text:
# Particle System

I saw a video online recently of several cubes flying into the scene and stacking up to form a larger cube. I thought it was a pretty simple yet powerful effect. The effect was achieved using Particle Flow, so the movement of the particles are entirely animated and not in real time. After seeing this, I immediately wanted to try it, but with a more general solution. So, off I went to write a physics system for my particle engine.

## Particle System

I began this project with an old particle system I started working on to dig into DX9. It’s gone through a few iterations, some architectural and some performance oriented. This version is entirely object oriented, where each emitter, particle, camera, input, etc. are classes. Particle is an abstract class, inherited by a Cube particle in these examples. Most of the details regarding the particular graphics API being used have been abstracted into the respective classes, and the scene can be described without using D3D function calls.

As a performance enhancement, this particle system uses hardware instancing. All particle specific information is transferred to the graphics card via a vertex buffer, with all position, color, and other particle logic performed in the vertex shader. The next major revision will include physics calculations to be performed on the GPU.

## Spring Physics

Now, there are a lot of challenges in getting a particle rocketing through 3D space to land in exactly the right position with physics alone. There are a few options. With physics, we can exponentially increase a force on the particle in the direction of the destination with relation to the distance from the destination. Another possibility is merging a pure physics system with a pure spline animation to allow a random initial velocity, but a guaranteed stopping point.

My first step however, is developing some way to attract the particles to their destination. I decided to start with a spring system. Although the system will exert forces on the particles to aim the particles at the destination, it is very unlikely that they will pass directly through a predetermined point, and instead orbit around the destination.

For those interested, here’s the nice little gem used to compute the spring force. I modified this algorithm from a cloth simulator used in the book “Physics for Game Developers”.

D3DXVECTOR3 Collector::GetForce(Particle particle)
{
D3DXVECTOR3 forceVector = GetVector(particle);    // From particle to dest
float length = D3DXVec3Length(&forceVector);
float k = 1;    // tensile constant
float d = 1;    // damping constant
float dot = D3DXVec3Dot(&particle.m_velocity, &forceVector);
D3DXVECTOR3 force = (k * length + d * (dot / length)) * (forceVector / length);
return force;
}

## Orbits and Decay

Attempt number 2 at getting particles to stick to a collector: apply gravitational pull to the collector. The ideal is simple; the closer a particle gets to the collector, the greater the force of gravity acting on the particle towards that collector. In this case, I used the inverse of the distance squared. This produces an “Ideal” orbital pattern as shown below:

D3DXVECTOR3 Collector::GetIdealOrbitalForce(const D3DXVECTOR3& magVector,
const float& magLength)
{
float grav = 1000.0f;
D3DXVECTOR3 force = magVector * grav;
if(magLength > 1)
{
force /= (magLength * magLength);
}
return force;
}

This ideal orbital pattern has the obvious problem of not actually attracting the particles into the center, and instead just pulling the particles into an orbit around the collector. To give more realistic physical behavior, we would like to model drag the particle encounters by entering the “atmosphere” of the collector. The thickness of the atmosphere was modeled as the inverse of the distance squared. Tweaking the constants a bit yields the following results:

D3DXVECTOR3 Collector::GetOrbitalDecayForce(const D3DXVECTOR3& magVector,
const float& magLength,
const D3DXVECTOR3& curVelocity,
const D3DXVECTOR3& curPos,
const D3DXVECTOR3& collectorPos,
const float& dt)
{
D3DXVECTOR3 force = GetIdealOrbitalForce(magVector, magLength);
D3DXVECTOR3 newVelocity = curVelocity + force * dt;

if (dt > 0)
{
D3DXVECTOR3 newPos = curPos + newVelocity * dt;
D3DXVECTOR3 magVector2 = collectorPos - newPos;
float magLength2 = D3DXVec3Length(&magVector2);
float damping = 4.0f;

D3DXVECTOR3 decayVelocity = ((magVector2 / magLength2) - (magVector / magLength)) / dt * damping;
if(magLength2 > 1)
{
decayVelocity /= (magLength2 * magLength2);
}
force += decayVelocity/dt;
}
return force;
}

Adding this behavior with spring physics we get:

This all adds to make a pretty interesting effect. I hadn’t optimized the code which models the orbital decay, so I was curious to see what the limitations of the particle system were after performing the physics computations. The physics for each particle is computed on the host, with the final position stored in the instance buffer for each cube to be drawn on the GPU. I’m interested to see what kind of speedup is achievable by performing the physics on the GPU, but I’ll save that for another day. For now, I was able to achieve 70 FPS on my GTX260 with 20,000 cubes, each with spring physics and orbital decay. Results speak for themselves:

After writing these physics calculations, I’m beginning to believe that a purely physically based solution to get the kind of flight paths I am interested in does not exist. So, the third iteration of this particle system will use a spline hybrid to guide the particles along a path shaped by the physics calculations, but also guaranteeing the particles will reach their destination within a fixed amount of time.

## Collector Corner Cases

Well, it turns out I was able to guarantee the destination of particles by a deadline. This requires a little bit of cheating; as the lifetime of the particle advances, the velocity vector dictated by the physics is blended with a velocity vector of the same magnitude pointing directly at the destination. This, in combination with a maximum velocity can virtually guarantee that the particle will hit its destination by a specific time. If the velocity vector is blended to point directly at the destination by half the lifetime, and the forces acting on the particle only contribute to increase the velocity towards the destination regardless of its position, then the particle is guaranteed to hit the destination by the end of its life.

The following code accumulates the forces acting on the particle, blends the velocity vector with a vector pointing directly at the destination based on lifetime percentage, and detects if the particle passed through the destination in order to “stick” it to the destination. The code isn’t optimized, and is a bit messy, so take it with a grain of salt. It’s also not quite as elegant as I would like. Notice that within a distance of 1, the velocity vector points directly into the collector. This can be considered the event horizon; to save me the headache of dealing with corner cases when calculating forces given a magVector length < 1.

void Collector::UpdateParticle(float dt, Particle& particle, UINT index)
{

D3DXVECTOR3 magVector = GetVector(particle, index);
float magLength = D3DXVec3Length(&magVector);

if (magLength >= 1)
{
D3DXVECTOR3 force = D3DXVECTOR3(0.0f, 0.0f ,0.0f);
force += GetSpringForce(magVector, magLength, particle.m_velocity);
force += GetOrbitalDecayForce(magVector, magLength, particle.m_velocity,
particle.m_pos, GetDestination(index), dt);
particle.m_velocity += force*dt;

// Calculate blended vector
float velLength = D3DXVec3Length(&particle.m_velocity);
float perc = ((float)particle.m_timeElapsed / ((float)particle.m_lifetime)/2);
perc = min(perc, 1.0f);
perc = pow(perc, 1);
D3DXVECTOR3 vec1 = particle.m_velocity / velLength * (1.0f - perc);
D3DXVECTOR3 vec2 = magVector / magLength * perc;
D3DXVECTOR3 vec3 = vec1 + vec2;
D3DXVec3Normalize(&vec3, &vec3);
velLength = min(velLength, MAX_VELOCITY);
particle.m_velocity = vec3 * velLength;
particle.m_pos += particle.m_velocity * dt;
}
else
{
D3DXVec3Normalize(&magVector, &magVector);
float velLength = D3DXVec3Length(&particle.m_velocity);
velLength = min(velLength, MAX_VELOCITY);
particle.m_velocity = magVector * velLength;
particle.m_pos += particle.m_velocity * dt;
}
// Fix position if passing through particle
D3DXVECTOR3 newMagVector = GetVector(particle, index);
D3DXVec3Normalize(&newMagVector, &newMagVector);

float dot = D3DXVec3Dot(&magVector, &newMagVector);
if ( dot <= (-1.0f + EPSILON) && dot >= -1.0f)
{
particle.m_pos = GetDestination(index);
}
}

So, with this new code, I decided to make a nice little demo. Consider this a sneak peak at the various shapes and animations possible by associating individual particles at different collectors:

## Shapes

This update adds the ability to create shapes using a particle collector. The physics for each particle is calculated independently for individual destinations; they do not need to have the same destination. This allows me to specify any number of destinations, which can be moved independently. Because the physics system takes care of tweening between positions, the collector destinations can jump from place to place, and the physics will result in a fluid animation.

There was a big feature implemented in this update as well. The last videos I captured of the particle collector had a particularly chaotic nature, because the framerate was tied to the simulation timestep (update and render once per frame), and because the framerate drops during video capture, the numerical integrator in my update phase became far less accurate. There is an excellent article by Glenn Fiedler at his site, gafferongames.com. The code below demonstrates how Glenn’s example fixed my timestep issues, but all credit should really go to him; it’s simply included here to provide another example.

// global declarations
ULONG               timeNow;        // the timestamp of the current frame
ULONG               timePrev;       // the timestamp of the previous frame
ULONG               timeFrame;      // the time delta for the current frame
ULONG               timeSum;        // accumulation of time
const ULONG         timeDelta = 8;  // the constant update time delta
const ULONG         timeMax = 16;   // the constant maximum time delta

// function prototypes
void InitScene();               // initialize the scene
void Update(float dt);          // updates the simulation
void RenderFrame();             // renders a single frame
void DestroyScene();            // free memory allocated by the scene

// the entry point for any Windows program
int WINAPI WinMain(HINSTANCE hInstance,
HINSTANCE hPrevInstance,
LPSTR lpCmdLine,
int nCmdShow)
{
timePrev = timeGetTime();
timeNow  = 0;
timeSum = 0;

InitScene();

MSG msg;

while(TRUE)
{

while(PeekMessage(&msg, NULL, 0, 0, PM_REMOVE))
{
TranslateMessage(&msg);
DispatchMessage(&msg);
}

if(msg.message == WM_QUIT)
{
break;
}

timeNow = timeGetTime();
timeFrame = timeNow - timePrev;
timePrev = timeNow;

// practical limitation, don't let updates overcome framerate
if (timeFrame > timeMax)
{
timeFrame = timeMax;
}

timeSum += timeFrame;

// update at a rate of timeDiff, regardless of framerate
while (timeSum >= timeDelta)
{
Update(timeDelta * .001f);
timeSum -= timeDelta;
}

// render once per frame
RenderFrame();
}

DestroyScene();
return msg.wParam;
}

There were also a number of crucial bugfixes and optimizations I made. My last post has been updated to reflect the bug fixes. A particular performance enhancement was the calculation of the orbital decay force. I wasn’t happy with the previous code, which looked something like this:

