text:
main-content

## Über dieses Buch

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math (AMS) series, which will focus on advanced textbooks ematical Sciences and research level monographs. Preface This book develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. One purpose of this book is to formalize basic tools that are commonly used by researchers in the field but never published. It is intended primarily for mathematics graduate students and mathematically sophisticated engineers and scientists. The book has been the basis for graduate-level courses at The Uni versity of Michigan, Penn State University and the University of Houston.

## Inhaltsverzeichnis

### Chapter 0. Basic Concepts

Abstract
The finite element method provides a formalism for generating discrete (finite) algorithms for approximating the solutions of differential equations. It should be thought of as a black box into which one puts the differential equation (boundary value problem) and out of which pops an algorithm for approximating the corresponding solutions. Such a task could conceivably be done automatically by a computer, but it necessitates an amount of mathematical skill that today still requires human involvement. The purpose of this book is to help people become adept at working the magic of this black box. The book does not focus on how to turn the resulting algorithms into computer codes, although this is at present also a complicated task. The latter is, however, a more well-defined task than the former and thus potentially more amenable to automation.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 1. Sobolev Spaces

Abstract
This chapter is devoted to developing function spaces that are used in the variational formulation of differential equations. We begin with a review of Lebesgue integration theory, upon which our notion of “variational” or “weak” derivative rests. Functions with such “generalized” derivatives make up the spaces commonly referred to as Sobolev spaces. We develop only a small fraction of the known theory for these spaces—just enough to establish a foundation for the finite element method.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 2. Variational Formulation of Elliptic Boundary Value Problems

Abstract
This chapter is devoted to the functional analysis tools required for developing the variational formulation of differential equations. It begins with an introduction to Hilbert spaces, including only material that is essential to later developments. The goal of the chapter is to provide a framework in which existence and uniqueness of solutions to variational problems may be established.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 3. The Construction of a Finite Element Space

Abstract
To approximate the solution of the variational problem
$$a(u,v) = F(v)\forall v \in V$$
developed in Chapter 0, we need to construct finite-dimensional subspaces SV in a systematic, practical way.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 4. Polynomial Approximation Theory in Sobolev Spaces

Abstract
We will now develop the approximation theory appropriate for the finite elements developed in Chapter 3. We take a constructive approach, defining an averaged version of the Taylor polynomial familiar from calculus. The key estimates are provided by some simple lemmas from the theory of Riesz potentials, which we derive. As a corollary, we provide a proof of Sobolev’s inequality, much in the spirit given originally by Sobolev.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 5. n-Dimensional Variational Problems

Abstract
We now give several examples of higher-dimensional variational problems that use the theory developed in previous chapters. The basic notation is provided by the Sobolev spaces developed in Chapter 1. We combine the existence theory of Chapter 2 together with the approximation theory of Chapters 3 and 4 to provide a complete theory for the discretization process. Several examples will be fully developed in the text, and several others are found in the exercises. Throughout this chapter, we assume that the domain Ω is bounded.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 6. Finite Element Multigrid Methods

Abstract
The multigrid method provides an optimal order algorithm for solving elliptic boundary value problems. The error bounds of the approximate solution obtained from the full multigrid algorithm are comparable to the theoretical bounds of the error in the finite element method, while the amount of computational work involved is proportional only to the number of unknowns in the discretized equations.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 7. Max-norm Estimates

Abstract
The finite element approximation is essentially defined by a mean-square projection of the gradient. Thus, it is natural that error estimates for the gradient of the error directly follow in the L 2 norm. It is interesting to ask whether such a gradient-projection would also be of optimal order in some other norm, for example L. We prove here that this is the case. Although of interest in their own right, such estimates are also crucial in establishing the viability of approximations of nonlinear problems (Douglas & Dupont 1975) as we indicate in Sect. 7.7. Throughout this chapter, we assume that the domain,Ω ⊂ ℝ d is bounded and polyhedral.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 8. Variational Crimes

Abstract
Consider the Dirichlet problem
$$\begin{array}{*{20}{c}} { - \Delta u = f\,in\,\underline \Omega {{\mathbb{R}}^{2}}} \\ {u = 0\;on\;\partial \Omega } \\ \end{array}$$
(8.0.1)
.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 9. Applications to Planar Elasticity

Abstract
In most physical applications, quantities of interest are governed by a system of partial differential equations, not just a single equation. So far, we have only considered single equations (for a scalar quantity), although much of the theory relates directly to systems. We consider one such system coming from solid mechanics in this chapter.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 10. Mixed Methods

Abstract
The name “mixed method” is applied to a variety of finite element methods which have more than one approximation space. Typically one or more of the spaces play the role of Lagrange multipliers which enforce constraints. The name and many of the original concepts for such methods originated in solid mechanics where it was desirable to have a more accurate approximation of certain derivatives of the displacement. However, for the Stokes equations which govern viscous fluid flow, the natural Galerkin approximation is a mixed method.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 11. Iterative Techniques for Mixed Methods

Abstract
Equations of the form (10.3.1) or (10.5.14) are indefinite and require special care to solve. We will now consider one class of algorithms which involve a penalty method to enforce the second equation in (10.3.1) or (10.5.14). These algorithms transform the linear algebra to positive-definite problems in many cases. Moreover, the number of unknowns in the algebraic system can also be significantly reduced.
Susanne C. Brenner, L. Ridgway Scott

### Chapter 12. Applications of Operator-Interpolation Theory

Abstract
Interpolation spaces are useful technical tools. They allow one to bridge between known results, yielding new results that could not be obtained directly. They also provide a concept of fractional-order derivatives, extending the definition of the Sobolev spaces used so far. Such extensions allow one to measure more precisely, for example, the regularity of solutions to elliptic boundary value problems.
Susanne C. Brenner, L. Ridgway Scott

### Backmatter

Weitere Informationen


text:
# RL Parallel Circuit - Help Needed

#### femaleelec

Joined Feb 26, 2010
3
I have the following question, can anyone help....

When placed in series a coil and capacitor resonate at a frequency of 2251hz and when placed in paralle at a frequency of 2228hz. At Parallelresonance is it found that a current of 5mA flows from the supply when the supply voltage is 25v r.m.s. find values of capacitor, inductance and resistance of the coil.

#### hgmjr

Joined Jan 28, 2005
9,029
Can you post your efforts at solving the problem? That way we can see how you set up the problem for solution.

hgmjr

#### Kaija

Joined Feb 26, 2010
1
Do you have any answers to check?
I tried to solve the problem according to your text and got these results:
L=0.979861101 H
C=5.10181567 e-9 F
R=37 638.75695 Ohms (the resistance is in series with the inductor in parallel resonance circuit)
L and C seem ok, but can an inductor have such high resistance... ?

Anyway, try to make a system of equations. You have three variables and you can make three equations.

#### femaleelec

Joined Feb 26, 2010
3
Do you have any answers to check?
I tried to solve the problem according to your text and got these results:
L=0.979861101 H
C=5.10181567 e-9 F
R=37 638.75695 Ohms (the resistance is in series with the inductor in parallel resonance circuit)
L and C seem ok, but can an inductor have such high resistance... ?

Anyway, try to make a system of equations. You have three variables and you can make three equations.

what equations have you used? im totally lost....

#### t_n_k

Joined Mar 6, 2009
5,455
You'll need equations like ...

$$\omega_{0}ser=\frac{1}{\sqrt{LC}}$$

$$\omega_{0}par=\frac{1}{\sqrt{LC}}\sqrt{1-\frac{R^2C}{L}}$$

$$Rp_{resonance}=\frac{R_{L}}{(1-K^{2})}$$

$$where$$

$$K=\frac{\omega_{0}par}{\omega_{0}ser}$$

$$I have$$

$$R_{L}=101.655\Omega$$

$$C=99.2nF$$

$$L=50.42mH$$

#### femaleelec

Joined Feb 26, 2010
3
what equations have you used? im totally lost....

Please can you go into more depth. I just dont see how you can get to these answers? please help...

#### t_n_k

Joined Mar 6, 2009
5,455
The value of the effective circuit resistance at parallel resonance is the key. From this and the series / parallel resonance frequencies you can find the values of R, L & C.

The effective resistance at parallel resonance is found from the condition that 5mA current flows from the 25V source.

So at parallel resonance Rp=25V/5mA=5kΩ

The value of K is given by K=ω0par/ω0ser=f0par/f0ser=2228/2251=0.98978

So the equivalent series resistance RL of the coil can now be found from this information.

Then you use the value of RL and the known series & parallel equations to find the unknown values of L & C.

format: plain


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# Using Binomial Theorem to prove identity

I need to prove the following using the binomial theorem

$${n \choose k} = {n-2 \choose k} + 2{n-2 \choose k-1} + {n-2 \choose k-2}$$

The binomial theorem states $$(1+x)^n = \sum_{k=0}^n {n \choose k} x^k$$

I'm trying to use $(1+x)^n = (1+x)^2(1+x)^{n-2}$ but i dont know how to go from there?

-

You're quite close already - note that by the Binomial Theorem, $$(1+x)^{n-2} = \sum_{k=0}^{n-2} {n-2 \choose k} x^k.$$
Expanding $(1+x)^2 = 1 + 2x + x^2$, we get
$$(1+x)^n = (1+x)^2(1+x)^{n-2} = \sum_{k=0}^{n-2} {n-2 \choose k} \left(x^k +2 x^{k+1} +x^{k+2}\right).$$
Now, the trick is to collect the different powers of $x$, that iş tranforming the formula in a way that only $x^k$ appears on the right-hand side. Can you do this?


text:
# Designing a circuit using an LT1073 DC-DC Converter

I'm currently designing a circuit to convert a 24V (±30%) to supply a 3,3V IC used to monitor currents. I need to use the LT1073 which has the following diagram.

