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## Research and analysis of the modified intake system for the Formula Student vehicle

### Krzysztof Retka

#### Abstract

The aim of the present study is to evaluate the impact of the intake system modification, which consists in installing the turbocharger with air cooling, on the motor fuel parameters. The desire to increase the power output and torque is related to the formula student competition in which the car will be involved, that is why all the work done on the vehicle comply with the SAE regulations. Changes made to the intake system resulted in the need of modifiying other engine systems, such as the lubrication and cooling system. Due to the direct connection of all of the working systems, the subject of their interaction was also raised. The next stage of research is empirical studies of a bolide on a chassis dynamometer, in order to illustrate the impact of the intake system modification on engine parameters, such as power output or torque. At the same time, due to the registered data it was possible to calculate the fuel consumption in a rotation speed function. This made it possible to assess the influence of the hypertension boost on the amount of consumed mixture. Due to the obtained data regarding course of the derivative rotational speed in regard to time during the acceleration of the car, it was possible to assess the impact of the boost pressure on the dynamic properties of the vehicle.
Diploma type
Engineer's / Bachelor of Science
Diploma type
Engineer's thesis
Author
Krzysztof Retka (FACME) Krzysztof Retka,, Faculty of Automotive and Construction Machinery Engineering (FACME)
Title in Polish
Badania i analiza modyfikacji układu dolotowego do pojazdu typu Formuła Student
Supervisor
Zdzisław Chłopek (FACME/IV) Zdzisław Chłopek,, Institute of Vehicles (FACME/IV)Faculty of Automotive and Construction Machinery Engineering (FACME)
Jarosław Seńko (FACME/IV) Jarosław Seńko,, Institute of Vehicles (FACME/IV)Faculty of Automotive and Construction Machinery Engineering (FACME)
Certifying unit
Faculty of Automotive and Construction Machinery Engineering (FACME)
Affiliation unit
Institute of Automotive Engineering (FACME/IV)
Study subject / specialization
, Mechanika i Budowa Maszyn
Language
(pl) Polish
Status
Finished
Defense Date
25-01-2016
Issue date (year)
2016
Reviewers
Jarosław Seńko (FACME/IV) Jarosław Seńko,, Institute of Vehicles (FACME/IV)Faculty of Automotive and Construction Machinery Engineering (FACME) Piotr Orliński (FACME/IV) Piotr Orliński,, Institute of Vehicles (FACME/IV)Faculty of Automotive and Construction Machinery Engineering (FACME)
Keywords in Polish
1. Formuła student 2. Turbodoładowanie 3. Układ dolotowy
Keywords in English
1.Formula student 2.Turbocharging 3.Intake system
Abstract in Polish
File
• File: 1
Krzysztof Retka.pdf
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Identyfikator pracy APD: 8923

Uniform Resource Identifier
https://repo.pw.edu.pl/info/bachelor/WUT6021329b8f764a91b259d6d603c67192/
URN
urn:pw-repo:WUT6021329b8f764a91b259d6d603c67192

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## Toward Autonomic Grids: Analyzing the Job Flow with Affinity Streaming

author: Xiangliang Zhang, Laboratory for Computer Science, University of Paris-Sud 11
published: Sept. 14, 2009,   recorded: June 2009,   views: 4320
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# Description

The Affinity Propagation (AP) clustering algorithm proposed by Frey and Dueck (2007) provides an understandable, nearly optimal summary of a dataset, albeit with quadratic computational complexity. This paper, motivated by Autonomic Computing, extends AP to the data streaming framework. Firstly a hierarchical strategy is used to reduce the complexity to ${\cal O}(N^{1+\e})$; the distortion loss incurred is analyzed in relation with the dimension of the data items. Secondly, a coupling with a change detection test is used to cope with non-stationary data distribution, and rebuild the model as needed. The presented approach StrAP is applied to the stream of jobs submitted to the EGEE Grid, providing an understandable description of the job flow and enabling the system administrator to spot online some sources of failures.

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6.

3 . 7. Kinetic theory explains the behaviour of gases based on the idea that the gas consists of rapidly moving atoms or molecules. Kinetic energy of the molecule in directly proportional to absolute temperature. The temperature of a gas is a measure of the mean kinetic energy of the gas.

The kinetic theory of gases explains the three macroscopic properties of a gas in terms of the microscopic nature of atoms and molecules making up the gas. 5. a plot of PV/RT against Pext gives a lower value than the ideal at moderate pressures and a higher value than the ideal at very high pressures for most gases. select all the statements that correctly account for this behavior-at low pressures gas particles repel each other In a collection of gas particles, the average kinetic energy Related terms: Nucleation The collisions between gas molecules or molecule and wall of container are completely elastic meaning no energy is gained or lost from collisions. This state is called molecular chaos. KINETIC THEORY OF AN IDEAL GAS. The following list State the kinetic theory of gases.

The model, called the kinetic theory of gases, assumes that the molecules Physical Chemistry. This model of gases explains some of the physical properties of gases. The kinetic theory of gases has developed a model that explains the behavior of molecules, which should further explain the behavior of an ideal gas. The kinetic theory of gases describes this state of matter as composed of tiny particles in constant motion with a lot of distance between the particles. Statistical Description of a Gas 5 1.1. V = Volume of gas. Kinetic Molecular Theory can be used to explain both Charles' and Boyle's Laws.

All state changes can be defined in terms of heat absorbing and releasing which ultimately is affecting the kinetic energy of particles. 4.

-All collisions between gas molecules are perfectly elastic; all kinetic energy is conserved. Further, under suitable Postulates or assumptions of kinetic theory of gases. The kinetic theory of gases, developed in the 19th century, describes gases as assemblages of tiny particles ( atom s or molecule s) in constant motion and contributed much to an understanding of their behaviour. Although the assumptions are not completely true, they are approximately true at low 4. The volume of molecule is very small in comparison to the volume occupied by gas i.e., the size of molecule is infinitesimally small. Introducing Boltzmann's constant k = R / NA , it is easy to check from our result for the All gases at a given temperature have same Kinetic Energy. State the ideas of the kinetic molecular theory of gases.

The Kinetic Equation: Maxwell derived an equation on the basis of assumptions Of Kinetic Theory Of gases as. Kinetic Theory and Gas Pressure. The kinetic particle theory explains the properties of solids, liquids and gases. Postulates of Kinetic Gas Theory. The following are the postulates of the Kinetic gas theory: Gases are made up of many small tiny, and discrete particles called molecules. Molecules of gas are well separated from each other. The volume occupied by molecules of a gas is negligible compared to the volume of gas. Ideal and real gases. Physics DF025 Chapter 14 Consider 1 mole of gas at standard temperature and pressure (S.T.P. In the 19th century Maxwell, Boltzmann, and others proposed the theory of kinetics. (2) The molecules of a gas are in a state of Following are the five postulated of the kinetic theory of gases: Gas is a composition of a large number of molecules that are constantly in a random movement. Kinetics means the study of motion, and in this case motions of gas Energy8 1.3. There are energy changes when changes in state occur.

0 votes . ), T = 273.15 K, P = 101.3 kPa and Vm = 0.0224 m3 From equation (3), 3 PVm 101.3 10 0.0224 R T 1 273.15 1 R 8.31 J K mol where R is called molar gas constant and its value is the same for all gases. answered Jun 21, 2021 by anonymous.

Motion of particles is random . The constituent particles of a kind of matter are identical in all respects. STUDY GUIDE: Kinetic Theory of Gases . Validity of the Classical Limit21 2.3.1. Gases are made up of large number of the minute particles. 0 votes . The kinetic molecular theory (KMT) describes the behavior of ideal gases at the particle level.

Ans: The kinetic molecular theory of gases is stated in the following principles: 1. Equation of State and Temperature19 2.3.

1. Kinetic theory is based on an atomic model of matter.

Kinetic theory explains the behaviour of gases based on the idea that the gas consists of rapidly moving atoms or molecules. b) The intermolecular force of attraction between gas molecules are negligible. State the ideas of the kinetic molecular theory of gases. The kinetic theory of gases states that the mean free path between collisions of atoms or molecules in a gas is given by L=kBT/(21/2 PD2), where D is the atomic or molecular diameter of the species (Herman and Sitter, 1989); From: Reference Module in Materials Science and Materials Engineering, 2016. The kinetic theory of gases is the study that relates the microscopic properties of gas molecules (like speed, momentum, kinetic energies etc..)with the macroscopic properties of gas molecules velocity changes but the molecular density is constant in steady state. Usually, the physical

They continue in a straight line until they collide with somethingusually each other or the walls of their This will clear students doubts about any question and improve application skills while preparing for board exams. The k inetic theory states that every substance consists of atoms or molecules and that the atom or molecule moves continuously carelessly. Using the kinetic molecular theory, explain how an increase in the number of moles of gas at constant volume and temperature affects the pressure. Actual volume of the gaseous molecule very small. The kinetic theory of gases is a set of three assumptions about the behavior of ideal gases. The kinetic theory of gases is concerned with molecules in motion and with the microscopic and macroscopic consequences of such motion in a gas. State the kinetic theory of gases. with solved examples & diagrams! Kinetic Theory of Gases. Kinetic Molecular Theory states that gas particles are in constant motion and exhibit perfectly elastic collisions. 5. Kinetic theory of gases relates the macroscopic property of the gas, like Temperature, Pressure, Volume to the microscopic property of the gas, like speed, momentum, position. It states that equal volumes of all gases contain an equal number of molecules if Temperature and Pressure are the same. The kinetic theory of gases is concerned with molecules in motion and with the microscopic and macroscopic consequences of such motion in a gas.