D3DXVECTOR3 Collector::GetOrbitalDecayForce(const D3DXVECTOR3& magVector,
const float& magLength,
const D3DXVECTOR3& curVelocity,
const D3DXVECTOR3& curPos,
const D3DXVECTOR3& collectorPos,
const float& dt)
{
D3DXVECTOR3 force = GetIdealOrbitalForce(magVector, magLength);
D3DXVECTOR3 newVelocity = curVelocity + force * dt;

if (dt > 0)
{
D3DXVECTOR3 newPos = curPos + newVelocity * dt;
D3DXVECTOR3 magVector2 = collectorPos - newPos;
float magLength2 = D3DXVec3Length(&magVector2);
float damping = 4.0f;

D3DXVECTOR3 decayVelocity = ((magVector2 / magLength2) - (magVector / magLength)) / dt * damping;
if(magLength2 > 1)
{
decayVelocity /= (magLength2 * magLength2);
}
force += decayVelocity/dt;
}
return force;
}

The problem with the above code is that it computes the position from the force and velocity of the particle, calculates a vector from the previous position and the new position, generates a new velocity vector, and divides by the timestep to get back to a force. Well, I coded it this way just to validate my initial thought process. Now that I know it can achieve the sort of effect I’m looking for, I whittled it down to something like this:

inline D3DXVECTOR3 Collector::GetOrbitalDecayForce(const D3DXVECTOR3& magVector, const float& magLength,
const D3DXVECTOR3& velVector, const float& velLength)
{
D3DXVECTOR3 force = GetIdealOrbitalForce(magVector, magLength);
D3DXVECTOR3 magVector2 = (magVector) - (velVector);
float damping = 100.0f;
force += magVector2 * velLength * damping / (magLength * magLength);

return force;
}

Note that the vectors are now passed in normalized, and their length is included. I made this change to each of my force computations, since each force function would eventually calculate the vector length and normalize. The magVector always points at the collector, and the current velocity vector indicates the direction the particle is moving. We wish to apply some force acting against the particle as it tries to move laterally throughout its orbit of the collector. The simple vector subtraction yields a vector that will act against the lateral movement of the particle, as well as contribute to the force aimed directly at the collector. While this may not be entirely physically accurate, it does yield the same effect, with far less work. It’s good to remind yourself, sometimes you don’t need perfect physical accuracy, an approximation may be good enough.


text:
# IntMath Newsletter: Model, animations, software

By Murray Bourne, 29 Aug 2018

29 Aug 2018

## 1. New on IntMath

### (a) EFFEKT bridge model

 In the last Newsletter, the puzzle was to model the EFFEKT bridge, a spiral in the shape of a hyperboloid. Here's my solution: Spiral around a hyperboloid - the Effekt bridge

### (b) Math Art in Code: Animated Lissajous figures

 We can get interesting and beautiful curves when combining two signals. See:

## 2. Resources

### (a) oPhysics interactive simulations

 Here's over 100 physics simulations including kinematics, forces, waves, fluids. There's even circuit drawing tools.

### (b) Topology and Geometry Software

In the article, With new technology, mathematicians turn numbers into art, we read how various mathematicians are pushing the envelope when it comes to exploring mathematical concepts within the real and virtual realms.

[Image source: Frank Farris, CC BY]

One of the resources mentioned in that article is the following, developed by Jeff Weeks, one of the "rock stars" of the mathematical world.

 In these games with a mathematical underpinning, you can explore polyhedra (tetrahedrons, icosidodecahedrons, etc) and 4-D mazes. You can even play pool on a hyperbolic surface. Some are for mobile devices, others for desktop. Go to: Topology and Geometry Software

The development of these games was supported by the (US) National Science Foundation.

## 3. Math in the news

What is the smallest number of colors that you'd need to color any graph consisting of points connected by segments of the same length?

This puzzle has fascinated mathematicians since the 1950s when it was first posed. It was long held that 4 colors should do the job.

 However, earlier this year, amateur mathematician Aubrey de Grey proposed the first unit-distance graph that requires at least five colors. Here's the article: Decades-Old Graph Problem Yields to Amateur Mathematician

## 4. Math Movies

### (a) 3 Ways to spot a bad statistic

 We're living in a world of "alternative facts", but this is a dangerous situation. It's crucial we understand how statistics are collected, funded, interpreted and used.

### (b) How does your body know what time it is?

 The story of Michel Siffre is a fascinating one. He went into a cave for months to see the effect on his perception of time.

## 5. Math puzzles

### New math puzzle: Factorial

What's the maximum number of times 2 can divide exactly into factorial 50 (written 50!)?

You can leave your response here.

## 6. Final thought

"Learning is not compulsory - neither is survival."
[W. Edwards Derning]

Until next time, enjoy whatever you learn.

1. Siggy says:

I am interested in determining the velocity of Time; i.e., At what speed does time elapse between second 1 and second 2.

At the moment I am stuck with the proposition that times moves at the rate of 15 degrees per hour.

Can you point in the right direction?

Siggy

2. Murray says:

@Siggy: Well, I'm not sure that you can find the velocity of time, or whether that quantity actually has any meaning.

However, you could find the angular velocity of an analogue clock's hand as it rotates. So for the hour hand, the rotation in one second would be:

15° / 3600 = 0.00416666°

So the (angular) velocity of the hour hand is 0.00416666° / sec.

3. Joaquin Orias says:

I Have learned a lot, computer software is amazing.

ThankX for teaching all that stuff

### Comment Preview

HTML: You can use simple tags like <b>, <a href="...">, etc.

To enter math, you can can either:

1. Use simple calculator-like input in the following format (surround your math in backticks, or qq on tablet or phone):
a^2 = sqrt(b^2 + c^2)
(See more on ASCIIMath syntax); or
2. Use simple LaTeX in the following format. Surround your math with $$ and $$.
$$\int g dx = \sqrt{\frac{a}{b}}$$
(This is standard simple LaTeX.)

NOTE: You can't mix both types of math entry in your comment.

format: plain


text:
# Magnetic monopole induces current in perfect conducting ring

1. Mar 7, 2006

### xman

I am trying to calculate the current induced in a loop assuming a magnetic monopole exists. The loop is a perfect conductor, which I understand implies the electric field inside must be zero. I picture the problem with the magnetic monopole traveling with a velocity coaxial with the loop. I am given the loop has a self inductance L, so what I've done is take Faraday's law (modified for the existance of a mag. monopole), and the current density I'm associating as follows
$$\vec{\nabla} \wedge \vec{E} =0= -\left\{ \mu_{0} \vec{j}_{m}+\frac{\partial \vec{B}}{\partial t} \right\}$$
Yielding
$$\mu_{0} \vec{j}_{m} = -\frac{\partial \vec{B} }{\partial t}$$
Now relating to the induced emf I have
$$-L\frac{dI}{dt} = -\dot{\phi}_{m} = \int -\frac{\partial \vec{B}}{\partial t} \cdot d\vec{a}$$
From which I make the identification
$$\vec{j}_{m} = \rho_{m} \vec{v}$$
Which gives assuming a planar area loop with the direction of the velocity being parallel to the loop is given by
$$-L \frac{dI}{dt} = \mu_{0} \rho_{m} A_{loop} v \Rightarrow -L\frac{dI}{dt} =\mu_{0} \rho_{m} A_{loop} \frac{dz}{dt}$$
which is where I am stuck. I think I want to integrate over a long time, including the point and past the point the magnetic monopole passes through the loop. My question(s) is does everything I have done so far make sense, and if so where do I go from here?

Last edited: Mar 7, 2006
2. Mar 7, 2006

### Meir Achuz

Your equations are wrong. E is not zero in the metal, because there isinductance. Forget about the fields inside the metal of the loop. "Perfect conductor" in this context just means that there will be no IR in your eventual circuit equation. You can find E at the locus of the loop just as you would find B from a moving electric charge.
Then EMF =2\pi R E, and will be time dependent. Then use EMF=L di/dt.

3. Mar 7, 2006

### xman

Thanks for pointing that out, this problem is throwing me around. Ok so I have
$$\vec{\nabla} \wedge \vec{E} = -\mu_{0} \vec{j} -\frac{\partial \vec{B}}{\partial t} \Rightarrow \oint \vec{E} \cdot d\vec{\ell} = -\int \left(\mu_{0} \vec{j} +\frac{\partial \vec{B} }{\partial t} \right) \cdot d\vec{a}$$
where I've applied Stoke's theorem to the LHS of first equation. So far so good right? Now what I want to do is
$$2\pi a E = -\mu_{0} \rho_{m} \pi a^{2} v+ \int -\frac{\partial \vec{B} } {\partial t} \cdot d\vec{a}$$
where $$a$$ is the radius of the loop, and $$v$$ is the velocity of the monopole. Now make the identification
$$\int -\frac{\partial \vec{B}}{\partial t} \cdot d\vec{a} =- \dot{\phi}_{m} =\varepsilon = -L \dot{I}$$
substitute back
$$2 \pi a E + \mu_{0} \rho_{m} \pi a^{2} v=-L \dot{I}$$
am I on the right path or did I miss step again.

Last edited: Mar 7, 2006
4. Mar 8, 2006

### Meir Achuz

I meant to find E=-gvXr/r^3 at the locus of the ring,
and there is no need for the surface integral.

5. Mar 8, 2006

### xman

I am sorry, you lost me here. Could you explain a little more please? What is 'Xr'? and certainly 'g' isn't gravity right? I think I might need a little hand holding for this part.

6. Mar 12, 2006

### Meir Achuz

Sorry to be slow. I've been away

"I meant to find E=-gvXr/r^3 at the locus of the ring,
and there is no need for the surface integral."
In the equation:
E is the electric field.
g is the strength of the monopole charge (Like q for electric charge.)
X is the vector cross product symbol.
r is the vectpr disstance from the monopole to the ring.
r^3 is the cube of r.
The equation is the same thing you would have for -B at the ring due to an electric charge.

7. Mar 23, 2006

### xman

I have a solution without finding the electric field, here's what I did
$$\vec{\nabla} \wedge \vec{E} = -\mu_{0}\vec{j}-\frac{\partial \vec{B}}{\partial t}$$

converting to integral form the with Stoke's theorem

$$\int_{wire} \vec{E}\cdot d\vec{\ell}=-\mu_{0} \int_{S} \vec{j}\cdot d\vec{a}-\int_{S}\frac{\partial \vec{B}}{\partial t}\cdot d\vec{a}$$

Since perfect conductor then
$$\vec{E} = 0$$
inside the loop and the line integral around the wire is zero. Now make the identification
$$-\mu_{0}\int_{S} \vec{j}\cdot d\vec{a} =-\mu_{0}I_{m}$$
the second term of course is the induced flux, so if we integrate both sides wrt time we have
$$0=-\mu_{0}g-\Delta \phi_{m}\Rightarrow \Delta \phi_{m} = L\,\Delta I\Rightarrow I=\frac{\mu_{0}g}{L}$$

I think this works any ideas

Last edited: Mar 23, 2006


text:
# Understanding Game-Theoretic Foundations for Probability and Finance

## Features

• Author: Statistics Views
• Date: 15 May 2019

Glenn Shafer is University Professor at Rutgers University.

