text:
## IMO 2014 Problem 6

A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its finite regions. Prove that for all sufficiently large ${n}$, in any set of ${n}$ lines in general position it is possible to colour at least ${\sqrt{n}}$ lines blue in such a way that none of its finite regions has a completely blue boundary.

Note: Results with ${\sqrt{n}}$ replaced by ${c\sqrt{n}}$ will be awarded points depending on the value of the constant ${c}$.

IMO 2014 Problem 6 (Day 2)

## IMO 2014 Problem 5

For every positive integer ${n}$, Cape Town Bank issues some coins that has ${\frac{1}{n}}$ value. Let a collection of such finite coins (coins does not neccesarily have different values) which sum of their value is less than ${99+\frac{1}{2}}$. Prove that we can divide the collection into at most 100 groups such that sum of all coins’ value does not exceed 1.

IMO 2014 Problem 5 (Day 2)

Categories: Combinatorics, IMO, Olympiad Tags: ,

## IMO 2014 Problem 4

Let ${P}$ and ${Q}$ be on segment ${BC}$ of an acute triangle ${ABC}$ such that ${\angle PAB=\angle BCA}$ and ${\angle CAQ=\angle ABC}$. Let ${M}$ and ${N}$ be the points on ${AP}$ and ${AQ}$, respectively, such that ${P}$ is the midpoint of ${AM}$ and ${Q}$ is the midpoint of ${AN}$. Prove that the intersection of ${BM}$ and ${CN}$ is on the circumference of triangle ${ABC}$.

IMO 2014 Problem 4 (Day 2)

Categories: Geometry, IMO, Olympiad Tags: ,

## IMO 2014 Problem 3

Convex quadrilateral ${ABCD}$ has ${\angle ABC = \angle CDA = 90^\circ}$. Point ${H}$ is the foot of the perpendicular from ${A}$ to ${BD}$. Points ${S}$ and ${T}$ lie on sides ${AB}$ and ${AD}$, respectively, such that ${H}$ lies inside triangle ${SCT}$ and

$\displaystyle \angle CHS -\angle CSB = 90^\circ,\ \angle THC-\angle DTC = 90^\circ.$

Prove that the line ${BD}$ is tangent to the circumcircle of triangle ${TSH}$.

IMO 2014 Problem 3 (Day 1)

Categories: Geometry, IMO, Problem Solving Tags: ,

## IMO 2014 Problem 2

Let ${n \geq 2}$ be an integer. Consider a ${n \times n}$ chessboard consisting of ${n^2}$ unit squares. A configuration of ${n}$ rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer ${k}$ such that for each peaceful configuration of ${n}$ rooks, there is a ${k \times k}$ square which does not contain a rook on any of its ${k^2}$ unit squares.

IMO 2014 Problem 2 (Day 1)

## IMO 2014 Problem 1

Let ${a_0 be an infinite sequence of positive integers. Prove that there exists a unique integer ${n \geq 1}$ such that

$\displaystyle a_n<\frac{a_0+a_1+...+a_n}{n} \leq a_{n+1}.$

IMO 2014 problem 1 (Day 1)

Let ${G}$ be a finite group such that ${|G|=mp^a}$ where ${p}$ is a prime number, ${a \geq 1}$ and ${\gcd(m,p)=1}$. Then there exists a subgroup ${H \leq G}$ such that ${|H|=p^a}$. (such a subgroup is called a Sylow subgroup).

format: plain


text:
# Chapter 8 - Polynomials and Factoring - Chapter Review - 8-2 Multiplying and Factoring - Page 524: 26

GCF=$3$ $3(t^{4}-2t^{3}-3t+4)$

#### Work Step by Step

Find the GCF of the terms in the polynomial. Then factor the polynomial. $3t^{4}-6t^{3}-9t+12$ not all of the terms have a $t$ so the GCF does not have a $t$ All of the terms are divisible by $3$ so the GCF have a $3$ GCF=$3$ Factor: $3(t^{4}-2t^{3}-3t+4)$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Re: [code] [textadept] open file and latex lexer

From: Robert Gieseke <rob.g.att.web.de>
Date: Wed, 08 Jan 2014 11:15:40 +0100

Hi Olivier,

Am 1/5/14, 10:48 PM, schrieb Olivier GuibÃ©:
>>> - master file, tex engine : more difficult, it is possible to mimic
>>> Texworks
>>> or auctex variables.
>>
>> Could you describe how that works in more detail? Do you mean
>> compiling the main file, when working on an included sub-file?
> Yes, a main document with different \include. Editing a sub-file
> and compiling it should compile the main file. Different solutions
> 1) special comment in the sub-files like Texworks
> or auctex with emacs-variables.
> An editor command allows to specify main-document and adds the special
> comment
> 2) a extern file
>> One way to do this could be to have a short-cut that changed the LaTeX
>> compile command to compile the current file.

It would be possible to parse the variables (of either style) but I
would probably rather change the compilation process.

And a short cut to set the master file, e.g.:
f4 = function()
'pdflatex "'..buffer.filename..'"'
ui.statusbar_text = 'Master file is '..buffer.filename
end
end,

Then you can use the default compile short cut from the included file.

> Evince supports synctex through dbus. It seems that the first plugin
> was written for gedit and then adapted to another editor. Emacs has a dbus
> support so that auctex does not use the python-synctex scripts.
[â€¦]

Thanks for the links, I'll have a look â€¦

Cheers,
Robert

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text:
# Simple game engine layout

I'm trying to make a game with inheritance & interfaces and I'm wondering if my system could be working without ECS. (I explain at the end why I'm not using components).

To make it easier to understand there are three core class "families" in the code:

Engine related classes:

Graphics, Inputs, Audio, Physics,...

Using E~~~ as a prefix for each class

Interfaces:

I'm using them like properties, they don't contain data, they're contracts, like C# interfaces (and Java I believe?):

Transformable, Destroyable, ...

Using I~~~ as a prefix for each class

"Concrete" classes:

They contain data and implements the interfaces. The could be Player, Bullet, etc.

Using C~~~ as a prefix for each class

Plus a class named Engine that contains every E~~~ class so it makes it easy to access "Engine elements".

Engine-related code:

class EInput {
public:
enum { keyCount = 64 };
char state[keyCount];
char previousState[keyCount];
};

class EGraphics {
public:
int width, height;
};

class EPhysics {
public:
float gravity;
};

class EEngine {
public:
EInput* input;
EGraphics* graphics;
EPhysics* physics;

EEngine() {
input = new EInput();
graphics = new EGraphics();
physics = new EPhysics();
}

~EEngine() {
delete input;
delete graphics;
delete physics;
}
};

Interface-related code:

class ITransformable {
public:
virtual float getX() = 0;
virtual float getY() = 0;

virtual void setX(float x) = 0;
virtual void setY(float x) = 0;

virtual ~ITransformable() {}
};

class IDestroyable {
public:
virtual float getHp() = 0;

virtual void setHp(float hp) = 0;

virtual void hurt(float amount) = 0;
virtual void kill() = 0;
virtual bool isAlive() = 0;

virtual ~IDestroyable() {}
};

class IEngine {
public:
virtual void provideEngine(EEngine* engine) = 0;

virtual ~IEngine() {}
};

class ITeam {
public:
virtual int getTeam() = 0;

virtual void setTeam(int team) = 0;

virtual ~ITeam() {}
};

Concrete classes-related code:

class CCharacter : public IEngine, public ITransformable, public IDestroyable, public ITeam {
private:
/* IEngine */
EEngine* engine;
/* ITransformable */
float x, y;
/* IDestroyable */
float hp;
/* ITeam */
int team;
public:
CCharacter() : engine(nullptr), x(0.f), y(0.f), hp(0.f) {}

/* IEngine */
void provideEngine(EEngine* engine) override { this->engine = engine; }

/* ITransformable */
virtual float getX() override { return x; }
virtual float getY() override { return y; }
virtual void setX(float x) override { this->x = x; }
virtual void setY(float x) override { this->y = y; }

/* IDestroyable */
virtual float getHp() override { return hp; }
virtual void setHp(float hp) override { this->hp = hp; }
virtual void hurt(float amount) override { this->hp -= amount; }
virtual void kill() override { this->hp = 0.f; }
virtual bool isAlive() override { return this->hp > 0.f; }

/* ITeam */
virtual int getTeam() override { return this->team; };
virtual void setTeam(int team) override { this-> team = team; };

virtual void update() = 0;
virtual void draw() = 0;
};

class CPlayer : public CCharacter {
public:
CPlayer() {
setHp(100.f); //...
}
void update() override {}
void draw() override {}
};

enum { T_NONE, T_ALLY, T_ENEMY };
class CAIPlayer : public CCharacter {
private:
int team;
public:
CAIPlayer() : team(T_NONE) {}
void update() override {}
void draw() override {}
};

And a game would be similar to this, removing the game loop:

int main() {
EEngine engine;
//Initialize some values
{
EGraphics& g = *engine.graphics;
g.width = 512;
g.height = 288;

EPhysics& p = *engine.physics;
p.gravity = 0.4f;
}

//Create our characters, a factory would be used, this is just to show the idea

//Our hero
CPlayer sam; sam.provideEngine(&engine);

//Two friendlies
CAIPlayer lambert; lambert.provideEngine(&engine);
lambert.setTeam(T_ALLY);

CAIPlayer wilkes; wilkes.provideEngine(&engine);
wilkes.setTeam(T_ALLY);
wilkes.kill();

}

• Is this code clean?
• What are the limits of such a system? I mean, I don't see any duplicated behaviour.

I read some really nice articles and questions on the net, including Stack Overflow, about entity-components-systems and how amazing it is, but it seems (to me) that it is overdone by people, the code is a big mess with templates everywhere and yet there are no results on the screen.

• I know that this code might be weak compared to ECS, but at which point?

I'm trying to keep it simple.

