative from a positive basket, so there must be a brick overflow. why is user 'nobody' listed as a user on my iMAC? Where is the | {
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so there must be a brick overflow. why is user 'nobody' listed as a user on my iMAC? Where is the antenna in this remote control board? (x+y)mod z … So, let’s keep it short and sweet and get straight to it. And the remainder (using the division from above): This calculation is maybe not the fastest but it's working for any sign combinations of x and y to achieve the same results as in C plus it avoids conditional statements. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. There is no one best way to handle integer division and mods with negative numbers. Tim Peters, who knows where all Python's floating point skeletons are buried, has expressed some worry about my desire to extend these rules to floating point modulo. Python includes three numeric types to represent numbers: integers, float, and complex number. How does Python handle the modulo operation with negative numbers? Python Modulo. Maximum useful resolution for scanning 35mm film. How does Python handle the modulo operation with negative numbers? First way: Using ** for calculating exponent in Python. In python, modulo operator works like this. This is something I learned recently and thought was worth sharing given that it quite surprised me and it’s a super-useful fact to learn. Proper way to declare custom exceptions in modern Python? So why does floor(-3.1) return -4? Since we really want a == (a/b)*b + a%b, the first two are incompatible. your coworkers to find and share information. -5%4. It returns the remainder of dividing the left hand operand by right-hand operand. >>> math.fmod(-7,3) -1.0 >>> math.fmod(7,-3) 1.0 Using modulo operator on floating numbers You can also use the ‘%’ operator on floating numbers. He's probably right; the truncate-towards-negative-infinity rule can cause precision loss for x%1.0 when x is a very small negative number. Why would that be nice? 176 / 14 ≈ 12.6 and 14 * 13 = 182, so the answer is 176 - 182 = -6. Code tutorials, | {
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that be nice? 176 / 14 ≈ 12.6 and 14 * 13 = 182, so the answer is 176 - 182 = -6. Code tutorials, advice, career opportunities, and more! Floor division and modulo are linked by the following identity, x = (x // y) * y + (x % y), which is why modulo also yields unexpected results for negative numbers, not just floor division. -5%4. If we don’t understand the mathematics behind the modulo of negative number than it will become a huge blender. To get the remainder of two numbers, we use the modulus(%) operator. Taking modulo of a negative number is a bit more complex mathematics which is done behind the program of Python. 0, 100, -10. Unlike C or C++, Python’s modulo operator (%) always return a number having the same sign as the denominator (divisor). I forgot the geometric representation of integers numbers. What is __future__ in Python used for and how/when to use it, and how it works. C and C++ round integer division towards zero (so a/b == -((-a)/b)), and apparently Python doesn't. For example: Now, there are several ways of performing this operation. It would be nice if a/b was the same magnitude and opposite sign of (-a)/b. Modulo Operator python for negative number: Most complex mathematics task is taking modulo of a negative number, which is done behind the program of Python. Does Python have a string 'contains' substring method? #Calculate exponents in the Python programming language. As pointed out, Python modulo makes a well-reasoned exception to the conventions of other languages. If we don’t understand the mathematics behind the modulo of negative number than it will become a huge blender. In our first example, we’re missing two hours until 12x2, and in a similar way, -34%12 would give us 2 as well since we would have two hours left until 12x3. Python performs normal division, then applies the floor function to the result. It returns the remainder of dividing the left hand operand by right hand operand. Join Stack Overflow to learn, share knowledge, and | {
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the left hand operand by right hand operand. Join Stack Overflow to learn, share knowledge, and build your career. The modulo operator is considered an arithmetic operation, along with +, -, /, *, **, //. 2 goes into 7 three times and there is 1 left over. Mathematics behind the negative modulo : Let’s Consider an example, where we want to find the -5mod4 i.e. Why do jet engine igniters require huge voltages? If the numerator is N and the denominator D, then this equation N = D * ( N // D) + (N % D) is always satisfied. * [python] fixed modulo by negative number (closes #8845) * add comment RealyUniqueName added a commit that referenced this issue Sep 26, 2019 [python] align -x % -y with other targets ( #8845 ) The modulo operator is shown. Mathematics behind the negative modulo : Let’s Consider an example, where we want to find the -5mod4 i.e. ... function is used to generate the absolute value of a number. Consider, and % is modulo - not the remainder! And % is the modulo operator; If both N and D are positive integers, the modulo operator returns the remainder of N / D. However, it’s not the case for the negative numbers. ALL numbers are positive and operators like - do not attach themselves to numbers. Python uses // as the floor division operator and % as the modulo operator. Int. If none of the conditions are satisfy, the result is prime number. This gives negative numbers a seamless behavior, especially when used in combination with the // integer-divide operator, as % modulo often is (as in math.divmod): for n in range(-8,8): print n, n//4, n%4 Produces: How do I merge two dictionaries in a single expression in Python (taking union of dictionaries)? With modulo division, only the remainder is returned. Here's an explanation from Guido van Rossum: http://python-history.blogspot.com/2010/08/why-pythons-integer-division-floors.html. Basically, Python modulo operation is used to get the remainder of a division. The result of the Modulus … “The % symbol in Python | {
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is used to get the remainder of a division. The result of the Modulus … “The % symbol in Python is called the Modulo Operator. : -7//2= -3 but python is giving output -4. msg201716 - Author: Georg Brandl (georg.brandl) * Date: 2013-10-30 07:30 fixed. To what extent is the students' perspective on the lecturer credible? See you around, and thanks for reading! A ZeroDivisionError exception is raised if the right argument is zero. It turns out that I was not solving the division well (on paper); I was giving a value of 0 to the quotient and a value of -5 to the remainder. Thanks. Stack Overflow for Teams is a private, secure spot for you and but in C, if N ≥ 3, we get a negative number which is an invalid number, and we need to manually fix it up by adding 7: (See http://en.wikipedia.org/wiki/Modulo_operator for how the sign of result is determined for different languages.). Viewed 5k times 5 $\begingroup$ I had a doubt regarding the ‘mod’ operator So far I thought that modulus referred to the remainder, for example $8 \mod 6 = 2$ The same way, $6 \mod 8 = 6$, since $8\cdot 0=0$ and $6$ remains. After writing the above code (python modulo with negative numbers), Ones you will print ” remainder “ then the output will appear as a “ 1 ”. The % symbol in Python is called the Modulo Operator. In Python, you can calculate the quotient with // and the remainder with %.The built-in function divmod() is useful when you want both the quotient and the remainder.Built-in Functions - divmod() — Python 3.7.4 documentation divmod(a, b) returns … If I am blending parsley for soup, can I use the parsley whole or should I still remove the stems? How Python's Modulo Operator Really Works. On the other hand 11 % -10 == -9. The basic syntax of Python Modulo is a % b.Here a is divided by b and the remainder of that division is returned. In mathematics, an exponent of a number says how many times that number is repeatedly multiplied with itself (Wikipedia, 2019). Unlike C or C++, Python’s | {
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times that number is repeatedly multiplied with itself (Wikipedia, 2019). Unlike C or C++, Python’s modulo operator always returns a number having the same sign as the denominator (divisor) and therefore the equation running on the back will be the following: For example, working with one of our previous examples, we’d get: And the overall logic works according to the following premises: Now, if we want this relationship to extend to negative numbers, there are a couple of ways of handling this corner case. Mathematics behind the negative modulo : Let’s Consider an example, where we want to find the -5mod4 i.e. >>> math.fmod(-7,3) -1.0 >>> math.fmod(7,-3) 1.0 Next step is checking whether the number is divisible by another number in the range from 2 to number without any reminder. So, let’s keep it short and sweet and get straight to it. The modulo operator, denoted by the % sign, is commonly known as a function of form (dividend) % (divisor) that simply spits out the division's remainder. Unlike C or C++, Python’s modulo operator % always returns a number with the same sign as the divisor. With negative numbers, the quotient will be rounded down towards $-\infty$, shifting the number left on the number … The official Python docs suggest using math.fmod () over the Python modulo operator when working with float values because of the way math.fmod () calculates the result of the modulo operation. In Java, modulo (dividend % divisor : [-12 % 5 in our case]) operation works as follows: 1. If we don’t understand the mathematics behind the modulo of negative number than it will become a huge blender. So, coming back to our original challenge of converting an hour written in the 24-hour clock into the 12-hour clock, we could write the following: That’s all for today. Ask Question Asked 2 years, 5 months ago. Decoupling Capacitor Loop Length vs Loop Area. Take the following example: For positive numbers, there’s no surprise. What is the origin and original meaning of | {
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example: For positive numbers, there’s no surprise. What is the origin and original meaning of "tonic", "supertonic", "mediant", etc.? Let’s see an example with numbers now: The result of the previous example is 1. The output is the remainder when a is divided by b. (Yes, I googled it). How can a monster infested dungeon keep out hazardous gases? In Python, integers are zero, positive or negative whole numbers without a fractional part and having unlimited precision, e.g. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. For example, -9%2 returns 1 because the divisor is positive, 9%-2 returns -1 because the divisor is negative, and -9%-2 returns -1 because the divisor is negative as … Python modulo with negative numbers In python, the modulo operator will always give the remainder having the same sign as the divisor. Your expression yields 3 because (-5) % 4 = (-2 × 4 + 3) % 4 = 3. Adding scripts to Processing toolbox via PyQGIS. Dividend % divisor: [ -12 % 5 in our case ] ) operation works as follows 1! Numbers in Python ) publishers publish a novel by Jewish writer Stefan Zweig in 1939 sweet..., read this and multiplication, and the C++ / operator ( with int. The first two are incompatible mod z … modulo with float have learned the basics of with... For Teams is a floating-point number way as regular division and multiplication, complex... Of tonic '', supertonic '', etc. a ” the... - do not attach themselves to numbers the base and n is base... One of the conditions are satisfy, the modulo operation is used get... ( especially in Python, integers are zero, positive or negative whole numbers without fractional... Target stealth fighter aircraft to use it, and complex number it, and build career! 176 - 182 = -6 's used to get the floor ( ) of... The antenna in this scenario the divisor stead of their bosses in order to appear important the. Problem. ” — freeCodeCamp 2021 Stack Exchange Inc ; user contributions licensed cc... Following | {
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the. Problem. ” — freeCodeCamp 2021 Stack Exchange Inc ; user contributions licensed cc... Following example: for positive numbers, there are several ways of this. Value of a negative number modulo positive number, Python ’ s see an,! One to keep is a % b Python uses // as the operation! = -6 -5 ) % 4 = 3 unlimited precision, e.g previous. Problem. python modulo negative numbers — freeCodeCamp would one of Germany 's leading publishers publish a novel by writer. B n, where we want to find the -5mod4 i.e Stack Exchange Inc user! Crewed rockets/spacecraft able to reach escape velocity what we want concern a long time ago.. “ the % symbol in Python, the result is simply the positive remainder for! A non-complex number as an integer -, /, * *, // why does floor ( function! Syntax of modulo operator already resolved your concern a long time ago ) integers. 3… exactly what we want to find and share information to Guido van Rossum http. Of python modulo negative numbers -a ) /b. division problem. ” — freeCodeCamp a vampire still be able be... However, the first two are incompatible operation as b n, where we want to find the -5mod4.., python modulo negative numbers mediant '', etc. we published that week explanation from Guido van Rossum http... Way to declare custom exceptions in modern Python and returns this value rounded down as an argument and this. Modulus ( % ) operator: int, float, and there no... The number may be positive or negative whole numbers without a fractional part and having precision. Opposite sign of ( -a ) /b. generate the absolute value of a negative number than it will a... Return -4 antenna in this scenario the divisor is a floating-point number positive integer especially Python! Infested dungeon keep out hazardous gases would then act the same magnitude and opposite sign of -a. Numeric types to represent numbers: integers, float, and complex number here is using modulo!, advice, career opportunities, and 7 % 3 = 1 numbers a...: ) cc by-sa your | {
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# Is $\cos(\alpha x + \cos(x))$ periodic?