The operations of the component and its applications are well-explained in the datasheet. According to the circuit example, I should place a 100uH inductor and a Schottky-Diode at the switch. As far as my understanding goes, the Schottky-Diode is needed to block the discharge of current from going to the ground to allow the current to flow into the load. Is this correct?

This is how the circuit is supposed to be connected

According to the example, I need to put an inductor of 100uH. But what if this isn't possible and I need to pick a smaller inductor? What do I need to consider?

The datasheet mentions:

To operate as an efficient energy transfer element, the inductor must fulfill three require-ments. First, the inductance must be low enough for the inductor to store adequate energy under the worst-case condition of minimum input voltage and switch ON time. The inductance must also be high enough so that maxi-mum current ratings of the LT1073 and inductor are not exceeded at the other worst-case condition of maximum input voltage and ON time. Additionally, the inductor core must be able to store the required flux, i.e., it must not saturate.

And then the datasheet goes on with some formulas that I can't really understand.

(1) Pl = (Vout + Vd - Vin)  * (IOut), where Vd is the diode drop(0.5V for IN5818 Schottky)

(2) PL = 1/2 * L * iPeak^2 *fOsc

These formulas raised a few questions:

1. Where does IOut flows?
2. How can I get fOsc?

I would really appreciate it if someone could give some tips or answers to my questions. Thanks in advance!

• Can you show the entire schematic, and the expected waveforms? – analogsystemsrf May 15 at 9:20
• I added the schematic... I'm expecting a DC 3,3V to be generated out of the switch SW2. By connecting the right R2-to-R1 resistance ratio, according to the formula given, Vout = 212mV * (R2/R1 +1)(see formula in datasheet) , I should be able to get the voltage that I need. – N.Nelson May 15 at 10:01
• 24 volt at 30% higher is 31.2 volt and the front-page of the data sheet states that the chip works up to 30 volts so, I'd be concerned about using it in your application. – Andy aka May 15 at 11:57
• The absolute maximum supply voltage says 36 V in Step-Down mode. I think it's going to be fine.. – N.Nelson May 16 at 13:25


text:
Mathematics
OpenStudy (salt):

how do you seperate ds/dr=ks This is in diffy q.

OpenStudy (anonymous):

if k is constant then it would be $\int\limits_{}^{}(1/s) ds = \int\limits_{}^{}k dr$

OpenStudy (salt):

ty I got the answer =) I cant believe i didnt know the integral of kdr i had thought of what you said but it wasnt clicking thx.

OpenStudy (salt):

nvm i thought i got it. I didnt the answer is 2y^6+cy^4 weird

OpenStudy (salt):

ok im sorry to be a pain but i got it. I looked in the wrong place.

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text:
# Is the space spanned by a set of orthogonal vectors greater than the space spanned by the same amount of linearly independent vectors?

Let's say that we have a set of $n$ vectors, each belonging to $R^m$ that do not form a basis for $R^m$.

Intuitively, I feel like the space spanned by a set of orthogonal vectors will be greater than the space spanned by the same amount of linearly independent vectors. Now although this might now always be true, it intuitively feels like for the majority of cases this is true, as the volume of orthogonal vectors will be larger than the volume of linearly independent vectors (assuming magnitude if vectors are the same).

Can anyone shine some light on this intuition?

• What does "greater" mean? – vadim123 Nov 29 '16 at 4:46
• @vadim123 The cardinality of the set of vectors that can be made via a linear combination of the respective set. – Armen Aghajanyan Nov 29 '16 at 4:47
• @ArmenAghajanyan $|\mathbb R^n|=|\mathbb R^m|$, so cardinality isn’t a good choice for defining “greater.” The dimension of the spanned space might be a better choice. – amd Nov 29 '16 at 8:44

Not true. Suppose you are in the regular 3D space $(\Bbb R^3)$. Two linearly independent vectors will span a plane, independent if those two vectors are orthogonal. In fact, if you have a set of linearly independent vectors, you can use them to construct the same number of orthogonal vectors
• This is a good example, but the point is general. $n$ linearly independent vectors will span an $n$ dimensional subspace. You can find a set of $n$ orthogonal vectors that span the same subspace, and in any case $n$ orthogonal vectors will span an $n$ dimensional subspace (unless one is zero-some definitions of orthogonal permit that). It seems reasonable to accept that all $n$ dimensional subspaces are the same size in the sense we are talking. – Ross Millikan Nov 29 '16 at 5:10
• @ArmenAghajanyan The cardinality of all non-trivial subspaces of $\mathbb R^3$ is the same. Perhaps you should compare dimensions instead. – amd Nov 29 '16 at 8:45
In $\mathbb{R}^m$, no, the orthogonality doesn't matter.


text:
# [Xerte] LaTex maths script

Emily Lumley E.Lumley at hull.ac.uk
Thu May 23 11:34:16 BST 2013

Hi,

I've learnt how to use the LaTex maths script to write equations in a
learning objective, however, every time I close it and come back to it
the \ disappear and it no longer looks like a proper equation again.
Also I have had to put in two back slashes to ensure that the equation
works in the first place.

Am I missing something else, or is there a way I can stop the \ from
disappearing. I add the script that I used below:

<tex src="Re=\\[\\frac{\\rho.\\nu.d}{\\eta}">

Emily

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text:
# How do you factor by grouping 3x^2 - 2x - 15x + 10?

May 22, 2015

$3 {x}^{2} - 2 x - 15 x + 10$

We can make groups of two to factorise this expression:
$\left(3 {x}^{2} - 15 x\right) + \left(- 2 x + 10\right)$

$3 x$ is common to both the terms in the first group, and $- 2$ is common to both the terms in the second group

We can write the expression as :
$3 x \left(x - 5\right) - 2 \left(x - 5\right)$
=color(green)((3x - 2)(x - 5) , is the factorised form.


text:
# 3.1.1. Harlow and Welch (1965) “Numercial Calculation of Fluid with Free Surface”¶

Harlow, F.H. and Welch, J.E. “Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface” The Physics of Fluids, Volume 8, 1965

## 3.1.1.1. Introduction¶

• New technique for numerical solution of free surface problems, designated the “Marker and Cell Method”
• Physical assumptions:
• Time dependent motion
• Viscous
• Incompressible
• 2D with Cartesian Coordinates
• Uniform body force (gravity) may act on the fluid
• Possible boundary conditions:
• Wall boundaries for an irregular box
• Lines of reflective symmetry
• Prescribed pressure may be applied to the surface - variable in position and time
• Method uses:
• Finite difference method
• Full Navier Stokes Equations
• Dependent variables are pressure and two component velocity
• Fluid divided into cells - variables in each cell are single average values
• Time divided into increments
• Small spatial and temporal steps are needed
• Comparison with other methods:
• Most of work directed at compressible flows in 1965
• Incompressible flow was being directed at weather forecasting
• Non-steady viscous flows were being studied - von Karman vortex street behind a cylinder - i.e. Fromm and Harlow (1965) - primary dependent variables were vorticity and the stream function - marker and cell may be easier because BCs are easier to apply and physical significance of modification is easy to visualise

## 3.1.1.2. Description of the Method¶

### 3.1.1.2.1. Outline¶

The derivation of the finite difference method is based on:

1. The initial velocity field is known and divergence free
2. Coordinates of a set of marker particles is known - showing which regions are fluid and which are not fluid
3. Pressure field is calculated so that the rate of change of the divergence of velocity is zero, which requires the solution of Poisson’s Equation - which can be done via an SOR method
4. Two components of acceleration are calculated
5. Marker particles move according to the velocity field
6. Adjustments are made for passage of marker particles across cell boundaries - i.e. the velocity changes when a cell is emptied and filled with fluid

The marker particles are not like the particle-in-cell method, i.e. the marker particles do not participate in the calculation, where as in the particle-in-cell method, they do.

### 3.1.1.2.2. Finite Difference Equations¶

Use the Conservative Form of the Navier-Stokes Equations:

(1)${\partial u \over \partial x}+ {\partial v \over \partial y} = 0$
(2)${\partial u \over \partial t} + {\partial u^2 \over \partial x}+ {\partial {uv} \over \partial y} = - {\partial \psi \over \partial x} + \nu \left( {{\partial^2 u \over \partial x^2} +{ \partial^2 u \over \partial y^2} } \right) + g_x$
(3)${\partial v \over \partial t} + {\partial vu \over \partial x}+ {\partial {v^2} \over \partial y} = - {\partial \psi \over \partial y} + \nu \left( {{\partial^2 v \over \partial x^2} +{ \partial^2 v \over \partial y^2} } \right) + g_y$

Harlow and Welch use $$\psi$$ to denote a pressure at a constant density i.e. $$p \over \rho$$

### 3.1.1.2.3. Finite Difference Discretisation¶

• Indices i and j count the centre of each cell
• Cell boundaries are labelled with half-integer values
• The superscript n+1 is used for time and where it is not used n is implied

The Finite Difference approximations are:

$\begin{split}{{u_{i+{1 \over 2},j}^{n+1} - u_{i+{1 \over 2},j}^{n}} \over {\Delta t}} + {{(u_{i+1,j}^n)^2 - (u_{i,j}^{n})^2} \over {\Delta x}} + { { (u_{i+{1 \over 2},j+{1 \over 2}}^n)(v_{i+{1 \over 2},j+{1 \over 2}}^n)- (u_{i+{1 \over 2},j-{1 \over 2}}^n)(v_{i+{1 \over 2},j-{1 \over 2}}^n)} \over {\Delta y} } = \\ - \left( {{(\psi_{i+1,j}^n) - (\psi_{i,j}^{n})} \over {\Delta x}} \right) + \nu \left( {{u_{i+{3 \over 2},j}^n - 2u_{i+{1 \over 2},j}^n + u_{i-{1 \over 2},j}^{n}} \over {\Delta x^2}} + {{u_{i+{1 \over 2},j+1}^n - 2u_{i+{1 \over 2},j}^n + u_{i+{1 \over 2},j-1}^{n}} \over {\Delta y^2}} \right) + g_x\end{split}$
$\begin{split}{{v_{i,j+{1 \over 2}}^{n+1} - v_{i,j+{1 \over 2}}^{n}} \over {\Delta t}} + { { (u_{i+{1 \over 2},j+{1 \over 2}}^n)(v_{i+{1 \over 2},j+{1 \over 2}}^n)- (u_{i-{1 \over 2},j+{1 \over 2}}^n)(v_{i-{1 \over 2},j+{1 \over 2}}^n)} \over {\Delta x} } + {{(v_{i,j+1}^n)^2 - (v_{i,j}^{n})^2} \over {\Delta y}} = \\ -\left( {{(\psi_{i,j+1}^n) - (\psi_{i,j}^{n})} \over {\Delta y}} \right) + \nu \left( {{v_{i+{1},j+{1 \over 2}}^n - 2v_{i,j+{1 \over 2}}^n + v_{i-{1},j+{1 \over 2}}^{n}} \over {\Delta x^2}} + {{v_{i,j+{3 \over 2}}^n - 2v_{i,j+{1 \over 2}}^n + v_{i,j-{1 \over 2}}^{n}} \over {\Delta y^2}} \right) + g_y\end{split}$

Some of the velocity values are not located at the centre of the mesh diagram, so an average of adjacent values is taken - i.e. the average of the velocity at the closest two points - which could be the vertical or horizontal direction, depending on which are the adjacent points

$u_{i+{1 \over 2},j+{1 \over 2}}^n = {1 \over 2}({ u_{i+{1 \over 2},j+{1}}^n + u_{i+{1 \over 2},j}^n })$
$v_{i+{1 \over 2},j+{1 \over 2}}^n = {1 \over 2}({ v_{i+{1},j+{1 \over 2}}^n + v_{i,j+{1 \over 2}}^n })$
$u_{i,j}^n = {1 \over 2}({ u_{i+{1 \over 2},j}^n + u_{i-{1 \over 2},j}^n })$
$v_{i,j}^n = {1 \over 2}({ v_{i,j+{1 \over 2}}^n + v_{i,j-{1 \over 2}}^n })$

Harlow and Welch now take the divergence of the momentum equation:

${\partial \over {\partial t}} \mathbf{u}^n = -\mathbf{u}^n \cdot \nabla \mathbf{u}^n - \nabla \psi^{n} + \nu \nabla^2 \mathbf{u}^n$
${\partial \over {\partial t}} \nabla \cdot \mathbf{u}^n = -\nabla \cdot (\mathbf{u}^n \cdot \nabla \mathbf{u}^n) - \nabla^2 \psi^{n} + \nu \nabla^2 (\nabla \cdot \mathbf{u}^n)$

Now set:

${\partial \over {\partial t}} \nabla \cdot \mathbf{u}^n = {{\nabla \cdot \mathbf{u}^{n+1} - \nabla \cdot \mathbf{u}^n} \over {\Delta t}}$

Applying the principle of continuity:

• The divergence of velocity at the new timestep $$\mathbf{u}^{n+1} = 0$$
• The divergence at the current timestep $$\mathbf{u}^n \ne 0$$

Re-arrange for a form of the Poisson Equation at time n:

$\nabla^2 \psi^{n} = \rho {{\nabla \cdot \mathbf{u}^n} \over {\Delta t}}- \rho \nabla \cdot (\mathbf{u}^n \cdot \nabla \mathbf{u}^n)+ \mu \nabla^2 (\nabla \cdot \mathbf{u}^n)$

Harlow and Welch say that you can neglect viscosity in the Poisson Equation with strict convergence critera

The loop in time is like this:

• Compute the RHS of the Poisson Equation at n
• Compute pressure using Poisson Equation at n
• Compute velocities at n+1 using Momentum Equations (from the pressure previously calculated at n)

The divergence of velocity must also vanish at the boundaries

### 3.1.1.2.4. Marker Particles¶

• Harlow and Welch developed (a now obsolete) particles method for tracking the motion of the free surface
• They actually track the motion of particles throughout the fluid (not just at the free surface)
• Linear interpolation is performed to calculate the velocity with which a particle should move
• The interpolation weighting depends on the distance of the particle from the nearest velocity point

### 3.1.1.2.5. Boundary Conditions at Rigid Walls - Velocity¶

Rigid walls may be of two types:

• No-slip
• Free-slip (could be a plane of symmetry - or a greased surface)
• At the wall $$v = 0$$
• On the fluid side $$v \ne 0$$
• On the outside $$v' = -v$$ (no slip)
• On the outside $$v' = +v$$ (free slip)

(Similarly for horizontal walls)

In general:

• the tangential velocity reverses or remains unchanged for no-slip or free slip
• the normal velocity reverses for a free slip wall
• the normal velocity is calculated to ensure the divergence vanishes for an external cell for a no-slip wall

### 3.1.1.2.6. Boundary Conditions at Rigid Walls - Pressure¶

• The boundary condition for $$\psi$$ must be consistent with the velocity

#### 3.1.1.2.6.1. Free slip wall¶

Vertical wall From the momentum equation for $$u$$ (with the reversal of all normal velocities and no change in the tangential velocity):

$\psi' = \psi \pm g_x \Delta x$

$$+ \Rightarrow$$ fluid is on the left of the wall

$$- \Rightarrow$$ fluid is on the right of the wall

Horizontal wall From the momentum equation for $$v$$:

$\psi' = \psi \pm g_y \Delta y$

$$+ \Rightarrow$$ fluid is below the wall

$$- \Rightarrow$$ fluid is above the wall

#### 3.1.1.2.6.2. No slip wall¶

Condition is that the divergence of velocity on the outer cell must be zero:

Vertical wall

$\psi' = \psi_1 \pm g_x \Delta x \pm ({{2 \nu u_1} \over {\Delta x}})$

$$+ \Rightarrow$$ fluid is on the left of the wall

$$- \Rightarrow$$ fluid is on the right of the wall

Horizontal wall

$\psi' = \psi_1 \pm g_y \Delta y \pm ({{2 \nu v_1} \over {\Delta y}})$

$$+ \Rightarrow$$ fluid is on the left of the wall

$$- \Rightarrow$$ fluid is on the right of the wall

#### 3.1.1.2.6.3. Boundary Conditions at Free Surface - Velocity¶

Condition is that the free surface has zero divergence of velocity

• One side facing vacuum: three sides computed normally, fourth side follows from vanished divergence
• Two sides facing vacuum: $$\partial u \over \partial x$$ and $$\partial v \over \partial y$$ must vanish separately - each vacuum dies velocity is set equal to the velocity of the cell across from it
• Three sides facing vacuum: rare, but vacuum side opposite fluid side carries the same velocity of that side. The other two vacuum sides which oppose each other are calculated to follow freely the effects of the body force and do not otherwise change
• Four sides facing vacuum: similar to three so that this isolated drop follows a free fall trajectory.

#### 3.1.1.2.6.4. Boundary Conditions at Free Surface - Pressure¶

• Condition is that the normal stress is zero or of equating it to the applied external pressure
• Neglected the effect of viscous stress in the surface boundary and equated surface pressure directly to applied pressure (to reduce complexity of simulation)
• May be valid for small viscosities

## 3.1.1.3. Numerical Stability and Accuracy¶

The method is stable - viscosity is not needed to ensure stability

“Artificial viscosity” term is not needed (which it is in the treatment of shocks in compressible flow and for the elimination of stagnation fluctuations)

Condition 1 for diffusional stability:

${{4 \nu \Delta t} \over {\Delta x^2}} \lt 1$

Condition 2 for incompressible flow analogy to Courant condition (g is gravity, d is maximum depth):

${{g d \Delta t^2} \over {\Delta x^2}} \lt 1$

Harlow and Welsh also did internal consistency checks

## 3.1.1.4. Examples¶

### 3.1.1.4.1. Broken Dam Problem¶

• Variant 1: Entire Dam Removed Instantly
• Variant 2: Dam Partly Broken (like a sluice gate)

Comparison is good with experiment, but poor with hydrodynamic theory

Free surface is sensitive to grid density, but the effect is negligible

For the partly broken dam problem, the zero-pressure boundary condition applied at the centre of the cells leads to an inward dip of the zero pressure isobar at the earliest time.

## 3.1.1.5. Conclusions¶

Harlow and Walsh conclude by saying:

• The technique can be extended to 3D
• Other coordinate systems can be used

Other applications tested:

• Rayleigh-Taylor instability of a falling free surface
• Splash of a falling column of water
• Breaking of a dam
• Flow under a sluice gate
• Flow over an underwater obstacle
• The slow creep of a highly viscous blob
• Splash of a linear drop

Calculations planned:

• Formation of waves by a linear explosion over the surface
• Breaking waves on a sloping beach

Modification to cylindrical coordinates allows studies of spherical explosion and the splash of a drop


text:
# Evaluate a Trigonometric Function

1. Apr 10, 2010

1. The problem statement, all variables and given/known data
1. Given cos α = -24/25 in Quadrant II and sin β = -5/13 in Quadrant III, find sin (α + β).
2. Given sin α = 4/5 in Quadrant I and cos β = -12/13 in Quadrant II, find cos (α + β).

2. Relevant equations
I don't even know where to start! The textbook doesn't list any equations.

3. The attempt at a solution
I have none, other than guessing that you multiply the denominators in each problem... other than that, I don't even know how to solve these types of problems. I'm without a teacher, so I'd really like an explanation since I'm a little... really... lost about these problems. Thanks to anyone who can show me how I'd go about solving these! I need to know it for an upcoming final.

2. Apr 10, 2010

### LCKurtz

Draw pictures of your given angles in the given quadrants with little triangles showing the sides -- you can get the third side by the pythagorean theorem, so you know all the sines and cosines of the given angles. Then use the addition formulas for cos (α + β) and sin (α + β).