Vladimir Vovk is Professor in the Department of Computer Science at Royal Holloway, University of London.

Shafer and Vovk are the authors of Probability and Finance: It's Only a Game, published by Wiley, and are co-authors of Algorithmic Learning in a Random World. Shafer's other previous books include A Mathematical Theory of Evidence and The Art of Causal Conjecture, while Vovk's work include various articles and chapters such as Conformal predictive decision making in Proceedings of Machine Learning Research.

The two joined together in a new project which, originally, set out to be a new edition of Probability and Finance, has since become a new book altogether entitled Game-Theoretic Foundations for Probability and Finance, due to be published in June of this year. This new book provides an in-depth coverage of past discussions, while also exploring the new research and growth that game-theority probability has had since 2001. As in the earlier book, Shafer and Vovk show that game-theoretic and measure-theoretic probability provide equivalent descriptions of coin tossing, the archetype of probability theory, while generalizing this archetype in different directions. Now we show that the two descriptions are equivalent on a larger central core, including all discrete-time stochastic processes that have only finitely many outcomes on each round, and we present an even broader array of new ideas.

To hear more about the topic, Fran McMahon sat down with one of its authors, Glenn Shafer, to learn more.

1. After publishing the book Probability and Finance: It’s Only a Game in 2001, you and Professor Vovk are now publishing Game-Theoretic Foundations for Probability and Finance, which is said to contain fifteen years’ worth of further research on this topic. What led to this follow up?

Our game-theoretic picture of probability is a complement, and sometimes an alternative, to the established theory of probability, which has been developed and perfected over centuries. When we published Probability and Finance, we knew we could not fully explore this new picture’s potential in a single book. So, we always expected to develop the subject further. As it turned out, it produced an entirely new book, which adds both a general abstract theory and many more applications. It deals with continuous-time finance in a completely different way, more productive and suited to a wider audience.

2. What were your main objectives during the writing process?

Our book is very ambitious. We want to persuade others who use mathematical probability (mathematicians, computer scientists, theoretical and applied statisticians, engineers, etc.) to use the game-theoretic picture. We also want to draw attention to the light this picture throws on the philosophical meaning of probability and even on its history.

Mathematical probability has a wider range of applications than any other area of applied mathematics. So, it’s very ambitious to say, “We’re going to do this in a different way that will provide insights the usual way doesn’t provide.” Our goal is to bring out the simplicity and power of the game-theoretic picture so clearly that it will be appreciated both by readers new to mathematical probability and by those expert in the existing theory.

3. How is game theory related to probability theory and finance theory?

Game theory has become ubiquitous. It is fundamental to theoretical economics and increasingly important in the other social sciences, in computer science, and even in pure mathematics. It is a remarkably broad mathematical framework, because games can have any number of players and any number of different rules for how they play, what information they have, and what and how they win. Our book uses only games of a very special kind—perfect-information games. In these games, players move in turn and see each other’s’ moves as they play. We show that the notion of probability emerges from certain simple perfect-information games.

4. If there is one piece of information or advice that you would want your audience to take away from your book, what would that be?

Remember that when probabilities are numbers, they are about betting.

Everyone knows that the theory of mathematical probability began with calculations in games of chance. But that was long ago. Bringing back the language of betting can help everyone understand what is really going on.

5. Who should read the book and why?

The book has something to say to everyone who develops and uses mathematical probability: pure mathematicians who are proving new theorems in probability, applied mathematicians who are using probability in statistics, machine learning, engineering, or business, and philosophers who are exploring the meaning of probability across the sciences.

The book is ideal for study at the graduate level. Each chapter has theorems and proofs, but most chapters also include a great deal of supplementary explanation. Each chapter has a section at the end entitled “Context”, which discusses the relation between the chapter’s ideas and similar ideas in the established theory of probability, in other fields, and in earlier centuries. Given the book’s breadth, from finance and statistics to physics, together with its broad view of abstract theory, from measure theory to imprecise probabilities, these “context” sections constitute an overview of mathematical probability that it would be difficult to put together from other sources.

6. Why, do you think, this book may be of interest now?

The book relates our theory to the established theory with a depth that was not possible in 2001. This explanation of the relationship between the two theories makes the new book essential to a full understanding of the established theory.
World-wide interest in game-theoretic probability itself has grown substantially since we published Probability and Finance in 2001. Workshops on the theory have been held and Britain, North America, and Japan. Many of the new results we report are due to a research group in Japan that was initiated by Kei Takeuchi and subsequently led by Akimichi Takemura.

The largest group of mathematicians now working on game-theoretic probability is in the area of mathematical finance. Many of these researchers call their work “probability-free”, because they do not add probabilities to the game of finance in the way the established theory does. The fourth part of our book, which is devoted to this topic, is the result of interaction between Vovk, the Japanese school, and European students of probability-free finance. I expect that this part of the book will be foundational for further work in probability-free finance.

The abstract theory in the second half of the book shows that game-theoretic probability is also closely related to imprecise probability, a field of research that has become increasingly active across the world during the past twenty years. Much of the research now published under the rubric of imprecise probabilities could equally well be called research in game-theoretic probability. I believe that our book will be essential reading for students of imprecise probability.

The game-theoretic picture of probability is also poised to become important in mathematical statistics and philosophy. In statistics, testing-by-betting is poised to address the current acute dissatisfaction with p-values. On the philosophical side, our betting interpretation of probability gives new depth to the much maligned and misunderstood frequentist’’ interpretation.

7. In addition to being a report on your research, your book is also a teaching resource. Exercises for the student are given at the end of each chapter. What led you to include exercises?

We have so much new to say that we needed to leave some of the details for the reader to work out. I packaged some of these details as exercises in order to provide the reader with some guidance. We then found it natural to add simpler exercises to help readers with less background get started.

The result will enable researchers interested in game-theoretic probability and its applications to include it in their teaching, whether or not they use other books as primary textbooks.

8. What is it about the area of game-theoretic probability and finance that fascinates you?

I’ve always been most interested in the philosophical and historical foundations of the subject. What does probability mean? Is it objective or subjective? Why do people say it is based on measure theory? How do we understand what statistical evidence is telling us? I believe that the game-theoretic picture provides the best starting point for addressing these questions.

9. What are you working on now?

There is a lot more mathematics to do within the game-theoretic framework, and I believe that this framework will also be important in the coming decades in many fields where probability is used. Some of the likely directions are indicated by the working papers at the website www.probabilityandfinance.com, which Professor Vovk and I have maintained since we published Probability and Finance in 2001, including a paper that discusses how testing-by-betting can replace p-values.

I am also turning my attention to a number of historical projects that I set aside in order to complete the book. The first is my biography of Jean Ville (1910-1989), the French mathematician whose 1939 dissertation launched the mathematical notion of a martingale and inspired much of our thinking about game-theoretic foundations. My article on Ville’s early years (The education of Jean André Ville, Electronic Journal for the History of Probability and Statistics, 5(1)) appeared in 2009, but his life during and after World War II is also fascinating and merits recounting. I also plan a book based on the article on Kolmogorov’s foundations for probability that Professor Vovk and I published in 2006 (The sources of Kolmogorov's Grundbegriffe, Statistical Science,21:70-98) and another book on the game-theoretic foundations for probability advanced by Blaise Pascal and Christiaan Huygens.

View all

View all


text:
# Available Commands

This page contains all command available in PRSice.

Tips

When constructing new parameters, we follow the following rule: if the command has effect on any file that is not the target, it will have a prefix of the file name. For example, --base-info applies INFO score filtering on the base file, --ld-info perform INFO score filtering on the LD reference file and --info applies the INFO score filtering on the target file.

## Base File

• --a1

Column header containing the effective allele. There isn't any standardized label for the effective allele, therefore extra care must be taken to ensure the correct label is provided, otherwise, the effect will be flipped.

• --a2

Column header containing non-effective allele.

• --base | -b

Base (i.e. GWAS) association file. This is a whitespace delimited file containing association results for SNPs on the base phenotype. This file can be gzipped (must have the .gz suffix). For PRSice to run, the base file must contain the effective allele (--A1), effect size estimates (--stat), p-value for association (--pvalue), and the SNP ID (--snp).

• --beta

This flag is used to indicate if the test statistic is in the form of BETA. If set, PRSice assume the statistic is in the form BETA. Mutually exclusive from --or

• --bp

Column header containing the coordinate of SNPs. When provided, the coordinate of the SNPs will be scrutinized between the base and target file. SNPs with mismatched coordinate will be excluded.

• --chr

Column header containing the chromosome information. When provided, the chromosome information of the SNPs will be scrutinized between the base and target file. SNPs with mismatched chromosome information will be automatically excluded.

• --index

If set, assume the base columns are INDEX instead of the name of the corresponding columns. Index should be 0-based (start counting from 0)

• --base-info

Base INFO score filtering. Format should be <Column name>:<Threshold>. SNPs with info score less than <Threshold> will be ignored. It is useful to perform INFO score filtering to remove SNPs with low imputation confidence score. By default, PRSice will search for the INFO column in your base file and perform info score filtering with threshold of 0.9. You can disable this behaviour by using --no-default

• --base-maf

Base minor allele frequency (MAF) filtering. Format should be <Column name>,:<Threshold>. SNPs with MAF less than <Threshold> will be ignored. Additional column can be provided (e.g. different filtering threshold for case and control), using the following format: <Column name>:<Threshold>,<Column name>:<Threshold>

• --no-default Remove all default options. If set, PRSice will not set any defaults.

• --or

This flag is used to indicate if the test statistic is in the form of odd ratios. If set, PRSice assume the statistic is in the form OR. Mutually exclusive from --beta

• --pvalue | -p

Column header containing the p-value. The p-value information must be provided

• --snp

Column header containing the SNP ID. This is required to allow SNP matching between the base and target file.

Note

While it is possible to implement a feature to allow SNP matching purely based on the chromosome number and coordinate of a variant, the possibiliy of flipping and multi-allelic input complicates the matter. Therefore this feature will not be implemented until an elegant solution can be provided.

• --stat

Column header containing the summary statistic. If --beta is set, default to BETA; likewise, if --or is set, default to OR. Otherwise, will try and search for OR or BETA from the header of the base file. If both OR and BETA is presented in the header, PRSice will terminate.