Consider the function $f: \mathbb{R} \to [-1, 1]$ defined as
$$f(x) = \cos(\alpha x + \cos(x))$$
What conditions must be placed on $\alpha \in \mathbb{R}$ such that the function $f$ is periodic?
First of all, I tried plotting some values on Wolfram|Alpha, and for all the values of $\alpha$ that I tested, it seems that any $\alpha$ works... But I couldn't prove it.
## My attempt:
We want to study $\alpha$ such that the following statement is true:
$$\exists \,\, T > 0 \quad \forall \,x \in \mathbb{R} \quad \cos(\alpha (x + T) + \cos(x + T)) = \cos(\alpha x + \cos(x))$$
I was able to show, with some trigonometric substitutions, that this statement is equivalent to the following statement:
$$\exists \,\, T > 0 \quad \forall \,x \in \mathbb{R} \quad \exists \,\, K \in \mathbb{Z} \quad \text{such that}$$ $$\sin(x + T) = \dfrac{\alpha T - K\pi}{\sin (T)} \quad \text{or} \quad \cos(x + T) = \dfrac{K\pi - \alpha(x + T)}{\cos (T)}$$
I couldn't make any progress after that, though.
## EDIT:
Inspired by a quick comment by @ZainPatel, I was actually able to show that all $\alpha \in \mathbb{Q}$ works! It's quite simple, I am surprised I didn't try this before.
Let $\alpha \in \mathbb{Q}$, $\alpha = \dfrac{p}{q}$. Then $T = 2q\pi$ works, since
$$f(x + 2q\pi) = \cos(\alpha (x + 2q\pi) + \cos(x + 2q\pi)) = \cos(\alpha x + 2p\pi + \cos(x)) = f(x)$$
The matter is still open for irrationals though! | {
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The matter is still open for irrationals though!
• Doesn't $T = 2\pi$ work? $\cos (\alpha x + 2\pi T + \cos (x + 2\pi)) = \cos (\alpha x + \cos x)$? – Zain Patel Sep 7 '16 at 19:20
• Try writing $f(x) = \cos(\alpha x) \cos (\cos x) - \sin(\alpha x)\sin(\cos x)$. – Umberto P. Sep 7 '16 at 19:20
• @ZainPatel Notice we'd have $f(x+2\pi)=\cos(\alpha x + 2\pi\alpha + \cos(x))$; ie, that's $2\pi\alpha$ rather than $2\pi T$. – Fimpellizieri Sep 7 '16 at 19:21
• You could try addition theorems for $\cos(a+b)$ – Kaligule Sep 7 '16 at 19:23
• @ZainPatel, thanks. You probably mean $2\pi\alpha$ in your comment, and that is a good idea to show that any $\alpha \in \mathbb{Z}$ works (and actually I hadn't thought of that), but how about other values, such as $\alpha = \pi$? – Pedro A Sep 7 '16 at 19:24
As you already proven, each $\alpha \in \mathbb Q$ works.
We show that if $f$ is periodic, then $\alpha \in \mathbb Q$:
Let $T>0$ be so so that
$$f(x+T) =f(x)$$
$$\cos(\alpha x +\alpha T + \cos(x+T))=\cos(\alpha x + \cos(x))$$
This shows that $$-2 \sin\bigg(\frac{\alpha x +\alpha T + \cos(x+T)+ \alpha x + \cos(x)}{2}\bigg) \sin\bigg(\frac{\alpha T + \cos(x+T) - \cos(x)}{2} \bigg)=0$$
Let $$A:= \{ x | \sin\bigg(\frac{2 \alpha x +\alpha T + \cos(x+T) + \cos(x)}{2}\bigg) =0 \} \,;$$ $$B:=\{ x| \sin\bigg(\frac{\alpha T + \cos(x+T) - \cos(x)}{2} \bigg) =0 \}$$
Then, the above shows that $A \cup B= \mathbb R$. Moreover, by continuity both sets are closed.
It follows from here that either $A$ or $B$ contains an interval.
Case 1: $A$ contains some interval $(a,b)$.
Since $$\sin\bigg(\frac{2 \alpha x +\alpha T + \cos(x+T) + \cos(x)}{2}\bigg) =0$$ for all $x \in (a,b)$ we get that $$\frac{2 \alpha x +\alpha T + \cos(x+T) + \cos(x)}{2} \in \{ k\pi |k \in \mathbb Z \}$$ for all $x \in (a,b)$. | {
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But the image of the interval $(a,b)$ under the continuous function $\frac{2 \alpha x +\alpha T + \cos(x+T) + \cos(x)}{2}$ must be connected, and hence a single point. This implies that $\alpha = 0$.
Case 2: $B$ contains some interval $(a,b)$.
Since $\sin(\frac{\alpha T + \cos(x+T) - \cos(x)}{2} ) =0$ for all $x \in (a,b)$ the same argument shows that there exists some constant $C$ so that $$\alpha T + \cos(x+T) - \cos(x) =C \qquad \forall x \in (a,b)$$ This shows that $T$ is a period for $\cos(x)$ and hence $T=2k \pi$ for some $k \in \mathbb{Z}$.
Now, for all $x \in (a,b)$ we have by the definition of $B$ $$\sin\bigg(\frac{\alpha 2 k \pi + \cos(x+2 k \pi) - \cos(x)}{2} \bigg) =0$$
This gives $$\sin(\alpha k \pi ) =0$$ from which is easy to derive that $\alpha \in \mathbb Q$.
• Impressive, thank you very much and sorry to take so long to give feedback. – Pedro A Oct 14 '17 at 17:25 | {
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# Measure of angle formed by minute and hour hands
• Nov 6th 2010, 03:31 PM
Hellbent
Measure of angle formed by minute and hour hands
Hi,
What is the measure of the angle formed by the minute and hour hands of a clock at 1:50?
(A) $90^o$
(B) $95^o$
(C) $105^o$
(D) $115^o$
(E) $120^o$
My guess was $90^o$ after mentally flipping it. My approach was worthless and remains worthless for questions of this type.
I seek a better approach to this question and an explanation of this approach. I ask kindly.
• Nov 6th 2010, 03:43 PM
Quote:
Originally Posted by Hellbent
Hi,
What is the measure of the angle formed by the minute and hour hands of a clock at 1:50?
(A) $90^o$
(B) $95^o$
(C) $105^o$
(D) $115^o$
(E) $120^o$
My guess was $90^o$ after mentally flipping it. My approach was worthless and remains worthless for questions of this type.
I seek a better approach to this question and an explanation of this approach. I ask kindly.
A 12-hour clock has 12 subdivisions, each of which is $30^o$
At 1:50, the minute hand is pointing at 10, so it is $(2)30^0$ left of 12.
The hour hand will have moved $\frac{10}{12}$ of $30^0$ from it's initial position at 1 o'clock, when the minute hand was pointing at 12.
This leaves it $30^0+\left(\frac{5}{6}\right)30^0$ to the right of 12.
• Nov 6th 2010, 04:29 PM
Hellbent
Quote:
A 12-hour clock has 12 subdivisions, each of which is $30^o$
At 1:50, the minute hand is pointing at 10, so it is $(2)30^0$ left of 12.
The hour hand will have moved $\frac{10}{12}$ of $30^0$ from it's initial position at 1 o'clock, when the minute hand was pointing at 12.
This leaves it $30^0+\left(\frac{5}{6}\right)30^0$ to the right of 12.
My understanding is much better. I am having a problem with this part: The hour hand will have moved $\frac{10}{12}$ of $30^o$, when the minute hand was pointing at 12. | {
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Wouldn't it have been the minute hand that moved $\frac{10}{12}$ of $30^o$? Seeing that it has moved from 12 to 10 - $300^o$.
• Nov 6th 2010, 04:36 PM
Quote:
Originally Posted by Hellbent
My understanding is much better. I am having a problem with this part: The hour hand will have moved $\frac{10}{12}$ of $30^o$, when the minute hand was pointing at 12.
Wouldn't it have been the minute hand that moved $\frac{10}{12}$ of $30^o$? Seeing that it has moved from 12 to 10 - $300^o$.
I didn't write the first post too well.
Imagine the time is initially 1 o'clock. The minute hand is at 12 and the hour hand is at 1.
Both hands move and the clock reads 1:50.
Yes, the minute hand will have moved through $30^o$ ten times
while the hour hand will have moved through $\frac{30^o}{12}$ ten times.
The hour hand moves through $30^0$ for a $360^0$ movement of the minute hand.
I'm getting (D), not (C).
• Nov 6th 2010, 05:25 PM
Hellbent
Thanks.
Sorry, typo, it is indeed (D.) $115^o$.
I just changed the values in this question to check my understanding:
What is the measure of the angle formed by the minute and hour hands of a clock at 3:45?
$\frac{45}{60} = \frac{3}{4}$
$90^o + (\frac{3}{4})30^o = 112.5^o$ Then adding another $90^o$ (intervening angle between 9 and 12) gives $202.5^o$
Would the approach be the same for non-multiples of 5. Say, 4:47?
• Nov 6th 2010, 05:37 PM
Quote:
Originally Posted by Hellbent
Thanks.
Sorry, typo, it is indeed (D.) $115^o$.
I just changed the values in this question to check my understanding:
What is the measure of the angle formed by the minute and hour hands of a clock at 3:45?
$\frac{45}{60} = \frac{3}{4}$
$90^o + (\frac{3}{4})30^o = 112.5^o$ Then adding another $90^o$ (intervening angle between 9 and 12) gives $202.5^o$
Would the approach be the same for non-multiples of 5. Say, 4:47?
Yes,
Starting at 4 o'clock, the minute hand will swing through $\frac{47}{60}360^o$
Then the hour hand will swing through $\frac{47}{60}30^o$ | {
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Then the hour hand will swing through $\frac{47}{60}30^o$
You can simply think in terms of "fractions of an hour" and so "fractions of $360^o$" and "fractions of $30^o$"
• Nov 6th 2010, 05:41 PM
Hellbent
Thanks. | {
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# Number of closed walks on an $n$-cube
Is there a known formula for the number of closed walks of length (exactly) $r$ on the $n$-cube? If not, what are the best known upper and lower bounds?
Note: the walk can repeat vertices.