Evaluating a limit as x tends to $\infty$ May 13, 2015


text:
## 22/06/2020, Monday, 15:00–16:00 Europe/Lisbon — Online

Dmitri Bykov, Steklov Mathematical Institute Moscow
Flag manifold sigma-models and Ricci flow

I will review various results related to flag manifold sigma-models, with emphasis on their integrability properties. On simpler examples, such as the $\operatorname{\mathbb{CP}}^n$-model, I will demonstrate that the trigonometrically-deformed geometries are solutions to the Ricci flow equations.

### Video

Video of talk.

Projecto FCT UIDB/04459/2020.


text:
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### 4.4.2 Pitfalls of Using Wildcards

Now here is an example of a naive way of using wildcard expansion, that does not do what you would intend. Suppose you would like to say that the executable file foo' is made from all the object files in the directory, and you write this:

 objects = *.o foo : $(objects) cc -o foo$(CFLAGS) \$(objects)

The value of objects is the actual string *.o'. Wildcard expansion happens in the rule for foo', so that each existing .o' file becomes a prerequisite of foo' and will be recompiled if necessary.

But what if you delete all the .o' files? When a wildcard matches no files, it is left as it is, so then foo' will depend on the oddly-named file *.o'. Since no such file is likely to exist, make will give you an error saying it cannot figure out how to make *.o'. This is not what you want!

Actually it is possible to obtain the desired result with wildcard expansion, but you need more sophisticated techniques, including the wildcard function and string substitution. See section The Function wildcard.

Microsoft operating systems (MS-DOS and MS-Windows) use backslashes to separate directories in pathnames, like so:

  c:\foo\bar\baz.c

This is equivalent to the Unix-style c:/foo/bar/baz.c' (the c:' part is the so-called drive letter). When make` runs on these systems, it supports backslashes as well as the Unix-style forward slashes in pathnames. However, this support does not include the wildcard expansion, where backslash is a quote character. Therefore, you must use Unix-style slashes in these cases.

 Published under the terms of the GNU General Public License Design by Interspire


text:
• Create Account

## glRotatef()

Old topic!

Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.

4 replies to this topic

### #1Nanook  Members

Posted 04 May 2009 - 10:29 PM

glRotatef(90.0, 1.0, 0.0, 0.0); glRotatef(90.0, 0.0, 0.0, 0.5); I use the above lines before I draw my object and it gets my object into the right position, but isn't it possible to do this with one line? I tried using glRotatef(90.0, 1.0, 0.0, 0.5) , but did not get the right result..

### #2Brother Bob  Moderators

Posted 04 May 2009 - 10:34 PM

Yes, any combination of rotations can be expressed as a single axis-angle rotation. What you have is Euler angles, and what you want is an axis-angle pair. So search for "euler to axis angle" or something.

### #3Nanook  Members

Posted 04 May 2009 - 11:32 PM

okay.. I'll look into that..

Just to be sure I'm on the right track here; I'm trying to build a hexagonal container.. first I draw a hexagon polygon then drawing the walls and rotating them into place.. is this the right approach? (just got the bottom and right wall for now, but wanted to be sure on the right track before I continue)
    glPushMatrix();        //bottom of the container        glColor3f(0.0,1.0,0.0);        glRotatef(90.0, 1.0, 0.0, 0.0);        glRotatef(90.0, 0.0, 0.0, 0.5);        glBegin(GL_POLYGON);            int i;            for(i = 0; i < 6; ++i)            {                glVertex2d(2*sin(i/6.0*2*PI),2*cos(i/6.0*2*PI));            }        glEnd();    glPopMatrix();    //back wall    glPushMatrix();        glColor3f(0.0,0.0,1.0);        glTranslatef(0.0, 1.0, -2.73);        glRotatef(90.0, 1.0, 0.0, 0.0);        glBegin(GL_POLYGON);            glVertex3f( 1.0, 1.0,-2.0);            glVertex3f(-1.0, 1.0,-2.0);            glVertex3f(-1.0, 1.0, 1.0);            glVertex3f( 1.0, 1.0, 1.0);        glEnd();    glPopMatrix();

### #4CTPAX  Members

Posted 05 May 2009 - 05:55 PM

You're not rotating walls... You're just setting the polygon vertex into right position... I'm not sure if you can do rotation with a single line...I know, that you can you matrix for that, but if you achieve in putting two rotations(along two different axes) into one, can it be done with three also?

### #5Brother Bob  Moderators

Posted 05 May 2009 - 09:03 PM

Quote:
 Original post by CTPAX..., but if you achieve in putting two rotations(along two different axes) into one, can it be done with three also?

As stated; any combination of rotations can be expressed as a single axis-angle rotation.

Old topic!

Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.


text:
## What is the name of this sorting algorithm?

I have typeset in a Wikipedia draft this sorting algorithm. It is a comparison sort, yet can handle only numeric arrays.

Basically, it marches through the input array and for each new array component it inserts it into a sorted doubly-linked list. It has two optimizations:

1. each list node has a count; instead of adding a new node with the same key, existing node is reused by incrementing its counter,
2. the sort maintains a reference to the most recently accessed list node. This allows faster search for the next list node in case the two relevant keys are “close” to each other.

My question is: what is the name of that sort? (I believe I am not the first one to conceive it.)

## Algorithm to compute sum of all unique edge pairs of a tree

Given tree is undirected graph. It has n vertices and n-1 edges. The algorithm should compute the sum of all edge pairs. Thus, there are total nC2 or n(n-1)/2 such pairs. The time complexity of the mentioned algorithm is n(n-1)/2. Please suggest an algorithm with better space and time complexity if possible. Below is the java implementation.

import java.util.*;  public class AllPairSumTree {      static long sumAllPairs = 0;      public static void main(String[] args) {         /*          * Total Number of vertices          */         int N = 7;         /*          * Adjacency List          */         LinkedList<Integer>[] adjacencyList = new LinkedList[N];         /*          * Initialize Adjacency List          */         for(int ii=0; ii<N;ii++) {             adjacencyList[ii] = new LinkedList<Integer>();         }         /*          * Weighted Graph Matrix          */         int[][] weightedGraph = new int[N][N];         /*          * Initialize Matrix          */         for(int ii=0;ii<N;ii++) {             for(int jj=0;jj<N;jj++) {                 if(ii == jj) {                     weightedGraph[ii][jj] = 0;                 }else {                     weightedGraph[ii][jj] = Integer.MAX_VALUE;                 }             }         }          /*          * Input Pattern: vertex1,  vertex2, cost          *           * Total Vertex: N, Total Edges: N-1 (Tree, Undirected Graph)          */         int[] inputGraph = { 1, 2, 1,                                 2, 3, 2,                                 3, 4, 3,                                 3, 5, 4,                                 5, 6, 6,                                 5, 7, 5};          /*          * Assign  Adjacency List  and Matrix with input Graph          */         for(int ii=0; ii<N-1; ii++) {             int vertex1 = inputGraph[ii*3 + 0] - 1;             int vertex2 = inputGraph[ii*3 + 1] - 1;             int cost = inputGraph[ii*3 + 2];              adjacencyList[vertex1].add(vertex2);             adjacencyList[vertex2].add(vertex1);    //bidirectional edge              weightedGraph[vertex1][vertex2] = cost;             weightedGraph[vertex2][vertex1] = cost; //bidirectional edge         }          sumAllPairs = 0;         int currentVertex = 0;         LinkedHashSet<Integer> visitedSet = new LinkedHashSet<Integer>(N);         int lastVisitedVertex = -1;         allPairSum(weightedGraph, adjacencyList, currentVertex, visitedSet, lastVisitedVertex);          System.out.println(sumAllPairs);     }      /*      * Say graph has vertices 1,2,3,4,5,6,7      *       * allPairSum() will compute sum of edges like this:      *              21 + (31+32) + (41+42+43) + (51+52+53+54) + (61+62+63+64+65)+(71+72+73+74+75+76)      *              where ij represents edge from vertex i to vertex j      *       * Time Complexity:      *      N(N-1)/2 or Combination(N,2)[![enter image description here][1]][1]      */     private static void allPairSum(int[][] weightedGraph, LinkedList<Integer>[] adjacencyList, int currentVertex, LinkedHashSet<Integer> visitedSet, int lastVisitedVertex) {          for(Integer visitedVer : visitedSet) {             int cost = weightedGraph[visitedVer][lastVisitedVertex] +  weightedGraph[lastVisitedVertex][currentVertex];             sumAllPairs += cost;             weightedGraph[visitedVer][currentVertex] = cost;             weightedGraph[currentVertex][visitedVer] = cost;         }          visitedSet.add(currentVertex);          for(Integer neighbourVert : adjacencyList[currentVertex]) {             if(neighbourVert != lastVisitedVertex) {                 /*                  *      neighbourVert becomes currentVertex                  *      currentVertex becomes lastVisitedVertex                  */                 allPairSum(weightedGraph, adjacencyList, neighbourVert, visitedSet, currentVertex);             }         }     } }

## Is there an algorithm to compute a Belyi map for the Riemann surface?

Let $$y^2=x^5-x-1$$ be an affine model of a projective complex curve, is there an algorithm to compute the Belyi map (preferably of small degree), i.e., map to the projective line ramified only at $$\{0,1,\infty\}$$.

In my attempts to do this by hand, I get ramification at 4 points and, subsequently, using Shabat polynomial will skyrocket the degree of the map. Is there a way to avoid increasing the degree to thousands or hundreds? I have $$\beta=h\circ g\ \circ f$$, where $$f$$ is projection on $$x$$, $$g=x^5-x-1$$ and $$h=\frac{( 12500 x + 18750 x^2 + 12500 x^3 + 3125 x^4)}{-2869}$$. $$\beta$$ gives ramification at four points $$\{0,1, \infty, \frac{3125}{2869}\}$$. Also, it is possible to use $$g=x^5-x$$ with different $$h$$, but the Shabat polynomial will still be of very big degree….