b. The Theory was put up forward by Part II.Kinetic Theory 5 1. See also solid. 4. Kinetic theory of gases definition, a theory that the particles in a gas move freely and rapidly along straight lines but often collide, resulting in variations in their velocity and Whereas Avogadros theory of diatomic molecules was ignored for 50 years, the kinetic theory of gases was rejected for more than a century. Kinetic Theory of Gases is shared under a not declared license and was authored, remixed, and/or curated by Jim Clark. Ideal and real gases. The time it takes to collide is negligible . Other gases include those that make breads and This is possible as the inter-atomic forces, which are short range forces that are important for solids and liquids, can be neglected for gases. The kinetic theory is widely accepted explanation of the theory of gas behaviour. The Nature of Gases Three basic assumptions of the kinetic theory as it applies to gases: 1. 3. The kinetic theory of gases is the study that relates the microscopic properties of gas molecules (like speed, momentum, kinetic energies etc..)with the macroscopic properties of gas Basic kinetic theory ideas about solids, liquids and gases, and changes of state. and represents dynamic particle or The average kinetic energy of a collection of gas particles is directly proportional to absolute temperature only. Isotropic Distributions13 2. Molecules of gas attract on one another. - [Instructor] So I wanna talk to you a little more about the kinetic-molecular theory of gases. What this basically says is that the macroscopic properties of a gas, like the pressure or the volume or the temperature are just a result of the microscopic properties of the gas molecules, like the position and the speeds of these molecules.

Where P = Pressure of gas. The particles are moving all the time. Heavier particles move slower then lighter particles. Gases are literally all around usthe air that we breathe is a mixture of gases. Matter is composed of very small particles called atoms and molecules. Kinetic Theory of Gases: Avogadros Number, Degrees of Freedom, Molar Specific Heats, Gas Law, RMS Speed, Pressure, etc. Each one is a perfectly identical elastic sphere. Kinetic Calculation of Pressure11 1.5. volume (V) and temperature (T) of the given state for an ideal gas is known as the ideal gas equation or equation of states. 2. a) A gas consists of very large number of extremely small particles known as molecules. The kinetic theory states that matter is made up of particles and these particles are constantly in motion i.e. The five main postulates of the KMT are as follows: (1) the particles in a gas are in constant, random motion, (2) the combined volume of the particles is negligible, (3) the particles exert no forces on one another, (4) any collisions between the particles are completely elastic, and (5) Maxwells Distribution16 2.2. Assumptions of kinetic theory of gases. The kinetic theory of gases can be stated as three postulates: -The particles of a substance constantly move in random motion, and during this random motion, they collide with each other and with the walls of the container. 74 views. Postulates for the Kinetic Theory Objective 2 asks you to list the assumptions that define the microscopic model of an ideal gas. 1)Every gas is made up of a large number of extremely small particles called molecules. What are the assumptions made in the kinetic theory of gases? 1) All gases are made up of molecules that are constantly and persistently moving in random directions. 2) All the collisions between molecules and even between molecules and walls are considered to be elastic. 3) All the molecules in a certain gas sample obey Newtons laws of motion. Ideal gas equation is PV = nRT.

The molecules are in constant random motion, and there is an energy (mass x square of the velocity) associated Answers. Using the kinetic molecular theory, explain how an increase in the number of moles of gas at The actual atomic theory got established more than 150 years later.

Kinetic Molecular Theory states that gas particles are in constant motion and exhibit perfectly elastic collisions. Effect of gravity on gas molecule is neglected . This equation can easily be derived from the combination of Boyles law, Charless law, and Avogadros law. A gas consist of a very large number of molecules which are perfect elastic spheres and are identical in all respects for a given gas and are different for different gases. Mass: f a. According to the kinetic molecular theory, the molecules of two gases at the same temperature will possess the same average kinetic energy. All the molecules of a particular gas are identical in mass and size and differ in these from gas to gas. The kinetic theory of gases can define the general properties of gases at very low concentration. for n moles of gas, that is, n = N / NA , with NA Avogadro's number and R the gas constant. Kinetic theory of gasses is a theory based on a simplified molecular or particle description of gas and from this many gross properties of the gas can be derived.

Figure:Kinetic theory of gases:phase space! 1.1 Kinetic Theory of Gases Out of the three prominent states of matter, solid, liquid and gas, the kinetic theory o f gases is perhaps the most developed and complete. asked Jun 21, 2021 in Physics Form 3 by anonymous. The field requires a great amount of knowledge and practical use in the ideas of the kinetic theory of gases, and it links the kinetic theory of gases with the solid state physics through the study What are the five principles of the kinetic molecular theory of gases? The kinetic molecular theory of gases describes this state of matter as composed of tiny particles in constant motion with a lot of distance between the particles. Kinetic Molecular Theory states that gas particles are in constant motion and exhibit perfectly elastic collisions. to x + dx, y to y + dy, z to z + dz, and velocity components in the ranges u to u + du, v to v + dv, w to w + dw. It models the properties of matter in terms of continuous random motion of molecules. The molecules of a gas are in Kinetic Theory. Following are the three main components of the kinetic theory of gas:When molecules collide with each other, no energy is gained or lost.The space occupied by the molecules of gas in a container is very negligible.These molecules always have linear motion. Gaseous State : Kinetic theory of gases Kinetic theory of gases (1) Kinetic theory was developed by Bernoulli, Joule, Clausius, Maxwell and Boltzmann etc. The Kinetic-Molecular Theory Explains the Behavior of Gases, Part I. Recalling that gas pressure is exerted by rapidly moving gas molecules and depends directly on the number of molecules hitting a unit area of the wall per unit of time, we see that the KMT conceptually explains the behavior of a gas as follows: Amontonss law. It is the study of gas molecules at the macroscopic level. Kinetic Theory of Gases states that gaseous particles are in constant motion and undergo perfectly elastic collisions.

One-particle phase space: position x 2 Rd, momentum p 2Rd f (t;x;p) the probability density distribution. Gas is composed of particles- usually molecules or atoms Small, hard spheres

In this Kinetic Theory of Gases. to x + dx, y to y + dy, z Kinetic theory of matter states that Matter is made up of those substances or particles which are constantly moving.. 8. Introducing Boltzmanns constant k = R / NA, it is easy to check from our result for the pressure and the ideal gas law that the average molecular kinetic energy is proportional to The Kinetic-Molecular Theory Explains the Behavior of Gases, Part I. Recalling that gas pressure is exerted by rapidly moving gas molecules and depends directly on the number P V = 1 3 m n u 2. The main postulates of the kinetic theory of matter are: 1.

But here, we will derive the equation from the kinetic theory of gases. The British scientist James Clerk The average kinetic energy of a collection of gas particles is directly proportional to absolute temperature only. Kinetic Molecular Theory states that gas particles are in constant motion and exhibit perfectly elastic collisions.Kinetic Molecular Theory can be used to explain both Charles' and Boyle's Laws. Postulates of Kinetic theory of gases.

Gases consist of particles, which have the following properties: The particles are so small compared to the distances between them that the The detailed, step-by-step solutions will help you understand the concepts better Because most of the Various texts list these in various orders and forms. 2.1: Prelude to The Kinetic Theory of Gases.

At the macroscopic level, it is the study of gas molecules. 2.

The space between Chemistry Gases Kinetic Theory of Gases. The basic assumptions of kinetic theory are : A gas consist of particles called molecules which move randomly in all directions. gas laws; 1 Answer. The kinetic theory of gases correlates between macroscopic properties and microscopic phenomena.

Written by a former Princeton University professor specializing in the thermal properties of matter, this monograph and text was designed for first-year students of physical chemistry who require further details of kinetic theory. There is no loss of kinetic energy.

If v 1 and v 2 are the average velocities of the The kinetic theory of gases, developed in the 19th century, describes gases as assemblages of tiny particles ( atom s or molecule s) in constant motion and contributed much to an Calculate the rms speed of CO 2 at 40C. The volume of the molecules is negligible as the distance between the gas molecules is greater than the size of the molecules. This can then be used to analyse the interactions found in each of the states of matter. Boyle's Law and Kinetic Theory of Gases. m = mass of one kinetic theory of gases, a theory based on a simplified molecular or particle description of a gas, from which many gross properties of the gas can be derived. Because most of a gas is empty space, a gas has a low density and can expand or contract under the appropriate influence. The kinetic theory of gases describes this state of matter as composed of tiny particles in constant motion, with a lot of distance between the particles. Kinetic molecular theory (also known as particle theory) states that all matter is made up particles and these particles are always in motion. Kinetic Theory Of Gases: The common characteristics of gases can be explained by assuming that gases consist of molecules which are far apart and are in a state of constant Basic kinetic theory ideas about solids, liquids and gases, and changes of state. To be more precise, this theory and formula help determine macroscopic properties of a gas, if you already know the velocity value or internal molecular energy of the compound in question. Pressure is exerted by a gas 3. Middle School Praxis Kinetic Theory 3 Boyles Law Boyles law (named after Robert Boyle and his work in 1662) states that the pressure and volume of a gas are inversely proportional to one another, or PV = k, where P is pressure, V is volume, and k is a constant of proportionality. Kinetic molecular theory is useful in The kinetic theory relates the independent motion If a gas sample is left for a sufficient time, it eventually comes to a steady-state. 4. Ans: According to the kinetic theory of gases, the average kinetic energy of gas only depends on its absolute temperature and is independent of the nature of the gas. Introduction5 1.2. Here P0 is the pressure of the gas at t = 0 and k the time constant of the gauge head (see Section 2.3.6 ). The term gas can also mean gasoline, natural gas, or the anesthetic nitrous oxide. 6. The ideal gas equation. The kinetic theory states that: All matter is made up of particles. Gases consist of tiny particles of matter that are in constant motion. This model of gases explains they have kinetic/movement energy. The ideal gas law can be expressed in terms of the mass of the gass molecules and $$\bar{v^2}$$, the average of the molecular speed squared, instead of the temperature.

c) Molecules are always in the state of random motion, i.e., they are moving in all possible directions with all possible velocities. It describes how interactions between molecules influence gas characteristics such as temperature and Classical Ideal Gas in Equilibrium 15 2.1. Brownian motion is with solved examples & diagrams! The model, called the kinetic theory of gases, assumes that the molecules are very small relative to the distance between molecules. According to kinetic theory: Gases consist of particles in constant, random motion.