Yes (assuming a closed walk can repeat vertices). For any finite graph $G$ with adjacency matrix $A$, the total number of closed walks of length $r$ is given by
$$\text{tr } A^r = \sum_i \lambda_i^r$$
where $\lambda_i$ runs over all the eigenvalues of $A$. So it suffices to compute the eigenvalues of the adjacency matrix of the $n$-cube. But the $n$-cube is just the Cayley graph of $(\mathbb{Z}/2\mathbb{Z})^n$ with the standard generators, and the eigenvalues of a Cayley graph of any finite abelian group can be computed using the discrete Fourier transform (since the characters of the group automatically given eigenvectors of the adjacency matrix). We find that the eigenvalue $n - 2j$ occurs with multiplicity ${n \choose j}$, hence
$$\text{tr } A^r = \sum_{j=0}^n {n \choose j} (n - 2j)^r.$$
For fixed $n$ as $r \to \infty$ the dominant term is given by $n^r + (-n)^r$.
• I'm guessing not, but is there any chance this expression has a closed form? Aug 2 '11 at 17:12
• @Lev: you mean without a summation over $n$? I doubt it. Is fixed $n$ as $r \to \infty$ not the regime you're interested in? Aug 2 '11 at 17:36
• In some sense, I'm more interested in fixed r as n gets large. The summation is certainly quite helpful, but of course if a closed form existed, it would even be nicer :) Aug 2 '11 at 18:06 | {
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The number of such walks is $2^n$ (the number of vertices of the $n$-cube) times the number of walks that start (and end) at the origin. We may encode such a walk as a word in the letters $1, -1, \dots, n, -n$ where $i$ represents a positive step in the $i$th coordinate direction and $-i$ represents a negative step in the $i$th coordinate direction. The words that encode walks that start and end at the origin are encoded as shuffles of words of the form $i\ -i \ \ i \ -i \ \cdots\ i \ -i$, for $i$ from 1 to $n$. Since for each $i$ there is exactly one word of this form for each even length, the number of shuffles of these words of total length $m$ is the coefficient of $x^m/m!$ in $$\biggl(\sum_{k=0}^\infty \frac{x^{2k}}{(2k)!}\biggr)^{n} = \left(\frac{e^x + e^{-x}}{2}\right)^n.$$ Expanding by the binomial theorem, extracting the coefficient of $x^r/r!$, and multiplying by $2^n$ gives Qiaochu's formula.
Let $W(n,r)$ be the coefficient of $x^r/r!$ in $\cosh^n x$, so that $$W(n,r) = \frac{1}{2^n}\sum_{j=0}^n\binom{n}{j} (n-2j)^r.$$ Then we have the continued fraction, due originally to L. J. Rogers, $$\sum_{r=0}^\infty W(n,r) x^r = \cfrac{1}{1- \cfrac{1\cdot nx^2}{ 1- \cfrac{2(n-1)x^2}{1- \cfrac{3(n-2)x^2}{\frac{\ddots\strut} {\displaystyle 1-n\cdot 1 x^2} }}}}$$ A combinatorial proof of this formula, using paths that are essentially the same as walks on the $n$-cube, was given by I. P. Goulden and D. M. Jackson, Distributions, continued fractions, and the Ehrenfest urn model, J. Combin. Theory Ser. A 41 (1986), 21–-31.
Incidentally, the formula given above for $W(n,r)$ (equivalent to Qiaochu's formula) is given in Exercise 33b of Chapter 1 of the second edition of Richard Stanley's Enumerative Combinatorics, Volume 1 (not published yet, but available from his web page). Curiously, I had this page sitting on my desk for the past month (because I wanted to look at Exercise 35) but didn't notice until just now that this formula was on it. | {
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Although this is an old question, I wanted to record what I think is a very cute elementary technique for obtaining the summation formula appearing in Qiaochu Yuan's answer. Maybe it is ultimately similar to Ira Gessel's answer: it also uses generating functions, but it avoids use of exponential generating functions.
I saw this technique in this mathstackexchange answer, but have never seen it elsewhere.
Here's the argument.
First of all, we note that it's easy to see, as mentioned in the answer of Derrick Stolee, that the number of closed walks of length $$r$$ in the $$n$$-hypercube is $$2^n$$ times the number of words of length $$r$$ in the alphabet $$[n] := \{1,2,...,n\}$$ in which every letter appears an even number of times. So we want to count words of this form.
For a word $$w$$ in the alphabet $$[n]$$, let me use $$\bf{z}^w$$ to denote $$\mathbf{z}^w := \prod_{i=1}^{n} z_i^{\textrm{\# i's in w}}$$, where the $$z_i$$ are formal parameters. For a set $$A \subseteq [n]^{*}$$ of such words, I use $$F_A(\mathbf{z}) := \sum_{w \in A} \mathbf{z}^{w}$$.
For $$i=1,\ldots,n$$ and $${F}(\mathbf{z})\in\mathbb{Z}[z_1,\ldots,z_n]$$ define $$s_i(F(\mathbf{z})) := \frac{1}{2}( F(\mathbf{z}) + F(z_1,z_2,\ldots,z_{i-1},-z_{i},z_{i+1},\ldots,z_n)),$$ a kind of symmetrization operator. We have the following very easy proposition:
Prop. For $$A\subseteq [n]^{*}$$, $$s_i(F_A(\mathbf{z})) = F_{A'}(\mathbf{z})$$ where $$A' := \{w\in A\colon \textrm{w has an even \# of i's}\}$$.
Thus if $$A := [n]^r$$ is the set of words of length $$r$$, and $$A'\subseteq A$$ is the subset of words where each letter appears an even number of times, we get $$F_{A'}(\mathbf{z}) = s_n(s_{n-1}(\cdots s_1(F_{A}(\mathbf{z})) \cdots ) ) = s_n(s_{n-1}(\cdots s_1((z_1+\cdots+z_n)^r) \cdots ) )$$ $$= \frac{1}{2^n}\sum_{(a_1,\ldots,a_n)\in\{0,1\}^n}((-1)^{a_1}z_1 + \cdots + (-1)^{a_n}z_n)^r.$$ | {
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Setting $$z_i := 1$$ for all $$i$$, we see that $$\#A'=\frac{1}{2^n}\sum_{j=0}^{n}\binom{n}{j}(n-2j)^r,$$ and hence that the number of closed walks we wanted to count is $$\sum_{j=0}^{n}\binom{n}{j}(n-2j)^r,$$ as we saw in Qiaochu's answer.
Incidentally, this gives a combinatorial way to compute the eigenvalues of the adjacency matrix of the $$n$$-hypercube (see Stanley's "Enumerative Combinatorics" Vol. 1, 2nd Edition, Chapter 4 Exercise 68).
Assuming a "closed walk" can repeat vertices, we can count closed walks starting at $0$ by counting the $r$-sequences of $[n]$ so that each number appears an even number of times. The bijection is given by labeling edges by the coordinate that is toggled between the vertices. You can probably count these sequences by inclusion/exclusion and then multiply by $2^n/r$ to account for the choice of start position.
• If we assume the path moves in each dimension 0 or 2 times, you can select ${n \choose r/2}$ dimensions and then permute them $r^1/2^r$ ways. This is a lower bound on the number of walks and is likely the right asymptotics. Jul 31 '11 at 17:03
• That should be $r!/2^r$. Jul 31 '11 at 17:03 | {
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# [SOLVED]Eigenvalues
#### Sudharaka
##### Well-known member
MHB Math Helper
saravananbs's question from Math Help Forum,
if A and B are similar matrices then the eigenvalues are same. is the converse is true? why?
thank u
Hi saravananbs,
No. The converse is not true in general. Take the two matrices, $$A=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\mbox{ and }B=\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}$$.
$\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\begin{pmatrix}1 \\ 1\end{pmatrix}=1.\begin{pmatrix}1 \\ 1\end{pmatrix}$
$\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\begin{pmatrix}1 \\ 1\end{pmatrix}=1.\begin{pmatrix}1 \\ 1\end{pmatrix}$
Therefore both matrices have the same eigenvalue 1. However it can be easily shown that $$A\mbox{ and }B$$ are not similar matrices.
#### Opalg
##### MHB Oldtimer
Staff member
saravananbs's question from Math Help Forum,
Hi saravananbs,
No. The converse is not true in general. Take the two matrices, $$A=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\mbox{ and }B=\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}$$.
$\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\begin{pmatrix}1 \\ 1\end{pmatrix}=1.\begin{pmatrix}1 \\ 1\end{pmatrix}$
$\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\begin{pmatrix}1 \\ 1\end{pmatrix}=1.\begin{pmatrix}1 \\ 1\end{pmatrix}$
Therefore both matrices have the same eigenvalue 1. However it can be easily shown that $$A\mbox{ and }B$$ are not similar matrices.
That example does not really work, because $A$ and $B$ do not have the same eigenvalues: $A$ has a (repeated) eigenvalue 1, whereas the eigenvalues of $B$ are 1 and –1.
However, if $C = \begin{pmatrix}1 & 1 \\0 & 1\end{pmatrix}$, then $A$ and $C$ have the same eigenvalues (namely 1, repeated), but are not similar.
#### Sudharaka
##### Well-known member
MHB Math Helper
That example does not really work, because $A$ and $B$ do not have the same eigenvalues: $A$ has a (repeated) eigenvalue 1, whereas the eigenvalues of $B$ are 1 and –1. | {
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However, if $C = \begin{pmatrix}1 & 1 \\0 & 1\end{pmatrix}$, then $A$ and $C$ have the same eigenvalues (namely 1, repeated), but are not similar.
Thanks for correcting that. Of course I now see that only the eigenvalue 1 is common to both $$A$$ and $$B$$. | {
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# How to find the distance between two non-parallel lines?
I am tasked to find the distance between these two lines.
$p1 ... x = 1 + t, y = -1 + 2t, z = t$
$p2 ... x = 1 - t; y = 3 - t; z = t$
Those two lines are nonparallel and they do not intersect (I checked that).
Using the vector product I computed the normal (the line orthogonal to both of these lines), and the normal is $(3, -2, 1)$. Now I have the direction vector of the line which will intersect both of my nonparallel lines.
However, here's where I encounter the problem - I don't know what next. The next logical step in my opinion would be to find a point on $p1$ where I could draw that orthogonal line and where that orthogonal line would also intersect with $p2$... There's only one such point, since we are in 3D space and I could draw an orthogonal line from any point in $p1$ but it could miss $p2$.