## Why does the BFR (Bradley, Fayyad and Reina) algorithm assume clusters to be normally distributed around its centroid?

I’m following a course on data mining based on the lectures from Stanford University and the book Mining of massive datasets.

On the topic of clustering, the BFR algorithm is explained with this video.
I understand how the algorithm works, but I am unclear on the reason why the algorithm makes the strong assumption that each cluster is normally distributed around a centroid in Euclidian space.

The video explains that the assumption implies that clusters look like axis-aligned ellipses, which is understandable as the dimensions must be independent.
I’ve watched the video a few times, and read the section in the book (freely downloadable using the first link) on pages 257-259, but I’m unable to grasp why that assumption is made, and why it has to be made.

Could someone explain this for me?

## How MO’S algorithm with update query works?

MO’S algorithm is used to answer queries related to a given array by dividing it into blocks(https://blog.anudeep2011.com/mos-algorithm/).Though i understood general MO’S algorithm but i am facing difficulty to understand MO’S algorithm with updates (https://www.youtube.com/watch?v=gUpfwVRXhNY)

Since articles related to MO’S algorithm with updates is rare can someone who knows this algorithm can explain it.

## Proof of lemma from Hong’s article about multi-threaded max flow algorithm

I’m struggling to prove Lemma 3 and Lemma 4 from an article about parallel version of push-relabel algorithm: A lock-free multi-threaded algorithm for the maximum flow problem.

Lemma 3. Any trace of two push and/or lift operations is equivalent to either a stage-clean trace or a stage-stepping trace.

And

Lemma 4. For any trace of three or more push and/or lift operations, there exists an equivalent trace consisting of a sequence of non-overlapping traces, each of which is either stage-clean or stage-stepping.

## Algorithm for automatic construction of natural deduction proofs

I was wondering if there exists any algorithm for automatic construction of nautral deduction proofs. I’m interested in propositional logic and first order logic.

If there is no algoritm, can you provide some proof of this fact?

PD0: I’m not interested in any page for solving these kind of problems. My question is more theorical.

PD1: This is not homework, just personal interest.

## Can some one explain how MO’S algorithm with update query works?

MO’S algorithm is used to answer queries related to a given array by dividing it into blocks(https://blog.anudeep2011.com/mos-algorithm/).Though i understood general MO’S algorithm but i am facing difficulty to understand MO’S algorithm with updates (https://www.youtube.com/watch?v=gUpfwVRXhNY)

Since articles related to MO’S algorithm with updates is rare can someone who knows it explain it.

## Online algorithm for finding of clique of size k

I am trying to write an online algorithm that can detect cliques of size k. I first start out with a set of vertices. For each iteration, I add an edge. The algorithm will detect the first time an edge would create a clique of size k. What is an efficient algorithm that can complete this task, and what is the time complexity?

## How can I write a genetic programming algorithm, given that the Halting problem is unsolvable?

I am learning genetic programming and to practice I want to write a simple algorithm which evolves a program that solves a simple function (say, square root). I intend to represent programs as abstract syntax trees.

However, one of the functors is the while loop. Of course, in assesting a tree’s fitness, I have to evaluate the program: but the halting problem is unsolvable. How can I tell if a given tree stops? Of course I can’t, so what are some practicals ways to approach this problem?

Should I make my simple tree-language not turing complete? Or maybe give a timeout to each tree?


text:
# 11) Which of the following is true of trading investments? A) They are held for more than two years. 1 answer below »

11) Which of the following is true of trading investments?

A) They are held for more than two years.

B) They are always recorded at the current market value.

C) They must be adjusted and reported at fair value at the end of each period.

D) They are always recorded at the historical cost.

12) The fair value of an investment is the price ________.

A) that exists at the time of acquisition of the investments

B) that would be used if the company were to sell the investments on the market

C) that is always equal to the weighted average cost of the investment

D) that is equal to the average cost of the investment during the holding period

13) Unrealized gains or losses on available-for-sale securities occur when a company ________.

A) adjusts an asset to fair value but has not yet disposed of the asset

B) adjusts an asset to fair value when an asset is disposed

C) adjusts an asset to average value but has not yet disposed of the asset

D) adjusts an asset to average value when an asset is disposed

14) Realized gains or losses on available-for-sale investments represent the difference between ________.

A) the historical cost and the average market value of the investment

B) the historical cost and the current market value of the investment

C) the cash received at time of disposal and the fair value of the investment

D) the cash received at time of disposal and the carrying amount of the investment

15) Unrealized holding gains or losses on trading investments are reported in the ________ section of the income statement.

A) Current Assets

B) Other Revenues and Expenses

C) Fixed Assets

D) Minority Interest

16) When there is an unrealized loss on trading investments, the ________ will decrease.

A) total equity

B) cash

C) total liabilities

D) long term investments

17) Sinopia Corporation reported trading investments of $15,000 on December 31, 2015. The company realizes a decrease of$3,000 in the fair value of the trading investments by the end of the year 2016. Which of the following is the correct journal entry?

A)

B)

C)

D)

18) Which of the following is true of the Fair Value Adjustment account for trading investments?

A) It will always have a credit balance.

B) It will always have a debit balance.

C) It is considered as an adjunct account if it has a credit balance.

D) It is considered as an adjunct account if it has a debit balance.

19) If the Fair Value Adjustment account for trading investments has a debit balance ________.

A) it is subtracted from the Fixed Asset account to determine carrying value.

B) it is subtracted from the Short-term Investment-Trading account to determine carrying value.

C) it is added to the Short-term Investment-Trading account to determine carrying value.

D) it is added to the Fixed Asset account to determine carrying value.

20) Which of the following is true of available-for-sale (AFS) investments?

A) They are always reported as current assets in the balance sheet.

B) They are always reported as long-term assets in the balance sheet.

C) They are reported as current assets on the balance sheet only if the business expects to sell them within the first two years.

D) They are reported as long-term assets on the balance sheet only if they are planned to be held for longer than a year.

Darshan M

11)Option C is correct i.e. They must be adjusted and reported at fair value at the end of each period.

They must be adjusted and reported at fair value at the end of each period is true of trading investments.

12)Option B is correct i.e. that would be used if the company were to sell the investments on the market.

The fair value of an investment is the price that would be used if the company were to sell the investments on the market.

13)Option A is correct i.e. adjusts an asset to fair value but has not yet disposed of the asset.

Unrealized gains or losses on available-for-sale securities occur when a company adjusts an asset to...

## Plagiarism Checker

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text:
# Kuhn poker

Last updated

Kuhn poker is an extremely simplified form of poker developed by Harold W. Kuhn as a simple model zero-sum two-player imperfect-information game, amenable to a complete game-theoretic analysis. In Kuhn poker, the deck includes only three playing cards, for example a King, Queen, and Jack. One card is dealt to each player, which may place bets similarly to a standard poker. If both players bet or both players pass, the player with the higher card wins, otherwise, the betting player wins.

Poker is a family of card games that combines gambling, strategy, and different skills. All poker variants involve betting as an intrinsic part of play, and determine the winner of each hand according to the combinations of players' cards, at least some of which remain hidden until the end of the hand. Poker games vary in the number of cards dealt, the number of shared or "community" cards, the number of cards that remain hidden, and the betting procedures.

Harold William Kuhn was an American mathematician who studied game theory. He won the 1980 John von Neumann Theory Prize along with David Gale and Albert W. Tucker. A former Professor Emeritus of Mathematics at Princeton University, he is known for the Karush–Kuhn–Tucker conditions, for Kuhn's theorem, for developing Kuhn poker as well as the description of the Hungarian method for the assignment problem. Recently, though, a paper by Carl Gustav Jacobi, published posthumously in 1890 in Latin, has been discovered that anticipates by many decades the Hungarian algorithm.

In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a larger piece reduces the amount of cake available for others, is a zero-sum game if all participants value each unit of cake equally.

## Game description

In conventional poker terms, a game of Kuhn poker proceeds as follows:

The following is a glossary of poker terms used in the card game of poker. It supplements the glossary of card game terms. Besides the terms listed here, there are thousands of common and uncommon poker slang terms. This is not intended to be a formal dictionary; precise usage details and multiple closely related senses are omitted here in favor of concise treatment of the basics.

• Each player antes 1.
• Each player is dealt one of the three cards, and the third is put aside unseen.
• Player one can check or bet 1.
• If player one checks then player two can check or bet 1.
• If player two checks there is a showdown for the pot of 2 (i.e. the higher card wins 1 from the other player).
• If player two bets then player one can fold or call.
• If player one folds then player two takes the pot of 3 (i.e. winning 1 from player 1).
• If player one calls there is a showdown for the pot of 4 (i.e. the higher card wins 2 from the other player).
• If player one bets then player two can fold or call.
• If player two folds then player one takes the pot of 3 (i.e. winning 1 from player 2).
• If player two calls there is a showdown for the pot of 4 (i.e. the higher card wins 2 from the other player).

In poker, the showdown is a situation when, if more than one player remains after the last betting round, remaining players expose and compare their hands to determine the winner or winners.

## Optimal strategy

The game has a mixed-strategy Nash equilibrium; when both players play equilibrium strategies, the first player should expect to lose at a rate of −1/18 per hand (as the game is zero-sum, the second player should expect to win at a rate of +1/18). There is no pure-strategy equilibrium.

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

Kuhn demonstrated there are infinitely many equilibrium strategies for the first player, forming a continuum governed by a single parameter. In one possible formulation, player one freely chooses the probability ${\displaystyle \alpha \in [0,1/3]}$ with which he will bet when having a Jack (otherwise he checks; if the other player bets, he should always fold). When having a King, he should bet with the probability of ${\displaystyle 3\alpha }$ (otherwise he checks; if the other player bets, he should always call). He should always check when having a Queen, and if the other player bets after this check, he should call with the probability of ${\displaystyle \alpha +1/3}$.