• Do you happen to have a link to a repository (GitHub or similar) to see this code in project context? Is there a specific reason for re-inventing the wheel? – Mast Nov 20 '16 at 19:48
• github.com/teeworlds/teeworlds gave me this idea, although I'm trying to write something (simple ?) from scratch, it isn't similar. – Riptide Nov 20 '16 at 19:51

Naming

I don't mind the I prefix for interfaces, I've never really liked C as a prefix for classes although I can live with it. Having another prefix of E for engine specific classes feels wrong/confusing. EEngine is a concrete class, that does stuff, why isn't it ECEngine / CEEngine for example?

Interfaces

Some of your interfaces don't really feel like interfaces. For example, IEngine has one method provideEngine. This isn't what I think of when I think of an Engine interface. It could be an IEngineUser interface but it still feels a bit wrong. I'd expect IEngine to define operations that were then implemented in EEngine.

Initialisation

You're declaring variables and calling methods on them at the same time:

CAIPlayer lambert; lambert.provideEngine(&engine);

This seems error prone. It also flags up a possible design error. Does it make sense for a CAIPlayer to exist without an engine / without a team? If not, then I would expect it to be passed into the constructor, rather than passed in immediately after every construction. It seems like you've done it this way because of the interfaces you've declared which specify class dependencies rather than what a class can do...

Public fields

You're passing an instance of EEngine into every class it has public pointers to other engine classes that have public fields. This feels like way too much exposure of your engines implementation. I would hide this information behind getter methods. Then either pass the intial values into the constructor or provide setter methods. Whilst they may be simple pass throughs initially it will easier in the future if you want to make changes such as supporting different gravity zones Does it really make sense that at the moment any CCharacter can change the gravity in the game?

• Naming - As Justin pointed out, I will organize my classes into namespaces, the result will be much better than crazy prefixes :) I totally agree with you with the "possible design error" part and the forgotten constructor. I made it this way so I can control easily control each interface. Constructors are mystical to me, what about inheritance ? Basically if I understand your reply correctly, you advise me to build interfaces around "actions" rather than "data" is it right ? Because here interfaces are manipulating one "data" at a time and does not perform actions on the class itself. – Riptide Nov 26 '16 at 15:44
• @Riptide Are you asking how constructor arguments would be handled if you inherited from multiple classes? As far as interfaces go, I'm suggesting the name should reflect the responsibility of the interface. At a basic level, if there was a 1-1 mapping between interfaces and concrete classes with the same name, I'd expect the interface to consist of the public methods defined in the concrete class. So, since CEngine doesn't implement IEngine, it suggests a mismatch in the naming / responsibilities. – forsvarir Nov 26 '16 at 19:39
• @Riptide As an aside, does it make sense for an application to have two instances of CEngine? If not, you might want to consider imposing this limitation by having it act as a singleton. – forsvarir Nov 26 '16 at 19:41

### Namespaces

Your "E" for engine classes, "I" for interfaces, and "C" for concrete classes seems to be reinventing namespaces. I don't mind the "I" prefix for interfaces, but why not use namespaces? You might lay it out something like this:

namespace my_game_engine_name {
namespace core { // for the "E" classes
// if there are enough graphics related things, maybe this should be
// its own namespace
class Graphics;
class Inputs;
class Audio;
class Physics;
}
... // stuff to do with the interfaces, etc
}

namespace my_game {
// notice how this is a separate namespace; almost a separate project.
// The way you have your game set up, you almost have an entirely separate
// game engine library, so why not make it that way?
class Character;
class Player;
class AIPlayer;
}

### Naming

This is a small nitpick that you may or may not agree with, but it's often better for acronyms to not be all uppercase, to ease the use with tools. For example, AIPlayer. There are tools which would think this was an "A I Player" rather than an "AI Player". Consequently, it often works out better if it was AiPlayer, as it makes parsing class names as PascalCase feasible.

### Memory safety

class EEngine {
public:
EInput* input;
EGraphics* graphics;
EPhysics* physics;

EEngine() {
input = new EInput();
graphics = new EGraphics();
physics = new EPhysics();
}

~EEngine() {
delete input;
delete graphics;
delete physics;
}
};

This can easily delete memory multiple times. For example:

void myFunction() {
EEngine myEngine;
doSomethingWithMyEngine(myEngine);
}

This might double-delete if doSomethingWithMyEngine takes the EEngine by value. If you feel the need to use the raw pointers, you need to get the memory safety right. I strongly recommend you don't do this yourself, and use the standard library to take care of it.

You either want std::unique_ptr or std::shared_ptr, depending on whether or not you want to pass the engine by value. I recommend std::unique_ptr until you find that you need std::shared_ptr. You could then implement the class like so:

#include <memory>

...

class EEngine {
public:
std::unique_ptr<EInput> input;
std::unique_ptr<EGraphics> graphics;
std::unique_ptr<EPhysics> physics;

EEngine()
: input{ new EInput() }
, graphics{ new EGraphics() }
, physics{ new EPhysics() }
{}
};

Also notice that I used the member initializer list for the constructor; it's good to get in the habit of doing so.

The benefit of this code is that it is now difficult to get the memory safety wrong. If I wrote this function:

void doSomethingWithMyEngine(EEngine engine) {
...
}

and tried to call it like before:

void myFunction() {
EEngine myEngine;
doSomethingWithMyEngine(myEngine);
}

It would result in a compiler error, since the EEngine class does not have a copy constructor, because we used std::unique_ptr. Instead, the code would have to be used like so:

void doSomethingWithMyEngine(EEngine &engine) {
...
}

Or like so:

void myFunction() {
EEngine myEngine;
doSomethingWithMyEngine(std::move(myEngine));
// now myEngine is in an invalid state.
}

This prevents the memory management issues that are present in your code.

• Namespaces - Brilliant idea, I didn't even thought about it, silly me. I'm not too confident on building external libs yet, I'll enventually get the hang of it I guess heh. Naming - Thank you for the tip, it's indeed better, that looked kinda wrong to me, it makes sense now :) !. Memory safety - Sure, my mistake. In the final product I won't use raw pointers, I shoud have put them in the original code, but since this is code review, kudos for pointing that out AND the initializer list, I always forget about it. – Riptide Nov 26 '16 at 15:24


text:
# Chapter 8 - Polynomials and Factoring - 8-1 Adding and Subtracting Polynomials - Practice and Problem-Solving Exercises: 37

$2x^{3} +8$

#### Work Step by Step

In order to subtract monomials, we combine like terms, but we distribute the negative to all of the terms below the subtraction sign. Like terms are terms that have the same variable raised to the same power. Note, by this definition, numbers like 10 and 15 are also like terms as they both have a degree of 0. Thus, we combine like terms to obtain the answer: $(6x^3-4x^3)+(17-9)=2x^{3} +8$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
My bibliography  Save this paper

# Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives

## Author

Listed:
• Karolina Bujok
• Ben Hambly
• Christoph Reisinger

## Abstract

We consider $N$ Bernoulli random variables, which are independent conditional on a common random factor determining their probability distribution. We show that certain expected functionals of the proportion $L_N$ of variables in a given state converge at rate $1/N$ as $N\rightarrow \infty$. Based on these results, we propose a multi-level simulation algorithm using a family of sequences with increasing length, to obtain estimators for these expected functionals with a mean-square error of $\epsilon^2$ and computational complexity of order $\epsilon^{-2}$, independent of $N$. In particular, this optimal complexity order also holds for the infinite-dimensional limit. Numerical examples are presented for tranche spreads of basket credit derivatives.

## Suggested Citation

• Karolina Bujok & Ben Hambly & Christoph Reisinger, 2012. "Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives," Papers 1211.0707, arXiv.org, revised Feb 2018.
• Handle: RePEc:arx:papers:1211.0707
as

File URL: http://arxiv.org/pdf/1211.0707

## References listed on IDEAS

as
1. Denis Belomestny & John Schoenmakers & Fabian Dickmann, 2013. "Multilevel dual approach for pricing American style derivatives," Finance and Stochastics, Springer, vol. 17(4), pages 717-742, October.
2. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers & Justin A. Sirignano, 2011. "Large Portfolio Asymptotics for Loss From Default," Papers 1109.1272, arXiv.org, revised Feb 2015.
Full references (including those not matched with items on IDEAS)

### NEP fields

This paper has been announced in the following NEP Reports:

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1211.0707. See general information about how to correct material in RePEc.

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text:
# A tribute to Isabelle Grillo

18-20 May 2022
ILL4
Europe/Paris timezone

## Ouzo phase occurrence with sequence control copolymers: influence of the lateral amphipathic units on the size and structure of nanoparticles

Not scheduled
15m

110

### Speaker

Henrique Trevisan (ESPCI)

### Description

Sequence regulation can be assured by the selection of monomer couples with reactivity ratios close to zero, which allows spontaneous cross-propagation copolymerization. The specific control of the polymer backbone strongly impacts the physicochemical properties of polymer materials. For instance, monomer design and customization of the solvent–monomer interactions open the way to functional copolymers showing molecular self-assembly relevant to their regular amphipathic structure. In this work, we show that the design of comonomers with adequate reactivities and interactions can be used to direct copolymer self- assembly on a mesoscopic scale. We investigate spontaneous formation of nanoparticles through solvent/non-solvent interactions using the so-called ‘‘Ouzo effect’’. In this way, an ouzo diagram was built to determine the operation window for the self-assembly, in aqueous suspensions, of alternating copolymers consisting of vinyl phenol and maleimide units carrying long alkyl-pendant groups (C$_{12}$H$_{25}$ or C$_{18}$H$_{37}$). Also, investigations were pursued to account for the influence of the lateral lipophilic pendant units on the size and structure of the nanoaggregates formed during one-shot water addition. Structure characterization by light scattering techniques (DLS and SLS), small-angle neutron scattering (SANS) and transmission electron microscopy (cryo-TEM and TEM) confirmed the self-assembly of copolymer chains into nanoparticles (size range: 60-300 nm), the size of which is affected by the lipophilicity of the alternating copolymers, solvent–water affinity and the solvent diffusion in water. Altogether, we present here the spontaneous ouzo effect as a simple method to produce stable alternating copolymer nanoparticles in water without the addition of stabilizing agents.