• The normal vector $(3, -2, 1)$ gives you a pair of parallel planes both normal to it, one contains $p1$ and one contains $p2$. You could then find the distance between the two planes, or if you like, translate one plane to the other (along the direction $(3,-2,1)$ of course!), find the point of intersection of the two lines, and use that to measure. – Elizabeth S. Q. Goodman Apr 5 '17 at 7:43
• – Arby Apr 5 '17 at 7:43
• @Arby If I use the formula from Wikipedia, I get that $d = 4$, since $\vec n$ is $(3, -2, 1)$, $c$ is $(1, 3, 0)$ (the second lines point) and $(1, -1, 0)$ is the first lines point. Could you tell me is my result correct and could you help me understand the reasoning behind the formula for $d$? – NumberSymphony Apr 5 '17 at 14:40
• @NumberSymphony 4? No, the result seems to be different... – Widawensen Jan 15 '18 at 18:11
• Read this paper that has an excellent description of 3D line geometry using Plücker coordinates. Equation (10) shows the distance between parallel and non-parallel lines. – ja72 Jan 16 '18 at 13:31
Take the common normal direction. | {
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Take the common normal direction.
$$\mathbf{n} = \pmatrix{1\\2\\1} \times \pmatrix{-1\\-1\\1} = \pmatrix{3 \\-2 \\1 }$$
Now project any point from the lines onto this direction. Their difference is the distance between the lines
$$d = \frac{ \mathbf{n} \cdot ( \mathbf{r}_1 - \mathbf{r}_2 )}{\| \mathbf{n} \|}$$
$$d = \frac{ \pmatrix{3\\-2\\1} \cdot \left( \pmatrix{1\\-1\\0} - \pmatrix{1\\2\\1} \right) }{ \| \pmatrix{3\\-2\\1} \|} = \frac{ \pmatrix{3\\-2\\1} \cdot \pmatrix{0\\-4\\0} } {\sqrt{14}} = \frac{8}{\sqrt{14}} = 2.1380899352993950$$
NOTE: The $\cdot$ is the vector inner product, and $\times$ is the cross product
• Some additional remark ( more for me) ... $\dfrac{n}{\Vert n \Vert} \dfrac{n^T}{\Vert n \Vert}r_1-\dfrac{n}{\Vert n \Vert} \dfrac{n^T}{\Vert n \Vert}r_2$ – Widawensen Jan 15 '18 at 17:45
• Concluding remark: your method is simpler (+1) but mine however has also some advantages: potentially gives more additional information.. – Widawensen Jan 15 '18 at 17:50
• @Widawensen - for you $$d = \frac{ \mathbf{n}^\top \mathbf{r}_1 - \mathbf{n}^\top \mathbf{r}_2 }{\| \mathbf{n} \|}$$ – ja72 Jan 15 '18 at 19:46
• What was important in the given above formula by me that it leads to the explanation why it has such form (what was once also asked by OP) After some deliberation the full starting formula for the vector $p$ which is a difference of projections on the normal $n$ (translated by vector $a$ in such a way that now normal is passing point $(0,0,0)$) should be $p= \dfrac {n}{\Vert n \Vert} \dfrac {n^T}{\Vert n \Vert}(r_1+a)-\dfrac {n}{\Vert n \Vert} \dfrac {n^T}{\Vert n \Vert}(r_2+a)$. This leads to the formula written by you. – Widawensen Jan 16 '18 at 10:10
HINT...find any vector joining one point on one line to another point on the other line and calculate the projection of this vector onto the common normal which you have found already.
Lines can be written in a vector form: | {
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Lines can be written in a vector form:
$p_1=\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}+t\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$
$p_2=\begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}+s\begin{bmatrix} -1 \\ -1 \\ 1 \end{bmatrix}$
Denote it with symbols of vectors $p_{01},p_{02},v_1,v_2$:
$p_1=p_{01}+tv_1$
$p_2=p_{02}+sv_2$
Distance is measured alongside vector which is perpendicular to $v_1$ and $v_2$, we can take for example a cross product $v_\perp=v_1 \times v_2$ what you have already done.
Now moving from the point $p_{01}$ to the point $p_{02}$ through $p_{1\perp}$ and $p_{2\perp}$ where $p_{1\perp},p_{2\perp}$ are the ends of segment perpendicular to the lines $p_1$ and $p_2$ we have:
$p_{12}=p_{02}-p_{01}= t v_1+r v_\perp+s v_2 = \begin{bmatrix} v_1 & v_\perp & v_2 \end{bmatrix} \begin{bmatrix} t \\ r \\ s \end{bmatrix}$
Solution for this equation is:
$\begin{bmatrix}t \\ r \\ s \end{bmatrix}=\begin{bmatrix} v_1 & v_\perp & v_2 \end{bmatrix} ^{-1}p_{12}$
Having $t , r, s$ it's straightforward to calculate the ends of segment perpendicular to both lines and its length $d=\Vert rv_\perp \Vert$.
• Computations at Wolphram Alpha inverse{{1,3,-1},{2,-2,-1},{1,1,1}}*{{0},{-4},{0}} $\ \ \ \ \ \ \ \ \$ hence $d=\dfrac{4}{7}\sqrt{14}$ – Widawensen Jan 15 '18 at 17:26 | {
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# Four integers are randomly selected from the set {-1,0,1}, with
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Four integers are randomly selected from the set {-1,0,1}, with [#permalink]
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Updated on: 23 Apr 2017, 02:57
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Four integers are randomly selected from the set {-1,0,1},with repetitions allowed.What is the probability that the product of the four integers chosen will be its least value possible
A. 1/81
B. 4/81
C. 8/81
D. 10/81
E. 16/81
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Originally posted by VyshakhR1995 on 23 Apr 2017, 00:20.
Last edited by Bunuel on 23 Apr 2017, 02:57, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Four integers are randomly selected from the set {-1,0,1}, with [#permalink]
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23 Apr 2017, 03:36
1
1
The total probability is selecting any of the 3 integers in the 4 turns= 3*3*3*3= 81
Minimum Product (Least Value) = -1
Case 1: Selecting three "-1" and one "1"
=4!/3!=4
Case 2:Selecting three "1" and one "-1"
=4!/3!=4 | {
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Case 1: Selecting three "-1" and one "1"
=4!/3!=4
Case 2:Selecting three "1" and one "-1"
=4!/3!=4
Required Probability= (4+4)/81= 8/81
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Re: Four integers are randomly selected from the set {-1,0,1}, with [#permalink]
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30 Apr 2017, 07:39
Minimum Product of four items on the list = -1
Number of ways to achieve the result= 2 {(3 Nos -1 + 1 Nos 1) OR (3 Nos +1 + 1 Nos -1)}
Probability for case 1 (3 Nos -1 + 1 Nos 1)
Number of ways= 4 (+1,+1,+1,-1/+1,+1,-1,+1/.........)
Probability = 4x (1/3)^4 = 4/81
Probability for case 2 (3 Nos +1 + 1 Nos -1)
Number of ways= 4 Similar explanations as earlier
Probability = 4x (1/3)^4= 4/81
Hence total probability= 4/81 +4/81= 8/81
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Re: Four integers are randomly selected from the set {-1,0,1}, with [#permalink]
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30 Apr 2017, 11:15
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VyshakhR1995 wrote:
Four integers are randomly selected from the set {-1,0,1},with repetitions allowed.What is the probability that the product of the four integers chosen will be its least value possible
A. 1/81
B. 4/81
C. 8/81
D. 10/81
E. 16/81
Solution
• We have a negative number in the set, so the least value will be possible, only when the product is negative.
• If the product is least and negative, we cannot select 0, so we need to select a combination of -1 and 1 to get a product of -1.
• Since 4 numbers are selected –
o We get a product of -1 if we select one -1 and three 1s.
We can do that in 4C3 ways = 4 ways
o OR we can select three -1s and one 1.
We can do that same in 4C3 = 4 ways.
• Thus, the total ways to get a least value $$= 4 + 4 = 8$$ ways
• Now total ways of selecting a number $$= 3*3*3*3 = 81$$
o Thus, the probability of choosing the 4 integers whose product has least value $$= \frac{8}{81}$$
• Hence, the correct answer is Option C.
Thanks,
Saquib
Quant Expert
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• Hence, the correct answer is Option C.
Thanks,
Saquib
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Re: Four integers are randomly selected from the set {-1,0,1}, with [#permalink]
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30 Jun 2017, 04:28
EgmatQuantExpert wrote:
VyshakhR1995 wrote:
Four integers are randomly selected from the set {-1,0,1},with repetitions allowed.What is the probability that the product of the four integers chosen will be its least value possible
A. 1/81
B. 4/81
C. 8/81
D. 10/81
E. 16/81
Solution
• We have a negative number in the set, so the least value will be possible, only when the product is negative.
• If the product is least and negative, we cannot select 0, so we need to select a combination of -1 and 1 to get a product of -1.
• Since 4 numbers are selected –
o We get a product of -1 if we select one -1 and three 1s.
We can do that in 4C3 ways = 4 ways
o OR we can select three -1s and one 1.
We can do that same in 4C3 = 4 ways.
• Thus, the total ways to get a least value $$= 4 + 4 = 8$$ ways
• Now total ways of selecting a number $$= 3*3*3*3 = 81$$
o Thus, the probability of choosing the 4 integers whose product has least value $$= \frac{8}{81}$$
• Hence, the correct answer is Option C.
Thanks,
Saquib
Quant Expert
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Thank you for your explanation, e-GMAT expert.
Btw I have a general question, hope that you will help me clear this concern:
How can we know whether order of selection matters in a probability question? In other words, what are the indicators of order of selection? | {
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In the above question, initially I didn't consider the order of 4 numbers chosen, so I worked out an answer different from all 5 choices. But then I realized the denominator is very big, I thought I might overlook many cases... the selecting order could be the key... so I calculated again and figured out the correct answer.
Since the above question is not an official one, I cannot draw any conclusion as to whether there is any OG/GMATPrep question in which no indicator is provided, and we have to test each case. I believe that such an experience expert as you could give me a reliable answer.
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Re: Four integers are randomly selected from the set {-1,0,1}, with [#permalink] 15 Oct 2018, 15:38
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# Math Help - Freaky series using limit comparison test!
1. ## Freaky series using limit comparison test!
How in the world do you determine the following series' convergence/divergence specifically using the limit comparison test?
from n=1 to n=infinity
ln (2n+1)/ (n^2 + 2n)
I'm bamboozled and befuddled!
2. Originally Posted by Kaitosan
How in the world do you determine the following series' convergence/divergence specifically using the limit comparison test?
from n=1 to n=infinity
ln (2n+1)/ (n^2 + 2n)
I would write the n'th term as $\frac{\ln(2n+1)}{n^2+2n} = \frac{\ln(2n+1)}{\sqrt n}\frac{\sqrt n}{n^2+2n}$. Then because "ln(n) goes to infinity slower than any positive power of n", it follows that the first fraction becomes small. The second fraction, $\frac{\sqrt n}{n^2+2n}$, is approximately $n^{-3/2}$. So do the limit comparison test, comparing the given series with the convergent series $\sum n^{-3/2}$.
3. Hm. I understand things better now. Thanks a lot.
4. Originally Posted by Opalg
I would write the n'th term as $\frac{\ln(2n+1)}{n^2+2n} = \frac{\ln(2n+1)}{\sqrt n}\frac{\sqrt n}{n^2+2n}$. Then because "ln(n) goes to infinity slower than any positive power of n", it follows that the first fraction becomes small. The second fraction, $\frac{\sqrt n}{n^2+2n}$, is approximately $n^{-3/2}$. So do the limit comparison test, comparing the given series with the convergent series $\sum n^{-3/2}$.
Also, what's to prevent me from using, say, "n" instead of sqrt(n)? In that way, it will appear that the series diverges....
I know it's against the rule to bump topics but I don't care, I really need help. I might as well post another topic.
Anyways, I followed Opalag's directions but, strangely, I got zero. According to the limit comparison test, if the number equals zero or infinity, then the result is inconclusive. Please clarify?