Thermodynamic Limit9 1.4. Kinetic Theory of Gases: Avogadros Number, Degrees of Freedom, Molar Specific Heats, Gas Law, RMS Speed, Pressure, etc. 9. volume (V) The ideal gas equation. 7. The kinetic theory of gases is used to explain the behavior of gas molecules. Kinetic theory of What is the kinetic theory of gases? (1) A gas consists of a very large number of molecules. ideal gas law. 2. Balbharati solutions for Physics 12th Standard HSC Maharashtra State Board chapter 3 (Kinetic Theory of Gases and Radiation) include all questions with solution and detail explanation. Kinetic Theory of Gases I Ideal Gas The Ideal Gas Law Pressure and Temperature Internal Energy Mean Free Path Molecules collide elastically with other molecules Molar Specific Heat Constant Volume Adiabatic Process Equipartition of Energy Monatomic Gases Kinetic Theory of Gases I Ideal Gas The Ideal Gas Law Pressure and Temperature Internal Energy Mean Free Path

The kinetic theory of gases states that the time rate of change of pressure p of an ideal gas in a vessel of volume V at constant temperature, evacuated by a pump of constant volumetric speed S = dV/dt, is. Boyle's Law and Charles' Law. Preview. Calculate the rms speed of CO 2 at 40C. The Kinetic Theory of gases states that the gases are made of small particles (atoms or molecules) which are in random motion. physical law that relates the pressure and volume of a gas, far from liquefaction, to the number of gas molecules or number of moles of gas and the temperature

The continuous bombardment of the gas molecules against the walls of the container results in an increase in the gas pressure. This helps us to determine whether that matter is in a solid, liquid, or gas state. The kinetic theory of gases also describes properties such as temperature, coherence, and thermal conductivity. 2. The basic assumption of kinetic theory is that the measurable properties of gases, liquids, and solids reflect the combined actions of countless numbers of atoms and molecules. Kinetic theory of gases. It is a theory that derived that The five postulates of the kinetic theory of gases are as follows: Gas is made up of a vast number of molecules that are constantly moving at random. Kinetic theory is the atomic description of gases as well as liquids and solids. These particles have space or gaps between them which is known as interparticular or intermolecular space. Gas molecules are identical. This assumption of kinetic theory matches the The energy level of the particles depends upon the temperature possessed by the matter. Explanation: 1) Gas formed by point-like particles (#"volume"~~0#); 2) No intermolecualar attractions between the molecules of the gas; 3) Random motion; 4) Elastic collisions. Kinetic Molecular Theory of Gases.


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# Cars plates

How many different licence plates can country have, given that they use 3 letters followed by 3 digits?

n =  17576000

### Step-by-step explanation:

$n=2{6}^{3}\cdot 1{0}^{3}=17576000$

Did you find an error or inaccuracy? Feel free to write us. Thank you!

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How many 3 letter "words" are possible using 14 letters of the alphabet? a) n - without repetition b) m - with repetition
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From the numbers 1, 2, 3, 4, 5 create three-digit numbers that digits not repeat and number is divisible by 2. How many numbers are there?
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# Math Help - Asymptote of polar curve

1. ## Asymptote of polar curve

Find the asymptote of

2r^2=tan2α
How to find it?

2. Originally Posted by roshanhero
Find the asymptote of

2r^2=tan2α
How to find it?
Linear oblique asymptotes of the form $\alpha = \beta$ occur for values of $\alpha$ that make $\tan (2 \alpha)$ undefined.

And you know when $\tan (2 \alpha)$ is undefined, right .....?

3. Hello, roshanhero!

Find the asymptotes of: . $2r^2\:=\:\tan2\theta$

In polar coordinates, asymptotes occur where $r \to \infty$

We have: . $r^2 \;=\;\frac{1}{2}\tan2\theta$

We see that asymptotes occur where: $2\theta \:=\:\frac{\pi}{2} + \pi n \quad\Rightarrow\quad \theta \:=\:\frac{\pi}{4} + \frac{\pi}{2}n$

4. I am terribly sorry,but I simply could not get it at all.I think I can easily find the horizontal,vertical and oblique asymptotes of the cartesian curve but I just don't have any concept regarding of finding asymptotes of the polar curve.Please explain it to me from the scratch.

5. Originally Posted by roshanhero
I am terribly sorry,but I simply could not get it at all.I think I can easily find the horizontal,vertical and oblique asymptotes of the cartesian curve but I just don't have any concept regarding of finding asymptotes of the polar curve.Please explain it to me from the scratch.
What part of the discussion do you not understand? Do oyu know how to solve a trigometric equation such as $\cos (2 \theta) = 0$ ?

It may be that you're weak in the mathematical background assumed for these sorts of problems. In which case you're best advised to go back and revise the prerequisite mathematical concepts.

6. We have been teached to find the asymptotes of polar curve by using formula-
Rsin(q -a )=1/f'(a )
Where a denotes the roots of f (q )=0
I appiled it but failed.
Can we show me how to do it by using this formula.


text:
My Math Forum How to find the work in parabolic trajectory when the unknowns are angle and speed?
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 March 5th, 2019, 01:19 PM #1 Member     Joined: Jun 2017 From: Lima, Peru Posts: 97 Thanks: 1 Math Focus: Calculus How to find the work in parabolic trajectory when the unknowns are angle and speed? I'm confused about this question from my book. Apparently it is supposed to be simple but upon reading it several times I don't know how to proceed. I need also assistance with some theoretical or conceptual knowledge on how to find the work in a parabolic trajectory. The problem is as follows: A soccer player kicks a ball from rest over the head of a rival. The ball falls $\textrm{14 m}$ from the player who kicked the ball after $\textrm{1.4 s}$. At a TV studio the sports commentator decides to calculate the work done by the player in that pass. The sports programme tells the audience that the official ball has a mass of $\textrm{400 grams}$ and the trajectory is a parabola. Find the work calculated by the sports show commentator. Consider the gravity is $10\frac{m}{s^{2}}$ $\begin{array}{ll} 1.&\textrm{27.25 J}\\ 2.&\textrm{32.8 J}\\ 3.&\textrm{35 J}\\ 4.&\textrm{17.5 J}\\ 5.&\textrm{29.8 J}\\ \end{array}$ From the knowledge I have. In order to find the work done in this case it would be from the fact that this is given by the difference in the kinetic energy of the ball. The reason for this part of my judgment is due that when the ball touches ground there is no potential energy but there is kinetic energy. There is the fact that the energy of the soccer player which is needed to find the work, is the same that the energy he uses to kick the ball and that's the energy the ball has. Hence it would meant that what is needed to be found is the difference in kinetic energy the ball has so the work can be found. In my attempt to find the launching angle I tried this: $y = y_{o}+v_{o}\sin\omega-\frac{1}{2}gt^2$ Since it is given $t=1.4$ and $g=10$, (for brevity purposes I'm omitting units) $0=v_{o}(1.4)\sin\omega-\frac{1}{2}(10)(1.4)^2$ $v_{o}\sin\omega=(5)(1.4)$ The other known is given, when $x=14$: $x=v_{o}t\cos\omega$ $14=v_{o}(1.4)\cos\omega$ Thus: $v_{o}\cos\omega=10$ From this can be obtained by dividing both equations: $\frac{5(1.4)}{10}=\tan\omega$ Therefore: $\tan\omega=\frac{7}{10}$ From this I believe the initial speed can be calculated from knowing the $\cos\omega$ which can be obtained from the previous equation as follows: $\cos\omega=\frac{10}{\sqrt{149}}$ Therefore: $v_{o}=\frac{10}{\frac{10}{\sqrt{149}}}=\sqrt{149}$ Now here's where I'm stuck at: How do I make the right interpretation for finding the Work? Would I go on this route? $W=\frac{1}{2}mv^2=\frac{1}{2}(400)(10^{-3})(\sqrt{149})^2=(2)(10^{-1})(149) = \textrm{29.8 J}$ So the work found by the sportscaster would be $29.8\,J$ This answer does seem to check with one of the alternatives which is the fifth and my book tells this is the right answer, but I'm not very convinced if that would be the right one. Needless to say If what I'm doing is correct. Looking at my intuition I have these other questions. Does the kinetic energy in the x-axis is always zero?. Why to bother finding it anyway? The ball's motion in x-axis doesn't change as it remains constant so there is no change in kinetic speed hence the work on $\textrm{x-axis}$ is zero. Isn't it? Is there any change of kinetic energy in the $\textrm{y-axis}$? There is change in kinetic energy in $\textrm{y-axis}$ I presume. But If the ball goes up in the apex of the trajectory it will have zero in its speed. So, when the ball touches ground will it have the same speed when it departed?. Can this be proved? I hope somebody can help me with these doubts and more importantly offer some other alternative to easily solve this problem. By continuing in this problem I think the height can be found knowing the $v_{o}=\sqrt{149}$ and $\sin\omega=\frac{7}{\sqrt{149}}$. $y = y_{o}+v_{o}t\sin\omega-\frac{1}{2}gt^2$ Assuming kicking from ground so $y_{o}=0$ $h= 7t - 5t^2$ Using derivatives: $7-10t=0$ $t=\frac{7}{10}$ (although it was known from the problem that since it took $1.4$ seconds to get the whole trajectory then the apex would had been half of that time hence $0.7$ seconds). $y=7(\frac{7}{10})-5\left(\frac{7}{10}\right)^{2}$ $y=\frac{49}{10}-\frac{5\times 49}{10\times 10}$ $y=\frac{49}{10}\left(1-\frac{1}{2}\right)$ $y=\frac{49}{10}\left(\frac{1}{2}\right)$ $y=\frac{49}{20}$ Therefore the height attained by the ball would be $\frac{49}{20}$ meters. If I decide to use this to find the speed which will have the ball when reaches bottom I would use: This would be: $v^{2}_{f}=v^{2}_{o}-2g(y)$ $v^{2}_{f}=0^{2}-2(10)\left(-\frac{49}{20}\right)$ $v^{2}_{f}=49$ $v_{f}=\sqrt{49}=7$ Which it would meant 7 meters per second and this doesn't seem any close with what I assumed, in other words if the ball is kicked at some speed would it touch ground at that same speed. However it checks if I would use: $v_{f}=v_{o}-gt=0-(10)(0.7)=-7$ So in absolute value both are the same. Overall can somebody explain to me what's happening here? and clear out the doubts which I mentioned above?.
 March 5th, 2019, 02:24 PM #2 Math Team     Joined: Jul 2011 From: Texas Posts: 2,949 Thanks: 1555 In a parabolic trajectory under the influence of gravity, the work done by the player is the initial kinetic energy given to the ball since it started from rest. The ball lands 14m from the spot it was kicked after 1.4 sec ... $\Delta x = v_x \cdot t \implies v_x = \dfrac{\Delta x}{t} = \dfrac{14 \, m}{1.4 \, sec} = 10 \, m/sec$ $v_x$, the horizontal component of velocity remains constant since only the force of gravity acting in the vertical direction is considered (drag forces due to air resistance are ignored) Since the ball completes its flight in 1.4 sec, it takes 0.7 sec to reach its highest point above the ground. At that point, velocity in the vertical direction is zero ... $v_{yf} = v_{y0} - gt \implies v_{y0} = v_{yf} + gt = 0 + (10 \, m/sec^2)(0.7 \, sec) = 7 \, m/s$ Initial velocity of the ball is $v_0 = \sqrt{v_x^2 + v_{y0}^2} = \sqrt{10^2 + 7^2} = \sqrt{149} \, m/sec$ $KE = \dfrac{1}{2}mv_0^2 = \dfrac{1}{2}(0.4 \, kg)(149 \, m^2/sec^2) = 29.8 \, kg-m/sec^2 = 29.8 \, J$ Thanks from topsquark
March 6th, 2019, 05:37 PM   #3
Member