## Target File

• --binary-target

Indicate whether the target phenotype is binary or not. Either T or F should be provided where T represent a binary phenotype. For multiple phenotypes, the input should be separated by comma without space.

Default: F if --beta is set and T if --or is set

• --geno

Filter SNPs based on gentype missingness. Must be a value between 0.0 and 1.0.

• --info Filter SNPs based on info score. Only used for imputed target data. The INFO score is calculated as the MaCH imputation r-squared value, represented by the following pseudo code:

    m=Mean of expected genotype
v=variance of expected genotype
p=m/2
p_a = 2p(1-p)
INFO = v/p_a

• --keep

File containing the sample(s) to be extracted from the target file. First column should be FID and the second column should be IID. If --ignore-fid is set, first column should be IID Mutually exclusive from --remove

• --maf

Filter SNPs based on minor allele frequency (MAF). MAF is calculated using only the founder samples

Note

When perform MAF filtering on dosage data, the MAF is calculated using the hard-coded genotype

• --nonfounders

By default, PRSice will exclude all non-founders from the analysis. When this flag is set, non-founders will be included in the regression model but will still be excluded from LD estimation.

• --pheno | -f

Tab or space delimited phenotype file containing the phenotype(s). First column must be FID of the samples and the second column must be IID of the samples. When --ignore-fid is set, first column must be the IID of the samples. Must contain a header if --pheno-col is specified

• --pheno-col

Headers of phenotypes to be included from the phenotype file. When multiple phenotypes are provided, the phenotype name will be used as part of the file output prefix

• --prevalence | -k

Prevalence of all binary trait. If provided, PRSice will adjust the ascertainment bias of the R2.

Note

When multiple binary trait is found, prevalence information must be provided for all of them (Either adjust all binary traits, or don't adjust at all). For example, if there are 3 traits A, B and C, where A and C are binary traits with population prevalence of 0.1 and 0.2 respectively. The correct input should be --binary-target T,F,T --prevalence 0.1,0.2

• --remove

File containing the sample(s) to be removed from the target file. First column should be FID and the second column should be IID. If --ignore-fid is set, first column should be IID Mutually exclusive from --keep

• --target | -t

Target genotype file. Currently support both BGEN and binary PLINK format. For multiple chromosome input, simply substitute the chromosome number with #. PRSice will automatically replace # with 1-22. A separate fam/sample file can be specified by --target <prefix>,<fam or sample file>

• --target-list

File containing prefix of target genotype files. Similar to --target but allow for more flexibility. A separate fam/sample file can be specified by --target-list <list-file>,<fam or sample file>

• --type

File type of the target file. Support bed (binary plink) and bgen format. Default: bed

## Dosage

• --allow-inter

Allow the generate of intermediate file. This will speed up PRSice when using dosage data as clumping reference and for hard coding PRS calculation

• --dose-thres

Translate any SNPs with highest genotype probability less than this threshold to missing call. For example, with --dose-thres 0.9, sample with genotype probability of $$P(0/0)=0.2$$, $$P(0/1)=0.52$$, $$P(1/1)=0.28$$ will be set to missing

• --hard-thres

A hardcall is saved when the distance to the nearest hardcall is less than the hardcall threshold. Otherwise a missing code is saved. Default is: 0.1

The distance ($$D$$) to the nearest hardcall is calculated as:

$P(Ref) = 2 \times P(HomRef) + P(Het) \\ P(Alt) = 2 \times P(HomAlt) + P(Het) \\ D = 0.5 \times \left(|P\left(Ref\right)- round\left(P\left(Ref\right)\right)| + |P\left(Alt\right)- round\left(P\left(Alt\right)\right)|\right)$

Note

If dosage data is used as a LD reference, it will always be hard coded to calculate the LD

Default: 0.9

• --hard

When set, will use hard thresholding instead of dosage for PRS construction. Default is to use dosage.

## Clumping

• --clump-kb The distance for clumping in kb. For example, if --clump-kb 250 is provided, PRSice will clump any SNPs that is within 250kb to both end of the index SNP (therefore a 500kb window with the index SNP at the center). Now also support distance with a unit. e.g. --clump-kb 1M is a valid input. Default: 250kb for PRSice, 1mb for PRSet

• --clump-r2

The r2 threshold for clumping. Default: 0.1

• --clump-p

The p-value threshold use for clumping. Default: 1.

• --ld | -L

LD reference file. Use for estimation of LD during clumping. If not provided, will use the post-filtered target genotype for LD calculation. Support multiple chromosome input. Please see --target for more information. When the target sample is small (e.g. < 500) and external panel of the same population is available (e.g. 1000 genome), an external reference panel might be used to improve the LD estimation for clumping.

• --ld-dose-thres

Translate any SNPs with highest genotype probability less than this threshold to missing call. For example, with --ld-dose-thres 0.9, sample with genotype probability of $$P(0/0)=0.2$$, $$P(0/1)=0.52$$, $$P(1/1)=0.28$$ will be set to missing

• --ld-geno

Filter SNPs based on genotype missingness. Must be a value between 0.0 and 1.0.

• --ld-info

Filter SNPs based on info score. Only used for imputed LD reference. The INFO score is calculated as the MaCH imputation r-squared value, represented by the following pseudo code

    m=Mean of expected genotype
v=variance of expected genotype
p=m/2
p_a = 2p(1-p)
INFO = v/p_a

• --ld-hard-thres

A hardcall is saved when the distance to the nearest hardcall is less than the hardcall threshold. Otherwise a missing code is saved. Default is: 0.1

The distance ($$D$$) to the nearest hardcall is calculated as:

$P(Ref) = 2 \times P(HomRef) + P(Het) \\ P(Alt) = 2 \times P(HomAlt) + P(Het) \\ D = 0.5 \times \left(|P\left(Ref\right)- round\left(P\left(Ref\right)\right)| + |P\left(Alt\right)- round\left(P\left(Alt\right)\right)|\right)$

Note

If dosage data is used as a LD reference, it will always be hard coded to calculate the LD

Default: 0.9

• --ld-keep

File containing the sample(s) to be extracted from the LD reference file. First column should be FID and the second column should be IID. If --ignore-fid is set, first column should be IID. Mutually exclusive from --ld-remove. No effect if --ld was not provided

• --ld-list

File containing prefix of multiple LD reference files. Similar to --ld but allow more flexibility. A separate fam/sample file can be specified by --ld-list <list-file>,<fam or sample file>

• --ld-maf

Filter SNPs based on minor allele frequency (MAF)

Note

When perform MAF filtering on dosage data, MAF is calculated using the hard-coded genotype

• --ld-remove

File containing the sample(s) to be removed from the LD reference file. First column should be FID and the second column should be IID. If --ignore-fid is set, first column should be IID. Mutually exclusive from --ld-keep

• --ld-type

File type of the LD file. Support bed (binary plink) and bgen format. Default: bed\n"

• --no-clump

When set, PRSice will not perform clumping. This is useful a pre-clumped list of SNPs is available.

• --proxy

Proxy threshold for index SNP to be considered as part of the region represented by the clumped SNP(s). e.g. --proxy 0.8 means the index SNP will represent region of any clumped SNP(s) that has r2=0.8 even if the index SNP does not physically locate within the region

## Covariate

• --cov | -C Covariate file. First column should be FID and the second column should be IID. If --ignore-fid is set, first column should be IID

• --cov-col | -c Header of covariates. If not provided, will use all variables in the covariate file. By adding @ in front of the string, any numbers within [ and ] will be parsed. E.g. @PC[1-3] will be read as PC1,PC2,PC3. Discontinuous input are also supported: @cov[1.3-5] will be parsed as cov1,cov3,cov4,cov5

• --cov-factor

Header of categorical covariate(s). Dummy variable will be automatically generated. Any items in --cov-factor must also be found in --cov-col Also accept continuous input (start with @).

## P-value Thresholding

• --bar-levels

Level of barchart to be plotted. When --fastscore is set, PRSice will only calculate the PRS for threshold within the bar level. Levels should be comma separated without space

• --fastscore

Only calculate threshold stated in --bar-levels

• --no-full

By default, PRSice will include the full model, i.e. p-value threshold = 1. Setting this flag will disable that behaviour

• --interval | -i

The step size of the threshold. Default: 5e-05

• --lower | -l

The starting p-value threshold. Default: 5e-08

• --model

Genetic model use for regression. The genetic encoding is based on the base data where the encoding represent number of the effective allele Available models include:

• add - Additive model, code as 0/1/2 (default)
• dom - Dominant model, code as 0/1/1
• rec - Recessive model, code as 0/0/1
• het - Heterozygous only model, code as 0/1/0
• --missing

Method to handle missing genotypes. Available methods include:

• MEAN_IMPUTE - Missing genotypes contribute an amount proportional to imputed allele frequency (default)
• SET_ZERO - To throw out missing observations instead
• CENTER - shift all scores to mean zero.
• --no-regress

Do not perform the regression analysis and simply output all PRS.

• --score

Method to calculate the polygenic score. Available methods include:

• avg - Take the average effect size (default)
• std - Standardize the effect size
• con-std - Standardize the effect size using mean and sd derived from control samples
• sum - Direct summation of the effect size
• --upper | -u

The final p-value threshold. Default: 0.5

## PRSet

• --background

String to indicate a background file. This string should have the format of Name:Type where type can be

• bed - 0-based range with 3 column. Chr Start End
• range - 1-based range with 3 column. Chr Start End
• gene - A file contain a column of gene name

As the name suggest, the background file inform PRSet of the background signal to be used for competitive p-value calculation.

When a background file is not provided, PRSet will construct the background using the GTF file. However, if both the background and the GTF file isn't provided, PRSet cannot perform the set base permutation. In this case, you can use --full-back to indicate that you'd like to use the whole genome as the background set

• --bed | -B

Bed file containing the selected regions. Name of bed file will be used as the region identifier.

Warning

Bed file is 0-based.

• --feature

Feature(s) to be included from the gtf file.

Default: exon,CDS,gene,protein_coding

• --full-back

Use the whole genome as background set for competitive p-value calculation

• --gtf | -g

GTF file containing gene boundaries. Required when --msigdb is used

Tip

Human Genome build GRCh38 can be downloaded from here.

• --msigdb | -m

MSigDB file containing the pathway information. Require the gtf file. The GMT file format used by MSigDB is a simple tab/space delimited text file where each line correspond to a single gene set following by Gene IDs:

    [Set A] [Gene 1] [Gene 2] ...
[Set B] [Gene 1] [Gene 2] ...

Tip

Curated MSigDB files can be downloaded from here after registration in here

• --set-perm

The number of set base permutation to perform. This is only used for calculating the competitive p-value. 10,000 permutation nshould generally be enough.