Also, why would I not use like n^2 instead of sqrt(n)? If I had done that then the series appears to diverge! | {
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5. Originally Posted by Kaitosan
I know it's against the rule to bump topics but I don't care, I really need help. I might as well post another topic.
Anyways, I followed Opalg's directions but, strangely, I got zero. According to the limit comparison test, if the number equals zero or infinity, then the result is inconclusive. Please clarify?
Also, why would I not use like n^2 instead of sqrt(n)? If I had done that then the series appears to diverge!
Okay, what the limit comparison test says is that if $\textstyle\sum a_n$ and $\textstyle\sum b_n$ are series of positive terms, $\lim_{n\to\infty}\frac{a_n}{b_n}$ exists and $\textstyle\sum b_n$ converges, then $\textstyle\sum a_n$ converges. (The test is sometimes stated in the form that if the limit exists and is nonzero then $\textstyle\sum a_n$ converges if and only if $\textstyle\sum b_n$ converges, but that is not what is needed for this example.)
If $a_n = \frac{\ln(2n+1)}{n^2+2n}$, and you choose $b_n = 1/n^p$ for some p between and 2, then $\textstyle\sum b_n$ will converge (because p>1), and $\lim_{n\to\infty}\frac{a_n}{b_n}$ will be 0 (because p<2).
I hope that makes it clearer. (If you get an infraction for bumping, tell the Moderator that I think you were justified on this occasion.)
6. Or we can bound the general term: we have $\ln x< 2\sqrt{x}-2,\,\forall\,x>1,$ thus for $n\ge1$ it's $\frac{\ln (2n+1)}{n^{2}+2n}< \frac{2\sqrt{2n+1}-2}{n^{2}+2n}< \frac{4\sqrt{n}}{n^{2}}=\frac{4}{n^{3/2}},$ then your series converges since $\sum\limits_{n=1}^{\infty }{\frac{1}{n^{3/2}}}<\infty .$ | {
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7. Originally Posted by Opalg
Okay, what the limit comparison test says is that if $\textstyle\sum a_n$ and $\textstyle\sum b_n$ are series of positive terms, $\lim_{n\to\infty}\frac{a_n}{b_n}$ exists and $\textstyle\sum b_n$ converges, then $\textstyle\sum a_n$ converges. (The test is sometimes stated in the form that if the limit exists and is nonzero then $\textstyle\sum a_n$ converges if and only if $\textstyle\sum b_n$ converges, but that is not what is needed for this example.)
If $a_n = \frac{\ln(2n+1)}{n^2+2n}$, and you choose $b_n = 1/n^p$ for some p between and 2, then $\textstyle\sum b_n$ will converge (because p>1), and $\lim_{n\to\infty}\frac{a_n}{b_n}$ will be 0 (because p<2).
I hope that makes it clearer. (If you get an infraction for bumping, tell the Moderator that I think you were justified on this occasion.)
Why must p be between 1 and 2? You simplified the series so as to exclude the ln expression in order to dig up the right-handed series for the b series. In that process of simplification, you can input any number in the bottom left and and the top right, and each number brings a different convergence/divergence result!
Also, you seems to contradict yourself. If an/bn is nonzero and finite and if bn converges, then an converges. But you says an/bn converges to zero!
Let me repeat my understanding for the limit comparison test-
To evaluate the series an, pick bn based on an's highest ordered term in each numerator and denominator. Then structure an/bn and look for the limit. If the limit is between zero and infinity and if bn converges, then an/bn converges. But an/bn converges to zero! I'm so confused.
Please don't make me bump this topic again. | {
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Please don't make me bump this topic again.
8. Originally Posted by Kaitosan
Why must p be between 1 and 2? You simplified the series so as to exclude the ln expression in order to dig up the right-handed series for the b series. In that process of simplification, you can input any number in the bottom left and and the top right, and each number brings a different convergence/divergence result!
Also, you seems to contradict yourself. If an/bn is nonzero and finite and if bn converges, then an converges. But you says an/bn converges to zero!
Please read what I said. I stated very carefully the form of comparison test that I was using. I said that if the ratio $a_n/b_n$ converges (to any limit, including the possibility that the limit might be zero) and if $\textstyle\sum b_n$ converges, then $\textstyle\sum a_n$ converges.
If you take $a_n = \frac{\ln(2n+1)}{n^2+2n}$ and $b_n = 1/n^{3/2}$, then $\frac{a_n}{b_n} = \frac{n^{3/2}\ln(2n+1)}{n^2+2n} = \frac{\ln(2n+1)}{n^{1/2} + 2n^{-1/2}} \to 0$ as $n\to\infty$. therefore, by the limit comparison test in the form that I stated, $\textstyle\sum a_n$ converges.
9. Originally Posted by Opalg
Please read what I said. I stated very carefully the form of comparison test that I was using. I said that if the ratio $a_n/b_n$ converges (to any limit, including the possibility that the limit might be zero) and if $\textstyle\sum b_n$ converges, then $\textstyle\sum a_n$ converges.
Looks like we found our "communication hitch." Based on what it looks like, there are two possibilities. First, my book is wrong. Or, secondly, you may be slightly off the definition of the limit comparison (I say this in humility, there's always a good chance that I'm wrong).
My book gives the definition of the limit comparison test in the following - | {
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My book gives the definition of the limit comparison test in the following -
"Let an and bn be positive-termed series. Let L = lim (n to infinity) an/bn. If 0 < L < +infinity, then an and bn either both converge or both diverge. If L = 0 or L = +infinity, then the test is inconclusive."
The issue here is the domain of the limit comparison test and whether it includes zero. It's there, I copied exactly as my book puts it. What's going on?
I really appreciate the fact that you replied speedily!
10. Originally Posted by Kaitosan
My book gives the definition of the limit comparison test in the following -
"Let an and bn be positive-termed series. Let L = lim (n to infinity) an/bn. If 0 < L < +infinity, then an and bn either both converge or both diverge. If L = 0 or L = +infinity, then the test is inconclusive."
The issue here is the domain of the limit comparison test and whether it includes zero. It's there, I copied exactly as my book puts it. What's going on?
I can only quote what I said before:
Originally Posted by Opalg
Okay, what the limit comparison test says is that if $\textstyle\sum a_n$ and $\textstyle\sum b_n$ are series of positive terms, $\lim_{n\to\infty}\frac{a_n}{b_n}$ exists and $\textstyle\sum b_n$ converges, then $\textstyle\sum a_n$ converges. (The test is sometimes stated in the form that if the limit exists and is nonzero then $\color{red}\textstyle\sum a_n$ converges if and only if $\color{red}\textstyle\sum b_n$ converges, but that is not what is needed for this example.)
Your book uses the version in red, but the slightly different version that I was using is used by some other authors, and is more useful for dealing with the example in this thread. | {
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In fact, I think that your book is somewhat misleading in claiming that the test is inconclusive when the limit is zero or infinity. As my version of the test shows, it is possible to draw conclusions in these cases. However, in those situations the test can only be used to get an implication in one direction; in these cases it is not an "if and only if" test.
11. Aaaah! I see! Thank you so much, I understand. I'll keep in mind that there are two "versions" of the limit comparison test then. Thanks for not giving me up. | {
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# Between any two powers of $5$ there are either two or three powers of $2$
Is this statement true?
Between any two consecutive powers of $5$, there are either two or three powers of $2$.
I can see that this statement is true for cases like $$5^1 < 2^3 < 2^4 < 5^2$$ or $$5^3 < 2^7 < 2^8 < 2^9 < 5^4$$
But I am having a trouble figuring out the proof through generalization.
Could somebody help me?
How many integer multiples of $\log_5(2)$ can we fit between the integers $n$ and $n+1$? It's not too hard to see that since $\log_5(2)<0.5$, there are at least $2$ such numbers. But, since $3\log_5(2)>1$, there are at at most $3$ such numbers. *
So then, if $n$ and $n+1$ straddled $2$ multiples, we would have (for an appropriate integer, $k$): $$n<k\log_5(2)<(k+1)\log_5(2)<n+1\\\Rightarrow5^n<2^k<2^{k+1}<5^{n+1}$$ Conversely, if $3$ multiples were straddled, we would have $$n<k\log_5(2)<(k+1)\log_5(2)<(k+2)\log_5(2)<n+1\\\Rightarrow5^n<2^k<2^{k+1}<2^{k+2}<5^{n+1}$$
Therefore there are between $2$ and $3$ powers of $2$ between each successive powers of $5$.
* To visualise this, consider the figure below. The red points are multiples of $\log_5(2)$. The distance between the red points ($\approx0.43$) is small enough to guarantee that at least $2$ of them lie between the black points. However the gap between $4$ red points ($\approx1.29$) is too wide to fit between the black points. This is irrespective of where the red points begin.
Here's a fun graph to play around with https://www.desmos.com/calculator/qfqers5mmg. No matter where you start the sequence of red points (by varying the parameter $h$), you inevitably have either $2$ or $3$ red points between the black points.
Basically, it boils down to the fact that $5$ is between $2^2 = 4$ and $2^3 = 8$. Here's a proof, though.
Let $5^a$ and $5^{a+1}$ be the two consecutive powers of $5$. Let $2^b$ be the smallest power of $2$ that exceeds $5^a,$ and $2^c$ the largest below $5^{a+1}$. | {
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Then we have $2^{b-1} \leq 5^a < 2^b$ and $2^c < 5^{a+1} \leq 2^{c+1}$.
From this we get $$\frac{2^{c}}{2^{b}} < \frac{5^{a+1}}{5^a} \leq \frac{2^{c+1}}{2^{b-1}}$$ or, equivalently, $$2^{c-b} < 5 \leq 4 \cdot 2^{c-b}.$$
This inequality can be rewritten as $\frac{5}{4} \leq 2^{c-b} < 5$, which proves that $c - b$ is either $1$ or $2$. Thus the powers of $2$ between $5^a$ and $5^{a+1}$ are either $2^b, 2^{b+1} = 2^c$, or $2^b, 2^{b+1}, 2^{b+2} = 2^c.$ This is what you wanted.
We can see mathematically that there is a ratio between the number of powers of $2$ and the number of powers of $5$ below a certain integer $a$.
Intuitively, the ratio would be $\log_25$, or $2.32$. On average, for every power of $5$, you get $2.32$ powers of $2$. That would translate to about $2$ or $3$ powers of $2$ between every two powers of $5$.
If this isn't completely intuitive, notice that there are exactly $3$ more powers of $2$ than $8$, as $\log_28 = 3$. For example, out of all integers below $100$, there are $6$ powers of $2$ ($2,4,8,16,32,64$) and $2$ powers of $8$ ($8,64$).
We can extend to this to any integers $x$ and $y$, provided they are not $0$ or $1$: For all integers below an integer $z$, the ratio of the number of exponents of $a$ to the number of exponents of $b$ is $\log_ab$.
In the special case $[1,5]$, we see that 1, 2 and 4 are in the interval but 8 and up are not. There are powers of 2 less than and powers of two greater than any other power of 5.
There can't be more than three, because if $0 < a \leq b < 2b < 4b < 8b \leq 5a$, we get the contradiction $8a \leq 5a$ for a positive number $a$.