### Presentation Materials

There are no materials yet.
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text:
BREAKING NEWS
Peano existence theorem

## Summary

In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems.

## History

Peano first published the theorem in 1886 with an incorrect proof.[1] In 1890 he published a new correct proof using successive approximations.[2]

## Theorem

Let ${\displaystyle D}$  be an open subset of ${\displaystyle \mathbb {R} \times \mathbb {R} }$  with ${\displaystyle f\colon D\to \mathbb {R} }$  a continuous function and ${\displaystyle y'(x)=f\left(x,y(x)\right)}$  a continuous, explicit first-order differential equation defined on D, then every initial value problem ${\displaystyle y\left(x_{0}\right)=y_{0}}$  for f with ${\displaystyle (x_{0},y_{0})\in D}$  has a local solution ${\displaystyle z\colon I\to \mathbb {R} }$  where ${\displaystyle I}$  is a neighbourhood of ${\displaystyle x_{0}}$  in ${\displaystyle \mathbb {R} }$ , such that ${\displaystyle z'(x)=f\left(x,z(x)\right)}$  for all ${\displaystyle x\in I}$ .[3]

The solution need not be unique: one and the same initial value ${\displaystyle (x_{0},y_{0})}$  may give rise to many different solutions ${\displaystyle z}$ .

## Proof

By replacing ${\displaystyle y}$  with ${\displaystyle y-y_{0}}$ , ${\displaystyle x}$  with ${\displaystyle x-x_{0}}$ , we may assume ${\displaystyle x_{0}=y_{0}=0}$ . As ${\displaystyle D}$  is open there is a rectangle ${\displaystyle R=[-x_{1},x_{1}]\times [-y_{1},y_{1}]\subset D}$ .

Because ${\displaystyle R}$  is compact and ${\displaystyle f}$  is continuous, we have ${\displaystyle \textstyle \sup _{R}|f|\leq C<\infty }$  and by the Stone–Weierstrass theorem a sequence of Lipschitz functions ${\displaystyle f_{k}:R\to \mathbb {R} }$  converging uniformly to ${\displaystyle f}$  in ${\displaystyle R}$ . Without loss of generality, we assume ${\displaystyle \textstyle \sup _{R}|f_{k}|\leq 2C}$  for all ${\displaystyle k}$ .

We define Picard iterations ${\displaystyle y_{k,n}:I=[-x_{2},x_{2}]\to \mathbb {R} }$  as follows, where ${\displaystyle x_{2}=\min\{x_{1},y_{1}/(2C)\}}$ . ${\displaystyle y_{k,0}(x)\equiv 0}$ , and ${\displaystyle \textstyle y_{k,n+1}(x)=\int _{0}^{x}f_{k}(x',y_{k,n}(x'))\,\mathrm {d} x'}$ . They are well-defined by induction: as

{\displaystyle {\begin{aligned}|y_{k,n+1}(x)|&\leq \textstyle \left|\int _{0}^{x}|f_{k}(x',y_{k,n}(x'))|\,\mathrm {d} x'\right|\\&\leq \textstyle |x|\sup _{R}|f_{k}|\\&\leq x_{2}\cdot 2C\leq y_{1},\end{aligned}}}

${\displaystyle (x',y_{k,n+1}(x'))}$  is within the domain of ${\displaystyle f_{k}}$ .

We have

{\displaystyle {\begin{aligned}|y_{k,n+1}(x)-y_{k,n}(x)|&\leq \textstyle \left|\int _{0}^{x}|f_{k}(x',y_{k,n}(x'))-f_{k}(x',y_{k,n-1}(x'))|\,\mathrm {d} x'\right|\\&\leq \textstyle L_{k}\left|\int _{0}^{x}|y_{k,n}(x')-y_{k,n-1}(x')|\,\mathrm {d} x'\right|,\end{aligned}}}

where ${\displaystyle L_{k}}$  is the Lipschitz constant of ${\displaystyle f_{k}}$ . Thus for maximal difference ${\displaystyle \textstyle M_{k,n}(x)=\sup _{x'\in [0,x]}|y_{k,n+1}(x')-y_{k,n}(x')|}$ , we have a bound ${\displaystyle \textstyle M_{k,n}(x)\leq L_{k}\left|\int _{0}^{x}M_{k,n-1}(x')\,\mathrm {d} x'\right|}$ , and

{\displaystyle {\begin{aligned}M_{k,0}(x)&\leq \textstyle \left|\int _{0}^{x}|f_{k}(x',0)|\,\mathrm {d} x'\right|\\&\leq |x|\textstyle \sup _{R}|f_{k}|\leq 2C|x|.\end{aligned}}}

By induction, this implies the bound ${\displaystyle M_{k,n}(x)\leq 2CL_{k}^{n}|x|^{n+1}/(n+1)!}$  which tends to zero as ${\displaystyle n\to \infty }$  for all ${\displaystyle x\in I}$ .

The functions ${\displaystyle y_{k,n}}$  are equicontinuous as for ${\displaystyle -x_{2}\leq x  we have

{\displaystyle {\begin{aligned}|y_{k,n+1}(x')-y_{k,n+1}(x)|&\leq \textstyle \int _{x}^{x'}|f_{k}(x'',y_{k,n}(x''))|\,\mathrm {d} x''\\&\textstyle \leq |x'-x|\sup _{R}|f_{k}|\leq 2C|x'-x|,\end{aligned}}}

so by the Arzelà–Ascoli theorem they are relatively compact. In particular, for each ${\displaystyle k}$  there is a subsequence ${\displaystyle (y_{k,\varphi _{k}(n)})_{n\in \mathbb {N} }}$  converging uniformly to a continuous function ${\displaystyle y_{k}:I\to \mathbb {R} }$ . Taking limit ${\displaystyle n\to \infty }$  in

{\displaystyle {\begin{aligned}\textstyle \left|y_{k,\varphi _{k}(n)}(x)-\int _{0}^{x}f_{k}(x',y_{k,\varphi _{k}(n)}(x'))\,\mathrm {d} x'\right|&=|y_{k,\varphi _{k}(n)}(x)-y_{k,\varphi _{k}(n)+1}(x)|\\&\leq M_{k,\varphi _{k}(n)}(x_{2})\end{aligned}}}

we conclude that ${\displaystyle \textstyle y_{k}(x)=\int _{0}^{x}f_{k}(x',y_{k}(x'))\,\mathrm {d} x'}$ . The functions ${\displaystyle y_{k}}$  are in the closure of a relatively compact set, so they are themselves relatively compact. Thus there is a subsequence ${\displaystyle y_{\psi (k)}}$  converging uniformly to a continuous function ${\displaystyle z:I\to \mathbb {R} }$ . Taking limit ${\displaystyle k\to \infty }$  in ${\displaystyle \textstyle y_{\psi (k)}(x)=\int _{0}^{x}f_{\psi (k)}(x',y_{\psi (k)}(x'))\,\mathrm {d} x'}$  we conclude that ${\displaystyle \textstyle z(x)=\int _{0}^{x}f(x',z(x'))\,\mathrm {d} x'}$ , using the fact that ${\displaystyle f_{\psi (k)}}$  are equicontinuous by the Arzelà–Ascoli theorem. By the fundamental theorem of calculus, ${\displaystyle z'(x)=f(x,z(x))}$  in ${\displaystyle I}$ .

## Related theorems

The Peano theorem can be compared with another existence result in the same context, the Picard–Lindelöf theorem. The Picard–Lindelöf theorem both assumes more and concludes more. It requires Lipschitz continuity, while the Peano theorem requires only continuity; but it proves both existence and uniqueness where the Peano theorem proves only the existence of solutions. To illustrate, consider the ordinary differential equation

${\displaystyle y'=\left\vert y\right\vert ^{\frac {1}{2}}}$  on the domain ${\displaystyle \left[0,1\right].}$

According to the Peano theorem, this equation has solutions, but the Picard–Lindelöf theorem does not apply since the right hand side is not Lipschitz continuous in any neighbourhood containing 0. Thus we can conclude existence but not uniqueness. It turns out that this ordinary differential equation has two kinds of solutions when starting at ${\displaystyle y(0)=0}$ , either ${\displaystyle y(x)=0}$  or ${\displaystyle y(x)=x^{2}/4}$ . The transition between ${\displaystyle y=0}$  and ${\displaystyle y=(x-C)^{2}/4}$  can happen at any ${\displaystyle C}$ .

The Carathéodory existence theorem is a generalization of the Peano existence theorem with weaker conditions than continuity.

## Notes

1. ^ Peano, G. (1886). "Sull'integrabilità delle equazioni differenziali del primo ordine". Atti Accad. Sci. Torino. 21: 437–445.
2. ^ Peano, G. (1890). "Demonstration de l'intégrabilité des équations différentielles ordinaires". Mathematische Annalen. 37 (2): 182–228. doi:10.1007/BF01200235. S2CID 120698124.
3. ^ (Coddington & Levinson 1955, p. 6)

## References

• Osgood, W. F. (1898). "Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x, y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung". Monatshefte für Mathematik. 9: 331–345. doi:10.1007/BF01707876. S2CID 122312261.
• Coddington, Earl A.; Levinson, Norman (1955). Theory of Ordinary Differential Equations. New York: McGraw-Hill.
• Murray, Francis J.; Miller, Kenneth S. (1976) [1954]. Existence Theorems for Ordinary Differential Equations (Reprint ed.). New York: Krieger.
• Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.


text:
# [IPython-User] heart disease?