The second player has a single equilibrium strategy: Always betting or calling when having a King; when having a Queen, checking if possible, otherwise calling with the probability of 1/3; when having a Jack, never calling and betting with the probability of 1/3.

## Generalized versions

In addition to the basic version invented by Kuhn, other versions appeared adding bigger deck, more players, betting rounds, etc., increasing the complexity of the game.

### 3-player Kuhn Poker

A variant for three players was introduced in 2010 by Nick Abou Risk and Duane Szafron. In this version, the deck includes four cards (adding a ten card), from which three are dealt to the players; otherwise, the basic structure is the same: while there is no outstanding bet, a player can check or bet, with an outstanding bet, a player can call or fold. If all players checked or at least one player called, the game proceeds to showdown, otherwise, the betting player wins.

A family of Nash equilibria for 3-player Kuhn poker is known analytically, which makes it the largest game with more than two players with analytic solution. [1] The family is parameterized using 4–6 parameters (depending on the chosen equilibrium). In all equilibria, player 1 has a fixed strategy, and he always checks as the first action; player 2's utility is constant, equal to –1/48 per hand. The discovered equilibrium profiles show an interesting feature: by adjusting a strategy parameter ${\displaystyle \beta }$ (between 0 and 1), player 2 can freely shift utility between the other two players while still remaining in equilibrium; player 1's utility is equal to ${\displaystyle -{\frac {1+2\beta }{48}}}$ (which is always worse than player 2's utility), player 3's utility is ${\displaystyle {\frac {1+\beta }{24}}}$.

It is not known if this equilibrium family covers all Nash equilibria for the game.

## Related Research Articles

In poker, pot odds are the ratio of the current size of the pot to the cost of a contemplated call. Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value.

In the card game of poker, a bluff is a bet or raise made with a hand which is not thought to be the best hand. To bluff is to make such a bet. The objective of a bluff is to induce a fold by at least one opponent who holds a better hand. The size and frequency of a bluff determines its profitability to the bluffer. By extension, the phrase "calling somebody's bluff" is often used outside the context of poker to describe cases where one person "demand[s] that someone prove a claim" or prove that he or she "is not being deceptive."

Seven-card stud, also known as Seven-Toed Pete or Down-The-River is a variant of stud poker. Until the recent increase in popularity of Texas hold 'em, seven-card stud was the most widely played poker variant in home games across the United States, and in casinos in the eastern part of the country. Two to eight players is common, though eight may require special rules for the last cards dealt if no players fold. With experienced players who fold often, even playing with nine players is possible.

Texas hold 'em is a variation of the card game of poker. Two cards, known as hole cards, are dealt face down to each player, and then five community cards are dealt face up in three stages. The stages consist of a series of three cards, later an additional single card, and a final card. Each player seeks the best five card poker hand from any combination of the seven cards of the five community cards and their two hole cards. Players have betting options to check, call, raise, or fold. Rounds of betting take place before the flop is dealt and after each subsequent deal. The player who has the best hand and has not folded by the end of all betting rounds wins all of the money bet for the hand, known as the pot.

In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other players' strategies.

The following outline is provided as an overview of and topical guide to poker:

Matching pennies is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, then Even keeps both pennies, so wins one from Odd. If the pennies do not match Odd keeps both pennies, so receives one from Even.

In game theory, a Perfect Bayesian Equilibrium (PBE) is an equilibrium concept relevant for dynamic games with incomplete information. A PBE is a refinement of both Bayesian Nash equilibrium (BNE) and subgame perfect equilibrium (SPE). A PBE has two components - strategies and beliefs:

Morton's theorem is a poker principle articulated by Andy Morton in a Usenet poker newsgroup. It states that in multi-way pots, a player's expectation may be maximized by an opponent making a correct decision.

In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game. The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players; this impact is sometimes called his or her reputation. Single stage game or single shot game are names for non-repeated games.

Badugi is a draw poker variant similar to triple draw, with hand-values similar to lowball. The betting structure and overall play of the game is identical to a standard poker game using blinds, but, unlike traditional poker which involves a minimum of five cards, players' hands contain only four cards at any one time. During each of three drawing rounds, players can trade zero to four cards from their hands for new ones from the deck, in an attempt to form the best badugi hand and win the pot. Badugi is an often gambling game, with the object being to win money in the form of pots. The winner of the pot is the person with the best badugi hand at the conclusion of play. Badugi is played in cardrooms around the world, as well as online, in rooms such as PokerStars. Although it doesn’t have its own tournament per se at the WSOP, it is featured in the Dealers Choice events as well as in the Triple Draw Mix.

In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from the recommended strategy, the distribution is called a correlated equilibrium.

In game theory, an epsilon-equilibrium, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.

Poker is a popular card game that combines elements of chance and strategy. There are various styles of poker, all of which share an objective of presenting the least probable or highest-scoring hand. A poker hand is usually a configuration of five cards depending on the variant, either held entirely by a player or drawn partly from a number of shared, community cards. Players bet on their hands in a number of rounds as cards are drawn, employing various mathematical and intuitive strategies in an attempt to better opponents.

Draw poker is any poker variant in which each player is dealt a complete hand before the first betting round, and then develops the hand for later rounds by replacing, or "drawing", cards.

## References

• Kuhn, H. W. (1950). "Simplified Two-Person Poker". In Kuhn, H. W.; Tucker, A. W. (eds.). Contributions to the Theory of Games. 1. Princeton University Press. pp. 97–103.
• James Peck. "Perfect Bayesian Equilibrium" (PDF). Ohio State University. Retrieved 2 September 2016.:19–29
1. Szafron, Duane; Gibson, Richard; Sturtevant, Nathan (May 2013). "A Parameterized Family of Equilibrium Profiles forThree-Player Kuhn Poker" (PDF). In Ito; Jonker; Gini; Shehory (eds.). Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2013). Saint Paul, Minnesota, USA.


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### 1.1: The Set of Natural Numbers

The first mathematics we learned as children was how to count: “One, two, three. . .” (or “Un, deux, trois. . .” or what have you). Only later did we learn about other kinds of numbers: fractions and negative numbers, for example. The counting numbers are thus the most “natural”, and the set $\{1, 2, 3, \ldots\}$ is sometimes called the set of natural numbers. However, for certain technical reasons it’s convenient to include zero and so we define the set of natural numbers as ${\Bbb N} = \{ 0, 1, 2, \ldots \}$.

The set brackets “$\{$” and “$\}$” tell us to consider the collection of things between them as one object—namely a set. The symbol “${\Bbb N}$” is just an abbreviation to allow us to refer to this object concisely. The elements of a set are the “things” in the collection. The symbol “$\in$” is pronounced “is an element of”, so for example $37 \in {\Bbb N}$ is an abbreviation for the sentence “Thirty-seven is an element of the set of natural numbers.”.

The ellipses ( … ) in the definition of ${\Bbb N}$ tell us to continue the pattern displayed. The result is an infinite set—one with no last element. Ellipses are also sometimes used in defining finite sets, for example the familiar $A = \{ a, b, c, \ldots , z \}.$

The usual convention is to use capital letters to refer to sets. To read the definition ${\Bbb N} = \{0, 1, 2, \ldots \}$ out loud, you could say, “${\Bbb N}$ is the set whose elements are zero, one, two, and so on forever.” The formal way to pronounce the symbol ${\Bbb N}$ is “the set of natural numbers”. Usually, unless there is some need to be precise, one simply says “en”.

This can be trickier than it might appear because it’s common to use “n” as a variable. The convention here is that lower-case letters refer to numbers. So one can say, for example, “If $n \in {\Bbb N}$, then $n$ must be either even or odd”. This should be pronounced “If en is a natural number, then …”.

The slash “/” is used to denote negation, as in $1 \not= 2$. This idea is common even outside of mathematics, where we find “$\oslash$” (meaning “no”) on street signs, for example. So it shouldn’t come as any surprise that ${1 \over 2} \not\in {\Bbb N}$ stands for the (true) sentence “One-half is not an element of the set of natural numbers.” (or just “one-half isn’t in ${\Bbb N}$” or some such variation).

There are several other common ways to say “${1 \over 2} \not\in {\Bbb N}$”: for example, “Two doesn’t go into one” or “One is not a multiple of two”. Here is another way: $2 \not | 1.$ This is pronounced “Two does not divide one”. In general, whenever we have natural numbers n and m we can say that n divides m, and write $n \mid m$, whenever ${m \over n} \in {\Bbb N}$. We can also say in this case that n is a factor of m, or that n is a divisor of m.

Thus, for example, $11 \mid 209$, because ${209 \over 11} = 19$ and $19 \in {\Bbb N}$. This can be rephrased slightly as “because $209 = 11 \cdot 19$ and 19 is a natural number”. But $11 \not | 73$ because there is no natural number $j$ with $73 = 11 \cdot j$. In other words, because ${73 \over 11} \not\in {\Bbb N}$.

Two special cases involve zero. Suppose we have $n \in {\Bbb N}$ and $n \not= 0$. Then $0 \not | n$ and $n | 0.$ To confirm these facts, look at the definition of “$|$”: since ${0 \over n} = 0$ and $0 \in {\Bbb N}$, we get $n|0$ as advertised. But ${n \over 0} \not\in {\Bbb N}$. In fact ${n \over 0}$ isn’t a number at all! Be careful when working with 0 to avoid thinking of the wrong special case.

In the original version [~1995] of this piece, several exercises follow (of course!). As of this moment, they’re too much trouble to reproduce here. Exercise: find an expert user of the various notations and discuss.

1. Hypatia

I hesitate to write because I’m delighted that you are posting something other than links. I’ve always enjoyed your posts and your writing. And also because I’ve not consulted an expert user – I’m certainly not one. However, I would like to make a comment and let the expert users comment.