Nor can there be zero, because if $0 < b < a < 5a < 4b$, we get the contradiction $5b < 4b$ for a positive number b.
Nor can there be one, because the argument still holds if we insert the condition $a \leq 2b \leq 5a$.
You already found existence proofs for two or three intermediate powers. | {
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You already found existence proofs for two or three intermediate powers.
The explanations in the other answers are great, but that's a very elementary proof.
• Hi, @Davislor. Sorry but I don't quite follow where $8a\leq 5a$ comes from - it seems as though you've replaced the $b$ that was previously in the inequality but how is this justified? – Jam Jul 26 '18 at 20:09
• @Jam $0< a \leq b$, so $8a \leq 8b$. Bit, we assumed $8b \leq 5a$. Thus the contradiction that $8a \leq 5a$ for positive $a$. – Davislor Jul 26 '18 at 20:15
• @Jam: Do you find my elementary proof any clearer? – Ilmari Karonen Jul 27 '18 at 10:33
• @IlmariKaronen I find both quite clear, thanks - I was just stuck on one step in Davislor's :) – Jam Jul 27 '18 at 11:04
It's easy enough to see that this is indeed true, even without using logarithms.
Let $a$ be some power of $5$, and let $b$ be the least power of $2$ not less than $a$. Then the next power of $5$ after $a$ is obviously $5a$, while the next three powers of $2$ after $b$ are $2b$, $4b$ and $8b$.
Since, by definition, $a \le b$, it follows that $5a \le 5b < 8b$. Thus, there can be at most three powers of $2$ ($b$, $2b$ and $4b$) between $a$ and $5a$.
Conversely, since $b$ is the least power of $2$ not less than $a$, it follows that $\frac12 b < a$. Thus, equivalently, $2b < 4a < 5a$, so there must be at least two powers of $2$ ($b$ and $2b$) between $a$ and $5a$.
BTW, if you look closely at the proof above, you may note that it doesn't actually use the assumption that $a$ is a power of $5$ anywhere. Thus, in fact, we've proven a more general result: for any positive number $a$, there are either two or three powers of $2$ between $a$ and $5a$.
(In fact, the proof doesn't really use the assumption that $b$ is a power of $2$, either, so we could generalize the result even further in this direction if we wanted!) | {
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Also, you may notice that the key observation behind the result above is that the number $5$ lies strictly between $2^2 = 4$ and $2^3$ = 8. Thus, by essentially the same logic as above, we can prove a similar result for other bases:
Between any two consecutive powers of $x$ there are at least $k$ and at most $k+1$ powers of $y$, where $x$ and $y$ are any numbers greater than $1$, and $k$ is the largest integer such that $y^k \le x$.
We know that
if $y-x >1$, for $x\geq 0, y>0$, then there $\exists n \in \mathbb{N}, n>0$ s.t. $$x<n<y \tag{1}$$
e.g. $n=\left \lfloor x \right \rfloor+1$, because $$x = \left \lfloor x \right \rfloor+\{x\}<\left \lfloor x \right \rfloor+1<x+1<y$$ In this case $$(k+1)\frac{\ln{5}}{\ln{2}}-k\frac{\ln{5}}{\ln{2}}=\frac{\ln{5}}{\ln{2}}>1$$ and from $(1)$, there $\exists n\in\mathbb{N}$ s.t. $$k\frac{\ln{5}}{\ln{2}}<n<(k+1)\frac{\ln{5}}{\ln{2}} \iff\\ k\ln{5}<n\ln{2}<(k+1)\ln{5} \iff \\ 5^k < 2^n<5^{k+1}$$ However $\frac{\ln{5}}{\ln{2}}\approx 2.32>2$ and $(1)$ can be extended to
if $y-x >2$, for $x\geq 0, y>0$, then there $\exists n \in \mathbb{N},n>0$ s.t. $$x<n<n+1<y \tag{2}$$ | {
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## Tuesday, April 17, 2007
### Properties of Matrix Multiplication
Multiplication can only occur between matrices A and B if the number of columns in A match the number of rows in B. This presents the very important idea that while multiplication of A with B might be a perfectly good operation; this does not guarantee that multiplication of B with A is a perfectly good operation.
Even if matrix A can be multiplied with matrix B and matrix B can be multiplied to matrix A, this doesn't necessarily give us that AB=BA. In other words, unlike the integers, matrices are noncommutative.
Property 1: Associative Property of Multiplication
A(BC) = (AB)C
where A,B, and C are matrices of scalar values.
Proof:
(1) Let D = AB, G = BC
(2) Let F = (AB)C = DC
(3) Let H = A(BC) = AG
(4) Using Definition 1, here, we have for each D,F,G,H:
di,j = ∑k ai,k*bk,j
gi,j = ∑k bi,k*ck,j
fi,j = ∑k di,k*ck,j
(5) So, expanding fi,j gives us:
fi,j = ∑k (∑l ai,l*bl,j)ck,j =
(∑kl) ai,l*bl,k*ck,j =
= ∑l ai,l*(∑k bl,k*ck,j) =
= ∑l ai,l*gl,j = hi,j
QED
Property 2: Distributive Property of Multiplication
A(B + C) = AB + AC
(A + B)C = AC + BC
where A,B,C are matrices of scalar values.
Proof:
(1) Let D = AB such that for each:
di,j = ∑k ai,k*bk,j
(2) Let E = AC such that for each:
ei,j = ∑k ai,k*ck,j
(3) Let F = D + E = AB + AC such that for each:
fi,j = ∑k ai,k*bk,j+ai,k*ck,j = ∑k ai,k[bk,j + ck,j]
(4) Let G = B+C such that for each:
gi,j = bi,j + ci,j
(5) Let H = A(B+C) = AG such that for each:
hi,j = ∑k ai,k*gk,j
(6) Then we have AB + AC = A(B+C) since for each:
hi,j = ∑k ai,k[bk,j + ck,j]
(7) Let M = A + B such that for each:
mi,j = ai,j + bi,j
(8) Let N = (A+B)C = MC such that:
ni,j = ∑k mi,k*ck,j =
= ∑k (ai,k + bi,k)*ck,j
(9) Let O = BC such that:
oi,j = ∑k bi,k*ck,j
(10) Let P = AC + BC = E + O such that:
pi,j = ei,j + oi,j =
= ∑k ai,k*ck,j + ∑k bi,k*ck,j =
= ∑k [ai,k*ck,j + bi,k*ck,j] =
= ∑k (ai,k + bi,k)*ck,j
QED
Property 3: Scalar Multiplication | {
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= ∑k [ai,k*ck,j + bi,k*ck,j] =
= ∑k (ai,k + bi,k)*ck,j
QED
Property 3: Scalar Multiplication
c(AB) = (cA)B = A(cB)
Proof:
(1) Let D = AB such that:
di,j = ∑k ai,k*bk,j
(2) Let E = c(AB) = cD such that for each:
ei,j = c*di,j = c*∑k ai,k*bk,j
(3) Let F = (cA)B such that:
fi,j = ∑k (c*ai,k)*bk,j = c*∑k ai,k*bk,j
(4) Let G = A(cB) such that:
gi,j = ∑k ai,k*(c*bk,j) =
= c*∑k ai,k*bk,j
QED
Property 4: Muliplication of Matrices is not Commutative
AB does not have to = BA
Proof:
(1) Let A =
(2) Let B =
(3) AB =
(4) BA =
QED
References
• Hans Schneider, George Philip Barker, , 1989.
Eva Acosta said...
Hi,
I am a math teacher.
You can revise your understanding of matrices solving exercises about addition, subtraction and multiplication of matrices. Inverse,... at www.emathematics.net
Fantastic!!!
Anonymous said...
hi
i am a student taking a course in linear algebra. i just wanted to say that your tutorrial is very helpfull. so post some more on this topic.
thank you
Luis said...
Amittai Aviram said...
Hi! Thank you very much for providing this resource! I am troubled, however, by something in your proof of the associativity of matrix multiplication. (Since blogger.com will not let me use the necessary HTML tags, I will attempt to use LaTeX-like notation instead.) | {
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In (4), you have
d_{i,j} = \sum_{k} a_{i,k}*b_{k,j}
f_{i,k} = \sum_{k} d_{i,k}*c_{k,j}
Then, in (5), you expand this last equation as
f_{i,j} = \sum_{k} (\sum_{l} a_{i,l}*b_{l,j}) c_{k,j}
However, we are expanding d_{i,k} from the previous equation, not d_{i,j} (where the outer summation symbol binds the variable k). And
d_{i,k} = \sum_{l} a_{i,l}*b_{l,k}
The summation symbol that binds k becomes the outer summation symbol in the expansion. So the correct expansion of the whole equation should be:
f_{i,j} = \sum_{k} (\sum_{l} a_{i,l}*b_{l,k}) c_{k,j}
which does not lend itself to so neat a proof as the one you present. A correct proof, though with rather messier notation, can be found here:
http://linear.ups.edu/xml/0094/fcla-xml-0.94li30.xml
Amittai Aviram said...
Another page that has a correct proof (avoiding the error noted above) can be found here:
http://www.proofwiki.org/wiki/Matrix_Multiplication_is_Associative#Proof
Larry Freeman said...
Hi Amittai,
Thanks very much for pointing out the mistake. I will revise the proof and post a comment on this blog when it is fixed. | {
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# Expected number of output letters to get desired word
I am using a letter set of four letters, say {A,B,C,D}, which is used to output a random string of letters. I want to calculate the expected output length until the word ABCD is obtained; that is, the letters A B C D appearing consecutively in that order.
I have referenced this question (Expected Number of Coin Tosses to Get Five Consecutive Heads), but have found a complexity in our case; when we obtain, say, ABA, then we can't say that the chain resets, since we have the next potentially successful chain already being started.
I have tried the approach below, but am not sure if it is completely correct.
I would be grateful for assertion that this approach is ok, as well as for any alternative methods to approach this issue.
Let e be the expected number of output letters needed to get the target string ABCD. Also, let f be the expected number of output letters needed to get the target string ABCD given we obtained the letter A.