Joined: Jun 2017
From: Lima, Peru

Posts: 97
Thanks: 1

Math Focus: Calculus

Quote:
 Originally Posted by skeeter In a parabolic trajectory under the influence of gravity, the work done by the player is the initial kinetic energy given to the ball since it started from rest. The ball lands 14m from the spot it was kicked after 1.4 sec ... $\Delta x = v_x \cdot t \implies v_x = \dfrac{\Delta x}{t} = \dfrac{14 \, m}{1.4 \, sec} = 10 \, m/sec$ $v_x$, the horizontal component of velocity remains constant since only the force of gravity acting in the vertical direction is considered (drag forces due to air resistance are ignored) Since the ball completes its flight in 1.4 sec, it takes 0.7 sec to reach its highest point above the ground. At that point, velocity in the vertical direction is zero ... $v_{yf} = v_{y0} - gt \implies v_{y0} = v_{yf} + gt = 0 + (10 \, m/sec^2)(0.7 \, sec) = 7 \, m/s$ Initial velocity of the ball is $v_0 = \sqrt{v_x^2 + v_{y0}^2} = \sqrt{10^2 + 7^2} = \sqrt{149} \, m/sec$ $KE = \dfrac{1}{2}mv_0^2 = \dfrac{1}{2}(0.4 \, kg)(149 \, m^2/sec^2) = 29.8 \, kg-m/sec^2 = 29.8 \, J$
Thanks skeeter for the explanation, but there are still unattended questions from my original post which I do hope you may help me to clear out.

First off, why isn't considered the $v_{o}\cos\omega$ in any of your equations for the x-axis and why $v_{o}\sin\omega$ is also left out from your analysis?.

Does it makes sense to consider that the Kinetic energy solely on the $x-axis$ to be zero?. I mean since there isn't any change of the speed, wouldn't it mean that the $KE$ in this axis is zero?. Can be found the $KE$ on the $y-axis$ alone or should for this part the Energy must be calculated using the Gravitational potential energy $U$?

You've found the $v_{oy}$ by using the equation in that axis, but why it doesn't consider the $\sin\omega$ function?. When we do analyze this situation isn't implied to use the trigonometrical functions for both axis?.

I believe a graph or a sketch of the parabola with the right triangle would help me to understand why you took the square root to find the initial speed?.

It makes sense from using the Pythagorean relationship but I want to know where are the directions of the vectors because this part I'm kind of confused, so can you please add some drawing?.

March 6th, 2019, 06:36 PM   #4
Math Team

Joined: Jul 2011
From: Texas

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Thanks: 1555

Note ...

$v_x = v_0 \cos{\theta_0}$

$v_{y0} = v_0 \sin{\theta_0}$
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text:
# How to formulate case distinctions in AMPLs objective function?

This is my first real optimisation problem I formulated and now trying to solve by using AMPL.

The following objective function is from a linear 0-1 LP means all variables $$x_i^b\in\{0,1\}$$, with $$i\in[1,n]$$ and $$b$$ referring to the type of the node, which means $$0$$ and $$1$$ say a node is or is not of a certain type.

The objective function is as follows with $$A$$ a set of nodes, $$N[a]$$ a set of all neighbours of $$a$$ including $$a$$, and $$n$$ a given number of types:

$$\min - \sum_{a\in A}f\left(\frac1n\sum\limits_{i=1}^nf\left(\sum\limits_{b\in N[a]}x_i^b\right)\right)$$

with

$$f:\mathbb{R}\mapsto\{0,1\}\\ f(x):=\begin{cases}0,&x<1\\1,&x\geq 1\end{cases}$$

I already heard that AMPL doesn't support the definition of functions/methods and that I have to use the Big M method to create this objective function. I couldn't really figure out yet how to use the method to replace my case distinctions...

Let $$\rho$$ be some small value. \begin{align}M \times f(x) &\geq x - 1 + \rho\\M \times (1 - f(x)) &\geq 1 - x - \rho\end{align}

Here is the small working code in Python pulp

import pulp as pl

prob = pl.LpProblem("Problem", pl.LpMinimize)
x = 1
f = pl.LpVariable("f_{0}", 0, 1, pl.LpBinary)
prob += f
M = 100
prob += M * f >= x - 1 + 0.001
prob += M * (1 - f) >= 1 - x - 0.0001

print(prob)
prob.solve()

for v in prob.variables():
print(v.name, "=", v.varValue)

• Can I ask for another detail - I am very interested in the "WHY" so why is it not possible to solve objective functions with case distinctions directly? – baxbear Feb 14 '20 at 12:31


text:
# How can I add birthday celebrations with automated age to Google Calendar?

Exists any kind of means to add a birthday celebration to my Google Calendar as if the age is upgraded instantly?

As an example, if I was born upon May 2nd, 1980, after that I would certainly such as the access for 2011 to claim, "Nathan is 31st birthday", and also for 2012 to claim, "Nathan is 32nd birthday", and more.

Can this be done?

0
2019-12-02 02:55:20
Source Share

You can do this yet you first require to inhabit the Birthday area in your calls.

After you have actually done this, most likely to Google Calendar :

• In the left hand pane click Add > Browse Interesting Calendars
• Click the More tab
• Select Contacts' birthdays and events
• Click Subscribe
• Click Back to Calendar


text:
Question

# The magnetic flux linked with a coil varies as $$\phi =3t^2+4t+9$$. The magnitude of the emf induced at $$t=2$$ seconds is?

A
8 V
B
16 V
C
32 V
D
64 V

Solution

## The correct option is B $$16$$ V$$\phi =\dfrac{d\theta}{dt}$$$$=6t+4=6(2)+4$$$$=16V$$.PhysicsNCERTStandard XII

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text:
Today we will try to tackle the Numerai tournament using the fastai deep learning library. However, as the results likely depend on many different hyperparameters, let’s take advantage of the weights and biases library and their sweeps API. Sweeps are hyperparameter runs which test out different combinations of your model’s hyperparameters.

## What is Numerai?

Numerai is a hedge fund which trades stocks in a market neutral fashion. That means that they try to make money without having a lot of risk for their customers. Basically, whenever they buy or call a stock, they sell or short another one to counter-balance the risk.

The fun thing about Numerai is that they take advantage of an ensemble of data science predictions. To do so, they provide you with anonymized features (1050 currently) and a target column. It is a regression problem, so the target is between 0 and 1. You have a training, a validation and a test set. For the training and validation set, the targets are available. For the test set, you only have the features and need to predict the targets.