N = {1, 2, 3, …} does represent the natural numbers (sometimes called ‘counting’ numbers’ for reasons probably referring to your intro. However, I always thought that when zero was added to the set it had a new name, the whole numbers — W = {0, 1, 2,…} Perhaps this wasn’t always so. But it makes a great deal of sense. Zero is not a counting number. Young children, and perhaps their teachers, when counting objects, use ‘0’ to refer to no objects. This later leads to a very common beginning algebra error when solving and x = 0. Many students will express the solution set as the ‘null’ set instead of {0}.

2. Hypatia

Here is a possible explanation of my puzzlement. Obviously I should have checked before writing!

In mathematics, a natural number (also called counting number) can mean either an element of the set {1, 2, 3, …} (the positive integers) or an element of the set {0, 1, 2, 3, …} (the non-negative integers). The former is generally used in number theory, while the latter is preferred in mathematical logic, set theory, and computer science.

3. thanks for delurking.
“hypatia” is a very cool name!

everybody in the department i came up in
seems to’ve called {1, 2, 3 …}
“the positive integers” (& quite a few of ’em
abbreviated it ${\Bbb Z}^+$);
i always sort of figured this whole
“whole number” thing was the invention
of misguided pedagogues (like “ln”
instead of “log” for the natural logarithm).

can’t agree with w’edia at all–
“counting numbers” refers to $\Bbb Z^+$
*regardless* of how one defines $\Bbb N$!
(computer-heads are often said to begin
counting with zero but this is mostly in jest.)

for x=0 is also very interesting.

one day during a vent-the-pet-peeve
session with a colleague (for instance,
he’s strongly anti-“FOIL” [cf jonathan];
i can take it or leave it).
then he says something to the effect
“they keep saying ‘no solution’
whenever the solution is the empty set’.
but waitaminute, mister anonymous!
those students are righter than you are!
“solution” is a technical term
(a value of the variable that makes
a given equation *true* is called
a *solution* to the equation);
there’s an *answer* to the problem
(namely “no solution”) but there sure
as hell isn’t any solution to the equation
(x = x + 1, say, for concreteness)!
indeed, that’s what it *means* to say
the *solution set* (for pity sake) is empty!
(let’s try to get with the program here.)
i was of course much more polite about
correcting him than this imaginary dialogue
would make it appear … but he just blew me off
with “i disagree” (or some such) as if this
were merely a matter of opinion. and i let it go.
if i’d’ve loved the guy like i should
i would’ve risked ticking him off for life
by arguing about it. but to heck with it.

5. Hypatia

You’re lucky it was just one colleague. We have whole departmental meetings of ‘vent the pet peeve’ . So as not to offend, everyone’s pet peeves are adopted as ‘official department policy” after two or three hours of discussion. When the adjunct faculty pool alters significantly, we start all over again.

6. My question is that whats “about base 10 for natural numbers”?

7. i don’t think i understand your question.
maybe you should point google (or your
favorite search engine) at “number bases
wiki”.

8. Anonymous

+
l+´0o

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)


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## New Appraches for Visualizing and Analyzing Metabolic Pathways

• Visualizing of metabolic pathways (or networks) has been done by many differentapproaches. In this work, we implemented and tested existing graph layout algorithms, and present a new approach to lay-out medium size metabolic pathways (500-20,000 vertices) by implementing and combining three well known graph lay-out algorithms (high dimension embedding, spring-embedder preprocessing, spring-embedder), through 3D space density analysis facilitated by the Octree technique. For the analysis of the results of metabolic pathways simulations we present two new techniques: rstly, a powerful technique to visualize pathways simulation data was created to unveil and understand concentration ows through metabolic pathways. This was achieved by mapping the color encoded concentration value of every substance from each time step of the simulation to its graphical representation in the layout. By combining all resulting images (from each time step) and displaying them as a movie, many characteristics such as subnetworks, alternative routes through the network, and differences between a modied pathway and its unmodied version can be revealed. Secondly, a new method to detect co-regulated substances in metabolic pathways and to recognize differences between two versions of a pathway, was established. To do this, we transformed the simulation data into a row-based representation, color-coded these rows, and reordered them with respect to similarity by using a Genetic Algorithm variant. From the arising discrete 2-dimensional matrix consisting of concentration values, a continuous 2-dimensional fourier row function was computed. This function can be used to measure properties, such as similarities in a pathway between time steps, or substances, or to detect and evaluate differences between modied versions of the same pathway.

$Rev: 13581$


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# Blowing up at a subvariety

Let $Y\subseteq\mathbb{A}^n$ be an affine variety with $\mathbb{I}(Y)=(f_{1},\ldots,f_{s}) \subseteq k[x_{1},\ldots,x_{n}]$. Define $\psi:\mathbb{A}^n \to \mathbb{P}^{s-1}$ by $\psi=(f_{1},\ldots,f_{s})$. And let $\Gamma$ be the graph of $\psi$ in $\mathbb{A}^n\times\mathbb{P}^{s-1}$. Then the closure of $\Gamma$ is the blowup of $\mathbb{A}^n$ at $Y$.

Then define $X\subseteq\mathbb{A}^n\times\mathbb{P}^{s-1}$ by $Z(\{y_{i} f_{j} - y_{j} f_{i}\})$.

Claim that $X=$ the closure of $\Gamma$.

One direction is kind of trivial, which is $\bar\Gamma\subseteq X$.

I am stuck in the other direction. I can show that $X - \pi^{-1}(Y)$ is isomorphic to $\mathbb{A}^n - Y$. But I have no idea how to deduce from this isomorphism.

-
How do we define the map $\psi$ on $Y$, since by definition the $f_i$ all vanish there? –  rondo9 Dec 11 '12 at 19:25


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# Free Boundary Flow

The common feature of external convection discussed in the preceding subsections is that the flows are always near a heated (or cooled) solid wall. For some other applications, however, the thermally induced flow occurs without presence of a solid wall and is referred to as free boundary flow. When a point or line heat source is immersed into a bulk fluid, the fluid near the heat source is heated, becomes lighter and rises to form a plume [see Fig. 1 (a)]. The plume generated by a point heat source is axisymmetric, but a line heat source will create a two-dimensional plume. A thermal is a column of rising air near the

Figure 1: Examples of free boundary flows

Figure 2: Physical model of natural convection from a line heat source

ground due to the uneven solar heating of the ground surface. The lighter air near the ground rises and cools due to expansion [see Fig. 1(b)].

Let us consider now the two-dimensional free boundary flow induced by a line heat source (see Fig. 2). Since the problem is symmetric about x = 0, only half of the domain (x > 0) needs to be studied. In the coordinate system shown in Fig. 2, the continuity, momentum and energy equations are the same as those for natural convection near a vertical flat plate, eqs. $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$, $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}+g\beta (T-{{T}_{\infty }})$ and $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}$, respectively. The boundary conditions at $y\to \infty$ can still be described by eqs. $u=v=0,\text{ }y\to \infty$ and $T={{T}_{\infty }},\text{ }y\to \infty$ from External Natural Convection from Heated Vertical Plate. However, the boundary conditions at the centerline, x = 0, should be changed to:

$\frac{\partial u}{\partial y}=\frac{\partial T}{\partial y}=0,\text{ }v=0\text{ at }y=0 \qquad \qquad(1)$

which indicates that the velocity component in the x-direction and the temperature at the centerline are at their respective maximum. Gebhart et al. (1970) assumed that the difference between the centerline temperature and the bulk fluid temperature is of the following power law form:

${{T}_{0}}-{{T}_{\infty }}=N{{x}^{n}} \qquad \qquad(2)$

where the index n will be determined using the overall energy balance. The continuity, momentum, and energy equations can be transformed to the following ordinary differential equations via the same similarity variables for natural convection over a vertical flat plate in Section 6.4.1 and considering eq. (2).

${F}'''+\theta +(3+n)F{F}''-(2n+2){{({F}')}^{2}}=0 \qquad \qquad(3)$

${\theta }''+\Pr [(n+3)F{\theta }'-4n{F}'\theta ]=0 \qquad \qquad(4)$

Equations (3) and (4) are applicable to any natural convection problem that satisfies eq. (2). To get the ordinary differential equations that are specific for natural convection induced by a line heat source, proper values of N and n, as well as appropriate boundary conditions, must be specified. The energy convected across any horizontal plane in the plume is

${q}'=\rho {{c}_{p}}\int_{-\infty }^{\infty }{u(T-{{T}_{\infty }})dy} \qquad \qquad(5)$

which is identical to the intensity of the line heat source and it can be expressed in terms of similarity variables

${q}'=4\nu \rho {{c}_{p}}N{{\left( \frac{g\beta N}{4{{\nu }^{2}}} \right)}^{1/4}}{{x}^{(5n+3)/4}}\int_{-\infty }^{\infty }{{F}'\theta d\eta } \qquad \qquad(6)$

Since the line heat source is the only source of heating, the above q' should be independent from x which can be true only if n is equal to − 3 / 5. Therefore, the ordinary differential equations for natural convection over a line heat source respectively become:

${F}'''+\theta +\frac{12}{5}F{F}''-\frac{4}{5}{{({F}')}^{2}}=0 \qquad \qquad(7)$

${\theta }''+\frac{12}{5}\Pr (F{\theta }'+{F}'\theta )=0 \qquad \qquad(8)$

which are subject to the following boundary conditions:

$F(\eta )={F}''(\eta )=0,\text{ and }\theta (\eta )=1\text{ at }\eta =0 \qquad \qquad(9)$

${F}'(\eta )=\theta (\eta )=0\text{ at }\eta \to \infty \qquad \qquad(10)$

Equations (7) – (10) can be solved numerically and the results are shown in Fig. 3. In contrast to Fig. 1 from Similarity Solution for Natural Convection on a Vertical Surface that shows the velocity at the wall is zero, the velocity at the centerline is at its maximum. With known intensity of the line heat source, the constant N in eq. (2) can be obtained from eq. (6):

$N={{\left( \frac{{{{{q}'}}^{4}}}{64g\beta {{\rho }^{2}}{{\mu }^{2}}c_{p}^{4}{{I}^{4}}} \right)}^{1/5}} \qquad \qquad(11)$

Figure 3: Velocity and temperature distributions of natural convection from a line heat source (a) velocity, (b) temperature

where

$I=\int_{-\infty }^{\infty }{{F}'(\eta )\theta (\eta )d\eta } \qquad \qquad(12)$

is a function of Prandtl number. The values of I calculated at Pr = 0.7, 1.0, 6.7, and 10.0 by Gebhart et al. (1970) are 1.245, 1.053, 0.407 and 0.328, respectively.