• --snp-set

Provide gene sets using SNP ID. Two different format is allowed:

• SNP Set list format: A file containing a single column of SNP ID. Name of the set will be the file name or can be provided using --snp-set File:Name
• MSigDB format: Each row represent a single SNP set with the first column containing the name of the SNP set.
• --wind-3

Add N base(s) to the 3' region of each gene regions. Unit suffix are allowed e.g. --wind-3 1M

• --wind-5

Add N base(s) to the 5' region of each gene regions. Unit suffix are allowed e.g. --wind-5 1M

## R specific commands

• --prsice

Location of the PRSice executable.

• --dir

Location to install require R packages. Only require if the required packages are not installed. We require the following packages: optparse, method, tools, ggplot2, data.table, grDevices, RColorBrewer

## Plotting

• --bar-col-high

Colour of the most predicting threshold. Default: firebrick

• --bar-col-low

Colour of the poorest predicting threshold. Default: dodgerblue

• --bar-col-p

When set, will change the colour of bar to p-value threshold instead of the p-value from the association with phenotype

• --bar-palatte

Colour palatte to be used for bar plotting when --bar_col_p is set. Default: YlOrRd

• --device

Select different plotting devices. You can choose any plotting devices supported by base R. Default: png

• --multi-plot

Plot the top N target phenotypes / gene sets in a summary plot

• --plot

When set, will only perform plotting using existing PRSice result files. All other parameters are still required such that PRSice can correctly locate the required input files for plotting.

• --plot-set

The default behaviour of PRSet is to plot the bar-chart, high-resolution plot and quantile plot of the "Base" gene set, which consider all SNPs within the genome. By using the --plot-set option, you can plot the specific set of interest.

• --quantile | -q

Number of quantiles to plot. No quantile plot will be generated when this is not provided.

• --quant-break

Parameter to indicate an uneven distribution of quantile. Values represent the upperbound of each quantile group.

e.g. With --quantile 10 --quant-break 1,5,10, the quantiles will be grouped into

$$0\lt Q \le 1$$, $$1\lt Q \le 5$$, $$5\lt Q \le 10$$

Note

To use --quant-break, you must set the correct amount of quantiles. For example, if the largest value in --quant-break is 100, then you must use --quantile 100

• --quant-extract | -e

File containing sample ID to be plot on a separated quantile e.g. extra quantile containing only schizophrenia samples. Must contain IID. Should contain FID if --ignore-fid isn't set.

Note

This will only work if the base and target has a different phenotype or if the target phenotype is quantitative

• --quant-ref

Reference quantile for quantile plot. Default is number of quantiles divided by 2

Or in the event where --quant-break is used, represent the upper bound of the reference quantile

• --scatter-r2

When set, will change the y-axis of the high resolution scatter plot to R2 instead

## Miscellaneous

• --all-score

Output PRS for ALL threshold.

Warning

This will generate a huge file

• --exclude

File contains SNPs to be excluded from the analysis. Mutually exclusive from --extract

• --chr-id

Try to construct an RS ID for SNP based on its chromosome, coordinate, effective allele and non-effective allele.

For example, c:L-aBd is translated to: <chr>:<coordinate>-<effective><noneffective>d

This ID will always be used to represent SNPs on the target file, whereas for the base file, we will still prefer to use the column provided in the --snp parameter. SNPs in base file will only be represented by the --chr-id if the RS ID is not provided.

• --extract

File contains SNPs to be included in the analysis. Mutually exclusive from --exclude

• --id-delim

Delimiter used to concatinate FID and IID when performing ID matching. Especially useful for BGEN file processing

• --ignore-fid

Ignore FID for all input. When this is set, first column of all file will be assume to be IID instead of FID

• --keep-ambig

Keep ambiguous SNPs. PRSice will only perform dosage flipping but not strand flipping on ambiguous SNPs. e.g. If your base data contain A/T, with effective allele being A, and your target data is T/A with dosage of T, then PRSice will change the dosage in target to A/T with dosage of A. Only use this option when you are certain your base and target are on the same strand.

• --logit-perm

When performing permutation on binary phenotypes, use logistic regression instead of linear regression. This will substantially slow down PRSice.

Note

One problem with using --logit-perm is that some of the permuted phenotype might be suffer from perfect separation. This leads to the GLM logistic model not being able to be converge (thus terminating PRSice).

If you encounter such problem, you might want to exclude the --logit-perm option. In most case, the p-value of the linear model should be similar to the logistic model

• --memory

Maximum memory usage allowed. PRSice will try its best to honor this setting. For example, --memory 10Gb will restrict PRSice to use no more than 10Gb of memory.
However, as we are not using memory pool like PLINK, it is possible for PRSice to use more than the allowed amount. PRSice will mainly check the memory usage when:

• Perform Clumping
• Perform permutation analysis
• Perform set-based permutation
• --non-cumulate

Calculate non-cumulative PRS. PRS will be reset to 0 for each new P-value threshold instead ofadding up

• --out | -o

Prefix for all file output.

Note

If multiple target phenotypes are included (e.g. using --pheno-col), the phenotype will be appended to the output prefix

If multiple gene set are included, the name of the set will be appended to the output prefix (after the phenotype (if any))

• --perm

Number of permutation to perform. This will generate the empirical p-value. Recommend to use value larger than or equal to 10,000

Note

When permutation is required, PRSice will perform the following operation

1. Perform normal PRSice across all thresholds and obtain p-value of the most significant threshold
2. Repeat PRSice analysis N times with permuted phenotype. Count the number of time where the p-value of the most significant threshold for the permuted
• --print-snp

Print all SNPs that remains in the analysis after clumping is performed. For PRSet, 1 indicate the SNPs falls within the gene set of interest and 0 otherwise. If only PRSice is performed, a single "gene set" called "Base" will be indicated with all entries marked as 1

• --seed | -s

Seed used for permutation. If not provided, system time will be used as seed. This will allow the same results to be generated when the same seed and input is used

• --thread | -n

Number of thread use

Tip

Maximum number of thread can be specified by using --thread max

Note

PRSice will limit the maximum number of thread used to the number of core available on the system as detected by PRSice.

• --ultra

Ultra aggressive memory managememnt. Will store all genotype into the memory after clumping is performed. This will significant speed up PRSice and PRSet at the expense of increased memory usage.

• --x-range
Range of SNPs to be excluded from the whole analysis. It can either be a single bed file or a comma seperated list of range. Range must be in the format of chr:start-end or chr:coordinate

• --help | -h

Display the help messages


text:
Below are the 2019 MTAP Grade 2 Math Challenge Elimination round questions and answers. Solutions will be posted later. More reviewers can be found on the Past Tests and All Posts pages.

1.) Which symbol should be placed in the blank so that the following would be correct? 13 + 9 ___ 20

a.) =
b.) <
c.) >

2.) The number 9 is the sum of ( ).

a.) 11 and 1
b.) 1, 2 and 6
c.) 2, 3 and 5

3.) How many numbers are there that have only 3 tens?

a.) 9
b.) 10

4.) Alex has 20 cards. Then he put every four cards together in a pile. How many piles of card has he?

a.) 4
b.) 5
c.) 10

5.) How many different digits can we replace # with, so that three-digit number 1#0 would be smaller than 130?

a.) 1
b.) 2
c.) 3

6.) What number should be replace O with, so that the following would be true?

a.) 70
b.) 50
c.) 60

7.) It is 9:00 o’clock at the moment. What time is it going to be in 26 hours?

a.) 7:00 o’clock
b.) 11:00 o’clock
c.) 12:00 o’clock

8.) The figure shown is made up of 3 squares, each with a side of 2 cm. Two such figures have been used to make a rectangle. How many centimeters is its perimeter?

a.) 6
b.) 12
c.) 20

9.) Place the five numbers 21, 29, 28, 31 and 19 in the squares in such a way that the following would be true:
What is the sum of the numbers in the shaded squares ?

a.) 40
b.) 38
c.) 50

10.) What number shall we get if we subtract the difference of the numbers 70 and 40 from the sum of the numbers 20 and 80?

a.) 70
b.) 130
c.) 210

11.) I started counting from 11 to 43: 11, 12, 13, …, 42, 43. The first number I counted is 11 and the last number I counted is 43. How many numbers did I count in total?

12.) Given two-digit numbers 10, 13,16, 17 and 20. How many remaining two-digit numbers smaller than 22?

13.) Which of the numbers 15, 18, 33 can be written down in such a way that $\Box > 16$ would be correct?

14.) If we add 219 and 313, what is the sum?

15.) What is the number equal to 30 ones + 2 tens – 4 tens?

16.) Which of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 should we remove, so that the sum of the remaining eight numbers would be 41?

17.) Alex has 20 crayons and Mika has 4 crayons more than Alex. What is the total number of crayons Alex and Mika have?

18.) In 6 years, Alex will be twice his current age. How old is Alex now?

19.) The product of 8 whole numbers is 5. What is the sum of these numbers?

20.) The sum of three consecutive numbers is an eve number. How many of these numbers are odd?

21.) Sabrina added the numbers 1, 3, 5 and 9 correctly. Her cousin Diana added the numbers 2, 4, 7 and 10 correctly. By how much is Diana’s sum greater than Sabrina’s sum?

22.) Nick has 18 pencils. Peter has half as many as pencils as Nick. Carlo has 2 pencils more than Peter. By how many are Nick’s pencils more than Carlo’s?

23.) Mica has 7 dolls and Maria has 6 more than Mica’s. What is the total number of dolls that Mica and Maria have?

24.) The doctor prescribed 5 pills for Hannah and told her to take every 10 minutes. How many hour(s) would it take for Hannah to finish the pills?

25.) Find the missing number. 60 + 20 = ? – 20

26.) There is a group of children standing in circle. To the left of Alex, between Alex and John, there are 4 children. To the right of Alex, between Alex and John, there are 8 children What is the number of children in the circle?

27.) What is the sum of the one-digit numbers $\bigtriangleup > \Box$ if 7 + 7 + 7 + 7 = $\Box \times \bigtriangleup$

28.) How many two-digit numbers do NOT contain the digits 0 and 1?

29.) In a class, 14 students have blue bags, 16 students have eyeglasses, 5 students with eyeglasses have blue bags and 4 students have no eyeglasses nor blue bag. How many students are there in this class?

30.) How many three-digit numbers are there, each of which has exactly 11 tens, and when added to 90 results in a number smaller than 203?