You then upload your predictions to Numerai (not your model!) and can bet money on the quality of your predictions (in terms of the cryptocurrency Numeraire). If your model turns out to be good, you’ll earn some more money back, but if your predictions were not good, some of your money will be burned / lost. Here, we are not interested in bidding money, but you can also upload your predictions and just see how you do over time.

## Setting up fastai and wandb

First, we need to install our dependencies and import the basics from fastai and wandb:

!pip install wandb -q
!pip install fastai==2.5.3 -q
!pip install numerapi -q

import wandb
from fastai.tabular.all import *

Next, we download the data from Numerai, namely the training and the validation set. I download them in the _int8 format here to save some memory.

from numerapi import NumerAPI

napi = NumerAPI()
current_round = napi.get_current_round(tournament=8)  # primary tournament
"numerai_training_data_int8.parquet", "numerai_training_data.parquet")
"numerai_validation_data_int8.parquet", "numerai_validation_data.parquet")

## Data preparation for fastai

Then let’s prepare the data for fastai. In fastai we usually have a single data frame for training and validation data, so I’ll merge them together and pass the indexes of the validation data for the data split:

import pandas as pd

train_df = train_df.append(val_df).reset_index()
del val_df
val_idx = list(train_df[train_df['data_type'] == 'validation'].index)

feature_cols = [c for c in train_df if c.startswith("feature_")]

## Metrics for numerai

In Numerai we are interested in having a high correlation of our predictions with the target. In particular, we are looking at eras which are points in time (we don’t know about the details) and the goal is to have a model which is robust over eras, because they are trading with money, so it’s better to be consistent over time rather than having a few eras with good performance and then incurring losses in other eras.

Thus, two important metrics which are generally used are the mean correlation of the eras and the sharpe ratio which is basically the mean divided by the standard deviation and should indicate how attractive your investment is (it’s more attractive when it has a good mean, but low standard deviation).

These helper functions can be defined like this:

from scipy.stats import spearmanr

def corr(df: pd.DataFrame) -> np.float32:
corrs = df.groupby("era").apply(
lambda d: spearmanr(d['target'], d['prediction'])[0])
return corrs.mean()

def sharpe(df: pd.DataFrame) -> np.float32:
corrs = df.groupby("era").apply(
lambda d: spearmanr(d['target'], d['prediction'])[0])
return corrs.mean() / corrs.std()

## Defining the job to run for a weights and biases sweep

Now here comes the interesting part. we need to define a full training run which should also include the metric we want to optimize for.

As we want to see good results on the non-training data, we will of course use some validation metric to optimize for. One possibility would be to use the validation loss, but in this case, I’d rather go for one of the metrics we are actually interested in, so I chose the val_sharpe metric.

Note that I explicitly calculate this after each run and log it to wandb, using wandb.log({'val_sharpe': val_sharpe}).

So what exactly happens here?

We define a method called train() which can be called with a config object to define the hyperparameters which we want to have this run for. This will be called by wandb in the sweeps later on to test out many different hyperparams.

Inside the train() method, we initialize a wandb run using wandb.init() and make sure that wandb has the chance to override the config by using config = wandb.config.

Before and after each run we use the garbage collection and clean the cuda cache just to make sure that no unexpected memory leaks happen between runs.

We then use the set_seed(42, True) method from fastai to make the run reproducible. After all, otherwise, the randomization could have a larger effect than our hyperparameters between the runs. This is sth. you should also test and make sure that runs with the same hyperparameters in fact produce the same results.

Then, we initialize our data loaders, a tabular learner and fit using the one cycle method which fastai popularized. All the relevant hyperparameters are set using config.x, so for example config.epochs for the number of epochs to run for.

Finally, after the run is finished, we evaluate on our validation data and determine our correlation and sharpe values which we log to wandb.

import gc
from fastai.callback.wandb import *

def train(config=None):
with wandb.init(config=config):
config = wandb.config

gc.collect()
torch.cuda.empty_cache()

try:
set_seed(42, True)

train_df,
bs=config.batch_size,
cont_names=feature_cols,
y_names='target',
valid_idx=val_idx)

learn = tabular_learner(
dls,
wd=config.wd,
loss_func=MSELossFlat(),
cbs=[ShowGraphCallback(), SaveModelCallback(), WandbCallback()])

learn.fit_one_cycle(config.epochs, lr_max=config.learning_rate)

prediction, target = learn.get_preds()

era = dls.valid_ds.items['era']
eval_df = pd.DataFrame({
'prediction': prediction.numpy().squeeze(),
'target': target.numpy().squeeze(),
'era': era}).reset_index()

val_sharpe = sharpe(eval_df)
val_corr = corr(eval_df)

wandb.log({'val_sharpe': val_sharpe})
wandb.log({'val_corr': val_corr})

print((f'Mean correlation: {val_corr}. Sharpe: {val_sharpe}.'))

except Exception as e:
print(e)

del learn
gc.collect()
torch.cuda.empty_cache()

## Configuring wandb sweeps

The last step is now to tell wandb what to optimize for and which hyperparameters to try out.

In this case, I am using the bayes method which will try to figure out the optimal hyperparameters in a bayesian way (details are handled by wandb).

We define our val_sharpe as the metric with the goal to maximize it.

For the parameters we can either specify a concrete list of values like I did for the number of epochs or a range using min and max like for weight decay (wd).

For the sake of quickly running an example here, I am not using optimal values, but rather a small amount of epochs etc.

import wandb

sweep_config = {
"method": "bayes",
"metric": {
"name": "val_sharpe",
"goal": "maximize"
},
"parameters": {
"epochs": {
"values": [1, 2]
},
"wd": {
"min": 0.01,
"max": 0.1
},
"learning_rate": {
"min": 0.001,
"max": 0.01
},
"batch_size": {
"values": [2056, 4112]
}
}
}

sweep_id = wandb.sweep(sweep_config, project='numerai-sweeps-blog')

Create sweep with ID: q7zn7a0s
Sweep URL: https://wandb.ai/mpaepper/numerai-sweeps-blog/sweeps/q7zn7a0s

## Starting the weights and biases hyperparameter optimization

With this sweep_id we can now start agents to execute our runs - you could even do this from multiple machines to parallelize this. We pass our train function to the agent and the count of experiments to run. I chose 10 here for the count, so we can get an idea. Normally, you would probably run at least 100 or more experiments. You can omit the count, so the agent runs indefinitely until you stop it.

wandb.agent(sweep_id, function=train, count=10)

The output will look something like this:

epoch train_loss valid_loss time
0 0.049930 0.050241 01:08
Better model found at epoch 0 with valid_loss value: 0.05024105682969093.

Validation correlation is: 0.012763993933631422. Validation sharpe is: 0.4141087498927619.

Waiting for W&B process to finish, PID 860… (success).

…and then continues for many runs.

### Evalutating the results in weights and biases

Let’s take a look at some of the plots that we can get out of weights and biases to get insights about our problem. Again, here I ran with not the most sensible set of hyperparameters and only 10 runs, but we can probably already learn something.

The first nice thing we can get out of our runs is a feature importance weight table:

We can see here that it seems to be beneficial for our val_sharpe metric to run more than a single epoch, to have some weight decay and also to use a larger batch size. They are all positively correlated (green) with our target metric. In contrast, the learning_rate is negatively correlated, so a smaller learning rate was better in our experiment.

Another great tool is the parallel coordinates panel in which you can define all the aspects you are interested in and you get a plot which shows all the different runs and where they end up finally. This is interactive on the wandb website, but here I’ve exported it as a png:

We can see here that the run which finished with the highest val_sharpe value (yellow line) took advantage of a larger batch size and 2 epochs instead of 1 epoch while being somewhere in the middle of the weight decay and learning_rate scheme.

## Summary wandb with fastai

To sum up: we could quickly start an experiment and easily tune the hyperparameters without a lot of work. Fastai enables us to quickly build a model with sensible defaults and the wandb hyperparameter sweeps are a very nice tool to collect the metrics of your different runs and generate some insights from different hyperparameters. What I like most about it: you can code it and then let it run over night. Then the next morning, you have all the results available and nice visualization tools at your hand.


text:
# Function Composition

## Key Questions

Yes

#### Explanation:

Given composable functions $f$, $g$ and $h$

$\left(f \circ \left(g \circ h\right)\right) \left(x\right)$

$= f \left(\left(g \circ h\right) \left(x\right)\right) = f \left(g \left(h \left(x\right)\right)\right) = \left(f \circ g\right) \left(h \left(x\right)\right)$

$= \left(\left(f \circ g\right) \circ h\right) \left(x\right)$

So $f \circ \left(g \circ h\right) = \left(f \circ g\right) \circ h$

If $g : A \to B$ and $f : B \to C$, then the domain of $f \circ g$ is

${\overline{g}}^{- 1} \circ {\overline{f}}^{- 1} \left(C\right)$

using the notation described below...

#### Explanation:

If $g$ is a function that maps some elements of a set $A$ to elements of a set $B$, then the domain of $g$ is the subset of $A$ for which $g \left(a\right)$ is defined.

More formally:

$g \subseteq A \times B :$

$\forall a \in A \forall {b}_{1} , {b}_{2} \in B$

$\left(\left(a , {b}_{1}\right) \in g \wedge \left(a , {b}_{2}\right) \in g\right) \implies {b}_{1} = {b}_{2}$

Use the notation ${2}^{A}$ to represent the set of subsets of $A$ and ${2}^{B}$ the set of subsets of $B$.