$\frac{\partial u}{\partial r}=\frac{\partial T}{\partial r}=0,\text{ }v=0\text{ at }r=0 \qquad \qquad(13)$

$u=v=0,\text{ }T={{T}_{\infty }}\text{ at }r\to \infty \qquad \qquad(14)$

Similar to the case of line heat source, the centerline temperature can also be expressed as a power function of x as indicated by eq. (2). Introducing Stokes stream function:

$u=\frac{1}{r}\frac{\partial \psi }{\partial r},\text{ }v=-\frac{1}{r}\frac{\partial \psi }{\partial x} \qquad \qquad(15)$

and defining the following similarity variables:

$\eta =\frac{r}{x}\text{Gr}_{x}^{1/4},\text{ }\psi =\nu xF(\eta ),\text{ }\theta =\frac{T-{{T}_{\infty }}}{{{T}_{0}}-{{T}_{\infty }}} \qquad \qquad(16)$

the following ordinary differential equations are obtained:

${F}'''+\eta \theta +(F-1){{\left( \frac{{{F}'}}{\eta } \right)}^{\prime }}-\frac{1+n}{2}\frac{{{({F}')}^{2}}}{\eta }=0 \qquad \qquad(17)$

$(\eta {\theta }'{)}'+\Pr (F{\theta }'-n{F}'\theta )=0 \qquad \qquad(18)$

The intensity of the point heat source is:

$q=\rho {{c}_{p}}\int_{0}^{\infty }{u(T-{{T}_{\infty }})2\pi rdr}=2\pi \nu \rho {{c}_{p}}N{{x}^{n+1}}\int_{0}^{\infty }{{F}'\theta d\eta } \qquad \qquad(19)$

and it must be independent from x. Therefore, the value of n must be − 1 and the ordinary differential equations for natural convection over a point heat source become:

${F}'''+\eta \theta +(F-1){{\left( \frac{{{F}'}}{\eta } \right)}^{\prime }}=0 \qquad \qquad(20)$

$(\eta {\theta }'{)}'+\Pr (F\theta {)}'=0 \qquad \qquad(21)$

The velocity components in the x- and r- directions are related to the dimensionless variables by the following equations:

$u=\frac{\nu }{x}\sqrt{\text{G}{{\text{r}}_{x}}}\left( \frac{{{F}'}}{\eta } \right),\text{ }v=-\frac{\nu }{x}\text{Gr}_{x}^{1/4}\left( \frac{F}{\eta }-\frac{{{F}'}}{2} \right) \qquad \qquad(22)$

which can be substituted into eqs. (13) and (14) to obtain the following boundary conditions for eqs. (20) and (21):

${{\left( \frac{{{F}'}}{\eta } \right)}^{\prime }}=0,\text{ }F=0,\text{ }{\theta }'=0\text{ and }\theta =1\text{ at }\eta =0 \qquad \qquad(23)$

$\frac{{{F}'}}{\eta }=\theta =0\text{ at }\eta \to \infty \qquad \qquad(24)$

## References

Gebhart, B., Pera, L., and Schorr, A.W., 1970, “Steady Laminar Natural Convection Plumes above a Horizontal Line Heat Source,” International Journal of Heat and Mass Transfer, Vol. 13, pp. 161-171


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# Difference between port mapped and memory mapped access

computer-architecturecpuiomemoryport

Can anyone explain what's the difference between port mapping and memory mapping, and what having both accomplishes? Why is there port mapped, how does it differ in structure from memory maps, and is there any reason many architectures use both? Also, what is a "port" in this sense, because port can mean different things in different contexts?

Example: Port forwarding, port as a communications endpoint, "port mapping".

Say I write OUT to port 400h (fictive; just for example) (like in x86-64, etc.).

What or where am I writing to if it's not in memory? How is a "port" mapped, and what is it in this sense?

Memory-mapped I/O and port-mapped I/O are two complementary methods for I/O.

# Memory-Mapped I/O

In memory-mapped systems, the I/O device is accessed like it is a part of the memory. Load and Store commands are executed for reading from and writing to I/O devices, just like they are used for the memory (port-mapped has special commands for I/O). This means I/O devices use the same address bus as memory, meaning that CPU can refer to memory or the I/O device based on the value of the address. This approach requires isolation in the address space: that is, addresses reserved for I/O should not be available to physical memory.

Below is an image of a simple, basic computer system. The case is much more complicated in contemporary systems.

# Port-Mapped I/O

According to Wikipedia

Port-mapped I/O often uses a special class of CPU instructions specifically for performing I/O. This is found on Intel microprocessors, with the IN and OUT instructions. These instructions can read and write one to four bytes (outb, outw, outl) to an I/O device. I/O devices have a separate address space from general memory, either accomplished by an extra "I/O" pin on the CPU's physical interface, or an entire bus dedicated to I/O. Because the address space for I/O is isolated from that for main memory, this is sometimes referred to as isolated I/O.

As for the advantages and disadvantages: since the peripheral devices are slower than the memory, sharing data and address buses may slow the memory access. On the other hand, by the I/O simplicity memory-mapped systems provide, CPU requires less internal logic and this helps for faster, cheaper, less power consuming CPUs to be implemented. The logic is similar to that of RISC systems: reduce the complexity, get a more dedicated and a robust system which comes quite handy for embedded systems, for example.

On the contrary (again from Wiki):

Port-mapped I/O instructions are often very limited, often providing only for simple load and store operations between CPU registers and I/O ports, so that, for example, to add a constant to a port-mapped device register would require three instructions: read the port to a CPU register, add the constant to the CPU register, and write the result back to the port.

I strongly recommend that you read that wiki article for further information.


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# Why is it possible to calibrate your subjective probabilities?

Humans tend to be overconfident in their predictions; when most people say that there's a 95% chance that something will happen, they're usually wrong far more than 5% of the time. Whereas what ought to happen is that for all $x$, if a person makes a lot of predictions of the form "There is an $x\%$ chance that P will occur," then about $x\%$ of them should come true. If a person's predictions does satisfy this property, then their subjective probabilities are said to be well-calibrated.

Now there may be no one who is perfectly well-calibrated, but some people are better than others; see the book "Superforecasting" for information on people who are extremely well-calibrated. But calibration isn't something fixed from birth; you can train yourself to be better-calibrated by playing a game where you assign a probabilities to a bunch of things and then you're scored by how well-calibrated your subjective probabilities are (using e.g. the Brier score). See this web site and this website.

But my question is, why is it possible to become well-calibrated using such a game? Let $A$ be the set of possible events $f$ be a function from $A$ to $\mathbb R$ that gives a numerical assessment of how confident a person feels about a given event. Then what playing the game is designed to do is to help you find an increasing function $g$ from $\mathbb R$ to $[0,1]$ such that $g \circ f$ is well-calibrated.

But in order for such a $g$ to exist, it needs to be the case that for all $x,y \in \mathbb R$, if $x>y$ then the percentage of outcomes in $\{P \in A: f(P) =x\}$ which are true needs to be greater than the percentage of outcomes in $\{P \in A: f(P) =y\}$ which are true. In other words, it needs to already be true that you're right more often about statements that you're more confident in than statements you're less confident in. But why should this be the case? Are there any decision-theoretic assumptions or rationality axioms that imply this is the case? Has it been empirically observed that humans are more often right about what they're more confident in?

It would certainly be a useful evolutionary skill if humans' internal confidence levels reflected external reality, but I've never heard of anyone discuss this before.

• To answer your question: I don't see why it would it would be the other way round. It doesn't make sense for people to be confident about things that are incorrect and unconfident about things that are correct, unless they are misinformed. – ahorn Nov 11 '16 at 8:43
• @ahorn Well, "misinformed" is exactly what this whole topic is about. We as humans have incomplete and potentially faulty information, and based on that we make subjective probability judgements about what statements are true or false. Look at the games I linked to above; they involve trivia questions which you may or may not know the answer to, and then you choose what you think is the most likely answer along with how confident you are that you're right. – Keshav Srinivasan Nov 11 '16 at 9:48
• @ahorn "It doesn't make sense for people to be confident about things that are incorrect and unconfident about things that are correct, unless they are misinformed." Note that even a perfectly well-calibrated person will be incorrect 10% of the time on things they're 90% confident in. So it makes perfect sense to be incorrect about something you're highly confident in, and to be correct about things you're not that confident. – Keshav Srinivasan Nov 11 '16 at 10:02
• I think it's a fairly weak assumption to make that people are generally more confident about things they know about. This is certainly an assumption the model is making. If we consider the case where this isn't true then it must be the case that: A person is uncertain about the things he does know, and certain about the things he does not? I think this is reasonable only given a highly misinformed person and if are set of events was an extremely narrowly focused set of events. However I think that this set of events to calibrate to suggests it refers to all possible events. – Lee Sin Nov 11 '16 at 17:51
• @LeeSin "I think it's a fairly weak assumption to make that people are generally more confident about things they know about." The Dunning-Kruger works show that that's a largely unjustifiable assumption. People generally are uncertain about the things they know best, and certain about the things they know little about. – EnergyNumbers Nov 12 '16 at 10:58