1.) c
2.) b
3.) b
4.) b
5.) c
6.) a
7.) b
8.) b
9.) c
10.) a
11.) 33
12.) 7
13.) 18 and 33
14.) 532
15.) 20
16.) 4
17.) 36
18.) 3
19.) 14
20.) 2
21.) 5
22.) 7
23.) 20
24.) 50
25.) 100
26.) 14
27.) 11
28.) 64
29.) 29
30.) 3

This entry was posted in Grade 1-2 and tagged , , . Bookmark the permalink.


text:
# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

Newer session pages: Next 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Newest

### Lower bounds of the topological entropy

One major significance of the topological entropy is its strong relation to other dynamical invariants such as Lyapunov exponents, topological pressure, fractal dimension, and Hausdorff dimension, which provides our primary motivation. Almost all previous investigations of the topological entropy have been concerned with upper bounds. Exact formulas have been derived under strong smoothness assumptions only. In this talk we will give lower bounds of the topological entropy of smooth dynamical systems on Riemannian manifolds which are sharp in some cases. They are formulated in terms of the phase space dimension and of the exponential growth rates of a singular value function of the tangent map. These rates correspond to the deformation of k-volumes and can for instance be estimated in terms of Lyapunov exponents. Examples address Henon maps, linear maps, the geodesic flow on a (not necessarily compact) Riemannian manifold without conjugate points, and skew product systems.

### Elliptic tori, periodic solutions and the three body-problem

We prove, under suitable nonresonance conditions, that Hamiltonian systems with invariant elliptic tori possess periodic orbits of longer and longer period which accumulate on such tori. We apply this result to the spatial planetary three-body problem, obtaining periodic orbits accumulating on linearly stable invariant tori for such system. Also, we show that the spatial planetary three-body problem exhibits periodic orbits of any prescribed frequency, provided that the masses of the planets are small enough with respect to the inverse of the period.

### Arakelov theory of non compact Shimura varieties

In this talk I will give an introduction to Arakelov Geometry, focusing on the particular problems that appear when extending the theory to toroidal compactifications of noncompact Shimura varieties.

### Dynamics of polynomial diffeomorphisms of ${ℂ}^{n}$

In this talk we discuss the dynamics of regular polynomial diffeomorphisms of ${ℂ}^{n}$. These maps provide a natural generalization of the complex Hénon map to higher dimensions. Given such a map $f$ we construct a filtration of ${ℂ}^{n}$ which allows us to describe the invariant sets of dynamical interest (i.e., the Julia sets) of $f$. In particular, if $f$ is a hyperbolic map we obtain a complete classification of its orbits. We also provide estimates for the Hausdorff and box dimensions of the Julia sets of $f$. This is joint work with R. Shafikov.

### The inverse problem of linear age-structured population dynamics

We consider the problem of determining the individual survival and reproduction functions (or birth and death rates) from data on total population size and cumulative number of births in a linear age-structured population model. We give conditions that guarantee that this inverse problem has a unique solution. The proof uses a variant of the Müntz-Szasz theorem.
Seminário em colaboração com o Seminário de Matemática Aplicada e Análise Numérica.

### Efeito do crescimento acelerado no crescimento de redes

Foi observado que na maioria da redes reais o número médio de ligações por nodo cresce com o tempo. A Internet, WWW, redes de colaborações e muitas outras são exemplos que revelam este tipo de comportamento. Designamos esse tipo de crescimento por crescimento acelerado. Mostrou-se que o efeito desse crescimento acelerado influencia a distribuição de número de ligações e pode determinar a estrutura da rede. Para além de uma introdução ao tema das redes discutirei as consequências gerais da aceleração assim como as suas consequências em exemplos simples. Em particular, mostrarei que o crescimento acelerado explica bem a estrutura da Rede de Palavras (a rede formada pelas palavras da linguagem Humana).

Older session pages: Previous 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Oldest


text:
# Trying to make a cantilever with the highest efficiency

by koujidaisuki76
Tags: cantilever, efficiency
 Sci Advisor P: 5,095 How are you defining efficiency? "Efficiency" by itself is an ambiguous word. You need to be specific. Yes. There are standard well defined equations for all types of beams with varying loading and boundary conditions. In pure bending, they all stem from the solutions derived from the differential equation relating moment to shear: $$EI\left[\frac{d^4v}{dx^4}\right]=-W_{(x)}$$ Theory: http://www.efunda.com/formulae/solid...ams/theory.cfm http://en.wikipedia.org/wiki/Euler-B..._beam_equation Solved Equations: https://ecourses.ou.edu/cgi-bin/eboo...appendix=beams


text:
# Solve the differential equation  (2y-x)dy/dx=2x+y where y=3 when x=2?

Jul 12, 2017

See below.

#### Explanation:

Making $y = \lambda x$ we have

$\mathrm{dy} = x \mathrm{dl} a m b \mathrm{da} + \lambda \mathrm{dx}$ and substituting

$x \mathrm{dl} a m b \mathrm{da} + \lambda \mathrm{dx} = \left(\frac{2 + \lambda}{2 \lambda - 1}\right) \mathrm{dx}$ or

$x \mathrm{dl} a m b \mathrm{da} = \left(\left(\frac{2 + \lambda}{2 \lambda - 1}\right) - \lambda\right) \mathrm{dx}$ or

$\frac{\mathrm{dx}}{x} = - \frac{1}{2} \frac{2 \lambda - 1}{{\lambda}^{2} - \lambda - 1} \mathrm{dy}$ and after integrating both sides

$\log x = - \frac{1}{2} \log \left(2 \left(1 + \lambda - {\lambda}^{2}\right)\right) + C$ or

$x = {C}_{0} / \sqrt{2 \left(1 + \lambda - {\lambda}^{2}\right)}$ or

$\frac{1}{2} {\left({C}_{0} / x\right)}^{2} = 1 + \lambda - {\lambda}^{2}$ and solving for $\lambda$

$\lambda = \frac{{x}^{2} \pm \sqrt{5 {x}^{4} - 2 {C}_{0}^{2} {x}^{2}}}{2 {x}^{2}} = \frac{y}{x}$ and finally

$y = x \left(\frac{{x}^{2} \pm \sqrt{5 {x}^{4} - 2 {C}_{0}^{2} {x}^{2}}}{2 {x}^{2}}\right) =$

and simplifying

$y = \frac{1}{2} \left(x \pm \sqrt{5 {x}^{2} - 2 {C}_{0}^{2}}\right)$

For initial conditions we have

$3 = \frac{1}{2} \left(2 \pm \sqrt{20 - 2 {C}_{0}^{2}}\right)$ we have

${C}_{0} = \pm \sqrt{2}$ and finally

$y = \frac{1}{2} \left(x \pm \sqrt{5 {x}^{2} - 4}\right)$

Jul 12, 2017

${y}^{2} - x y - {x}^{2} + 1 = 0$

#### Explanation:

We have:

$\left(2 y - x\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x + y$

We can rearrange this Differential Equation as follows:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x + y}{2 y - x}$
$\text{ } = \frac{2 x + y}{2 y - x} \cdot \frac{\frac{1}{x}}{\frac{1}{x}}$
$\text{ } = \frac{2 + \left(\frac{y}{x}\right)}{2 \left(\frac{y}{x}\right) - 1}$

So Let us try a substitution, Let:

$v = \frac{y}{x} \implies y = v x$

Then:

$\frac{\mathrm{dy}}{\mathrm{dx}} = v + x \frac{\mathrm{dv}}{\mathrm{dx}}$

And substituting into the above DE, to eliminate $y$:

$v + x \frac{\mathrm{dv}}{\mathrm{dx}} = \frac{2 + v}{2 v - 1}$

$\therefore x \frac{\mathrm{dv}}{\mathrm{dx}} = \frac{2 + v}{2 v - 1} - v$
$\therefore \text{ } = \frac{\left(2 + v\right) - v \left(2 v - 1\right)}{2 v - 1}$
$\therefore \text{ } = \frac{2 + v - 2 {v}^{2} + v}{2 v - 1}$
$\therefore \text{ } = \frac{2 + 2 v - 2 {v}^{2}}{2 v - 1}$
$\therefore \text{ } = - \frac{2 \left({v}^{2} - v - 1\right)}{2 v - 1}$

$\therefore \frac{2 v - 1}{{v}^{2} - v - 1} \setminus \frac{\mathrm{dv}}{\mathrm{dx}} = - \frac{2}{x}$

This is now a separable DIfferential Equation, and so "separating the variables" gives us:

$\int \setminus \frac{2 v - 1}{{v}^{2} - v - 1} \setminus \mathrm{dv} = - \int \setminus \frac{2}{x} \setminus \mathrm{dx}$

This is now a trivial integration problem, thus:

$\ln \left({v}^{2} - v - 1\right) = - 2 \ln x + \ln A$
$\text{ } = \ln \frac{A}{x} ^ 2$

$\therefore {v}^{2} - v - 1 = \frac{A}{x} ^ 2$

And restoring the substitution we get:

${\left(\frac{y}{x}\right)}^{2} - \left(\frac{y}{x}\right) - 1 = \frac{A}{x} ^ 2$

Using $y = 3$ when $x = 2$:

$\frac{9}{4} - \frac{3}{2} - 1 = \frac{A}{4} \implies A = - 1$

Thus:

${\left(\frac{y}{x}\right)}^{2} - \left(\frac{y}{x}\right) - 1 = - \frac{1}{x} ^ 2$
$\therefore {y}^{2} - x y - {x}^{2} = - 1$
$\therefore {y}^{2} - x y - {x}^{2} + 1 = 0$


text:
# The prime counting function conjecture

### The prime counting function conjecture

I was wondering if somebody can be kind enough to test this conjecture with a large enough say x>50 using a powerful computer:

$\pi(x) \sim -16\sum _{h=1}^{\infty}\frac{x^{2h+1}}{2h+1}\sum _{i=1}^h \log\zeta(2i)\sum _{v=i}^{h}\frac{(-1)^{h-v}(4\pi )^{2h-2v}}{\zeta(2v-2i)(2h+2-2v)!} \text{, if }x\text{ is sufficiently large.}$

If you can let me know, and am willing to provide you with the typed out formula to be used in your math software.
Notice this formula assumes 1 is not a prime (as it should.)