Then we can define the pre-image function:

${\overline{g}}^{- 1} : {2}^{B} \to {2}^{A}$ by ${\overline{g}}^{- 1} \left({B}_{1}\right) = \left\{a \in A : g \left(a\right) \in {B}_{1}\right\}$

Then the domain of $g$ is simply ${\overline{g}}^{- 1} \left(B\right)$

If $f$ is a function that maps some elements of set $B$ to elements of a set $C$, then:

${\overline{f}}^{- 1} : {2}^{C} \to {2}^{B}$ is defined by ${\overline{f}}^{- 1} \left({C}_{1}\right) = \left\{b \in B : f \left(b\right) \in {C}_{1}\right\}$

Using this notation, the domain of $f \circ g$ is simply

${\overline{g}}^{- 1} \left({\overline{f}}^{- 1} \left(C\right)\right) = \left({\overline{g}}^{- 1} \circ {\overline{f}}^{- 1}\right) \left(C\right)$

• To compose a function is to input one function into the other to form a different function. Here's a few examples.

Example 1: If $f \left(x\right) = 2 x + 5$ and $g \left(x\right) = 4 x - 1$, determine $f \left(g \left(x\right)\right)$

This would mean inputting $g \left(x\right)$ for $x$ inside $f \left(x\right)$.

$f \left(g \left(x\right)\right) = 2 \left(4 x - 1\right) + 5 = 8 x - 2 + 5 = 8 x + 3$

Example 2: If $f \left(x\right) = 3 {x}^{2} + 12 + 12 x$ and $g \left(x\right) = \sqrt{3 x}$, determine $g \left(f \left(x\right)\right)$ and state the domain

Put $f \left(x\right)$ into $g \left(x\right)$.

$g \left(f \left(x\right)\right) = \sqrt{3 \left(3 {x}^{2} + 12 x + 12\right)}$

$g \left(f \left(x\right)\right) = \sqrt{9 {x}^{2} + 36 x + 36}$

$g \left(f \left(x\right)\right) = \sqrt{{\left(3 x + 6\right)}^{2}}$

$g \left(f \left(x\right)\right) = | 3 x + 6 |$

The domain of $f \left(x\right)$ is $x \in \mathbb{R}$. The domain of $g \left(x\right)$ is $x > 0$. Hence, the domain of $g \left(f \left(x\right)\right)$ is $x > 0$.

Example 3: if $h \left(x\right) = {\log}_{2} \left(3 {x}^{2} + 5\right)$ and $m \left(x\right) = \sqrt{x + 1}$, find the value of $h \left(m \left(0\right)\right)$?

Find the composition, and then evaluate at the given point.

$h \left(m \left(x\right)\right) = {\log}_{2} \left(3 {\left(\sqrt{x + 1}\right)}^{2} + 5\right)$

$h \left(m \left(x\right)\right) = {\log}_{2} \left(3 \left(x + 1\right) + 5\right)$

$h \left(m \left(x\right)\right) = {\log}_{2} \left(3 x + 3 + 5\right)$

$h \left(m \left(x\right)\right) = {\log}_{2} \left(3 x + 8\right)$

$h \left(m \left(2\right)\right) = {\log}_{2} \left(3 \left(0\right) + 8\right)$

$h \left(m \left(2\right)\right) = {\log}_{2} 8$

$h \left(m \left(2\right)\right) = 3$

Practice exercises

For the following exercises: $f \left(x\right) = 2 x + 7 , g \left(x\right) = {2}^{x - 7} \mathmr{and} h \left(x\right) = 2 {x}^{3} - 4$

a) Determine $f \left(g \left(x\right)\right)$

b) Determine $h \left(f \left(x\right)\right)$

c) Determine $g \left(h \left(2\right)\right)$

Hopefully this helps, and good luck!


text:
# This is a love letter

I love McElreath’s Statistical Rethinking text. It’s the entry-level textbook for applied researchers I spent years looking for. McElreath’s freely-available lectures on the book are really great, too.

However, I prefer using Bürkner’s brms package when doing Bayeian regression in R. It’s just spectacular. I also prefer plotting with Wickham’s ggplot2, and coding with functions and principles from the tidyverse, which you might learn about here or here.

So, this project is an attempt to reexpress the code in McElreath’s textbook. His models are re-fit with brms, the figures are reproduced or reimagined with ggplot2, and the general data wrangling code now predominantly follows the tidyverse style.

## Why this?

I’m not a statistician and I have no formal background in computer science. Though I benefited from a suite of statistics courses in grad school, a large portion of my training has been outside of the classroom, working with messy real-world data, and searching online for help. One of the great resources I happened on was idre, the UCLA Institute for Digital Education, which offers an online portfolio of richly annotated textbook examples. Their online tutorials are among the earliest inspirations for this project. We need more resources like them.

With that in mind, one of the strengths of McElreath’s text is its thorough integration with the rethinking package. The rethinking package is a part of the R ecosystem, which is great because R is free and open source. And McElreath has made the source code for rethinking publically available, too. Since he completed his text, many other packages have been developed to help users of the R ecosystem interface with Stan. Of those alternative packages, I think Bürkner’s brms is the best for general-purpose Bayesian data analysis. It’s flexible, uses reasonably-approachable syntax, has sensible defaults, and offers a vast array of post-processing convenience functions. And brms has only gotten better over time. To my knowledge, there are no textbooks on the market that highlight the brms package, which seems like an evil worth correcting.

In addition, McElreath’s data wrangling code is based in the base R style and he made most of his figures with base R plots. Though there are benefits to sticking close to base R functions (e.g., less dependencies leading to a lower likelihood that your code will break in the future), there are downsides. For beginners, base R functions can be difficult both to learn and to read. Happily, in recent years Hadley Wickham and others have been developing a group of packages collectively called the tidyverse. These tidyverse packages (e.g., dplyr, tidyr, purrr) were developed according to an underlying philosophy and they are designed to work together coherently and seamlessly. Though not all within the R community share this opinion, I am among those who think the tydyverse style of coding is generally easier to learn and sufficiently powerful that these packages can accommodate the bulk of your data needs. I also find tydyverse-style syntax easier to read. And of course, the widely-used ggplot2 package is part of the tidyverse, too.

To be clear, students can get a great education in both Bayesian statistics and programming in R with McElreath’s text just the way it is. Just go slow, work through all the examples, and read the text closely. It’s a pedagogical boon. I could not have done better or even closely so. But what I can offer is a parallel introduction on how to fit the statistical models with the ever-improving and already-quite-impressive brms package. I can throw in examples of how to perform other operations according to the ethic of the tidyverse. And I can also offer glimpses of some of the other great packages in the R + Stan ecosystem, such as loo, bayesplot, and tidybayes.

If you’re looking at this project, I’m guessing you’re either a graduate student, a post-graduate academic, or a researcher of some sort. So I’m presuming you have at least a 101-level foundation in statistics. If you’re rusty, consider checking out Legler and Roback’s free bookdown text, Broadening Your Statistical Horizons before diving into Statistical Rethinking. I’m also assuming you understand the rudiments of R and have at least a vague idea about what the tidyverse is. If you’re totally new to R, consider starting with Peng’s R Programming for Data Science. And the best introduction to the tidyvese-style of data analysis I’ve found is Grolemund and Wickham’s R for Data Science, which I extensively link to throughout this project.

That said, you do not need to be totally fluent in statistics or R. Otherwise why would you need this project, anyway? IMO, the most important things are curiosity, a willingness to try, and persistent tinkering. I love this stuff. Hopefully you will, too.

## How to use and understand this project

This project is not meant to stand alone. It’s a supplement to McElreath’s Statistical Rethinking text. I follow the structure of his text, chapter by chapter, translating his analyses into brms and tidyverse code. However, some of the sections in the text are composed entirely of equations and prose, leaving us nothing to translate. When we run into those sections, the corresponding sections in this project will sometimes be blank or omitted, though I do highlight some of the important points in quotes and prose of my own. So I imagine students might reference this project as they progress through McElreath’s text. I also imagine working data analysts might use this project in conjunction with the text as they flip to the specific sections that seem relevant to solving their data challenges.

I reproduce the bulk of the figures in the text, too. The plots in the first few chapters are the closest to those in the text. However, I’m passionate about data visualization and like to play around with color palettes, formatting templates, and other conventions quite a bit. As a result, the plots in each chapter have their own look and feel. For more on some of these topics, check out chapters 3, 7, and 28 in R4DS, Healy’s Data Visualization: A practical introduction, or Wilke’s Fundamentals of Data Visualization.

In this project, I use a handful of formatting conventions gleaned from R4DS, The tidyverse style guide, and R Markdown: The Definitive Guide.

• R code blocks and their output appear in a gray background. E.g.,
2 + 2 == 5
## [1] FALSE
• Functions are in a typewriter font and followed by parentheses, all atop a gray background (e.g., brm()).
• When I want to make explicit the package a given function comes from, I insert the double-colon operator :: between the package name and the function (e.g., tidybayes::mode_hdi()).
• R objects, such as data or function arguments, are in typewriter font atop gray backgrounds (e.g., chimpanzees, .width = .5).
• You can detect hyperlinks by their typical blue-colored font.
• In the text, McElreath indexed his models with names like m4.1 (i.e., the first model of Chapter 4). I primarily followed that convention, but replaced the m with a b to stand for the brms package.

## You can do this, too

This project is powered by Yihui Xie’s bookdown package, which makes it easy to turn R markdown files into HTML, PDF, and EPUB. Go here to learn more about bookdown. While you’re at it, also check out Xie, Allaire, and Grolemund’s R Markdown: The Definitive Guide. And if you’re unacquainted with GitHub, check out Jenny Bryan’s Happy Git and GitHub for the useR. I’ve even blogged about what it was like putting together the first version of this project.

The source code of the project is available here.

I released the initial 0.9.0 version of this project in September 26, 2018. In April 19, 2019 came the 1.0.0 version. Some of the major changes were:

• All models were refit with the current official version of brms, 2.8.0.
• Adopting the seed argument within the brm() function made the model results more reproducible.
• The loo package was updated. As a consequence, our workflow for the WAIC and LOO changed, too.
• I improved the brms alternative to McElreath’s coeftab() function.
• I made better use of the tidyverse, especially some of the purrr functions.
• Particularly in the later chapters, there’s a greater emphasis on functions from the tidybayes package.
• Chapter 11 contains the updated brms 2.8.0 workflow for making custom distributions, using the beta-binomial model as the example.
• Chapter 12 received a new bonus section contrasting different methods for working with multilevel posteriors.
• Chapter 14 received a new bonus section introducing Bayesian meta-analysis and linking it to multilevel and measurement-error models.
• With the help of others within the community, I corrected many typos and streamlined some of the code (e.g., dropped an unnecessary use of the mi() function in section 14.2.1)
• And in some cases, I corrected sections that were just plain wrong (e.g., some of my initial attempts in section 3.3 were incorrect).

In response to some reader requests, we finally have a PDF version! Making that happen required some formatting adjustments, resulting in version 1.0.1. Noteworthy changes include:

• Major revisions to the LaTeX syntax underlying many of the in-text equations (e.g., dropping the “eqnarray” environment for “align*“)
• Adjusting some of the image syntax
• Updating the reference for the Bayesian $$R^2$$ (Gelman, Goodrich, Gabry, & Vehtari, 2018)

Though we’re into version 1.0.1, there’s room for improvement. There are still two models that need work. The current solution for model 10.6 is wrong, which I try to make clear in the prose. It also appears that the Gaussian process model from section 13.4 is off. Both models are beyond my current skill set and friendly suggestions are welcome. In addition to modeling concerns, typos may yet be looming and I’m sure there are places where the code could be made more streamlined, more elegant, or just more in-line with the tidyverse style. Which is all to say, I hope to release better and more useful updates in the future.

Before we move on, I’d like to thank the following for their helpful contributions: Paul-Christian Bürkner (@paul-buerkner), Andrew Collier (@datawookie), Jeff Hammerbacher (@hammer), Matthew Kay (@mjskay), TJ Mahr (@tjmahr), Stijn Masschelein (@stijnmasschelein), Colin Quirk (@colinquirk), Rishi Sadhir (@RishiSadhir), Richard Torkar (@torkar), Aki Vehtari (@avehtari).


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# Event

### Astronomy and Astrophysics Seminar: Seeking solutions for the Hubble tension

##### Meng-Xiang Lin (Chicago)
- | Zoom

Although the $\Lambda$CDM model has been remarkably successful in explaining a wide range of observational data, the model does not reconcile distance-redshift measurements when anchored at low redshift through the distance ladder and high redshift by cosmic microwave background (CMB) anisotropies. Specifically, if we assume the $\Lambda$CDM model, the Hubble parameter $H_0$ has a $4.1\sigma$ tension between the measurements from low redshift SH0ES project and the high redshift Planck CMB data. This "Hubble tension" has become an increasingly significant mystery to solve in recent years. In this seminar, I will talk about the currently proposed solutions for the Hubble tension. Acoustic Dark Energy (an Early Dark Energy-like) model could provide the high H0 solution with simple scalar field realization, at the cost of a higher S8 value. On the other hand, early modified gravity could provide a different mechanism to relieve Hubble tension with a low S8 value.

The possible paths to better solutions in the future will also be talked about.

https://upenn.zoom.us/j/97336621524?pwd=Rk9vNUIxTlFLbVNuazFkeFhtR3Y2UT09

Meeting ID: 973 3662 1524

Passcode: 800483


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main-content

## Weitere Artikel dieser Ausgabe durch Wischen aufrufen

12.11.2018 | short review | Ausgabe 4/2018 Open Access

# Short review of biparametric prostate MRI

Zeitschrift:
memo - Magazine of European Medical Oncology > Ausgabe 4/2018
Autoren:
Dr. Fabian Steinkohl, PhD PD Dr. Renate Pichler, PD Dr. Daniel Junker

## Introduction

Prostate carcinoma (PCa) is the most common cancer in men [1]. PCa is suspected because of a positive digital rectal examination of the prostate or because of high values of prostate-specific antigen. This antigen is produced by normal but also by malignant prostatic epithelium [2] and is widely used as a screening tool for PCa. Usually, a systematic biopsy of the prostate is performed to clarify this suspicion. This approach has a sensitivity of only 50% [3]. If the systematic biopsy of the prostate is negative but PCa is still suspected, magnetic resonance imaging (MRI) of the prostate can be performed. MRI can identify suspicious lesions with a high diagnostic accuracy [4]. The first study on the diagnostic value of pelvic MRI was published in 1983 [5]. According to Hricak et al., it was unclear whether MRI can distinguish neoplastic nodules from benign prostatitis. The field of prostate MRI has been developing rapidly since then. A milestone was the introduction of the Prostate Imaging Reporting and Data System (PI-RADS) in 2012 [6]. PI-RADS standardized prostate MRI protocols and standardized image interpretation and reporting of prostate MRI. PI-RADS is also used to communicate imaging findings between radiologists and the referring physician. PI-RADS assigns scores for lesions in the prostate from 1 to 5. PI-RADS 1 means “clinically significant PCa is highly unlikely,” PI-RADS 5 means “clinically significant cancer is highly likely” (Table 1).
Table 1
PI-RADS scores and their definition according to the ESUR panel
Definition of the ESUR panel
1
Clinically significant cancer is highly unlikely
2
Clinically significant cancer is unlikely
3
Presence of clinically significant cancer is equivocal
4
Clinically significant cancer is likely
5
Clinically significant cancer is highly likely
PI-RADS Prostate Imaging Reporting and Data System, ESUR European Society of Urogenital Radiology
In recent years, prostate MRI is increasingly used to visualize PCa [7]. A new, simplified version, PI-RADS v2 was introduced in 2012 [8]. PI-RADS v2, too, is based on multiplanar T2-weighted sequences (T2w), diffusion-weighted sequences (DWI), and dynamic contrast-enhanced sequences (DCE). The prostatic anatomy is evaluated in the T2w sequences. The prostatic zones can be discriminated and important extraprostatic structures, such as the neurovascular bundles, can be seen. DWI is the key sequence in the PI-RADS system. PCa has a higher cell density than the surrounding normal prostatic tissue. Hence, the Brownian motion within the tumor is limited. This diffusion restriction can be visualized with DWI, but the spatial resolution of DWI is low. Intravenous contrast agent, usually gadolinium, has to be administered for the DCE sequences. DCE shows the contrast enhancement of the prostate over the time. PCa is believed to have an early gadolinium uptake and an early wash-out of the contrast agent. In the currently used PI-RADS v2, DCE plays a minor role. An early contrast enhancement in the DCE can lead to an upgrading from PI-RADS 3 to PI-RADS 4. Combining the aforementioned different sequences in one MRI examination has become known as multiparametric MRI (mpMRI; Fig. 1). MpMRI is well evaluated, but it has several drawbacks. To overcome some of them, a number of groups have proposed a “biparametric prostate MRI” (bpMRI).
This short review summarizes the latest findings on bpMRI for radiologists and referring physicians.

## Benefits of omitting DCE

In contrast to multiparametric MRI protocols, biparametric MRI protocols do not include DCE. Therefore, it has three major advantages: examination times are shorter, costs are lower, and the risk of adverse events with contrast agents is eliminated.

### Examination time

Omitting an MRI sequence shortens the examination time. There are wide differences in the literature on how much time can be saved by using a biparametric approach. Obmann et al. found 11.9 min scanning time (and 15.7 min table time) for bpMRI and ±45 min for mpMRI [9]. This was confirmed by Dong Hoon Lee et al., who found an in-bore time of 15 min for bpMRI and 45 min for mpMRI [10]. Other authors found considerably shorter time differences between mpMRI and bpMRI. Junker et al. reported a time reduction of 12 min using bpMRI [11] and another group found a time reduction of only 2.30 min [12]. A very fast biparametric MRI protocol that takes only 8.45 min has been published by Kuhl et al. [13].

### Costs for mpMRI

Performing a DCE involves direct costs for contrast agents and peripheral venous catheters as well as indirect costs for personnel and longer scanning times. Data for costs diverge in the literature. Junker et al. reported additional costs of about € 56 for a 70-kg patient only for the contrast agent [11]. A Korean group calculated costs of about $600 for mpMRI and only$ 300 for bmMRI [10].

### Risk associated with contrast medium

The contrast medium used for DCE in mpMRI is gadolinium-based. The risk of immediate hypersensitivity reactions to a gadolinium-based contrast medium is low [14]. Until recently, gadolinium was considered to be safe [15], but it has emerged recently that gadolinium can form depositions in the brain [16]. Therefore, caution is called for when gadolinium is administered [17]. In view of these findings, it seems to be advisable to reconsider the necessity of DCE.