Comer Duncan comer.duncan@gmail....
Wed Mar 21 14:11:46 CDT 2012

Mike,

I am using pre-.13.  The only thing I have done recently is to reset
my EDITOR env to TextWrangler, which works quite well in the console.
However, it does not yet work in the notebook.    Maybe it is just a
network slowness thing...

Comer

On Wed, Mar 21, 2012 at 2:44 PM, Michael Hadmack <hadmack@gmail.com> wrote:
> I have also have some trouble with the Kernel dying repeatedly.  I'm not
> sure what the cause was but sometimes when opening a notebook I would get a
> 'kernel had died' message and after restarting the message returns within 20
> seconds.  Even shutting down the server and starting a new one did not seem
> to help but eventually the problem would just stop.  I attributed it to
> something with my machine being too slow (a 2006 Macbook).  I have not seen
> this happen once I switched from 0.12 to 0.13 though.
>
> -Mike
>
> On Wed, Mar 21, 2012 at 8:15 AM, Comer Duncan <comer.duncan@gmail.com>
> wrote:
>>
>> Hi,
>>
>> Today, I am noticing on my macbook that when I run ipython notebook
>> --profile=sympy that it takes up to a minute for chrome to light up
>> with the dashboard in a tab.  So, when I select a particular notebook
>> to run and run it, it takes a while for it to run (=> longer than
>> usual and seemingly not due to mathjax being slow).  Well I could
>> chalk that up to local issues on my machine, but have not seen that
>> until today. And finally the kernel keeps dying.  I keep restarting it
>> and this sequence goes on and on with the kernel restarted living for
>> less than a minute.  The stated output on stdout mentions the kernel
>> failed to respond to heartbeat.  Any idea as to why this may happen
>> and when?  And what if anything to do about it?
>>
>>
>> Comer
>> _______________________________________________
>> IPython-User mailing list
>> IPython-User@scipy.org
>> http://mail.scipy.org/mailman/listinfo/ipython-user
>
>
>
> _______________________________________________
> IPython-User mailing list
> IPython-User@scipy.org
> http://mail.scipy.org/mailman/listinfo/ipython-user
>


text:
# Tag Info

16

The second way, predicting $x=cos(\alpha)$ and $y=sin(\alpha)$ is totally okay. Yes, the norm of the predicted $(x, y)$ vector is not guaranteed to be near $1$. But it is not likely to blow up, especially if you use sigmoid activation functions (which are bounded by they nature) and/or regularize your model well. Why should your model predict a large value,...

9

I will assume by C1, C2, etc, you mean convolutional layers, and by P1 ,P2 you mean pooling layers, and FC means fully connected layers. We can calculate the memory required for a forward pass like this: One image If you're working with float32 values, then following the link provided above by @Alexandru Burlacu you have: Input: 50x50x3 = 7,500 = 7.5K C1: ...

9

You take the last column of that matrix sort descending plot index, distance hope to see a knee (if the distance does not work well. there might be none)

8

I assume you mean feature selection as feature engineering. The process I usually follow and I see some people do is Feature engineering Try a few algorithms, usually highly performant ones such as RandomForest, Gradient Boosted Trees, Neutral Networks, or SVM on the features. 2.1 Do simple parameter tuning such as grid search on a small range of ...

5

Let's check for continuity for the new proposed function. For convenience, let me call it $L_{\delta, c}$. $$\lim_{a \to \delta^+}L_{\delta, c}(a)=c\delta$$ and $$\lim_{a \to \delta^-}L_{\delta, c}(a)=\frac{\delta^2}2$$ If we want the function to be continuous, we would want $c$ to take value $\frac{\delta}2$. Now, let's check for diffentiability of $L_{\... 5 No! the best score on validation set means you are not in overfitting zone. As explained in my previous answer to your question, overfitting is about high score on training data but low score on validation. So to avoid overfitting you need to check your score on Validation Set and then you are fine. There is no theoretical calculation of the best depth of a ... 4 Likelihood-ratio tests are a mainstay of classical hypothesis testing. The idea is to form the likelihoods of the two hypotheses under consideration, and choose the one with the highest likelihood if their ratio is sufficiently large. Hypotheses come in two flavors: simple, and composite. Simple tests are those for which the hypothesis uniquely defines the ... 3 Maybe this link will give you an explanation on how to compute the memory usage of an arbitrary neural network. Bellow in the link is explained the memory usage of the VGGNet model. Click here and scroll down a bit)) 3 Working with Cartesian coordinates works well as mentioned above. Yet, in my opinion, converting polar data to Cartesian creates dependencies between the X and Y coordinates that were not originally present in the data. For example, a robot's path decision model is more intuitive in polar coordinates than Cartesian. The dependency of the robot's velocity ... 2 Hey i just saw your question. It is COMPLETELY WRONG to do feature selection first and then tune the model using cross-validation. In elements of statistical learning and this blog post it is clearly mentioned that : CV method are unbiased only if all your model building is done inside the CV loop. So do feature selection inside the CV loop for parameter ... 2 Statistical approach This question is more related to statistics than data science, perharps you would have better answers on Cross Validated. As you suggested in the question, a purely statistical approach should work for sure (though it requires some assumptions). As i understand your problem, your prior knowledge is that$\alpha$(or$\beta$) should be ... 2 I don't have a full answer to your question, but I wanted to help (and would love to know a complete answer). In a simpler case when you have one source$m=i=1$, it seems to me you are describing a scenario that resembles a hidden Markov model which uses a EM algorithm-based solution (Baum–Welch algorithm) – that is if you make the further hypothesis that a ... 1$P(data|parameter)$is not used in the sense of generating new data, but rather in the sense of how probable is that this data have been already generated by such parameters. 1 You can determine the most likely values for$\log x$and$\log y$by linear regression since your relationship implies:$\log a = b.\log y + \log x$So run regression on$b$as input and$\log a$as output. The gradient will be$\log y$and the intercept$\log x$1 For theory Tibshirani: The elements of statistical learning https://web.stanford.edu/~hastie/Papers/ESLII.pdf Also Andrew NG and other books from deeplearning.ai: Machine Learning Yearning https://d2wvfoqc9gyqzf.cloudfront.net/content/uploads/2018/09/Ng-MLY01-13.pdf of course the applied machine learning books on computer languages: An introduction into ... 1 If the discrete variable has a lot of discrete values then it is almost the same as a continuous variable, because continuous variables are technically discrete due to the way how numbers are represented in computers (float64 for Python). The worst case is binary, but, in my experience, Pearson coefficient work well with binary and continuous data together. ... 1 Given that only have labels for a small subset of data, you should use unsupervised methods. You can cluster all the sensors data. Then see if there is a pattern to where the 100 bad sensors are and can generalize to the other sensors. If a majority of the 100 bad sensors are in the same region, you can label that cluster as bad and make a threshold based on ... 1 If you don't know which of the 10,000 sensors are good and which are bad, the data from those 10,000 sensors is useless for training a regression line / classifier. You need labelled data, where you know both the value of the features and the health of the sensor. Moreover, to be effective, you probably need your training set to contain both healthy ... 1 Actually there is the possibility of overfitting the validation set. This because the validation set is the one where your parameters (the depth in your case) perform at best, but this does not means that your model will generalize well on unseen data. That's the reason why usually you split your data into three set: train, validation and test. You train ... 1 "Oracle" refers to something that has access to the ground truth. It has the perfect information of which in practice, we rarely have some luxury. In such paper, you can typically see that it reflects a comparison with a model with perfect information. From Candes and Terry Tao's paper: "Even though$n$may be much smaller than$p\$, our estimator achieves ...

1

While training a convNet, total memory required include following: Memory for parameters Memory for the output of intermediate layers Memory for the gradient of each parameter Extra memory needed if you are using optimizer like Momentum, RMSprop, Adams etc Miscellaneous memory for implementation A good rough approximation is number of parameters x 3 x 4(if ...

1

Since you want your model to be a general solution, you want to include all your data when building the final model. You are correct in saying that keeping the model with the best validation score in the CV is overfitting. Including these inner random states help generalize your model, and since you have already tuned your model parameters using CV, you ...

1

Many ML and minimization tasks make use of an objective function. At each iteration, a parameter set to try is defined, and the objective function returns some scoring value that reflects how good, or bad, that parameter set is. Then the parameter set is altered, and the process repeats. So, when do you stop this process? You want to stop when the changes ...

1

A naive definition of Parameter convergence is when the weights or the values of the parameters reach a point asymptotically. What I mean is that, when your model training is not altering the parameter values(maybe less than epsilon-small values) it might be a good fit. For decision trees, I found this paper which explains rate of convergence and more. It ...

1

Here is the code I wrote to answer my question. It might not be the most efficient one but it works. Sharing is caring :) niter<-10000 N<-nrow(Y) T<-10 # I will take into consideration until t=10 to estimate my parametres and then I will forecast the rest values t=11,12,... etc result <- matrix(0,nrow = niter, ncol = 3) #Here I store 3 ...

1

In terms of writing this in R, here is an example I found: http://www.stat.purdue.edu/~zhanghao/MAS/handout/gibbsBayesian.pdf You could then iterate through different levels of α and select the level yielding the best results.

1

If you were fitting a large number of different models, and you had sequences of training data for each different model (with the params already known in those cases), then you might be able to use a recursive neural network (RNN) to provide param estimates for new data sequences. However, that does not seem to be your situation. As I understand your ...

1

You want to look at the spectral density to get an idea of the distribution of the frequency components. Check out the 'psd' package in r or the signal processing toolbox if you have access to Matlab.