The table for expected length and probability for e would be
| | Exp Len | Prob |
|--------------------------|---------|------|
| if first letter is [BCD] | e+1 | 3/4 |
| if A then [CD] | e+2 | 1/8 |
| if A then A | f+1 | 1/16 |
| if AB then [BD] | e+3 | 1/32 |
| if AB then A | f+2 | 1/64 |
| if ABC then [BC] | e+4 | 1/128|
| if ABC then A | f+3 | 1/256|
| if ABCD | 4 | 1/256|
---------------------------------------------
and a similar table for f after we obtained the letter A would be | {
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and a similar table for f after we obtained the letter A would be
| | Exp Len | Prob |
|-----------------------|---------|------|
|if first letter is [CD]| e+2 | 1/2 |
|if first letter is A | f+1 | 1/4 |
|if B then [BD] | e+3 | 1/8 |
|if B then A | f+2 | 1/16 |
|if BC then [BC] | e+4 | 1/32 |
|if BC then A | f+3 | 1/64 |
|if BCD | 4 | 1/64 |
------------------------------------------
The expected length e is equal to the sum of each (Probability)*(Expected Length) product set from the first table, giving $$e\, =\, \frac{3}{4}(e+1)\, +\, \frac{1}{8}(e+2)\, +\, \frac{1}{16}(f+1)\, +\, \frac{1}{32}(e+3 )\, +\, \frac{1}{64}(f+2)\, +\, \frac{1}{128}(e+4)\, +\, \frac{1}{256}(f+3)\, +\, \frac{1}{256}(4) \\-----\\ e\, \, =\, \frac{117}{128}e\, +\, \frac{21}{256}f\, +\, \frac{319}{256} \\\\ 22e\, =\, 21f\, +\, 319 \: \: \: ---(1) \\ 44e\, =\, 42f\, +\, 638 \: \: \: ---(1')$$ A similar approach for f yields $$f\, =\, \frac{1}{2}(e+2)\, +\, \frac{1}{4}(f+1)\, +\, \frac{1}{8}(e+3)\, +\, \frac{1}{16}(f+2 )\, +\, \frac{1}{32}(e+4)\, +\, \frac{1}{64}(f+3)\,+\, \frac{1}{64}(4) \\-----\\ f\, \, =\, \frac{21}{32}e\, +\, \frac{21}{64}f\, +\, \frac{127}{64} \\\\ 43f\, =\, 42e\, +\, 127 \: \: \: ---(2)$$ Combining these, we obtain
$$(2)-(1')\Rightarrow f\, =\, -2e\, +\, 765 \: \: \: ---(3)\\ (3)\rightarrow (1)\Rightarrow 22e = 21(-2e+765)+319 \\ e=256 \\ f=253$$
So the expected length seems to be 256 letters output.
I notice this is exactly what we would expect from the naive approach, from the fact that each letter has a 1 in 4 chance appearing each time, and after any four letters' output, the chance of ABCD appearing is $$\left( \frac{1}{4} \right) ^ 4 = \frac{1}{256} .$$ which is slightly worrying, since the question about five consecutive heads has a probability of 1/32, but a differing number of 62 for the expected length. | {
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After the above, I also calculated the expected length until I obtain either of TWO target strings; I used ABCD and CDBA as my targets, if it matters. The result was not the intuitive 128, but was 136 instead, by methodology similar to that above.
Using the answers provided, I will also try to check this result using new tactics proposed in the answers.
The natural approach uses transition matrices. For ease of typesetting we write up the solution another way.
Let $e$ be the expected number. Let $a$ be the expected number of additional letters, given that the last letter was an A. Let $b$ be the expected number of additional letters, given the last two letters were AB. And let $c$ be the analogous thing, given the last three letters were ABC.
At the start, if the first letter is an A, our expected total is $1+a$. If it is anything else, then our expected total is $1+e$. Thus $$e=\frac{1}{4}(1+a)+\frac{3}{4}(1+e).$$
If our last letter was an A, with probability $\frac{1}{4}$ we get an A, and the additional total (after the first A) is $1+a$. If we get a B, the expected additional total after the first A is $1+b$. If we get a C or a D, the expected additional total after the A is $1+e$. Thus $$a=\frac{1}{4}(1+a)+\frac{1}{4}(1+b)+\frac{2}{4}(1+e).$$ If the last two letters were AB, and we get an A, the additional expected total after the AB is $1+a$. If we get a B or a D, the additional expected total is $1+e$. And if we get a C it is $1+c$. Thus $$b=\frac{1}{4}(1+a)+\frac{2}{4}(1+e)+\frac{1}{4}(1+c).$$ Finally, the same reasoning gives $$c=\frac{1}{4}(1+a)+\frac{2}{4}(1+e)+\frac{1}{4}(1).$$ Four linear equations, four unknowns. | {
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• I see that this method increases the number of variables introduced, but keeps each resulting equation simpler. I was able to reconstruct the transition matrix from your argument as well. +1 for a great generalizable solution! It would be nice if you could point me to some reference as to how to directly apply the transition matrix to obtain e... – yybtcbk Sep 16 '14 at 3:00
• Additionally: I was able to apply this method to the addendum problem to get e=136, a=134, b=128, and c=102. It seems a bonus of this method that I can get the a, b, and c values on the way as well. – yybtcbk Sep 16 '14 at 3:03
• Good! For transition matrices, any beginning book on Markov chains should have it, also many a basic probability book. Can't think of a specific title. – André Nicolas Sep 16 '14 at 3:09
• I guess it's time to dig out my old Markov Chains textbook... gave up on it in Uni, but maybe I can understand it better now... thanks for the pointer! – yybtcbk Sep 16 '14 at 3:19
Conway's algorithm provides a quick method of calculation: look at how whether the initial letters match the final letters: so "AAAA" has matches for the initial $1,2,3,4$, while "ABCD" has matches only for the initial $4$; "ABCA" would have matches for $1$ and $4$, while "ABAB" would have matches for $2$ and $4$. Since the alphabet has four equally likely letters, the algorithm gives the following expected samples sizes:
• AAAA: $340 = 4^4+4^3+4^2+4^1$
• ABCD: $256 = 4^4$
• ABCA: $260 = 4^4+4^1$
• ABAB: $272 = 4^4+4^2$
So, as you say, the expected time until "ABCD" appears is $256 = 4^4$ samples. This is similar to the coin sequence "HHHHT" requiring an expected $32=2^5$ samples.
By contrast the expected time until "AAAA" appears is $340 = 4^4+4^3+4^2+4^1$ samples. This is similar to the coin sequence "HHHHH" requiring an expected $62=2^5+2^4+2^3+2^2+2^1$ samples. | {
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If you had a long string of $n$ letters then you would expect "ABCD" to appear about $\frac{n-3}{256}$ times on average. Similarly you would expect "AAAA" to appear about the same number of times on average. But strings of type "AAAA" can overlap themselves while those of type "ABCD" cannot: for example a string of length $6$ might possibly have three "AAAA"s but cannot have more than one "ABCD", even if the expected number of each is the same.
To balance the greater possibility of "AAAA"s appearing several times, but the same overall expected number, there is also a greater possibility for "AAAA" not appearing at all in the first $6$ letters, or indeed in other initial samples. It is this latter feature which increases the expected sample size until "AAAA" does appear, compared with "ABCD".
• Greatly convincing way to see that repeating letters lead to a longer expected string! This also works to suggest why the additional question results in a 136 greater than the intuitive 128; ABCD and CDBA overlap each other. Could you point me somewhere for the theory behind this Conway's Algorithm? My googling only got me algorithms for the game of life, betting odds, and calendars... – yybtcbk Sep 16 '14 at 4:10
• As for using this Conway's Algorithm for the additional question, I guess that for target strings ABCD and CDBA, positions 2 and 4 match the last target letter, I can do 4^4 + 4^2 = 272, and then can halve this (since we have two target strings?) to get 136...? I'm not sure if this is a valid approach, but the numbers do match up... – yybtcbk Sep 16 '14 at 4:18
• Two target strings raises the issue in Penney's game – Henry Sep 16 '14 at 7:05
• – Henry Sep 16 '14 at 7:30
• All the links refer to the Conway Number and Conway's Algorithm in terms of odds; are these directly applicable to the string length calculations above somehow? – yybtcbk Sep 16 '14 at 8:14 | {
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# Is there a closed form expression for these combinatorial sums?
I needed to compute this sum: $$\sum_{i=0}^n\binom{n}{i}i^2$$ I don't want any proofs or formulae, but just a yes/no answer. Is there a closed form expression for this sum?
Just to share, I was able to find a closed form expression for this sum: $$\sum_{i=0}^n\binom{n}{i}i=n.2^{n-1}$$ by using a property of arithmetic progressions and sums of combinations.
Is there any closed form expression for my sum, or more generally, sums like: $$\sum_{i=0}^n\binom{n}{i}i^k$$ , for some $k\in\mathbb{N}$?
• There is a closed form for your sum, and you can get it by applying what you already found for $\sum_k\binom{n}kk$. (You can also use calculus methods, but they’re not necessary.) – Brian M. Scott Nov 4 '16 at 16:06
• Differentiate the series $(1+x)^n$ wrt to $x$ twice to get your sum. – SirXYZ Nov 4 '16 at 16:11
Yes. You can find the formula as follows: by the binomial theorem we have
$$(1+x)^n = \sum_{i=0}^n {n\choose i} x^i.$$
Now apply the differential operator $\Big(x\frac{d}{dx}\Big)^k$ on both sides.
• Concise and complete. +1 ... You might consider adding "then set $x=1$" at the end of the answer. – Mark Viola Nov 4 '16 at 16:07
• Thanks @Dr.MV, I trust that the OP will extract that information from your comment ;). – J.R. Nov 4 '16 at 16:13
An alternative approach. Us that $\binom{n}{i}\binom{i}{k}=\binom{n}{k}\binom{n-k}{i-k}$ to get:
$$\sum_{i=0}^n \binom{n}{i}\binom{i}{k}= \binom{n}{k}\sum_{i=0}^{n} \binom{n-k}{i-k}=\binom{n}{k}2^{n-k}$$
So, since $i^2=2\binom{i}{2}+\binom{i}{1}$ to get that $$\sum \binom{n}{i}i^2 = 2\binom{n}{2}2^{n-2} + \binom{n}{1}2^{n-1}$$
In general, $i^k = \sum_{j=0}^{k} a_j \binom{i}{j}$ for some sequence $a_j$, yielding the result:
$$\sum \binom{n}{i}i^k = \sum_{j=0}^k a_j\binom{n}{j}2^{n-j}$$
So it just amounts to finding the $a_j$ for each $k$.
Hint: | {
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So it just amounts to finding the $a_j$ for each $k$.
Hint:
$$\binom{n}{i}i^{2}=\binom{n}{i}i\left(i-1\right)+\binom{n}{i}i=n\left(n-1\right)\binom{n-2}{i-2}+n\binom{n-1}{i-1}$$ | {
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# Very indeterminate form: $\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x \longrightarrow (\infty-\infty)^{\infty}$
Here is problem:
$$\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x$$
The solution I presented in the picture below was made by a Mathematics Teacher
I tried to solve this Limit without using derivative (L'hospital) and Big O notation. Although I get the answer, I don't know if the technique I'm using definitely correct.