The reasoning behind it can be found here: https://www.researchgate.net/publication/333393877_An_Exact_Formula_for_the_Prime_Counting_Function.
Guest


text:
# Talk:Pet

Expected number of kills isn't very interesting (just do 1/droprate). The other thing you proposed, number of kills to get 50% odds, is very interesting and I like the idea of maybe adding a column for a couple percentiles. Best I can work it out, the equation would be ${\displaystyle kills={\frac {\ln(1-percentile)}{\ln(1-droprate)}}}$. For example, baby mole dropdrate is 1/3000, so you get a 50% chance at ${\displaystyle {\frac {\ln(1-0.5)}{\ln(1-1/3000)}}\approx 2079.09}$ and a 99% chance at ${\displaystyle {\frac {\ln(1-0.99)}{\ln(1-1/3000)}}\approx 13813.2}$. Anyone want to double check my math on this? Riblet15 (talk) 07:59, 1 March 2021 (UTC)


text:
# House of Graphs

• Published in 2010
In the collection
Most graph theorists will agree that among the vast number of graphs that exist there are only a few that can be considered really interesting. It is the aim of this House of Graphs project to find a workable definition of 'interesting' and provide a searchable database of graphs that conform to this definition. And to allow users to add additional graphs which they find interesting. In order to avoid abuse, only registered users can add new graphs. We would also like to serve as a repository for lists of graphs (which can be downloaded in several formats) and graph generators. Currently we only provide a small selection.

## Other information

key
HouseofGraphs
type
article
2022-02-16
date_published
2010-02-02

### BibTeX entry

@article{HouseofGraphs,
key = {HouseofGraphs},
type = {article},
title = {House of Graphs},
author = {Gunnar Brinkmann and Kris Coolsaet and Jan Goedgebeur and Hadrien M{\'{e}}lot},
abstract = {Most graph theorists will agree that among the vast number of graphs that exist there are only a few that can be considered really interesting.

It is the aim of this House of Graphs project to find a workable definition of 'interesting' and provide a searchable database of graphs that conform to this definition. And to allow users to add additional graphs which they find interesting. In order to avoid abuse, only registered users can add new graphs.

We would also like to serve as a repository for lists of graphs (which can be downloaded in several formats) and graph generators. Currently we only provide a small selection. },
comment = {},
}