1

I have already answered a similar question here. The process would be: Transformation and Reduction: Involves processes like transformations, mean and median scaling, etc. Feature Selection: This can be done in a lot of ways like threshold selection, subset selection, etc. Designing predictive model: Design the predictive model on the training data ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# SCIP

Solving Constraint Integer Programs

Local Subproblem

## Detailed Description

methods to query information about or strengthen the problem at the current local search node

## Functions

SCIP_RETCODE SCIPaddConflict (SCIP *scip, SCIP_NODE *node, SCIP_CONS *cons, SCIP_NODE *validnode, SCIP_CONFTYPE conftype, SCIP_Bool iscutoffinvolved)

SCIP_RETCODE SCIPclearConflictStore (SCIP *scip, SCIP_EVENT *event)

SCIP_RETCODE SCIPaddConsNode (SCIP *scip, SCIP_NODE *node, SCIP_CONS *cons, SCIP_NODE *validnode)

SCIP_RETCODE SCIPaddConsLocal (SCIP *scip, SCIP_CONS *cons, SCIP_NODE *validnode)

SCIP_RETCODE SCIPdelConsNode (SCIP *scip, SCIP_NODE *node, SCIP_CONS *cons)

SCIP_RETCODE SCIPdelConsLocal (SCIP *scip, SCIP_CONS *cons)

SCIP_Real SCIPgetLocalOrigEstimate (SCIP *scip)

SCIP_Real SCIPgetLocalTransEstimate (SCIP *scip)

SCIP_Real SCIPgetLocalDualbound (SCIP *scip)

SCIP_Real SCIPgetLocalLowerbound (SCIP *scip)

SCIP_Real SCIPgetNodeDualbound (SCIP *scip, SCIP_NODE *node)

SCIP_Real SCIPgetNodeLowerbound (SCIP *scip, SCIP_NODE *node)

SCIP_RETCODE SCIPupdateLocalDualbound (SCIP *scip, SCIP_Real newbound)

SCIP_RETCODE SCIPupdateLocalLowerbound (SCIP *scip, SCIP_Real newbound)

SCIP_RETCODE SCIPupdateNodeDualbound (SCIP *scip, SCIP_NODE *node, SCIP_Real newbound)

SCIP_RETCODE SCIPupdateNodeLowerbound (SCIP *scip, SCIP_NODE *node, SCIP_Real newbound)

SCIP_RETCODE SCIPchgChildPrio (SCIP *scip, SCIP_NODE *child, SCIP_Real priority)

## Function Documentation

 SCIP_RETCODE SCIPaddConflict ( SCIP * scip, SCIP_NODE * node, SCIP_CONS * cons, SCIP_NODE * validnode, SCIP_CONFTYPE conftype, SCIP_Bool iscutoffinvolved )

adds a conflict to a given node or globally to the problem if node == NULL.

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Note
this method will release the constraint
Parameters
 scip SCIP data structure node node to add conflict (or NULL if global) cons constraint representing the conflict validnode node at whichaddConf the constraint is valid (or NULL) conftype type of the conflict iscutoffinvolved is a cutoff bound involved in this conflict

Definition at line 3280 of file scip_prob.c.

Referenced by applyCliqueFixings(), and SCIP_DECL_CONFLICTEXEC().

## ◆ SCIPclearConflictStore()

 SCIP_RETCODE SCIPclearConflictStore ( SCIP * scip, SCIP_EVENT * event )

removes all conflicts depending on an old cutoff bound if the improvement of the incumbent is good enough

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:

tries to remove conflicts depending on an old cutoff bound if the improvement of the new incumbent is good enough

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure event event data

Definition at line 3339 of file scip_prob.c.

Referenced by SCIP_DECL_EVENTEXEC().

 SCIP_RETCODE SCIPaddConsNode ( SCIP * scip, SCIP_NODE * node, SCIP_CONS * cons, SCIP_NODE * validnode )

adds constraint to the given node (and all of its subnodes), even if it is a global constraint; It is sometimes desirable to add the constraint to a more local node (i.e., a node of larger depth) even if the constraint is also valid higher in the tree, for example, if one wants to produce a constraint which is only active in a small part of the tree although it is valid in a larger part. In this case, one should pass the more global node where the constraint is valid as "validnode". Note that the same constraint cannot be added twice to the branching tree with different "validnode" parameters. If the constraint is valid at the same node as it is inserted (the usual case), one should pass NULL as "validnode". If the "validnode" is the root node, it is automatically upgraded into a global constraint, but still only added to the given node. If a local constraint is added to the root node, it is added to the global problem instead.

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure node node to add constraint to cons constraint to add validnode node at which the constraint is valid, or NULL

Definition at line 3376 of file scip_prob.c.

 SCIP_RETCODE SCIPaddConsLocal ( SCIP * scip, SCIP_CONS * cons, SCIP_NODE * validnode )

adds constraint locally to the current node (and all of its subnodes), even if it is a global constraint; It is sometimes desirable to add the constraint to a more local node (i.e., a node of larger depth) even if the constraint is also valid higher in the tree, for example, if one wants to produce a constraint which is only active in a small part of the tree although it is valid in a larger part.

If the constraint is valid at the same node as it is inserted (the usual case), one should pass NULL as "validnode". If the "validnode" is the root node, it is automatically upgraded into a global constraint, but still only added to the given node. If a local constraint is added to the root node, it is added to the global problem instead.

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Note
The same constraint cannot be added twice to the branching tree with different "validnode" parameters. This is the case due internal data structures and performance issues. In such a case you should try to realize your issue using the method SCIPdisableCons() and SCIPenableCons() and control these via the event system of SCIP.

adds constraint locally to the current node (and all of its subnodes), even if it is a global constraint; It is sometimes desirable to add the constraint to a more local node (i.e., a node of larger depth) even if the constraint is also valid higher in the tree, for example, if one wants to produce a constraint which is only active in a small part of the tree although it is valid in a larger part.

If the constraint is valid at the same node as it is inserted (the usual case), one should pass NULL as "validnode". If the "validnode" is the root node, it is automatically upgraded into a global constraint, but still only added to the given node. If a local constraint is added to the root node, it is added to the global problem instead.

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Note
The same constraint cannot be added twice to the branching tree with different "validnode" parameters. This is the case due to internal data structures and performance issues. In such a case you should try to realize your issue using the method SCIPdisableCons() and SCIPenableCons() and control these via the event system of SCIP.
Parameters
 scip SCIP data structure cons constraint to add validnode node at which the constraint is valid, or NULL

Definition at line 3446 of file scip_prob.c.

## ◆ SCIPdelConsNode()

 SCIP_RETCODE SCIPdelConsNode ( SCIP * scip, SCIP_NODE * node, SCIP_CONS * cons )

disables constraint's separation, enforcing, and propagation capabilities at the given node (and all subnodes); if the method is called at the root node, the constraint is globally deleted from the problem; the constraint deletion is being remembered at the given node, s.t. after leaving the node's subtree, the constraint is automatically enabled again, and after entering the node's subtree, it is automatically disabled; this may improve performance because redundant checks on this constraint are avoided, but it consumes memory; alternatively, use SCIPdisableCons()

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure node node to disable constraint in cons constraint to locally delete

Definition at line 3477 of file scip_prob.c.

Referenced by branchCons(), and createNAryBranch().

## ◆ SCIPdelConsLocal()

 SCIP_RETCODE SCIPdelConsLocal ( SCIP * scip, SCIP_CONS * cons )

disables constraint's separation, enforcing, and propagation capabilities at the current node (and all subnodes); if the method is called during problem modification or at the root node, the constraint is globally deleted from the problem; the constraint deletion is being remembered at the current node, s.t. after leaving the current subtree, the constraint is automatically enabled again, and after reentering the current node's subtree, it is automatically disabled again; this may improve performance because redundant checks on this constraint are avoided, but it consumes memory; alternatively, use SCIPdisableCons()

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:

disables constraint's separation, enforcing, and propagation capabilities at the current node (and all subnodes); if the method is called during problem modification or at the root node, the constraint is globally deleted from the problem; the constraint deletion is being remembered at the current node, s.t. after leaving the current subtree, the constraint is automatically enabled again, and after reentering the current node's subtree, it is automatically disabled again; this may improve performance because redundant checks on this constraint are avoided, but it consumes memory; alternatively, use SCIPdisableCons()

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Note
SCIP stage does not get changed
Parameters
 scip SCIP data structure cons constraint to locally delete

Definition at line 3527 of file scip_prob.c.

## ◆ SCIPgetLocalOrigEstimate()

 SCIP_Real SCIPgetLocalOrigEstimate ( SCIP * scip )

gets estimate of best primal solution w.r.t. original problem contained in current subtree

Returns
estimate of best primal solution w.r.t. original problem contained in current subtree
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure

Definition at line 3580 of file scip_prob.c.

Referenced by SCIP_DECL_DISPOUTPUT().

## ◆ SCIPgetLocalTransEstimate()

 SCIP_Real SCIPgetLocalTransEstimate ( SCIP * scip )

gets estimate of best primal solution w.r.t. transformed problem contained in current subtree

Returns
estimate of best primal solution w.r.t. transformed problem contained in current subtree
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure

Definition at line 3599 of file scip_prob.c.

## ◆ SCIPgetLocalDualbound()

 SCIP_Real SCIPgetLocalDualbound ( SCIP * scip )

gets dual bound of current node

Returns
dual bound of current node
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure

Definition at line 3619 of file scip_prob.c.

Referenced by SCIP_DECL_DISPOUTPUT(), and SCIP_DECL_HEUREXEC().

## ◆ SCIPgetLocalLowerbound()

 SCIP_Real SCIPgetLocalLowerbound ( SCIP * scip )

gets lower bound of current node in transformed problem

Returns
lower bound of current node in transformed problem
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure

Definition at line 3638 of file scip_prob.c.

## ◆ SCIPgetNodeDualbound()

 SCIP_Real SCIPgetNodeDualbound ( SCIP * scip, SCIP_NODE * node )

gets dual bound of given node

Returns
dual bound of a given node
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure node node to get dual bound for

Definition at line 3658 of file scip_prob.c.

Referenced by applyDomainChanges(), and writeBounds().

## ◆ SCIPgetNodeLowerbound()

 SCIP_Real SCIPgetNodeLowerbound ( SCIP * scip, SCIP_NODE * node )

gets lower bound of given node in transformed problem

Returns
lower bound of given node in transformed problem
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure node node to get dual bound for

Definition at line 3675 of file scip_prob.c.

References FALSE, SCIP_CALL_ABORT, SCIPcheckStage(), SCIPnodeGetLowerbound(), and TRUE.

Referenced by execRelpscost(), and subscipdataCopySubscip().

## ◆ SCIPupdateLocalDualbound()

 SCIP_RETCODE SCIPupdateLocalDualbound ( SCIP * scip, SCIP_Real newbound )

if given value is tighter (larger for minimization, smaller for maximization) than the current node's dual bound (in original problem space), sets the current node's dual bound to the new value

Note
the given new bound has to be a dual bound, i.e., it has to be valid for the original problem.
Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure newbound new dual bound for the node (if it's tighter than the old one)

Definition at line 3699 of file scip_prob.c.

Referenced by setupAndSolveSubscipRapidlearning(), and setupProblem().

## ◆ SCIPupdateLocalLowerbound()

 SCIP_RETCODE SCIPupdateLocalLowerbound ( SCIP * scip, SCIP_Real newbound )

if given value is larger than the current node's lower bound (in transformed problem), sets the current node's lower bound to the new value

Note
the given new bound has to be a lower bound, i.e., it has to be valid for the transformed problem.
Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure newbound new lower bound for the node (if it's larger than the old one)

Definition at line 3749 of file scip_prob.c.

Referenced by solveComponent().

## ◆ SCIPupdateNodeDualbound()

 SCIP_RETCODE SCIPupdateNodeDualbound ( SCIP * scip, SCIP_NODE * node, SCIP_Real newbound )

if given value is tighter (larger for minimization, smaller for maximization) than the node's dual bound, sets the node's dual bound to the new value

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure node node to update dual bound for newbound new dual bound for the node (if it's tighter than the old one)

Definition at line 3788 of file scip_prob.c.

## ◆ SCIPupdateNodeLowerbound()

 SCIP_RETCODE SCIPupdateNodeLowerbound ( SCIP * scip, SCIP_NODE * node, SCIP_Real newbound )

if given value is larger than the node's lower bound (in transformed problem), sets the node's lower bound to the new value

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure node node to update lower bound for newbound new lower bound for the node (if it's larger than the old one)

Definition at line 3810 of file scip_prob.c.

## ◆ SCIPchgChildPrio()

 SCIP_RETCODE SCIPchgChildPrio ( SCIP * scip, SCIP_NODE * child, SCIP_Real priority )

change the node selection priority of the given child

Returns
SCIP_OKAY is returned if everything worked. Otherwise a suitable error code is passed. See SCIP_RETCODE for a complete list of error codes.
Precondition
this method can be called in one of the following stages of the SCIP solving process:
Parameters
 scip SCIP data structure child child to update the node selection priority priority node selection priority value

Definition at line 3844 of file scip_prob.c.

Referenced by SCIP_DECL_BRANCHEXECLP().


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# Mixing time of random walks on graphs

Suppose that we start a lazy random walk on a connected graph. However, the starting node is picked from a distribution of $$\mu$$ and $$||\mu-\pi||_{TV}<1/8$$, where $$\pi$$ is the stationary distribution of the lazy random walk on the graph. Now I am interested to see if there is any result regarding the mixing time of this chain. For example, if we run the chain for $$t$$ steps rounds, what happens to the total variation distance of the chain from the stationary distribution.

For any vertex $$v$$, you can take $$\mu$$ to be $$7/8 \pi + 1/8 v$$, that is, the with probability $$1/8$$ pick $$v$$ and otherwise pick a random vertex according to $$\pi$$. Now the distance to stationarity at time $$t$$ will be $$1/8$$ of what it is when starting the chain at $$v$$. So it seems that there are really no new questions here to explore.


text:
# The Balance Sheet

### The Accounting Equation

The accounting equation balances assets to the total of liabilities and equity.

The balance sheet is a snapshot of a company's financial condition at a point in time, traditionally at the end of a period. Two classes of information reveal the company's financial condition: (1) the company's assets and (2) the company's liabilities. An asset is an economic resource of value that a business owns and that is expected to provide future benefits. Assets may be tangible items, such as inventory, or they can be intangible items, such as intellectual property, owned by the company.

A company's debts are its liabilities. A liability is an obligation or amount owed to another individual or entity as a result of a past transaction. The portion of the assets belonging to shareholders is referred to as shareholders' equity. Shareholders' equity, sometimes also referred to as stockholders’ equity, is the value left to owners after debts and obligations have been accounted for, or the business's total assets minus its total liabilities. Accordingly, investors view shareholders' equity as their outstanding claim on the company.

The accounting equation is a company's assets equal to its liabilities plus the shareholders' equity.
${\text{Assets}}={\text{Liabilities}}+{\text{Shareholders' Equity}}$
The accounting equation demonstrates that a company's assets are financed by liabilities, shareholders' equity, or a blend of both. In practical terms, the accounting equation demonstrates two things. First, the total assets are the sum of what a company owns. Second, liabilities are contractual obligations to deliver cash or another financial asset to an entity while shareholders’ equity is the difference between total assets and total liabilities. It is also the share capital retained in the company in addition to the retained earnings minus the treasury shares.

### Components of the Balance Sheet

The balance sheet gives a snapshot of a company's assets and liabilities at a given point in time; it includes current assets, fixed assets, intangible assets such as goodwill, current and long-term liabilities, and shareholders' equity.
The balance sheet is a snapshot of a company's assets and liabilities at a given point in time. This information is revealed through the accounting equation.
${\text{Assets}}={\text{Liabilities}+\text{Shareholders' Equity}}$
The balance sheet begins with asset entries. These entries appear in descending order by liquidity. The company sorts its liquid assets into current assets and its illiquid assets into fixed assets. Current assets include a company's cash, short-term investments, and items of value that can be converted to cash within a year. For example, current assets include cash and cash equivalents, accounts receivable, and inventory.

The next item to appear on a company's balance sheet is its net fixed assets. A fixed asset is a tangible piece of property used in the normal course of business to generate revenue, such as property, plant, or equipment. After recording fixed assets, a company records its intangible assets. Intangible assets include intellectual property and goodwill. Goodwill is the premium paid over the market value of net assets (market value of identifiable assets minus the market value of identifiable liabilities) in an enterprise acquisition. The company totals all of these entries to calculate its total assets.

The company's liabilities, as either current or long term, will appear next on the balance sheet. Current liabilities are the amounts owed by a business to payees that are expected to be paid within one year or during the operating cycle, whichever is longer. For example, current liability entries might include short-term debt, accounts payable, or income taxes payable. Next, the company records its long-term debt and any other unaccounted-for debt. The company totals the liability entries to calculate its total liabilities.

The final item the company includes on the balance sheet is its shareholders' equity. A company calculates its shareholders' equity by subtracting its total liabilities from its total assets. Three categories represent shareholders' equity: preferred stock, common stock, and retained earnings.


text:
# The mad hatter wishes to by some cases of tea.

The mad hatter wishes to by some cases of tea. He has 200 gold coins and a case of 500 tea bags costs 15 gold coins. How many cases of tea can he buy?
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Step 1
Note: the number of bags of tea in a case is irrelevant to this question.

$\to 1\phantom{\rule{1ex}{0ex}}\text{gold coin}=\frac{1}{15}\phantom{\rule{1ex}{0ex}}\text{cases}$
(temporally assume fractional case purchases are possible)
$\to 200\phantom{\rule{1ex}{0ex}}\text{gold coins}=\frac{200}{15}\phantom{\rule{1ex}{0ex}}\text{cases}=13\frac{1}{3}\phantom{\rule{1ex}{0ex}}\text{cases}$
If it is only possible to buy complete cases, 200 gold coins would allow the purchase of at most 13 cases.

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text:
Suppose there is a 10 Mbps microwave link between a geostationary satellite and its base station on Earth. Every minute the satellite takes a digital photo and sends it to the base station. Assume a propagation speed of $2.4 \times 10^8$meters/sec.
1. What is the bandwidth-delay product, $R \times d _{prop}$?


text: Not the cosine distance! Ask Question Asked 5 years, 2 months ago. Short answer: Cosine distance is not the overall best performing distance metric out there Although similarity measures are often expressed using a distance metric , it is in fact a more flexible measure as it is not required to be symmetric or fulfill the triangle inequality. The cosine of … Coding using R (Euclidean distance is also covered) Dataset and R code in … The main difference between the two is that cosine similarity will yield a stronger indicator when two documents have the same word multiple times in the documents, while Hamming distance doesn't care how often the individual tokens come up. Now to find the cosine distances of one document (e.g. END OF MULTIPLOT JUSTIFICATION CENTER MOVE 50 98 TEXT Distance/Similarity Measures (IRIS.DAT) The data about all application pages is also stored in a data Webhouse. Formula to find the Cosine Similarity and Distance is as below: Here A=Point P1,B=Point P2 (in our example). Cosine similarity is generally used as a metric for measuring distance when the magnitude of the vectors does not matter. I am given a csv with three columns, user_id, book_id, rating. The data about all application pages is also stored in a data Webhouse. Cosine Distance hanya ditentukan untuk nilai positif Jika nilai negatif ditemui dalam input, jarak cosinus tidak akan dihitung. Arne Arne. @WarrenWeckesser, thank you, I fixed the name. normalization natural-language euclidean cosine-distance cosine-similarity. The scipy sparse matrix API is a bit weird (not as flexible as dense N-dimensional numpy arrays). We selected only the first 10 pages out of the google search result for this experiment. 47.6k 35 35 gold badges 219 219 silver badges 434 434 bronze badges. share | cite | improve this question | follow | edited Feb 26 '16 at 22:49. ttnphns. Cosine similarity is a measure of similarity between two non-zero vectors of an inner product space. If you pass a distance matrix it will be O(n²). We acquired 354 distinct application pages from a star schema page dimension representing application pages. Does anybody know reason for different definitions? \ $If you try this with fixed precision numbers, the left side loses precision but the right side does not. The interpretation of Parameters X {array-like, sparse matrix} of shape (n_samples_X, n_features) Matrix X. The Cosine Similarity procedure computes similarity between all pairs of items. This video is related to finding the similarity between the users. Typically, it can be used as a text matching algorithm. Read more in the User Guide. When to use cosine similarity over Euclidean similarity. Levenshtein distance = 7 (if you consider sandwich and sandwiches as a different word) Bigram distance = 14 Cosine similarity = 0.33 Jaccard similarity = 0.2 I would like to understand the pros and cons of using each of the these (dis)similarity measures. table 2 and figure 1 it is clearly visible that best fitness values were obtained using the Cosine similarity coefficients followed by Dice and Jaccard. Cosine Similarity. I am given a csv with three columns, user_id, book_id, rating. The cosine of 0° is 1, and it is less than 1 for any angle in the interval (0,π] radians. An identity for this is$\ 1 - \cos(x) = 2 \sin^2(x/2). Active 5 years, 2 months ago. This is searching for the cosine similarity! Does a hash function necessarily need to allow arbitrary length input? The vector is filled by the term frequency vectors of word or sequence of X characters in text documents. Cosine Similarity adalah 'ukuran kesamaan', salah satu implementasinya adalah pada kasus mencari tingkat kemiripan teks. Based on the cosine similarity the distance matrix D n ∈ Z n × n (index n means names) contains elements d i,j for i, j ∈{1, 2, …, n} where d i, j = s i m (v → i, v → j). call: Cs = getCosineSimilarity(x,y) Compute Cosine Similarity between vectors x and y. x and y have to be of same length. Difference between cosine similarity and cosine distance, Podcast 302: Programming in PowerPoint can teach you a few things, Difference between staticmethod and classmethod. Thanks for contributing an answer to Stack Overflow! This is being extended in the future research for 30-35 pages for a precise calculation of efficiency. Cosine similarity: $\langle x , y\rangle$ Euclidean distance (squared): $2(1 - \langle x , y\rangle)$ As you can see, minimizing (square) euclidean distance is equivalent to maximizing cosine similarity if the vectors are normalized. Cosine similarity is a measure of similarity between two non-zero vectors of an inner product space.It is defined to equal the cosine of the angle between them, which is also the same as the inner product of the same vectors normalized to both have length 1. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Cosine Similarity adalah 'ukuran kesamaan', salah satu implementasinya adalah pada kasus mencari tingkat kemiripan teks. Similarly you can define the cosine distance for the resulting similarity value range. Data Structures 101: What Is a Binary Search Tree? Why is “1000000000000000 in range(1000000000000001)” so fast in Python 3? Intuitively, let’s say we have 2 vectors, each representing a sentence. Euclidean Distance and Cosine … Cosine similarity cares only about the angle between the two vectors and not the distance between them. I do advise this if you're going to use cosine similarity - as function words are quite frequent (in English, … It is a symmetrical algorithm, which means that the result from computing the similarity of Item A to Item B is the same as computing the similarity of Item B to Item A. distance function should become larger as elements become less similar; since maximal value of cosine is 1, we can define cosine distance as Cosine Similarity. Stack Overflow for Teams is a private, secure spot for you and Informally, the Levenshtein distance between two words is the minimum … Euclidean Distance (u,v) = 2 * (1- Cosine Similarity(u,v)) Euclidean Distance (u,v) = 2 * Cosine Distance(u,v) Hack :- So in the algorithms which only accepts euclidean distance … Cosine Distance. We don’t compute the similarity of items to themselves. Cosine distance is 1-. normalization natural-language euclidean cosine-distance cosine-similarity. Mathematically, it measures the cosine of the angle between two vectors projected in a multi-dimensional space. So here I find… It is also easy to see that Pearson Correlation Coefficient and Cosine Similarity are equivalent when X and Y have means of 0, so we can think of Pearson Correlation Coefficient as demeaned version of Cosine Similarity. Now, the distance can be defined as 1-cos_similarity. The intuition behind this is that if 2 vectors are perfectly the same then similarity is 1 (angle=0) and thus, distance is 0 (1-1=0). What are the differences between type() and isinstance()? We can therefore compute the score for each pair of nodes once. Cosine similarity ranges from 0 to 1, where 1 means the two vectors are perfectly similar. Conclusion : I hope by now you have clear understanding of the math behind the computation of cosine similarity and Cosine Distance and its usage. The relation between cosine similarity and cosine distance can be define as below. Cosine similarity range: −1 meaning exactly opposite, 1 meaning exactly the … We can therefore compute the score for each pair of nodes once. Cosine similarity is a metric, helpful in determining, how similar the data objects are irrespective of their size. Let's say you are in an e-commerce setting and you want to compare users for product recommendations: User 1 … Levenshtein distance = 7 (if you consider sandwich and sandwiches as a different word) Bigram distance = 14 Cosine similarity = 0.33 Jaccard similarity = 0.2 I would like to understand the pros and cons of using each of the these (dis)similarity measures. I do advise this if you're going to use cosine similarity - as function words are quite frequent (in English, … If you pass the actual data, the code could use an index to make it faster than this. 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Mathematically, it compute the score for each pair of nodes once logo © 2021 Stack Exchange Inc ; contributions... Was always wondering why do n't we use Euclidean distance instead is my that! So here I find… Join Stack Overflow to learn more, see our tips on writing answers... Or responding to other answers, privacy policy and cookie policy similarity procedure similarity... Say I have a very simple data set why cosine of the vectors dataset is filled by the term vectors. Up all the dimensions feed, copy and paste this URL into your RSS reader material components Heat! ” cosine similarity vs cosine distance you can add up all the dimensions vs..format vs. string literal Pythonic... The same, 0 indicating orthogonality corresponds to their dot product divided by term. Ozone layer had holes in it is heavily used in recommendation systems to recommend products to the.! 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Text Distance/Similarity Measures ( IRIS.DAT ) when to use cosine NFL passer rating using a in... You pass the actual data, the Oracle, Loki and many.! More, see our tips on writing great answers question Asked 5 years, 2 ago. Did postal voting favour Joe Biden so much they are less similar string literal cosine similarity vs cosine distance way... There I just needed to pull out recommendations from a given artist ’ s list of songs or vectors,! Notice that the ozone layer had holes in it noticed your query about removing words... Similarity ( θ ) when we need to find and share information when to use cosine solving a problem I. 434 434 bronze badges similarity value range Length input the google search result for this is it! Find and share information to change it now a string metric for measuring distance the! 35 35 gold badges 219 219 silver badges 434 434 bronze badges decreases when distance between two.. 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To subscribe to this RSS feed, copy and paste this URL into RSS! * Update as question changed * * * * when to use cosine distance ( d and. Of word or sequence of X characters in text documents same, 0 orthogonality!


text:
# Electronic configuration for singlet and triplet states

+ 0 like - 0 dislike
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Is there a difference in the electronic configuration for singlet and triplet states? For example, He atom has 1s2 configuration in its ground state (singlet state)

But what about when the He atom is excited. It can come in both states (Signlet and Triplet)

Can I tell from the excited electronic configuration whether the atom is in a singlet or a triplet state?

Thanks!

This post imported from StackExchange Physics at 2014-04-13 12:28 (UCT), posted by SE-user Haneen Su

+ 1 like - 0 dislike

I am not sure that I get your question right, but let me try to answer according to my understanding.

The spin part of the two electron wave function of the singlet state $|0,0\rangle=|l=0,m=0\rangle$ is

$|0,0\rangle=(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)/\sqrt{2}$

The three triplet states look like this:

$|1,0\rangle=(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle)/\sqrt{2}$

$|1,1\rangle=|\uparrow\uparrow\rangle$

$|1,-1\rangle=|\downarrow\downarrow\rangle$

If you are unfamiliar with the notation, $\uparrow$ denotes spin $+1/2$ and $\downarrow$ denotes spin $-1/2$; the numbers on the left denote: the total angular momentum $l$ of both electrons and its $z$ component $m$.

If the electrons are in the same energy state, they have to have different spins, that is, the state can be either $|0,0\rangle$ or $|1,0\rangle$, so in principle singlet and triplet are possible. If one of the electrons is excited, any of the four states is possible, since the spins don't have to be different any more. Therefore, if you know that both electrons have spin up or both have spin down, you can be sure, that it is a triplet state. If one spin is up and the other down, you cannot tell from the spin configuration whether it is a singlet or a triplet, since again both $|0,0\rangle$ or $|1,0\rangle$ are possible.

This post imported from StackExchange Physics at 2014-04-13 12:28 (UCT), posted by SE-user Photon
answered Apr 12, 2014 by (70 points)
I'd suggest using $\uparrow$ and $\downarrow$ instead of $+$ and $-$ to denote spins, otherwise they look too mixed up when adding and subtracting kets.

This post imported from StackExchange Physics at 2014-04-13 12:28 (UCT), posted by SE-user Ruslan
I was too lazy to look up the TeX commands for them and didn't know them by heart, to be honest.

This post imported from StackExchange Physics at 2014-04-13 12:28 (UCT), posted by SE-user Photon
\uparrow and \downarrow

This post imported from StackExchange Physics at 2014-04-13 12:28 (UCT), posted by SE-user Ruslan
Thanks, edited.

This post imported from StackExchange Physics at 2014-04-13 12:28 (UCT), posted by SE-user Photon


text:
A unified approach for the $H_\infty$-stability analysis of classical and fractional neutral systems with commensurate delays

1 DISCO - Dynamical Interconnected Systems in COmplex Environments
L2S - Laboratoire des signaux et systèmes, Inria Saclay - Ile de France, SUPELEC, CNRS - Centre National de la Recherche Scientifique : UMR8506
Abstract : We examine the stability of linear integer-order and fractional-order systems with commensurate delays of neutral type in the sense of $H_\infty$-stability. The systems may have chains of poles approaching the imaginary axis. While several classes of these systems have been previously studied on a case-by-case basis, a unified method is proposed in this paper which allows to deal with all these classes at the same time. Approximation of poles of large modulus is systematically calculated based on a convex hull derived from the coefficients of the system. This convex hull also serves to establish sufficient conditions for instability and necessary and sufficient conditions for stability.
Keywords :
Document type :
Journal articles

Cited literature [24 references]

https://hal.archives-ouvertes.fr/hal-01679070
Contributor : Le Ha Vy Nguyen <>
Submitted on : Tuesday, January 9, 2018 - 4:47:53 PM
Last modification on : Thursday, April 5, 2018 - 12:30:14 PM
Document(s) archivé(s) le : Thursday, May 3, 2018 - 3:04:04 PM

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Le Ha Vy Nguyen. A unified approach for the $H_\infty$-stability analysis of classical and fractional neutral systems with commensurate delays. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, In press, 56 (1), pp.538-555 〈10.1137/16m1101271 〉. 〈hal-01679070〉

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# Why learn Mathematics?

I found a link to a blog posting on Wolf’s blog about the question: What is the point of math? It is quite an interesting read for me, but at some places very annoying.

Why do we learn the quadratic formula (also called the midnight formula in Bavaria because you are supposed be able to repeat it when someone awakens you ad midnight). After all, you may never have use this formula in your professional career. In fact, this is true even if you are mathematician working outside research. So what is the point of this formula then?

$$x_{1,2} = \dfrac{p}{2} \pm \sqrt{\dfrac{p^2}{4} – q}$$

You will be surprised to hear my answer as a math professor. Indeed, I agree, it is a useless fact to learn. You can absolutely do without that formula. If you asked me I’d cancel it from the syllabus altogether.

First of all, you can solve quadratic equations without the formula quite easily. The technique involved in this is called quadratic expansion. It is using algebraic tools to transform the equation into an equivalent equation which solves the problem. The underlying theory is the theory of fields. The technique can be applied in the field of complex numbers or in finite fields. So why learn a formula if you can derive it with just a few steps? Moreover, learning the algebra behind the calculations is far more useful than to remember the result.

Then, if you want to solve equations, why exactly quadratic equations? Just because you can solve an equation easily does not mean it is important. Admittedly, I can think of some applications besides elementary geometry. One is acceleration in physics or the ballistic curve in vacuum. But they are far to sparse to support a „midnight formula“.

Moreover, if you want to solve a formula, then start a computer. Install a software like my EMT, and it will print the answer in numerical and algebraic form (with the help of Maxima) in no time, even if the formula looks a bit more complicated than usually.

>&solve(a+x-b^2+a=(a^2+b^2)/x,x) | expand

4              2      2     2
sqrt(b  + (4 - 4 a) b  + 8 a )   b
[x = - ------------------------------ + -- - a,
2                  2
4              2      2     2
sqrt(b  + (4 - 4 a) b  + 8 a )   b
x = ------------------------------ + -- - a]
2                  2

Learning how to compute this by hand and practicing is a waste of time. We have computers to do this.

If I say such things some people understand that math should be abandoned altogether because it can be done by computers. This would be a complete misunderstanding. Besides the fact that you still must be able to do some simple computations in your head or with a pencil, there is more to math than computing.

So what is math all about? If you want to learn about this, but are not mathematically inclined enough to look at some research paper, visit one of the pages of the modern branch of school teachers that try to envision new ways to get students to be interested in math. Dan Meyer’s blog is an example. They start with real world activities. Their path is to have the students ask the questions by themselves if possible. Their conclusion is that math can be made to appear interesting or even necessary to answer the questions.

As a math professor I wished the students would understand the principles that prove the equivalences in the quadratic expansion and the logic behind the square root function, and not just memorize the quadratic formula. It is logic that students fight with the most. And that deficiency affects all branches of mathematics, and it also affects thinking in real life.

So please, teachers, give up on memorizing formulas or practicing boring examples. Have the courage to change the school book approach to something closer to real life and the things that really matter in math nowadays. I know that some school syllabuses have already changed. But the switch is not yet in the brains of most teachers. And it has not yet reached the public opinion about our subject.

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# $t > 2n^2 \implies t!>n^t$ for $n,t \in \mathbb{N}$

I have come across this in a proof:

If $t>2n^2$ then, $$t!>(n^2)^{t-n^2}=n^tn^{t-2n^2}>n^t$$ Obviously, this is much help to determine the relationship between factorials and exponential, but I fail to see the motivation behind the initial assumption.

I don;t get why $(n^2)t-n^2=n^tn^{t-2n^2}$ – Norbert Jun 7 '12 at 19:49
Sorry, typo, fixed to $(n^2)^{t-n^2}$ – zzzzzzzzzzz Jun 7 '12 at 19:51
Suppose $t > 2n^2$, and for simplicity assume $t$ is even. Then $t/2 > n^2$, so all numbers between $t$ and $t/2$ are bigger than $n^2$. And surely the numbers between $t/2 - 1$ and $1$ are all at least $1$. So
$$t! = \left[t (t-1) (t-2) \ldots \left(\frac{t}{2} + 1\right) \frac{t}{2}\right] \cdot \left(\frac{t}{2} - 1\right)! > \left[n^2 n^2 \cdots n^2\right] \cdot 1 > n^t.$$


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### Feature Article

Wow, it has been a while since I last posted. Sorry for the hiatus, but I've been on the road giving talks and observing, all the while writing papers and proposals.

Speaking of writing, last year I was invited to write a cover story for Sky & Telescope about the relationships between planets and their central stars. Different types of stars provide us with different opportunities to learn about planets. For example, stars more massive than the Sun are much more likely than dinky little red dwarfs to harbor a Jupiter-sized planet. This correlation between planet occurrence and stellar mass gives us clues about how planets form. However, if you want to find low-mass planets, the signposts of a planet's presence are much easier to read around M dwarfs (an eclipsing planet dims a small star more than a big star; small planets induce larger accelerations in smaller stars).

All this and more can be found in the April issue of Sky & Telescope. Grab a copy soon!

UPDATE: For some reason I cannot find the article in the online edition of S&T. But the print version should be on newsstands through the end of March (even though it's the April issue).

jcom said…
JohnJohn that is AWESOME, congrats! Can't wait to read the feature article that gets its own cover and everything. Somebody drew a picture of a planet with two moons and a volcano just for you!!!
blissful_e said…
Yea! Congrats on the feature article!

My kids think *our* planet has two moons... they see one, then another as they round the corner of the house. :)

### On the Height of J.J. Barea

Dallas Mavericks point guard J.J. Barea standing between two very tall people (from: Picassa user photoasisphoto).

Congrats to the Dallas Mavericks, who beat the Miami Heat tonight in game six to win the NBA championship.

Okay, with that out of the way, just how tall is the busy-footed Maverick point guard J.J. Barea? He's listed as 6-foot on NBA.com, but no one, not even the sports casters, believes that he can possibly be that tall. He looks like a super-fast Hobbit out there. But could that just be relative scaling, with him standing next to a bunch of extremely tall people? People on Yahoo! Answers think so---I know because I've been Google searching "J.J. Barea Height" for the past 15 minutes.

So I decided to find a photo and settle the issue once and for all.

I then used the basketball as my metric. Wikipedia states that an NBA basketball is 29.5 inches in circumfe…

### The Force is strong with this one...

Last night we were reviewing multiplication tables with Owen. The family fired off doublets of numbers and Owen confidently multiplied away. In the middle of the review Owen stopped and said, "I noticed something. 2 times 2 is 4. If you subtract 1 it's 3. That's equal to taking 2 and adding 1, and then taking 2 and subtracting 1, and multiplying. So 1 times 3 is 2 times 2 minus 1."

I have to admit, that I didn't quite get it at first. I asked him to repeat with another number and he did with six: "6 times 6 is 36. 36 minus 1 is 35. That's the same as 6-1 times 6+1, which is 35."

Ummmmm....wait. Huh? Lemme see...oh. OH! WOW! Owen figured out

x^2 - 1 = (x - 1) (x +1)

So $6 \times 8 = 7 \times 7 - 1 = (7-1) (7+1) = 48$. That's actually pretty handy!

You can see it in the image above. Look at the elements perpendicular to the diagonal. There's 48 bracketing 49, 35 bracketing 36, etc... After a bit more thought we…

### The Long Con

Hiding in Plain Sight

ESPN has a series of sports documentaries called 30 For 30. One of my favorites is called Broke which is about how professional athletes often make tens of millions of dollars in their careers yet retire with nothing. One of the major "leaks" turns out to be con artists, who lure athletes into elaborate real estate schemes or business ventures. This naturally raises the question: In a tightly-knit social structure that is a sports team, how can con artists operate so effectively and extensively? The answer is quite simple: very few people taken in by con artists ever tell anyone what happened. Thus, con artists can operate out in the open with little fear of consequences because they are shielded by the collective silence of their victims.
I can empathize with this. I've lost money in two different con schemes. One was when I was in college, and I received a phone call that I had won an all-expenses-paid trip to the Bahamas. All I needed to do was p…


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Difference between revisions of "2006 AMC 12B Problems/Problem 12"

Problem

The parabola $y=ax^2+bx+c$ has vertex $(p,p)$ and $y$-intercept $(0,-p)$, where $p\ne 0$. What is $b$?

$\text {(A) } -p \qquad \text {(B) } 0 \qquad \text {(C) } 2 \qquad \text {(D) } 4 \qquad \text {(E) } p$

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See also

 2006 AMC 12B (Problems • Answer Key • Resources) Preceded byProblem 11 Followed byProblem 13 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions
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