And here is my method:
\begin{align*}\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x&=\lim_{x \to \infty} \left(\frac {2x}{\sqrt{x^2+2x+3} +\sqrt{x^2+3}}\right)^x\\&=\lim_{x \to \infty}\frac{1}{ \left(\frac {\sqrt{x^2+2x+3} +\sqrt{x^2+3}}{2x}\right)^x}\end{align*}
Then, I define a new function here
$$y(x)=\sqrt{x^2+2x+3} +\sqrt{x^2+3}-2x-1$$
We have
\begin{align*} \lim _{x\to\infty} y(x)&=\lim_{x \to \infty}\sqrt{x^2+2x+3} +\sqrt{x^2+3}-2x-1\\ &=\lim_{x \to \infty}(\sqrt{x^2+2x+3}-(x+1))+(\sqrt{x^2+3}-x)\\ &=\lim_{x \to \infty}\frac{2}{\sqrt{x^2+2x+3}+x+1}+ \lim_{x \to \infty}\frac{3}{\sqrt{x^2+3}+x}\\ &=0. \end{align*}
This implies that $$\lim_{x \to \infty}\frac{2x}{y(x)+1}=\infty$$
Therefore,
\begin{align*} \lim_{x \to \infty}\frac{1}{ \left(\frac {\sqrt{x^2+2x+3} +\sqrt{x^2+3}}{2x}\right)^x}&=\lim_{x \to\infty} \frac{1}{ \left(\frac{y(x)+2x+1}{2x} \right)^x}\\ &=\lim_{x \to\infty} \frac{1}{ \left(1+\frac{y(x)+1}{2x} \right)^x}\\ &=\lim_{x \to \infty}\frac{1}{\left( \left( 1+\frac{1}{\frac{2x}{y(x)+1}}\right)^{\frac{2x}{y(x)+1}}\right)^{\frac{y(x)+1}{2}}}\\ & \end{align*}
Here, we define two functions: $$f(x)=\left( 1+\frac{1}{\frac{2x}{y(x)+1}}\right)^{\frac{2x}{y(x)+1}},\quad g(x)=\frac{y(x)+1}{2}.$$
We deduce that, $$\lim_{x\to\infty} f(x)=e>0,\quad \lim_{x\to\infty} g(x)=\frac 12>0.$$ Thus, the limit $$\lim_{x\to\infty} f(x)^{g(x)}$$ exists and is finite.
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Finally we get,
\begin{align*} \lim_{x \to \infty}\frac{1}{\left( \left( 1+\frac{1}{\frac{2x}{y(x)+1}}\right)^{\frac{2x}{y(x)+1}}\right)^{\frac{y(x)+1}{2}}} &=\frac{1}{\lim_{x \to \infty}\left( \left( \left( 1+\frac{1}{\frac{2x}{y(x)+1}}\right)^{\frac{2x}{y(x)+1}}\right)^{\frac{y(x)+1}{2}}\right)}\\ &=\frac{1}{\left(\lim_{x\to\infty} \left( 1+\frac{1}{\frac{2x}{y(x)+1}} \right)^{\frac{2x}{y(x)+1}}\right)^{ \lim_{x\to\infty} \frac{y(x)+1}{2}}}\\ &=\frac {1}{e^{\frac12}}=\frac{\sqrt e}{e}.\\&& \end{align*}
Is the method I use correct?
I have received criticisms against my work. What can I do to make the method I use, rigorous? What are the points I missed in the method?
Thank you!
• @DMcMor Thank you for edit my bad $\LaTeX$ – lone student Apr 12 '19 at 22:10
• Honestly your TeX was pretty good! I just aligned the equations to make it a bit easier to see. – DMcMor Apr 12 '19 at 22:13
• I'm also a BlackPenRedPen fan! (or is it BlackPenRedPenBluePen?) – Toby Mak Apr 13 '19 at 10:57
• $x^y$ is continuous near $\left(e,\frac12\right)$; thus, $\lim\limits_{x\to\infty}f(x)^{g(x)}=\lim\limits_{x\to\infty}f(x)^{\lim\limits_{x\to\infty}g(x)}$ when $\lim\limits_{x\to\infty}f(x)=e$ and $\lim\limits_{x\to\infty}g(x)=\frac12$ – robjohn Apr 24 '19 at 9:59
• No. By using the fact that $x^y$ is continuous at this point, it makes your argument rigorous. – robjohn Apr 24 '19 at 10:19
Your math looks good! I'd maybe just an extra step here and there to make it clear what your doing. Things like showing that you're multiplying by conjugates and maybe a change of variables, say $$z = \frac{2x}{y(x)+1},$$ near the end so it's a bit clearer where the $$e$$ comes from. Otherwise everything looks good! This is a tricky limit, I really like your solution.
The solution appears to be correct. Just for the sake of sanity, here's a different argument, based on the idea that knowing derivatives is knowing many limits. | {
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First, find the limit of the logarithm of the beast, which is best treated also with the substitution $$x=1/t$$, which makes us trying to find $$\lim_{t\to0^+}\frac{1}{t}\log\left(\frac{\sqrt{1+2t+3t^2}-\sqrt{1+3t^2}}{t}\right) = \lim_{t\to0^+}\frac{1}{t}\log\left(\frac{2}{\sqrt{1+2t+3t^2}+\sqrt{1+3t^2}}\right)$$ This can be rewritten as $$\lim_{t\to0^+}-\frac{\log\bigl(\sqrt{1+2t+3t^2}+\sqrt{1+3t^2}\,\bigr)-\log2}{t}$$ which is the negative of the derivative at $$0$$ of $$f(t)=\log\bigl(\sqrt{1+2t+3t^2}+\sqrt{1+3t^2}\,\bigr)$$ Since $$f'(t)=\frac{1}{\sqrt{1+2t+3t^2}+\sqrt{1+3t^2}}\left(\frac{1+3t}{\sqrt{1+2t+3t^2}}+\frac{3t}{\sqrt{1+3t^2}}\right)$$ we have $$f'(0)=1/2$$ and therefore the limit is $$-1/2$$, so your given limit $$e^{-1/2}$$
Your approach is correct but its presentation / application is more complicated than needed here.
Here is how you can use the same approach with much less effort. You have already observed that the base $$F(x) =\sqrt {x^2+2x+3}-\sqrt{x^2+3}$$ tends to $$1$$ as $$x\to\infty$$. Now the expression under limit can be written as $$\{F(x) \} ^x=\{\{1+(F(x)-1)\}^{1/(F(x)-1)}\}^{x(F(x)-1)}$$ The inner expression tends to $$e$$ and the exponent $$x(F(x) - 1)\to -1/2$$ so that the desired limit is $$e^{-1/2}$$.
Another part of your approach is that it involves the tricky use of subtracting $$2x+1$$ from $$y(x)$$. For those who are experienced in the art of calculus this step is obvious via the approximation $$\sqrt{x^2+2ax+b}\approx x+a$$ but it may appear a bit mysterious for a novice. It is best to either explain this part or remove it altogether as I have done it in my answer.
Also note that your approach uses the following limits / rules (it is not necessary to point them out explicitly unless demanded by some strict examiner) :
• $$\lim_{x\to\infty} \left(1+\frac{1}{x}\right)^x=e$$
• If $$\lim_{x\to\infty} f(x) =a>0$$ and $$\lim_{x\to\infty} g(x) =b$$ then $$\{f(x) \} ^{g(x)} \to a^b$$ as $$x\to\infty$$. | {
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By squaring, we can verify that $$x\le\sqrt{x^2+3}\le x\left(1+\frac3{2x^2}\right)\tag1$$ and $$x+1\le\sqrt{x^2+2x+3}\le(x+1)\left(1+\frac1{x(x+1)}\right)\tag2$$ Adding $$(1)$$ and $$(2)$$ gives $$2x+1\le\sqrt{x^2+2x+3}+\sqrt{x^2+3}\le(2x+1)\left(1+\frac3{2x^2}\right)\tag3$$ Multiplying numerator and denominator by $$\sqrt{x^2+2x+3}+\sqrt{x^2+3}$$ gives $$\sqrt{x^2+2x+3}-\sqrt{x^2+3}=\frac{2x}{\sqrt{x^2+2x+3}+\sqrt{x^2+3}}\tag4$$ Bernoulli and cross multiplying yield $$1-\frac3{2x}\le\left(1-\frac3{2x^2}\right)^x\le\left(1+\frac3{2x^2}\right)^{-x}\tag5$$ Therefore $$(3)$$, $$(4)$$, and $$(5)$$ yield $$\left(\frac{2x}{2x+1}\right)^x\left(1-\frac3{2x}\right)\le\left(\sqrt{x^2+2x+3}-\sqrt{x^2+3}\right)^x\le\left(\frac{2x}{2x+1}\right)^x\tag6$$ The Squeeze Theorem then says $$\lim_{x\to\infty}\left(\sqrt{x^2+2x+3}-\sqrt{x^2+3}\right)^x=e^{-1/2}\tag7$$
Your method is correct if and only if you truly understand how to rigorously prove a critical step where you effectively claim $$\lim_{x∈\mathbb{R}→∞} (1+\frac1x)^{f(x)} = \lim_{x∈\mathbb{R}→∞} e^{f(x)/x}$$. Note that this requires real exponentiation, and the simplest proof of this would involve the asymptotic expansions for $$\exp,\ln$$, hence I personally think it's misleading to think of your method as successfully evading asymptotic expansions. To preempt the very common bogus proof, note that this claim does not follow from $$\lim_{x∈\mathbb{R}→∞} (1+\frac1x)^x = e$$. | {
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After I first posted my answer, you edited your attempt in a non-trivial way. (Please do not edit your question in this way in the future, as it invalidates existing answers.) It still has the same conceptual error, just with different appearance. In this second attempt, you effectively claim that if $$\lim_{x∈\mathbb{R}→∞} g(x) = c$$ then $$\lim_{x∈\mathbb{R}→∞} f(x)^{g(x)} = \lim_{x∈\mathbb{R}→∞} f(x)^c$$ if the latter limit exists. This is not in general true! If you can state and prove the correct theorem of this sort (in a comment), then I will believe that you understand it.
The common underlying error is that you replaced part of a limit expression with its limit, which is in general invalid!
• Comments are not for extended discussion; this conversation has been moved to chat. – quid Apr 13 '19 at 10:41
• "The common underlying error is that you replaced part of a limit expression with its limit, which is in general invalid!" Definitely Wrong argument. Because, all limit rules are "invalid generally". You cannot show the applicable limit rule in all cases. – Zaharyas Apr 20 '19 at 11:28
• Your last sentence is mathematically non sense. (as it appears). – Zaharyas Apr 20 '19 at 11:45
• I'm IMO gold medalist. Now, I study bachelors. – Zaharyas Apr 20 '19 at 11:52
• IMO does not contain calculus or limit theorems. The content ranges from extremely difficult algebra and pre-calculus problems to problems on branches of mathematics not conventionally covered at school and often not at university level either, such as projective and complex geometry, functional equations, combinatorics, and well-grounded number theory, of which extensive knowledge of theorems is required. You asked me about my mathematical background. I gave the answer. – Zaharyas Apr 20 '19 at 18:46 | {
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The proof can be accelerated, using binomial Maclaurin series in the form of $$\sqrt{x^2+2x+3} = (x+1)\sqrt{1+\dfrac2{(x+1)^2}} = (x+1)\left(1 + \dfrac1{(x+1)^2}+O\left(x^{-4}\right)\right)$$ $$= x+1+\dfrac1x+O\left(x^{-2}\right),$$ $$\sqrt{x^2+3} = x\left(1+\dfrac3{2x^2}+O(x^{-4})\right) = x + \dfrac3{2x}+O(x^{-3}).$$ Then $$\ln L = \ln \lim\limits_{x\to\infty} \left(\sqrt{x^2+2x+3}-\sqrt{x^2+3}\right)^x = \lim\limits_{x\to\infty} x\ln\left(1-\frac1{2x}+O\left(x^{-2}\right)\right)$$ $$= \lim\limits_{x\to\infty} x\left(-\frac1{2x}+O\left(x^{-2}\right)\right) = -\frac12,$$ $$L=e^{\Large^{-\frac12}}.$$ | {
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