## Diagnostic performance of mpMRI and bpMRI

Although DCE is part of the PI-RADS v2 guidelines [8] and is considered a cornerstone of prostatic MRI by some authors [18], its role is controversial in the literature. Greer and colleagues found that DCE significantly improves PCa detection in the peripheral zone of the prostate [19]. Some older studies found an improvement of PCa detection due to the use of DCE [20, 21]. A limited incremental value of 3% was calculated for DCE [22]. Recent studies state that DCE has no or only a limited role in PCa detection [2327]. The suspicion was raised that DCE could potentially increase the number of false-positive examinations [13]. These findings resonate well in the scientific community. To date, 16 articles can be found on PubMed.gov using the search terms “biparametric prostate MRI” from January 2018.
Two large meta-analyses have been published this year. One analyzed 33 studies from the period 2007–2017 [28], another one analyzed 20 studies, all published before December 2017 [29]. Xiang-ke Niu and colleagues found a pooled sensitivity of 0.81 and a specificity of 0.77 for bpMRI for PCa detection [28]. Woo et al. reported a pooled sensitivity of 0.74 and a specificity of 0.90 for bpMRI and a pooled sensitivity of 0.76 and a specificity of 0.89 for mpMRI. Therefore, the authors conclude that bpMRI has the same diagnostic performance as mpMRI for the detection of PCa [29]. Recent single-center studies from 2018 published similar results [7, 9, 11, 12, 30].

## Reasons for using DCE

Although there are many good reasons for omitting the use of gadolinium-based contrast agents for prostate MRI, there are still situations in which DCE can be useful. DCE can be used to detect certain small PCa [24]. It can be used to monitor response to therapy after radical prostatectomy [31]. Additionally, we find that DCE helps to avoid typical pitfalls (e. g., in the anterior fibromuscular stroma) and to diagnose seminal vesicle infiltration or extracapsular extension.

## Future developments of prostate MRI

Current multiparametric MRI protocols are designed to answer a variety of clinical questions (see above; [13]). The demand for prostate MRI rises since prostate MRI is used as a PCa screening tool prior to biopsy. In view of the latest publications it seems feasible to use bpMRI in these cases. Scialpi and colleagues proposed a new and simpler version of PI-RADS [24] that is based on a bpMRI. In their simplified PI-RADS, DWI is the dominant sequence in the peripheral and transition zone. Furthermore, the maximal size of a category 3 lesion is 0.5 cm3. Another advantage is that bpMRI images are easier to read. While a large Norwegian study found a poor inter-reader agreement for mpMRI [32], an Italian group reported that there is excellent agreement between different readers with different levels of experience for bpMRI [7].
PI-RADS v2 will continue to evolve. We believe that the use of DCE is not necessary in every patient, but the indication for administering gadolinium-based contrast agents will have to be considered individually depending on the clinical question.

## Conclusion

During the past months, several studies have been published on biparametric prostate MRI protocols. All of them underline the feasibility of shorter biparametric protocols for PCa detection. In light of these findings, we believe that the need for administering contrast agents for prostate MRI has to be considered individually for each patient depending on the clinical question.

### Conflict of interest

F. Steinkohl, R. Pichler, and D. Junker declare that they have no competing interests.


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Can QAOA be considered as simulation of a quantum annealer on a gate-based quantum computer?

Quantum annealers are single purpose machines allowing to solve quadratic unconstrained binary optimization (QUBO) problems. QUBO problems have following objective function: $$F=-\sum_{i where $$x_i$$ is a binary varibale and $$h_i$$ and $$J_{ij}$$ are coefficients. Such objective function is equivalent to Ising Hamiltonian $$H_{\text{ISING}}=-\sum_{i where $$\sigma^z_i$$ is Pauli Z gate acting on $$i$$th qubit and there are identity operators on other qubits, tensor product $$\sigma^z_i\otimes\sigma^z_j$$ means that Z gates act on $$i$$th and $$j$$th qubits and there are identity operators on other qubits.

Quantum annealers physically implements simulation of Hamiltonian $$H(t)=\Big(1-\frac{t}{T}\Big)\sum_{i=1}^N h_i\sigma^x_i+\frac{t}{T}H_{\text{ISING}},$$ where $$t$$ is a time, $$T$$ total time of simulation and $$\sigma^x_i$$ is Pauli X gate acting on $$i$$th qubit. Initial state of a quantum annealer is equal superposition of all qubits which is ground state of the Hamiltonian $$H(0)$$.

Quantum Approximate Optimization Algorithm (QAOA) is described by an operator $$U(\beta, \gamma) = \prod_{i=1}^{p}U_B(\beta_i)U_C(\gamma_i),$$ where $$p$$ is number of iteration of QAOA, $$U_B(\beta) = \mathrm{e}^{-i\beta\sum_{i=1}^N \sigma^x_i},$$ and $$U_C(\gamma) = \mathrm{e}^{-i\gamma(\sum_{i,j=1}J_{ij}(\sigma^z_i\otimes\sigma^z_j)+\sum_{i=1}^N h_i\sigma^z_i)}.$$ Initial state for QAOA is $$H^{\otimes n}|0\rangle ^{\otimes n}$$, i.e. equally distributed superposition as in case of the quantum annealer.

Since time evolution of quantum system described by Hamiltonian $$H$$ from state $$|\psi(0)\rangle$$ to state $$|\psi(t)\rangle$$ is expressed by $$|\psi(t)\rangle = \mathrm{e}^{-iHt}|\psi(0)\rangle,$$ it seems that operator $$U(\beta, \gamma)$$ from QAOA is simply simulation of Hamiltonian $$H(t)$$ describing quantum annealer beacause exponents of $$\mathrm{e}$$ are sums in Hamiltonian $$H(t)$$.

However, $$H(t)$$ is composed of two term containing Pauli matrices X and Z and $$\mathrm{e}^{A+B}=\mathrm{e}^A\mathrm{e}^B$$ is valid only for commuting matrices $$[A,B]=O$$. But Pauli matrices X and Z fulfil anti-commutation relation $$\{X,Z\}=O$$, not the commutation one.

So my questions are these:

1. Can QAOA be realy considered as a simulation of quantum annealer on gate-based universal quantum computer?
2. What I am missing in discussion above concerning commutation of Pauli matrices? Or is there any condition for matrices $$A$$ and $$B$$ allowing equality $$\mathrm{e}^{A+B}=\mathrm{e}^A\mathrm{e}^B$$?

1. If you use infinite depth then QAOA can be consider as quantum annealer on gate-based. The authors of QAOA original paper probably deduce it from quantum annealing. What I mean by infinite depth is you take $$p \to \infty$$ in the operator
$$U(\beta, \gamma) = \Pi_{i=1}^p U_B(\beta_i)U_C(\gamma_i)$$
$$\lim_{p \to \infty} \big(e^{A/p}e^{B/p} \big)^p = e^{A+B}$$


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# Heat maps

## Recommended Posts

Does anyone have any good resources for explaining how to generating heat maps? http://en.wikipedia.org/wiki/Heat_map Bonus points for pseudo or C# code. I've sifting through a lot of sites from google that discuss topological implications and click maps, but haven't found any that tie the graphical representation into event systems. Meaning, I have the event and a location, how do I color the image in the appropriate manner?

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Nican    100
I made a Heat Map generator in Java a long time ago, and I am unsure that it still works, and it is a mess, but you can take a look at it.

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Thanks Nican. I appreciate it.

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Hodgman    51323
Every time an event happens at a location, you increment a counter for that location.
When you want to display those counters graphically, you first normalize the data:
* Find the minimum and maximum counter values for all locations
* for each counter, counter = (counter-min)/(max-min)
Now you've got a value for each location in the 0-1 range

Using an interpolation function you can build a gradient(e.g. linear interpolation):
color lerp( color a, color b, float fraction ){ return (1-fraction)*a + fraction*b; }

For a three colour gradient you can partition the range and scale each partition back into the 0-1 range for interpolation. e.g. for a green->blue->red gradient:
if( counter < 0.5 ) display = lerp( green, blue, counter*2 );else display = lerp( blue, red, (counter-0.5)*2 );

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Quote:
 Original post by HodgmanEvery time an event happens at a location, you increment a counter for that location.When you want to display those counters graphically, you first normalize the data:* Find the minimum and maximum counter values for all locations* for each counter, counter = (counter-min)/(max-min)Now you've got a value for each location in the 0-1 rangeUsing an interpolation function you can build a gradient(e.g. linear interpolation):color lerp( color a, color b, float fraction ){ return (1-fraction)*a + fraction*b; }For a two colour gradient, you feed in your normalised counter values and your two colours.For a three colour gradient you can partition the range and scale each partition back into the 0-1 range for interpolation. e.g. for a green->blue->red gradient:if( counter < 0.5 ) display = lerp( green, blue, counter*2 );else display = lerp( blue, red, (counter-0.5)*2 );

I see. Almost like how fractals have colors associated with a fixed range of values. I guess fall offs will occur naturally depending on the events. I'm worried about certain locations of the map with just a red dot and no fall off. But if that's the frequency of events then I guess the representation would be accurate. Thanks for the help.

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Emergent    982
Quote:
 Original post by GhostDogStudioI guess fall offs will occur naturally depending on the events. I'm worried about certain locations of the map with just a red dot and no fall off.

If that's a concern, you can, instead of incrementing just one pixel, add a several-pixel-radius smooth "bump" to your map whenever there's an event. E.g.,

w(r) = (1 - r^2/R^2)^2 if r < R; 0 if r >= R.

where r is the distance of a pixel in your map from the event and R is the desired bump radius.

This is mathematically equivalent to showing a blurred version of your map.

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Krypt0n    4721
Quote:
 * for each counter, counter = (counter-min)/(max-min)

but you need to be careful, sometimes peaks can screw up the whole scale (e.g. the spawn point in an multiplayer game would be the "hottest" place. far more than any other. so you might want to ignore peaks, e.g. processing some median filter over the data.