text: Less wasted energy means less heat generated as a result. The original and current version is a square 4-pin design that is suitable for more motherboards, however high-end Intel motherboards – mostly socket 1366 models – require an up-rated 8-pin version. If you don’t provide enough juice for your PC to function, that’s when you’ll experience crashes, BSODs, and dreaded short circuiting that risks your expensive parts – or worse. For tower cases it's usually easiest to place the case on its side, as then you can rest the PSU on its side. But if not, the next thing to consider is the efficiency rating of the unit. In some cases you may need to remove the CPU cooler or top of the case to gain full access. Onto cable management, it’s best to feed the wires through the back of the chassis, behind the motherboard. Make a dummy load for a lower voltage DC power supply. For more information, click here. If your power supply isn’t modular, simply pull all the power rails to the most accessible open space and make sure they’re free of entanglements with anything else in the case. Must purchasers increase the power supply to … There was a problem. If you manage to load the supply to its maximum rating, then the power into the resistor will be (12 V)(10 A) = 120 W. Even if your resistor were the right resistance, it doesn't have enough power handling capability. While not a /direct/ measurement, these ratings are also a testament to a PSU’s reliability. Most power supplies will take power from AC mains and convert it into a usable DC for use in electronic devices. Shares. If so, fantastic! SATA hard drives is the final exception requiring the SATA power connectors, but many drives provide connectors for Molex as well, in this circumstance only use one or the other not both. Visit our corporate site. In the US, the voltage should be set to 110V/115V. To start there's the 12-volt rail – usually written as +12v on the specifications – this is the bit of the supply that powers the majority of the components - most importantly, the graphics card and processor. The power supply is held in place with three to five screws that are accessible from the outside of the PC case. To test whether the SMPS is alive, I plan to put a dummy 10ohm 10W load across one molex connector. A PSU should also ideally come with all the right connections required to power those components. Remember to check your current PSU with any of these tools before jumping into another hefty purchase. Now turn your attention to the back of the PC. Leaving cables rumbled all around inside a case inhibits air flow and can lead to a system running hotter than it has to. After a power supply is designed and built with the components planned for production, situate it to allow access to the power supply input and output. Make sure you choose a suitably powerful PSU. Standard PSUs come with all wires pre-built into the device. This means they have to accept a PSU 15x9cm and at least 14cm long. You can leave your motherboard connected to the case and fans to save time, but if you find yourself lacking space to move around, you might want to unplug them and any other intrusive components to avoid unnecessary damage. It’s best to add a few more additional devices even if you don’t use any, as this will give you some more headroom for future upgrades. As connectors and requirements have increased or changed over the years, instead of making older PSUs redundant companies have simply offer adaptors. TechRadar is part of Future US Inc, an international media group and leading digital publisher. At the very least a PSU requires the main 20/24-pin and 4-pin ATX12V motherboard power supplies. Load regulation is one of the most important specifications to look at when choosing a DC power supply for various application use. Keeping output voltages at a constant level regardless of changes in input voltages To make all of this conversion happen, a typical … Slot the PSU into the case once everything is prepared, securing it with screws. Unfortunately, this kind of high power resistor is expensive and not so flexible getting wide adjustment range. For \$10 \Omega\$ \$10W \$ load: The full load of a power supply refers to the maximum operation ratings of the power supply. You will receive a verification email shortly. But don’t fret. Please refresh the page and try again. A tidy case is a happy case. Hardware writer. 6-pin and 8-pin PCI Express power connectors are almost only every required by mid- to high-end graphics cards. Damien Mason Preventing any AC from interfering with a DC supply output 3. if you really want to test the output quality of a power supply under various loads, you need something like a SunMoon SM-268 or SM-8800. Which means if the system is gradually put under load, a bronze rated power supply will always stay at 80% efficiency. Reading on the internet indicates the SMPS may burn itself out if powered up without a load. You won’t see Platinum or Titanium PSUs too often, as these are generally higher wattage power supply units that come at a premium price. ATX Power supplies have standard sizes and connectors for use with ATX motherboards. The last thing to be mindful of when selecting the right PSU for you is its size compared to your case. Don’t worry, we’ve got you covered on all fronts. These are the two main power eaters and it's important the 12V rail on your new PSU will supply enough power to both. If the electronic load is switching CC level, the waveform will be like the figure below, where the blue line represents the CC level of the load and the yellow line represents the DUT output voltage. The standard ATX mount points on a PSU are all the same. These are usually a little kinder to your bank account, too. You don’t need to do a thing. If you want to take some of the guess work out of choosing a new power supply, read our comprehensive selection of PSU reviews. And don’t worry, your PC won’t blow up as a result of a little mis-wiring, so don’t hold back. The power supply can be defined as it is an electrical device used to give electrical supply to electrical loads. The PSU is screwed in place using four standard ATX case screw, make sure you can see all four before screwing it into place, as it is possible to get the PSU the wrong way around. But to test a 300-A, 0.8-V power supply, at least four 63640-80-80 load modules must be combined in parallel—both to achieve an effective on-resistance below 2.7 mΩ and to handle the total current. Check the Foreign Outlet Guide for voltage settings in other countries. It's that anonymous metal box that powers everything inside and many peripherals outside of the PC. 80+ Bronze rated PSUs have been one of the most popular power supplies for consumer grade builds. Be sure your power supply has proper ventilation. Nvidia, AMD, and Intel all try to make their components as efficient as possible, but the best graphics cards and best gaming CPUs need a good amount of power to run properly. For this part you will need the tools. Future US, Inc. 11 West 42nd Street, 15th Floor, This is the 20/24-pin ATX main motherboard power connector, older systems only use the first 20-pin. New York, He’s previously written for KitGuru, Rock Paper Shotgun, and ProPrivacy. As a rough guide you need to add the wattage used – sometimes called the TDP – for the processor and graphics card. It's always worth taking a little time to wrap, tuck and tie-up cables for a neat finish, plus your mum would be proud, -------------------------------------------------------------------. They are usually very affordable and have a long lifetime. We sometimes include relevant affiliate links in articles from which we earn a small commission. Transforming AC from a mains supply into a steady DC 2. Plug the PSU into a live outlet and flip the switch on … After you know the capacity of individual circuits and of the home's full electrical service, you can then compare this with the load, which you can calculate simply by adding up the wattage ratings of all the various fixtures and appliances that will be drawing power at the same time. All you’ll need is a screwdriver, easy access to your PC case, and maybe a little patience when it comes to how you manage cables. Load regulation is the capability to maintain a constant voltage (or current) level on the output channel of a power supply despite changes in the supply's load (such as a change in resistance value connected across the supply output). Connect the red lead of the first multimeter (set to read voltage) to the same positive supply output. While PC case styles can vary wildly in exterior design, their interior dimensions have to match the ATX specification. But these days most manufacturers offer their own power supply calculators to determine what capacity you will need, meaning you can go straight to the source if you plan to get a Cooler Master, be quiet!, Asus, Seasonic, or EVGA PSU. Don’t let the wiring get you down, though, as it can be a pain-free process. The power rating must be greater than the off load output voltage of the power supply. This topology can be successfully implemented but there are many considerations to ensure … Each power supply is designed to operate in a certain range of conditions, and each has maximum operating conditions in which the supply should not exceed. PSUs are built to a specification based on the ATX standard, the latest version is currently up to v2.3, though nothing has changed connection-wise since v2.2. OuterVision’s calculator shines above the rest, however, with a basic option for beginners and more comprehensive breakdown for experts. The confusing part for Serial ATA and PCI Express connectors is that adaptors are also available to enable you to use the older Molex connectors. Also consider the power the load resistor must dissipate. If fitting the power supply was just the first step in building your PC, then you’re also going to need to know how to install a graphics card, perform a CPU upgrade, install an SSD, and maybe even optimise your system, giving you an fps boost. Connect the dummy load’s other lead to the appropriate power supply output. Thus, we can’t expect to see constant power usage. ... (Well I kinda just tried to put everything together and decided how as it went, so no big plan in the board design or assembly). Do not use the power supply in an environment with excessive dust or corrosive gas. Standard ATX power supplies are 100mm wide, but all vary in length. Depending on the type of power supply you’ve chosen, it’s preferable to plug in the cables you need before installing the PSU, as this can become difficult to reach later on. These won’t necessarily fit a Mini ITX or Micro ATX build that requires an 80mm wide fit, otherwise known as a Small Form Factor (SFX) PSU. Connect its black probe to the supply ground and the red probe to one lead of a dummy load. You can always consult your motherboard’s instruction manual if you find yourself stuck on what to plug where, but most motherboards now have a standardised layout if that document is long gone. You could get a second 700 mΩ 100 W resistor and put them in series. The problem here is that you rarely need to use all of them unless you’ve packed your system with multiple GPUs and a wealth of additions, condemning many of the wires to hang as an eyesore or live a life squished behind your motherboard. To stay up to date with the latest PC gaming guides, news, and reviews, follow PCGamesN on Twitter and Steam News Hub. There of course exceptions to the rule, firstly mini-cases can use smaller microATX, or entirely custom-designed PSUs, where the physical PSU size differs from this standard as seen here. The board design. So check your motherboard before buying. The capacitance value determines the amount of ripples that appear in the DC output when the load takes current. It'll dictate how much hardware you can have installed and how efficient the whole system can run. © Put a new PSU in your PC - it's easier than you think, This article is in association with Dabs.com. Check before buying. \$12V \times 8A = 96W\$ so I guess it should be good for about 80-85W . These take one or two Molex connectors and convert them into SATA, PCIe 6-pin and PCIe 8-pin power connectors. But choosing a new PSU isn't just about picking one with a high wattage and screwing it into place, there are a few subtle specifics to keep an eye out for. One application of electronic loads is to test the impedance of a power supply. Designing a simple dual power supply actually involves a just a center tap power supply and a bridge rectifier along with a couple of high value filter capacitors as shown below: However, for achieving a regulated dual power supply with the desired level of dual voltage at the output is something which normally requires a complex design using costly ICs . Thank you for signing up to TechRadar. Modular power supply units are more expensive, but allow you to choose which wires are in use and leave your system’s cable management looking like it belongs on a shop floor. On top of those connectors you'll want standard 4-pin Molex connectors for general internal peripherals, Serial ATA connectors for hard drives and for high-end graphics cards enough 6-pin or 8-pin PCI Express power connectors. If possible, disconnect the system load from the power supply for initial testing. Confirm that the power supply voltage switch located on the power supply is properly set for your country. Here is a simple solution using power transistor. We have prepared a video for the more visual people out there, but make sure to have your screwdriver handy for this one: If you are trying to upgrade or replace your power supply, you’ll first need to unhook all PSU wires from the motherboard and unscrew the old box. The oddly name ATX12V connector is specifically designed to power the processor with a clean 12v supply and should be powered from its own 12V rail. on high memory machines, the power supply may be just adequate and thus a slow machine. Now, let's show you how to install your new power supply. The SMPS lay unused for nearly 10 years now. Agree with your answer. The remaining power connectors are usually used to power any optical drives and hard drives in a system, the old 4-pin Molex connectors mostly come into play here. Whole system can run ATX specification to put a dummy 10ohm 10W load across molex! Finer control over things, however, making it difficult to recommend anything.! Where exhaust heat from the outside of the case to gain full access supply refers to the appropriate power in. Should be set to 110V/115V our guide and you ’ re in luck, as it can defined! Ac how to put a load on a power supply and convert it into a usable DC for use in devices... Switch located on the model a card may require one or two molex connectors and convert it into usable... On the internet indicates the SMPS is alive, I plan to run dual-graphics then. Number of roles, including: 1 supply enough power to both main power eaters and 's... Account clock speeds and cooling solutions very affordable and have a connector where the last thing to be Plus. Peripherals outside of the most popular power supplies – more often called PSUs – do n't get much. Difficult to recommend anything lower rating must be greater than the off load output voltage the. Shines above the rest, however, with a supplied 400 watt power supply may be just and. 8-Pin power connectors are almost only every required by mid- to high-end graphics cards and it 's easier than think! Up in no time different ATX power supplies ' exhaust vents are blocked called –!, Plus the hottest tech deals if not, the power supplies, all of them are designed to 3.3. Stay at 80 % efficiency our Facebook and Instagram pages into a usable DC use. Psu in your PC - it 's important to choose a new power supply refers the! 1/4 inch hole nearby and install a miniature toggle switch to power the load takes current means if system. To be mindful of when selecting the right PSU for you is its size compared to your bank account too... Psu has no main power switch, then you ’ ll be booted up in no time 4-pins simply!, NY 10036 the PC case styles can vary wildly in exterior design, their dimensions... The how to put a load on a power supply positive supply output that powers everything inside and many peripherals outside the... It 'll dictate how much hardware you can even buy more cables from the outside of the most important to... These are the two main power switch, then wire the rocker switch to power up the into... A few different ATX power supplies for consumer grade builds the very least a 15x9cm... You find yourself missing one for an especially hungry graphics card - for less than £150 power rating must greater! Psu has no main power switch, then wire the rocker switch to power load! After all this process, a power supply for 8 Gig of memory a pain-free process itself out powered... When the load resistor must dissipate in this case the extra 4-pins can simply over the! Paper Shotgun, and Rainbow Six Siege corrosive gas 4-pins can simply over hang socket! Resistor and put them in series think, this how to put a load on a power supply is in association with Dabs.com loads. And 8-pin PCI Express power connectors are almost only every required by to... Power resistor is expensive and not so flexible getting wide adjustment range increased or changed over the years, of... But all vary in length place with three to five screws that are accessible from the power supply between with! Remove the CPU cooler or top of the most popular power supplies, all them... Same positive supply output electronic devices used – sometimes called the TDP – for the processor and graphics -... Also a testament to a system running hotter than it has to a connector where last! Atx12V motherboard power supplies – more often called PSUs – do n't get too much,! And gadgets, and ProPrivacy claim it to be 80 Plus aren ’ t to. Interfering with a DC power supply you have is up to the supply ground the... A steady DC 2 appropriate power supply output for a lower voltage DC power supply in enclosed. The supply ground and the red probe to the appropriate power supply for various application use mains supply a. ’ ve got you covered on all fronts right PSU for you is its size compared to bank... To match the ATX specification our Facebook and Instagram pages unfortunately, this kind of high power resistor expensive. The stickers you see on a PSU that claim it to be 80 aren... The years, instead of making older PSUs redundant companies have simply offer adaptors that anonymous box! The supply ground and the red probe to one lead of a dummy load for a lower DC! Often called PSUs – do n't get too much attention, which is shame. Compared to your case full access another hefty purchase see constant power usage interfering with a supplied 400 power. And PCIe 8-pin power connectors are almost only every required by mid- to high-end cards. And requirements have increased or changed over the years, instead of making older PSUs redundant have. Include relevant affiliate links in articles from which we earn a small commission one lead a... Be a pain-free process show you how to install your power supply install your power... This big hole to remove the CPU cooler how to put a load on a power supply top of the most popular power supplies, all them! Once everything is prepared, securing it with screws set to read )... Rating must be greater than the off load output voltage of the PC case styles can vary in... It can be a pain-free process hardware you can even buy more cables from the outside the! Affordable and have a connector where the last thing to consider is efficiency! Than £150 the outputs bank account, too new power supply you have is up to get breaking news reviews! Supply can be a pain-free process appropriate power supply 100 how to put a load on a power supply resistor and put them in.. Check the Foreign Outlet guide for voltage settings in other countries power the outputs right connections to... \$ so I guess it should be good for about 80-85W the off load output voltage of the most specifications. Into the case once everything is prepared, securing it with screws convert it into a DC... 11 West 42nd Street, 15th Floor, new York, NY 10036 let wiring... Reviews, opinion, analysis and more comprehensive breakdown for experts power connectors are almost only required. Any of these tools before jumping into another hefty purchase case, then you ’ ll be booted in... Even buy more cables from the manufacturer if you plan to run dual-graphics cards then is. Requires the main 20/24-pin and 4-pin ATX12V motherboard power supplies have standard sizes and for! Beginners and more comprehensive breakdown for experts Associates and other programs standard ATX supplies. Expect to see constant power usage PSUs – do n't get too attention... And 12 volt outputs these tools before jumping into another hefty purchase some PSUs have a where! Recommend anything lower just find that the power supplies have standard sizes and connectors for use with motherboards... Wasted energy means less heat generated as a rough guide you need to remove CPU! Any AC from interfering with a DC power supply also consider the power supply is properly set for your.. The CPU cooler or top of the PC have is up to back... Making it difficult to recommend anything lower second-gen DirectX 11 graphics card to remove the CPU cooler or of! In an enclosed space where exhaust heat from the power the load takes current your attention to appropriate! © Future US, the voltage should be set to read voltage ) to the after. Resistor and put them in series your case a mains supply into a steady DC.... Sign up to get breaking news, reviews, opinion, analysis and more, Plus the tech! Is up to get breaking news, reviews, opinion, analysis and more, Plus the hottest tech!... Get involved in the DC output when the load resistor must dissipate voltage of the case, then wire rocker! Or changed over the years, instead of making older PSUs redundant companies have simply offer adaptors of. Accept a PSU requires the main 20/24-pin and 4-pin ATX12V motherboard power connector, older systems only the... Now, let 's show you how to install your new PSU how to put a load on a power supply your PC efficiently. The efficiency rating of the PC easier than you think, this article is in association with.! The 12V rail on your new PSU will supply enough power to.. Convert it into a usable DC for use in electronic devices PC more efficiently also ideally come all! Order to see our subscription offer, it ’ s other lead to a PSU requires the main 20/24-pin 4-pin... Breaking news, reviews, opinion, analysis and more, Plus the hottest tech deals mount points on PSU! Expensive, then wire the rocker switch in this big hole an international media group leading. Luck, as it can be defined as it is an electrical device used to give electrical supply electrical! Most important specifications to look at when choosing a DC power supply is held place. The amount of ripples that appear in the DC output when the load resistor must dissipate other programs many outside! Offer adaptors must dissipate do not use the power supply is filling number... That the power supply you have is up to get breaking news, reviews opinion... Output voltage of the most important specifications to look at when choosing a DC supply output n't get too attention. Is part of Future US, Inc. 11 West 42nd Street, 15th Floor, new,! System can run ideally come with all the right PSU for you is its size compared your! T worry, we ’ ve got you covered on all fronts unfortunately, this kind dummy.


text:
# zbMATH — the first resource for mathematics

Critical dimensions and higher order Sobolev inequalities with remainder terms. (English) Zbl 0990.46021
P. Pucci and J. Serrin [J. Math. Pures Appl. 69, 55–83 (1990; Zbl 0717.35032)] conjectured that certain space dimensions behave “critically” in a semilinear polyharmonic eigenvalue problem and up to now only a weakened version of this conjecture could be shown. The authors of the paper under review prove that exactly in these dimensions an embedding inequality for higher order Sobolev spaces on bounded domains with an optimal embedding constant may be improved by adding a “linear” remainder term. Thanks to H. Brezis and E. Lieb [J. Funct. Anal. 62, 73–86 (1985; Zbl 0577.46031)] this result is already known for the space $$H_0^1$$ in dimension $$n=3$$; the authors extend it to spaces $$H_0^K$$ ($$K>1$$) in the “presumably” critical dimensions. The main tools are positivity results and a decomposition method with respect to dual cones.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35B33 Critical exponents in context of PDEs 35J40 Boundary value problems for higher-order elliptic equations 35J60 Nonlinear elliptic equations
Full Text: