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# How do I define a constant expression with in a class?

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My program was written using GCC and now I''m trying to port it to MSVC++ 6.0 (which is proving to be a big headache). I''ve managed to fix most of the errors, but there is one big problem that VC++ bitches about that GCC doesn''t care about:

class MyClass {

static const int MY_CONST = 1234;

};

I get the errors :
error C2258: illegal pure syntax, must be ''= 0''
error C2252: ''MY_CONST'' : pure specifier can only be specified for functions

This can be fixed by putting the constant in the source file instead of the header file. But there is one case where I can''t move it:

class OtherClass{

static const int MY_CONST; // this is assigned a value in the source file instead of here

int arr[OtherClass::MY_CONST];

};

Now, if I try to compile it,
error C2057: expected constant expression
warning C4200: nonstandard extension used : zero-sized array in struct/union

Is there some way to fix this knot? Thanks for your help.

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class foo
{
enum { bar = 1; }

// ...
};

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First, static and const are redundant, and I believe static can only be used for functions, though I might be wrong. I would recommend whacking the static part of your code and trying again. That should fix the problem. Otherwise, use an enumeration (enum).

Then again, I''m one of the most hated people in all of computer programming...a Borland user. Ahh! Run for your lives!

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Whats''s wrong with Borland?

At least Borland lets you do:

for (int i=sometging; blah; blah)
for (int i=sometging; blah; blah)
for (int i=sometging; blah; blah)

But MS complains about the earlie declaration of i. Bah.

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If you want to use a static member it should be:

const static int MY_CONST = 1234;

not:

static const int My_CONST = 1234;

It should work fine then.

Lektrix
[ Google || ACCU || BarrysWorld || E-Mail Me ]

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quote:
Original post by __tunjin__
class foo
{
enum { bar = 1; }

// ...
};

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quote:
Original post by Racso
First, static and const are redundant, and I believe static can only be used for functions, though I might be wrong.

Actually, you''re wrong on both accounts. Static and const are two completely different things. He wants a single constant used by all instances of the class. If he did just static, then the value would be editable, if he did just const, then there would be an instance of it in every single object! He doesn''t want either of those he wants them both at the same time. Thinking static is only applicable to functions makes me think that you probably don''t understand what static means.

Anyways, the enum version is the best -- while it doesn''t make a constant in memory, it gives the same effect as what you want (unless at some point you want to use the value''s memory address because with an enum it won''t have one). Don''t put extra stuff in memory when you can avoid it. Enumerations are fine in this case.

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In a .h file

class foo
{
public:
static const int CONST;
}

in a .cpp

const int foo::CONST = 1234;

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quote:
Original post by Anonymous Poster
In a .h file

class foo
{
public:
static const int CONST;
}

in a .cpp

const int foo::CONST = 1234;

He knows he can do that, but his point is that he has another variable in the class that is dependent on the value. In this case, what you mentioned will not solve his problem. Using enum is what he should be doing.

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That''s realyl a bit of a language flaw isn''t it?
The enum thing works, but it''s a hack.

"That''s not a bug, it''s a feature!"
--me

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quote:
Original post by Racso
Then again, I''m one of the most hated people in all of computer programming...a Borland user. Ahh! Run for your lives!

Fear not - for you are not the only one. Besides, one should take great pride in the ability to resist Microsoft''s wicked contraptions most of the people seem to be so fond of. Borland rules - too bad they''re not too much of a hurdle for MS to jump over, thought - takes all the fun out of it .

PS - I''m not flaming MS, this is just a personal opinion - me against VC (I am a Windows user, so I can''t really flush that - garn...), if you like...

Crispy

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quote:
Original post by ChaosEngine
That''s realyl a bit of a language flaw isn''t it?

Nope, not at all. It actually makes perfect sense and is not a flaw. The reason it doesn''t work is because the compiler has to know how much data to be allocated for the array. It can''t do this if the constant is defined in another file because the value can be anything.

If he really wanted the same value for the constant he can still make it and set both that value and the length of the array to the same constant.

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quote:
Original post by __tunjin__
class foo
{
enum { bar = 1; }

// ...
};

This works well. Thanks for the suggestion.

quote:
Original post by Lektrix
If you want to use a static member it should be:

const static int MY_CONST = 1234;

not:

static const int My_CONST = 1234;

It should work fine then.

That is weird Is there such a thing as a static const? What is the difference? The const keyword is one that I find to be somewhat elusive because there are so many usages (like why would someone put const at the end of a function or method?).

quote:
Original post by Matt Calabrese
If he really wanted the same value for the constant he can still make it and set both that value and the length of the array to the same constant.

Isn't that the purpose of using constants is to get away from magic numbers all over the place?

quote:
Original post by Matt Calabrese
Anyways, the enum version is the best -- while it doesn't make a constant in memory, it gives the same effect as what you want (unless at some point you want to use the value's memory address because with an enum it won't have one). Don't put extra stuff in memory when you can avoid it. Enumerations are fine in this case.

So are enums similar to a #define statement?

For example:
#define MY_CONST 12enum { MY_CONST = 12 };

Both lines produce the same results?

Thanks for the information.

[edited by - spiffgq on November 18, 2002 9:52:54 PM]

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quote:
Original post by Lektrix
If you want to use a static member it should be:

const static int MY_CONST = 1234;

not:

static const int My_CONST = 1234;

It should work fine then.

There is no difference, it''s just a matter of choice. Both will do the exact same thing.

quote:
Isn''t that the purpose of using constants is to get away from magic numbers all over the place?

That''s also the purpose of enum and #define. The reason you''d use them over constants is because enum and #define don''t take up any space in memory. You can still use them as magic numbers, but theres no point in setting aside a special place in memory for them.

IE. enum Foo { Bar = 1 };

Doesn''t create a constant in memory, but rather, when you use the word "Bar," the compiler replaces it with the value 1 (plus you get added type-safety). However, if you did

const int Bar = 1;

Then, when you''d "Bar," the value would be checked at runtime. So it''s actually [very] slightly slower and takes up extra space in memory that''s not necissary.

Basically, use const for "magic data" when you are dealing objects, not integers, etc. Or when you actually need to pass the memory location of the magic number.

quote:
So are enums similar to a #define statement?

Very similar, but, as I said, there''s also that bonus of type-safety. When you do

enum Foo { Bar = 1 };

You are not only saying that Bar equals 1, but you are also making it of the new datatype "Foo." If you are using it simply as a magic number, then #define might make more sense (for instance if it represents gravity in a game, etc.), however, if it represents something like an enumerated value that represents a subclass, then you are better off using enum. It''s very subjective, but that''s generally the idea.

Reflecting on what we said earlier it really is probably better to use #define instead of enum in this case just because it really is just a "magic number" that''s not of it''s own datatype. Once again, though, it''s really just a choice that you can make and whether you use #define or enum doesn''t really make a difference.

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In short... #define is ''used'' as a text replacement thingy. Try the menu [Edit|Find and Replace] and you''ll know what I mean. enum is a better choice.
quote:

...
IE. enum Foo { Bar = 1 };

Doesn''t create a constant in memory...

Are you *very* sure about this? How does the program ''knows'' or ''remember'' this value in the compiled binary code? Think about it....

hmm... I remember there are few posts about these... anyone could point it out where?

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quote:

Are you *very* sure about this? How does the program ''knows'' or ''remember'' this value in the compiled binary code? Think about it....

Enum does not create an variable in memory. When the source is compiled into binary, the enum is worked out at compiled time and itis replaced by the value.
In other works it ends up acts as if you hard coded the magic numbers in there.

And enum are much more useful in generative programming where you want the compiler to evaluate a expression at compile time.

-----------------------------
Gamedev for learning.
libGDN for putting it all together.

quote:

Yes

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quote:
Original post by DerekSaw
How does the program ''knows'' or ''remember'' this value in the compiled binary code?
The same way as if you''d used #defines - it replaces each use of the constant with the corresponding magic number.

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quote:
Original post by spiffgq
That is weird Is there such a thing as a static const? What is the difference? The const keyword is one that I find to be somewhat elusive because there are so many usages (like why would someone put const at the end of a function or method?).

static and const are completely unrelated. const means that something isn't going to change. Putting const at the end of a function means that the function isn't going to alter the objects data.

This won't work:

    class foo {  public:   void bar(void) {}   foo() {} };int main(void) {  const foo bar;  bar.bar(); // you can't do this because the function might alter the object  return 0; }

This will:

  class foo {  public:   void bar(void) const {}   foo() {} };int main(void) {  const foo bar;  bar.bar(); // It works now. Yay!!!  return 0; }

[edited by - smart_idiot on November 18, 2002 11:11:28 PM]

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Just an interesting note, you can also declare variables to be mutable, in which you can change the variable even if whatever has it is constant.

  class foo {  public:   mutable int bar;   foo(): bar(0) {} };int main(void) {  const foo bar;  bar.bar = 4;   return 0; }

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quote:
Original post by Kurioes
Whats''s wrong with Borland?

At least Borland lets you do:

for (int i=sometging; blah; blah)
for (int i=sometging; blah; blah)
for (int i=sometging; blah; blah)

But MS complains about the earlie declaration of i. Bah.

You can do that with g++ but I don''t really recommend it because can get confusing.

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Yes, variables use memory... so do constants.
and yes again, constants are solved at compile time.
But, where does those ''magic numbers'' stored? Sure they need some reference... and the reference I''m very sure is in the .EXE... and if it''s in the .EXE, then it will used up portion of memory as well. Let''s say:

  #define SOME_DISTINCT_VALUE 0x12345678int main(){  cout << SOME_DISTINCT_VALUE << endl;  return 0;}

Try search for the value 0x12345678 (maybe it is byte sequence: 78 56 34 12) in your .exe file using some hex editor. It does use spaces in your .exe file. If it does use spaces in the .exe file, I''m sure it will use memory as well. If this one is not enough, you could try replacing it with string (for easy searching ). Dare to try it?

p/s: And if you #define it, but doesn''t use ''em... the compiler is smart enough to remove it from your compiled code.

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I had the same problem porting from Visual Studio .net to VC++ 6.

In light of this all I can say is that I agree with the others, using the " enum {...blah...} " syntax is probably your best bet.

Hey, there are about 7 different uses for the ''const'' keyword in C++, if I remember. As if one use isn''t confusing enough...

Anyway, I just wanted to post this for anyone using VS.net and having the same prob.

BTW, VS.net allows you to declare the same var type <name> multiple times:
i.e.
for (int i; blah; blah);
...code...
for (int i; blah; blah);
...code...
for (int i; blah; blah);
.
.
.

...as the Borland compiler does. Microsoft is learning...

-Q

"It appears you are living two lives [Mr. Anderson]."
- Mr. Smith (The Matrix)

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  class foo {  public:   void bar(void) {}   foo() {} };int main(void) {  const foo bar;  bar.bar(); // you can''t do this because the function might alter the object  return 0;}

What is "const foo bar;"? And "bar.bar()", when bar is a member function of foo?

And this:

  class foo {  public:   mutable int bar;   foo(): bar(0) {}};int main(void) {  const foo bar;  bar.bar = 4;  return 0;}

Is the same thing...and that isn''t what mutable does. Mutable is used like this:

  class foo{  int a;  mutable int b;public:  void DoSomething() const  {    a = 1;  //wrong, this function is const, cant modify object    b = 2;  //ok, since b was declared mutable  }};

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Neosmyle: Smart_pointer is correct.
You can obviouly name a variable bar no matter if it has member functions called bar.
Secondly, his use of mutable is also correct. As is yours.
You are both doing the same thing.

DerekSaw: By 'use up memory', we mean space on the stack or the heap. Technically speaking, magic constants are not using up memory - they are part of the program instruction data. Of course you could say that it's using up memory because it appears in the program. But then you might as well say that the assembly instructions are also taking up memory.
quote:

p/s: And if you #define it, but doesn't use 'em... the compiler is smart enough to remove it from your compiled code.

If you don't use it, it would never replace anything. And therefore it would not be used. Similar thinking applies to enum, because it is a compile time constant, it is just compiled into the code - So there is no 'cost' to it.

-----------------------------
Gamedev for learning.
libGDN for putting it all together.

[edited by - risingdragon3 on November 18, 2002 11:59:43 PM]

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# 2009-13 Distances between points in [0,1]^2

Let $$P_1,P_2,\ldots,P_n$$ be n points in {(x,y): 0<x<1, 0<y<1} (n>1). Let $$r_i=\min_{j\neq i} d(P_i,P_j)$$ where d(x,y) means the distance between two points x and y. Prove that $$r_1^2+r_2^2+\cdots+r_n^2\le 4$$.

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text:
# Existence of measures assigning positive values to all open sets

Let $K$ be a compact Hausdorff space. Does there exist a finite Borel measure on $K$, assigning positive values to all non-empty open sets of $K$?

-
Nate's answer here gives a counterexample (the Stone–Čech compactification of an uncountable discrete set). You have uncountably many open singletons which you can't charge all at the same time. –  t.b. Jun 27 '12 at 8:55
At least if $K$ is a topological group, the answer is yes: Haar measure on $K$. –  Xabier Domínguez Jun 27 '12 at 10:05
@XabierDomínguez, does this work if you take an uncountable power of the two-element group with the product topology? –  user16299 Jul 2 '12 at 9:27
@YemonChoi: If I don't seriously misunderstand something here, this is straightforward in this case. The measure you get is an extension of the usual coin-flipping measure. The open sets in the product topology are finite intersections of cylinder sets and $\mu\big(\pi_i^{-1}(U_i)\cap\pi_j^{-1}(U_j)\big)=\mu\big(\pi_i^{-1}(U_i))\mu(\pi_i^{-1}(U_i)\big)$. So an open set can be identified with a finite sequence of coin-flips and they have all positive measure. –  Michael Greinecker Jul 5 '12 at 6:56
@Yemon: Let $U$ be an open null set of a compact group $G$. By compactness we can cover $G$ by finitely many translates of $U$ and deduce that Haar measure is zero. Similarly you can show that no open set in a locally compact group has zero measure: you'd deduce first that Haar measure is zero on compact sets and then deduce that it's zero on every measurable set by tightness. Alternatively use $\tau$-additivity of Haar measure. –  t.b. Jul 8 '12 at 11:40


text:
# group.create command

Syntax

group create s

Create a group. The name s is given to the group. The group has no objects, however.

This command is provided for the infrequent but possible context where having a group name defined, for scripting purposes (FISH or Python), will avoid an error, even if the group itself has no members.

For the purpose of creating groups of model objects, specific commands for that purpose should be favored. See Groups for complete information.


text:
# How football betting odds work using Poisson distribution

Toward the end of the 19th century, Russia-born Polish economist, Ladislaus Bortkiewicz came up with a strategy to predict the incidence of deaths among Prussian soldiers from horse kicks....

Toward the end of the 19th century, Russia-born Polish economist, Ladislaus Bortkiewicz came up with a strategy to predict the incidence of deaths among Prussian soldiers from horse kicks.

And he did this how? He applied the Poisson distribution. It ended becoming a famous example by the way.

Fast forward a bit…

Poisson distribution can be used in many scenarios—importantly and interestingly in betting.

Sports betting is a global phenomenon, and it is estimated that this industry is worth between $700bn and$1tn globally.

And football betting is most popular among all sports.

But how does football betting odds work? how are football betting odds calculated?

It’s difficult to believe that a simple mathematical equation – Poisson distribution is used to calculate the odds for a football match.

Betting on a team winning or losing is done based on the calculation explaining the sports betting across a globe.

### What is ‘Football betting odds’ and how they define bets?

If you’ve ever tried placing a few pounds on your favorite team, you would have noticed these confusing numbers in front of you.

These numbers are called ‘odds’ and they define the probabilities of each possible outcome in an event.

The higher the value of these numbers, the less probable that particular event is.

Take the Real Madrid vs Roma match above as an example.

Since the probability of Madrid winning is higher, the odds against them winning is just 1.40. A draw, which is more unlikely, has odds of 4.75.

And the odds of a highly unlikely Roma win in 7.00.

How do these numbers impact your bet amount and their returns?

Simple. In the above example, if you bet 1 pound on a Real Madrid win and win the bet, you get back a total of 1.40 pounds (inclusive of the 1 pound that you originally bet).

Which is why winning bets on more unlikely events (like a Roma victory) gets you bigger returns.

In this case, you would have got back 7 pounds for every pound you bet on Roma if they ended up winning.

Let us now see how do bookmakers calculate football betting odds using a simple Poisson Distribution equation.

### What is Poisson distribution?

“Poisson distribution is the probability of the number of events that occur in a given interval when the expected number of events is known and the events occur independently of one another.

For instance, suppose you sit in a park for a few days and count the number of people who come to the park in a black T-shirt.

Using Poisson distribution, you can guess if the number of people coming to the park on a specific day in a black t-shirt will be 10, 11, etc.

### But how does it relate to football betting odds prediction?

If you are able to calculate the average attack and defense strength of the teams in a match over a certain period and calculate the Poisson distribution, you will be able to predict the odds of one team performing over other.

But if the data is too long the data would be irrelevant, and if it’s short, outliers might skew the data.

This means that not only external factors like transfers, home, and away from ground affect the odds, but also the duration of events which need to be taken into consideration for calculation.

Let’s see how football betting odds work using Poisson distribution

Before we apply Poisson, we need to get some mathematical figures.

Let’s use this method to calculate the odds for the Manchester United vs. Manchester City matches to be played on February 26, 2017.

First,  we need to find the attack and defense strength of these teams.

#### Calculating Average Attack and Defense for prediction

Before we identify a particular team strength and weakness, we need to find the average strength and weakness of all teams in the last playing season.

This can be calculated by dividing the total goals scored in particular season by a total number of games played in a particular season.

• Number of Goals Scored at Home / Number of Games  = 203/380 = 0.534
• Number of Goals Scored Away / Number of Games  = 158/380 = 0.415

The difference from the values above gives the “Attack intensity” of a team.

We will also need an average of goals conceded to know the weakness, which is simply the inverse of goals scored.

The average number of goals conceded at Home = 0.415

The average number of goals conceded away    = 0.534

This gives us the “ Defense intensity” of a team.

Now that we have the average strength and weakness of the teams, let’s take a look at the stats for Manchester United and Manchester City in 2015.

Based on these stats, we can calculate the Poisson distribution for the teams playing in February 2017, where Manchester United is the away team and Manchester City is the home team.

Predicting Manchester United Goals

Let’s see the possibility of Manchester United scoring at Manchester City’s home ground.

Number of Goals scored away = 22/19 = 1.15

Manchester United Goals = Number of goals scored/Season’s average goals scored away = 1.36/0.415 = 2.77

Manchester City Defense at home

This is calculated by dividing the number of goals conceded at home in the last season by the home team by the number of away games, which is  1.105 ( (21/19).

Manchester City Goals = Number of goals conceded/ Season’s average goals conceded = 1.105/0.415 = 2.66

Manchester United’s Goals = Manchester United’s Attack? Manchester City Defense? Average Number of Goals = 3.05

Predicting Manchester City Goals

Manchester City Attack at Home

Manchester City = Number of Goals scored home – 47/19 = 2.47

Manchester City Goals = Number of goals scored/Average goals scored in the last season =2.47/0.534 = 4.62

Manchester United Defense away

Take the number of goals conceded away last season by the away team and divide by the number of away games, which is equal to  1.05 ((20/19).

Manchester United Goals = Number of goals conceded/Average goals conceded in the last season =1.05/0.534 =1.966

Manchester City’s Goals = Manchester City’s Attack? Manchester United’s Defense? Average No. Goals = 1.10

Once we have the averages of Manchester United’s goals and Manchester City’s goals, we can use them to calculate the Poisson distribution for a number of goals scored by a particular team .  (various goals possibilities).

### Poisson Distribution betting – Predicting multiple match outcomes

The formula for Poisson distribution is:

P(x; ?) = (e-?) (?x) / x!

e = Euler’s constant = 2.718

! = Factorial

x = number of successes of the event

µ = mean distribution of the event

It can be coded as follows

You can also use the Poisson Distribution Calculator for such equations.

### Poisson Distribution prediction for match the between Manchester United and Manchester City

 Goals 0 1 2 3 4 5 Manchester United 4.73% 14.44% 22.02% 22.39% 17.07% 10.41% Manchester City 33.28% 36.61% 20.13% 7.38% 2.03% 0.44% Probability 1.5 5.28 4.43 1.65 0.34 0.23

The possibility that Manchester United and Manchester City would score 1 goal each is 14.44% and 36.61% respectively.

From the above distribution table we can see that the possibility of Manchester city scoring no goal is 4.73 and that of Manchester United is 33.28.

Scoring 2 goals, each for Manchester United and Manchester City is 22.02% and 20.13% respectively.

The possibilities decrease as the number of goals increases to 4 and 5 goals by the individual team.

Taking these number into consideration, the odds of Manchester City winning is high if the number of goals in a match is 0 or 1. But if Manchester United is able to score 2 goals it’s the probability of winning the match increases.

### So how is football betting odds made?

Based on the number by distribution table and the probability of these goals in a match, it is clear that a 1-1 draw has the highest possibility of 5.28, followed by 4.43 for a 2-2 draw. But the possibility of Manchester City winning with 1-0 or 2-1 also looks great.

Based on the previous match at Old Trafford, which is the home ground to Manchester United, Manchester City won by 2-1.

Not taking any sides (I support Algorithm), I would put my bet on 1-0 or 1-1 draw in favor of Manchester City.

The disadvantage of using the Poisson equation is that it doesn’t take into consideration external factors like the players/coach changed in transfer windows, the home ground factor, and injured players.

It helps you calculate the distribution only based on the averages of previous occurrences.

But then Elihu Feustel is one such person who makes a million dollars by betting using mathematical algorithms.


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text:
Properties

 Label 7623.2.a.cl Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7623 = 3^{2} \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7623.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$60.8699614608$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3x^{2} + x + 1$$ x^4 - x^3 - 3*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1 + 1) q^{2} + (\beta_{3} + \beta_1) q^{4} + 2 \beta_{2} q^{5} + q^{7} + ( - \beta_1 - 2) q^{8}+O(q^{10})$$ q + (-b2 - b1 + 1) * q^2 + (b3 + b1) * q^4 + 2*b2 * q^5 + q^7 + (-b1 - 2) * q^8 $$q + ( - \beta_{2} - \beta_1 + 1) q^{2} + (\beta_{3} + \beta_1) q^{4} + 2 \beta_{2} q^{5} + q^{7} + ( - \beta_1 - 2) q^{8} + ( - 2 \beta_1 - 2) q^{10} + ( - 2 \beta_{2} - 2 \beta_1) q^{13} + ( - \beta_{2} - \beta_1 + 1) q^{14} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{16} + (2 \beta_{3} - 4 \beta_1 + 2) q^{17} + (2 \beta_{3} - 2 \beta_1 + 2) q^{19} + (2 \beta_{3} + 4 \beta_1 - 2) q^{20} + ( - 3 \beta_{3} + 4 \beta_1 + 3) q^{23} + ( - 4 \beta_{3} + 4 \beta_{2} + 3) q^{25} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{26} + (\beta_{3} + \beta_1) q^{28} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{31} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{32} + (6 \beta_{3} - 2 \beta_1 + 2) q^{34} + 2 \beta_{2} q^{35} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 3) q^{37} + (4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{38} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{40} + (6 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{41} + (5 \beta_{3} - 6 \beta_1) q^{43} + ( - 7 \beta_{3} - 4 \beta_{2} - \beta_1 + 3) q^{46} + (6 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{47} + q^{49} + ( - 4 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{50} + ( - 2 \beta_{3} - 4 \beta_{2} - 8 \beta_1) q^{52} + ( - 5 \beta_{3} + 2 \beta_1 + 2) q^{53} + ( - \beta_1 - 2) q^{56} + ( - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 6) q^{58} + ( - 2 \beta_{3} - 2 \beta_{2} + 6) q^{59} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 2) q^{61} + (8 \beta_{2} + 8 \beta_1 - 6) q^{62} + ( - 2 \beta_{3} - 5 \beta_{2} - \beta_1 + 1) q^{64} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{65} + (3 \beta_{3} - 2 \beta_1 - 8) q^{67} + (4 \beta_{3} - 6 \beta_{2} - 4 \beta_1 - 2) q^{68} + ( - 2 \beta_1 - 2) q^{70} + (3 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 6) q^{71} + ( - 2 \beta_{2} + 2) q^{73} + ( - 5 \beta_{3} - 3 \beta_1 - 7) q^{74} + (4 \beta_{3} - 2 \beta_{2}) q^{76} + (\beta_{3} + 2 \beta_1 + 4) q^{79} + ( - 4 \beta_{3} + 2 \beta_{2} + 10) q^{80} + (8 \beta_{3} - 2 \beta_{2} - 12 \beta_1 - 6) q^{82} + ( - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 2) q^{83} + ( - 8 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 8) q^{85} + (11 \beta_{3} + \beta_{2} - 4 \beta_1) q^{86} + ( - 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 2) q^{89} + ( - 2 \beta_{2} - 2 \beta_1) q^{91} + (5 \beta_{2} + 8 \beta_1 + 1) q^{92} + (10 \beta_{3} - 4 \beta_1) q^{94} + ( - 4 \beta_{3} + 4 \beta_{2} + 4) q^{95} + ( - 6 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 8) q^{97} + ( - \beta_{2} - \beta_1 + 1) q^{98}+O(q^{100})$$ q + (-b2 - b1 + 1) * q^2 + (b3 + b1) * q^4 + 2*b2 * q^5 + q^7 + (-b1 - 2) * q^8 + (-2*b1 - 2) * q^10 + (-2*b2 - 2*b1) * q^13 + (-b2 - b1 + 1) * q^14 + (-b3 + 3*b2 + b1 - 2) * q^16 + (2*b3 - 4*b1 + 2) * q^17 + (2*b3 - 2*b1 + 2) * q^19 + (2*b3 + 4*b1 - 2) * q^20 + (-3*b3 + 4*b1 + 3) * q^23 + (-4*b3 + 4*b2 + 3) * q^25 + (2*b3 + 2*b2 + 4*b1 + 2) * q^26 + (b3 + b1) * q^28 + (-b3 - 4*b2 + 2*b1 + 2) * q^29 + (-2*b3 + 2*b2 - 2*b1 - 4) * q^31 + (-2*b3 + 2*b2 + 2*b1 - 1) * q^32 + (6*b3 - 2*b1 + 2) * q^34 + 2*b2 * q^35 + (-b3 + 4*b2 + 4*b1 - 3) * q^37 + (4*b3 - 2*b2 - 4*b1 + 2) * q^38 + (-2*b3 - 4*b2 - 2*b1 + 2) * q^40 + (6*b3 + 4*b2 - 2*b1 - 2) * q^41 + (5*b3 - 6*b1) * q^43 + (-7*b3 - 4*b2 - b1 + 3) * q^46 + (6*b3 - 2*b2 - 4*b1 - 2) * q^47 + q^49 + (-4*b3 + b2 + b1 - 1) * q^50 + (-2*b3 - 4*b2 - 8*b1) * q^52 + (-5*b3 + 2*b1 + 2) * q^53 + (-b1 - 2) * q^56 + (-3*b3 - 3*b2 + 2*b1 + 6) * q^58 + (-2*b3 - 2*b2 + 6) * q^59 + (2*b3 - 4*b2 - 2*b1 - 2) * q^61 + (8*b2 + 8*b1 - 6) * q^62 + (-2*b3 - 5*b2 - b1 + 1) * q^64 + (-4*b2 - 4*b1 - 4) * q^65 + (3*b3 - 2*b1 - 8) * q^67 + (4*b3 - 6*b2 - 4*b1 - 2) * q^68 + (-2*b1 - 2) * q^70 + (3*b3 - 4*b2 - 4*b1 + 6) * q^71 + (-2*b2 + 2) * q^73 + (-5*b3 - 3*b1 - 7) * q^74 + (4*b3 - 2*b2) * q^76 + (b3 + 2*b1 + 4) * q^79 + (-4*b3 + 2*b2 + 10) * q^80 + (8*b3 - 2*b2 - 12*b1 - 6) * q^82 + (-2*b3 - 2*b2 + 6*b1 + 2) * q^83 + (-8*b3 + 4*b2 - 4*b1 + 8) * q^85 + (11*b3 + b2 - 4*b1) * q^86 + (-4*b3 + 6*b2 + 4*b1 + 2) * q^89 + (-2*b2 - 2*b1) * q^91 + (5*b2 + 8*b1 + 1) * q^92 + (10*b3 - 4*b1) * q^94 + (-4*b3 + 4*b2 + 4) * q^95 + (-6*b3 + 4*b2 + 4*b1 + 8) * q^97 + (-b2 - b1 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + 3 q^{4} + 4 q^{5} + 4 q^{7} - 9 q^{8}+O(q^{10})$$ 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 4 * q^7 - 9 * q^8 $$4 q + q^{2} + 3 q^{4} + 4 q^{5} + 4 q^{7} - 9 q^{8} - 10 q^{10} - 6 q^{13} + q^{14} - 3 q^{16} + 8 q^{17} + 10 q^{19} + 10 q^{23} + 12 q^{25} + 20 q^{26} + 3 q^{28} - 18 q^{31} - 2 q^{32} + 18 q^{34} + 4 q^{35} - 2 q^{37} + 8 q^{38} - 6 q^{40} + 10 q^{41} + 4 q^{43} - 11 q^{46} - 4 q^{47} + 4 q^{49} - 9 q^{50} - 20 q^{52} - 9 q^{56} + 14 q^{58} + 16 q^{59} - 14 q^{61} - 11 q^{64} - 28 q^{65} - 28 q^{67} - 16 q^{68} - 10 q^{70} + 18 q^{71} + 4 q^{73} - 41 q^{74} + 4 q^{76} + 20 q^{79} + 36 q^{80} - 24 q^{82} + 6 q^{83} + 20 q^{85} + 20 q^{86} + 16 q^{89} - 6 q^{91} + 22 q^{92} + 16 q^{94} + 16 q^{95} + 32 q^{97} + q^{98}+O(q^{100})$$ 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 4 * q^7 - 9 * q^8 - 10 * q^10 - 6 * q^13 + q^14 - 3 * q^16 + 8 * q^17 + 10 * q^19 + 10 * q^23 + 12 * q^25 + 20 * q^26 + 3 * q^28 - 18 * q^31 - 2 * q^32 + 18 * q^34 + 4 * q^35 - 2 * q^37 + 8 * q^38 - 6 * q^40 + 10 * q^41 + 4 * q^43 - 11 * q^46 - 4 * q^47 + 4 * q^49 - 9 * q^50 - 20 * q^52 - 9 * q^56 + 14 * q^58 + 16 * q^59 - 14 * q^61 - 11 * q^64 - 28 * q^65 - 28 * q^67 - 16 * q^68 - 10 * q^70 + 18 * q^71 + 4 * q^73 - 41 * q^74 + 4 * q^76 + 20 * q^79 + 36 * q^80 - 24 * q^82 + 6 * q^83 + 20 * q^85 + 20 * q^86 + 16 * q^89 - 6 * q^91 + 22 * q^92 + 16 * q^94 + 16 * q^95 + 32 * q^97 + q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 3x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 1$$ v^2 - v - 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 2\nu + 1$$ v^3 - v^2 - 2*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 1$$ b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 3\beta_1$$ b3 + b2 + 3*b1

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.09529 −1.35567 0.737640 −0.477260
−2.39026 0 3.71333 2.58993 0 1.00000 −4.09529 0 −6.19059
1.2 0.162147 0 −1.97371 4.38705 0 1.00000 −0.644326 0 0.711349
1.3 1.45589 0 0.119606 −2.38705 0 1.00000 −2.73764 0 −3.47528
1.4 1.77222 0 1.14077 −0.589926 0 1.00000 −1.52274 0 −1.04548
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.cl 4
3.b odd 2 1 2541.2.a.bm 4
11.b odd 2 1 7623.2.a.ci 4
11.d odd 10 2 693.2.m.f 8
33.d even 2 1 2541.2.a.bn 4
33.f even 10 2 231.2.j.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.f 8 33.f even 10 2
693.2.m.f 8 11.d odd 10 2
2541.2.a.bm 4 3.b odd 2 1
2541.2.a.bn 4 33.d even 2 1
7623.2.a.ci 4 11.b odd 2 1
7623.2.a.cl 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + 7T_{2} - 1$$ T2^4 - T2^3 - 5*T2^2 + 7*T2 - 1 $$T_{5}^{4} - 4T_{5}^{3} - 8T_{5}^{2} + 24T_{5} + 16$$ T5^4 - 4*T5^3 - 8*T5^2 + 24*T5 + 16 $$T_{13}^{4} + 6T_{13}^{3} - 8T_{13}^{2} - 16T_{13} + 16$$ T13^4 + 6*T13^3 - 8*T13^2 - 16*T13 + 16

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} - 5 T^{2} + 7 T - 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 4 T^{3} - 8 T^{2} + 24 T + 16$$
$7$ $$(T - 1)^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 6 T^{3} - 8 T^{2} - 16 T + 16$$
$17$ $$T^{4} - 8 T^{3} - 20 T^{2} + 144 T + 304$$
$19$ $$T^{4} - 10 T^{3} + 24 T^{2} - 16$$
$23$ $$T^{4} - 10 T^{3} - 9 T^{2} + 190 T + 109$$
$29$ $$T^{4} - 79T^{2} + 29$$
$31$ $$T^{4} + 18 T^{3} + 68 T^{2} + \cdots - 1744$$
$37$ $$T^{4} + 2 T^{3} - 77 T^{2} - 218 T + 281$$
$41$ $$T^{4} - 10 T^{3} - 104 T^{2} + \cdots - 4496$$
$43$ $$T^{4} - 4 T^{3} - 103 T^{2} + \cdots + 1861$$
$47$ $$T^{4} + 4 T^{3} - 80 T^{2} - 568 T - 976$$
$53$ $$T^{4} - 51 T^{2} + 20 T + 209$$
$59$ $$T^{4} - 16 T^{3} + 72 T^{2} - 104 T + 16$$
$61$ $$T^{4} + 14 T^{3} + 16 T^{2} - 256 T + 16$$
$67$ $$T^{4} + 28 T^{3} + 273 T^{2} + \cdots + 1301$$
$71$ $$T^{4} - 18 T^{3} + 43 T^{2} + \cdots - 179$$
$73$ $$T^{4} - 4 T^{3} - 8 T^{2} + 24 T + 16$$
$79$ $$T^{4} - 20 T^{3} + 129 T^{2} + \cdots + 149$$
$83$ $$T^{4} - 6 T^{3} - 120 T^{2} + \cdots + 2864$$
$89$ $$T^{4} - 16 T^{3} - 48 T^{2} + \cdots - 304$$
$97$ $$T^{4} - 32 T^{3} + 268 T^{2} + \cdots - 8464$$


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## What is the best 2nd level spell to concentrate on to maximize damage?

In a party full of crowd control, the Magus wishes to dish out as much damage as possible. What concentration spell will allow them to do that, under the given circumstance:

• The Magus is 4th level, any race, any class/multiclass, feats available
• The damage is dealt out over 1 minute maximum
• The rest of the party is focused on their own goals, so any preparation or buffs on the Magus will have to come from themselves: (no time limit)
• Assume things go perfectly 100% of the time: ie enemies will always fail on saving throws, the Magus will always succeed on an attack roll, disregard advantage and disadvantage
• The only source of damage tracked comes from the Magus’ concentration spell; no need to worry about ensuring their action is free every round to cast another damaging spell
• Assume all damage is average: 1d6 = 3.5 damage, don’t round
• If a spell can hit multiple targets, there are 2 available each round
• These 2 enemies behave optimally for your damage: if an enemy must end its turn in the spells area to take its damage, then 2 enemies in a round will do so.

## What are the best ways to maximize int score? [closed]

I am playing a level 10 alchemist, and because alchemist needs as much as he can for int (bombs per day, bomb’s damage, craft alchemy, etc.), i want to maximize it. I need the cheapest items i can get, as i almost have no money, as well i need it to well. Any extracts that can do that is great too. As well feats and discoveries.

## how could you maximize food production with magic items and spells over a long period of time?

Over the course of years, how would you appropriately maximize food production for a town or city? If you assume the small city/large town and its very large (100-mile radius) surrounding area can be created with this goal in mind, what spells and magic items could be useful? This should be ruling out spells that are very small individual buffs(Haste, Enlarge/Reduce etc). These options should be more efficient than Plant Growth unless they are lower level/repeatable. Creative options like the Decanter of Endless Water and similar would also be helpful!

I am asking this as a player: the premise is creating a city. Any spells are acceptable, lower magic is preferred(1-5), but higher-level options are also good to know. This question has a broader application than this related question that is only about the plant growth spell’s applicability.

## Does Sword of Sharpness’s maximize damage rule apply to a Rogue’s Sneak Attack damage? [duplicate]

If a rogue hits someone with a scimitar of sharpness and the preconditions for sneak attack are in place, does the rogue also get to maximize the damage of sneak attacks?

For reference the Sword of Sharpness states:

When you attack an object with this magic sword and hit, maximize your weapon damage dice against the target. […]

## Where does it say that a warlock can stack empower and maximize spell like abilities in single round without effecting caster level?

I know that Warlock can maximize and empower spell like abilities but I am having trouble finding in which book/page this is mentioned.

## How does Maximize Spell Like ability feat affect items that boost your damage?

Assume a level 9 warlock has following damage dice:

-5d6 from base EB

-2d6 from Mortalbane (Feat, does not require CL)

-2d6 from Greater Chasuble of Fell Power (Item – Caster level 9th)

-2d6 from gloves of eldritch admixture (Item – Caster Level: 3rd)

Maximize spell like ability feat mentions:

The spell-like ability you wish to maximize can be chosen only from those abilities that duplicate a spell of a level less than or equal to 1/2 your caster level (round down), minus 2.

If this warlock wanted to maximize their attack using all of the dice above, would he have to re-calculate the Maximize Spell Level according to formula [(CL/2) – 2]? If that is the case, would you take the item that requires the highest CL or add up all the spells that requires CL (1 for EB + 9 for Chasuble + 3 for the gloves)

## If a player rolls 33-34 on the wild magic table (maximize damage), what happens if they miss the target on the damaging attack?

On the Wild Magic Sorcerer’s Wild Magic Table, (PHB p.104), getting a roll of 33-34 results in:

Maximize the damage of the next damaging spell you cast within the next minute.

What happens if the sorcerer casts an attack roll spell that deals damage, and misses the target? I am being confused by the term "damaging." Is a "Damaging Spell" a hidden category of spell that can be cast, and thus means casting the spell ends this effect? or does the spell need to apply damage in order to be damaging, and would therefore need to hit for the effect to happen?

## If one target of a Twinned Spell has 0 HP, does the Grave cleric’s Circle of Mortality feature maximize healing on a second target who’s not at 0 HP?

One PC is a multiclassed Grave Domain cleric/sorcerer. They cast a healing spell on a PC that has 0 health, and use the Twinned Spell Metamagic option to target another PC with the same spell.

Would the second PC also have its healing from the spell maximized by the Grave cleric’s Circle of Mortality feature (XGtE, p. 20), even if the second PC is not also at 0 HP?

## Best way to maximize Sneak Attack damage

Unchained Rogue

Sources allowed:

• Core Rulebook
• Ultimate Combat
• Ultimate Magic

I’m asking for how to maximize damage output using Sneak Attack. In theory for all levels, but more specifically at level 1 through 6 or so. Whether it is using feats, magic items, (something I am overlooking?) or a combination of all of that, the objective is maximized damage output.

I don’t really want to multiclass, and want to focus on melee attacks, not ranged attacks. My character will use a shortsword.

I am just looking for maximum damage output, not how to create a ‘sneakable’ condition.

## Maximize an expression with respect to a variable and minimize it with respect to other variables

I started to use Mathematica a few time ago. I want to minimize the following expression (function of $$l,p,q,r,c$$) with respect to variables $$l, p, q, r$$ and then maximize the result obtained with respect to variable $$c$$. However, when I try to obtain an expression function of $$c$$ to maximize later using Minimize, I do not get any result because it takes too long. How can I solve this issue?

``Minimize[{(((l^2/2)*(1-(1/4-c))+(l*p)*(1-1/4)+(l*q)*(1-(1/4+c))+(l*r)*(1-1/2)+(p^2/2)*(1-c)+(p*q)*(1-2*c)+(p*r)*(1-(1/4+c))+(q^2/2)*(1-c)+(q*r)*(1-1/4)+(r^2/2)*(1-(1/4-c)))/((l^2/2)*(1-(1/4-c))+(l*p)*(1-1/4)+(l*q)*(1-(1/4-c))+(l*r)*(1-0)+(p^2/2)*(1-c)+(p*q)*(1-(1/4-c))+(p*r)*(1-0)+(q^2/2)*(1-(1/4-c))+(q*r)*(1-1/4)+(r^2/2)*(1-0))), l >= 1, p >= 0, q >= 0, r >= 0, l + p + q + r == 1000000, 1/7<c<1/5}, {l, p, q, r}] ``


text:
Question 568206
First find the acceleration of the car. We have

*[tex \LARGE a = \frac{80 kph - 44kph}{10 s} = 3.6 \frac{kph}{s}] (kilometers per hour, per second). However we want to convert kilometers per hour to meters per second. 1 kph is roughly .2778 m/s, and it turns out that 3.6 kph/s is exactly 1 m/s^2. So a = 1 m/s^2.

We use F = ma to obtain

*[tex \LARGE F = (1500 kg)(1 \frac{m}{s^2}) = 1500 N]


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# Connected normal space can just be a single point?

Using Urysohn's Lemma, it can be shown that a connected normal space $X$ (with more than one point) is uncountable. But then how can it be that a connected normal space might just be a single point? Is this immediate from Urysohn's Lemma?

-
Trivially?${}{}$ –  Asaf Karagila Nov 11 '12 at 22:31
@AsafKaragila: Come again? –  Libertron Nov 11 '12 at 22:32
There is no way of decomposing a singleton into two non-empty disjoint sets... everything much hold vacuously. –  Asaf Karagila Nov 11 '12 at 22:33
Urysohn makes no (nonvacuous) statement about one-point spaces. But the very definition of normal is trivially verified for one-point space. –  Hagen von Eitzen Nov 11 '12 at 22:33


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General all possible expression trees

This topic is 4595 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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I would like to enumerate/generate all possible expression trees from expressions like "1+2*3-8". My idea goes something like: generateExpressions: for each operator in expression leftExpressionLists = generateExpressions( subexpression-left-of-operator) rightExpressionLists = generateExpressions( subexpression-right-of-operator) combinedExpressionLists = combineAllPermutations( leftExpressionLists, rightExpressionLists ) But, this seems non-trivial for a quiz problem. There must be some trick for a simpler algorithm?

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I think this is the result you are looking for. There are 5 of them in this case. I assume that you want to know how to write a program to produce these?
   *            +              +                  -              - +   -         1  *           1    -            *  8         +    81 2 3 8          2  -            *  8         +  3          1  *                   3 8          2 3          1 2              2 3
Edit: I did post the answer, but I have removed it, as I need you to confirm that this isn't homework of some kind first.

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I think we are speaking of the same straight-forward recursive method that I briefly outlined.

Thus now we are just looking for better ways, which is purely optional though self motivated.

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# Limit of the integral of $\frac{x^n+1}{x^n+2}$

Consider the following integral:

\begin{align} F(x) = \lim_{n\rightarrow\infty}\int f_n(x)dx = \lim_{n\rightarrow\infty}\int\frac{x^n+1}{x^n+2}dx \end{align}

How does one evaluate this integral explicitly? I have considered the uniform convergence criterion for the exchange of limit and integration. If we do this, the convergence of $f_n$ depends on the domain in an interesting way:

My question is then, How can we interpret the indefinite integral? Do we simply define a piecwise continuous function for this while ignoring the two points $\pm$1? or am I way off base here?? A naive capitulation to Wolfram suggests a Hypergeometric function is involved?? I would like to define the following:

What is wrong with this idea? Thx

• Integrate first, then take the limit. The fact that you converge to different limit points for different $x$ values turns out not to matter. – Riemann1337 Apr 10 '14 at 4:04
• It would be better you rewrite your question with a definite integral. – Felix Marin Apr 10 '14 at 4:33

Another approach with the Heaviside function :

Note : Concerning the integral (not at limit), i.e.: $F(X,n)$, in case of $X<0$ and $n$ odd it is understood as the Cauchy principal value.

• Nice way to look at it. Thanks – JEM Apr 10 '14 at 13:12

A different manner to answer to the question is to show a graphical representation to the function $f(x)$ for various values of $n$ and a graphical representation of the respective integrals $F(X)$. ( The old saw that “One little picture says more than a long speech”)

This makes more understandable the behavior of the function and integral for $n$ tending to infinity. Of course the aim of this approach is not to give a formal and due proof. Nevertheless, well understanding the behavior is a valuable help for further attemp to built a formal proof.

• Cauchy principle value sounds like the result is multi valued somehow. Is this the case and if so how is the "principle" value singled out? – JEM Apr 18 '14 at 20:49
• The Cauchy principal value allows an extension of the theory of integration in some particular cases of improper integrals: mathworld.wolfram.com/CauchyPrincipalValue.html It must not be confused with the "Principal Value" of multivalued functions in the field of complex analysis. The confusing terms of "Cauchy Principal Value" and of "Principal Value" are not related. – JJacquelin Apr 18 '14 at 21:46

The indefinite integral is given by

$$\int\frac{x^n+1}{x^n+2}dx = x-\frac{x}{2}{}_2F_1\left(1,\frac{1}{n}, 1+\frac{1}{n}, -\frac{x^n}{2} \right).$$ In the limit, we have $$\lim_{n\rightarrow\infty}{}_2F_1\left(1,\frac{1}{n}, 1+\frac{1}{n}, -\frac{x^n}{2} \right)=1,$$ independent of the value of $x$ (the fourth argument can limit to $0$, $\pm 1$, or $\pm\infty$ and the limit of the hypergeometric function is the same in all cases), so the solution is $$\lim_{n\rightarrow\infty}\int\frac{x^n+1}{x^n+2}dx = \frac{x}{2}.$$

• Yes, I saw this with Mathematica. Took the limit as $n\rightarrow\infty$ and observed the hypergeometric function go to 1 no matter what my $x$ value was. However, I guess I'm looking for a somewhat more Euclidean geometric explanation if possible. Thx – JEM Apr 10 '14 at 4:39
• Ah, don't bump me down then if you agree with the solution. Just wait for another solution that you like and select that as your preferred answer. – Riemann1337 Apr 10 '14 at 4:40
• I didn't bump you Riemann. Btw, integrating first is an elementary result? – JEM Apr 10 '14 at 4:42


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# Capacitors and resistors

What is the energy stored in the capacitor in $$\mu\text{J}?$$

Details and Assumptions

• The capacitor is a parallel plate capacitor.

• The bottom plate of the capacitor is grounded.

×


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## Friday, June 10, 2011

### Adobe Reader's Big Problem

I use PDFs often. Nothing excels them in fidelity of design to documents for which design is important. The Adobe Reader, in its 5.5 days and earlier, was the standard PDF reader of choice. But with version 6, serious bloat occurred, and Adobe Reader is only now, in its latest versions, recovering from that problem. I have a new problem with Adobe Reader, though. Take a look:

This is a document that appears to be made of scanned images placed in text pages, produced by $$\mathrm{\TeX}$$. Note how the scanned images appear as completely useless gray boxes. Here is how the document should look, and does look, in other PDF viewers:

I have searched and searched on this matter, but the results are few and old, and whenever I try to employ a solution I do find, it doesn't work.


text:
## Archive for April, 2012

### What about a Galois Prize for young European mathematicians ?

April 3, 2012

(The following text is a rewritten version, translated into english, of an opinion column that I had submitted elsewhere but which didn’t appear, for some reason which has now appeared.)

I would like to humbly call for the establishment of a European Galois Prize for Outstanding Research in Mathematics by an Undergraduate Student to be awarded yearly by the EMS, similarly to the Morgan Prize of the AMS.  Here is the thinking:

I have noticed, and also have been told, that there are, in the USA, several great programs designed to mentor students interested by doing some research in mathematics, both high-schoolers (see the MIT programs like RSI and PRIMES) and undergraduates (see the list of REU at dozens of universities).

Some particularly talented students sometimes come up with great ideas and papers, and win a prize: there are the Siemens and Intel prize for high-schoolers, and the Frank and Brennie Morgan Prize of the AMS for undergraduates since 1996.  For example the 2012 winner of that latter prize, 21 year old John Pardon (now a PhD student at Stanford), solved a 1983 conjecture of Gromov (a problem he found in a list online while in high school, and solved after a couple years of work) and his paper got published in Annals of Math, no less. Earlier winners are equally talented (Soundararajan, Bhargava, Lurie, Fox, Kane…), and I have noticed that most have attended one of the programs mentionned above.  So indeed, thanks to those programs, there is a regular influx of such talented undergraduates with early research accomplishments.

And in fact, this US trend to get youngsters work on research problems with a mentor even before undertaking a PhD is gathering momentum. For example, an earlier winner of the Morgan Prize, Scott Duke Kominers (who won for works on number theory, and is now an economics professor at Chicago) has just launched with some coworkers a book and a website aimed at providing tips and impetus to such high-schoolers.

Let’s now look at Europe. Unless mistaken, there’s not much happening yet.  A summer school for youngsters will take place for the second time this summer: a great opportunity to learn, but perhaps not quite designed as a mentoring program, and also it is held in english when most european students don’t have good enough fluency to express themselves, let alone think, in that language.   And finally, even if some European student came up with a good paper, there is currently no prize to recognize her/his work and serve as a further incentive for other students.

That’s why I would think both a prize, and several mentoring programs with a good mix of english and the student’s tongue are commendable. Since the name of Galois is not associated with a prize yet, and since it does embody quite well the idea of a talented young european mathematician, that’s a good option I would think.

### Math projects at the 2012 Intel Science Talent Search

April 3, 2012

A few weeks ago, the 2012 crop of the 10 winners of the Intel Science Talent Search have been announced.  As usual, these are great projects in a wide variety of fields (cancer research, robotics, geophysics, ecology, …).

Several math projects were in there too: 18 year old David Ding came 4th with his work on Infinitesimal Cherednik algebras of $\mathfrak{gl}_2$ (done during a PRIMES program at MIT— which seems to be a fantastic setting indeed for insterested high-schoolers — under the supervision of Sasha Tsymbaliuk).

And 17 year old Anirudh Prabhu came 7th for his first non-trivial analytic lower bound of odd perfect numbers (his arxiv papers are here). Apparently he has been taking accelerated courses for a number of years, and did his research on his own without a mentor.

That’s impressive in both cases, and shows that there is not one single recipe for success.

### Functional analysis in the middle ages

April 1, 2012

Exerpt of a 1458 copy of  De Figura seu imagine mundi  by Louis de Langle who first worked out the shapes of the 2-dim unit $\ell^p$ spheres for $p=1, 2, \infty$.


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# 4.Some A Houses regarding Protons, Neutrons, in addition to Electrons

• The number of electrons inside a neutral atom comes to the quantity of protons.
• The range of protons within the nucleus in the atom is the same as your nuclear quantity (Z).
• A party could be the features in the directory line with the occasional kitchen table. A time is all sun and rain from your side to side row.
• Valance electrons would be the electrons inside the outermost level of the Bohr label of a strong atom.
• The amount of protons will not ever transform. Atoms using negative or positive rates simply reveal an increase as well as decrease in electrons.
• Define fischer bulk model (amu).
• Neutrons certainly are a style of subatomic chemical devoid of ask for (these are natural). Similar to protons, neutrons are limited into your atom’s nucleus resulting from this formidable atomic drive.

In get for your atom to stay electronically basic your protons plus electrons need to equilibrium the other. A good atomic large model ($$\text$$) is understood to be one-twelfth the huge of a carbon-12 atom. A recurrent family table directories this atomic fat each element, that you can use to uncover large amount, With regard to hydrogen, by way of example, the actual nuclear excess weight can be A single.008. In any kind of atom, the amount of protons will equivalent how many electrons, only in a basic atom. You’ll should gather basic information about the weather to search for the range of protons, neutrons, in addition to electrons. The location of each and every will be Protons and also Neutrons fundamentally, plus Electrons within the cover **oribits across the core.** Protons plus Neutrons usually are similar throughout amount unless of course the particular atom/element is surely an isotope.

### If a good atom has got 12 protons how many neutrons and electrons should be obtained in them?

Particles use a definite large plus they occupy space or room. Regarding F ree p ( blank ) , the volume of protons (through the recurrent table) is usually Nine as well as the range of electrons will be: The large number is actually a matter in the volume of particles within an atom’s nucleus. It matters not precisely how suitable protons along with neutrons come in the actual atom so that it is a well-balanced framework provided that the phone number will be equal. Within a neutral atom, for example not necessarily ionised, there’s an even range of electrons 12 G, 15 D, in addition to Age 14 Chemical are generally isotopes involving h2o and (Z Means 7) and for that reason contain six to eight protons. The quantity of protons comes to the particular atomic variety.

To look for the number of neutrons in an atom, you have to discover the size number for every part. Protons have a very beneficial electric management of a single $$\left( +1 \right)$$ in addition to a bulk of one atomic bulk device $$\left( \text \right)$$, which can be about \(Just one.67 \times 10^ kgs. The opposite 2 categories tend to be protons as well as neutrons. The actual nuclear variety is the number based in the uppr left spot as well as atomic bodyweight could be the number on the underside, such as this intended for krypton: Atom’s would be the minutest allergens that’s undivided as well as include the really incurred protons in addition to negatively charged electrons. Since contrary electronic expenses appeal to the other, unfavorable electrons are attracted to a optimistic nucleus. Electrons (definitely not valence electrons) will be similar to protons and atomic amount.

• The atomic mass product (amu) is a product of bulk comparable to one-twelfth the particular large of your carbon-12 atom
• Describe the particular locations, fees, along with many a few primary subatomic particles.
• The variety of electrons within the part is the same as the amount of protons.
• Protons certainly are a kind of subatomic compound which has a constructive impose. Protons usually are likely jointly within the atom’s nucleus because of a formidable fischer pressure.
• Atoms are created from protons, neutrons, in addition to electrons.
• Determine the amount of protons and electrons within the atom.

Atom’s would be the littlest contaminants which is undivided in addition to include things like the particular confidently priced protons in addition to badly charged electrons. The number of neutrons equals the number of protons in the ordinary atom, on the other hand. And capability to take energy, swells have got 4 some other attribute attributes: velocity, consistency, wave length, plus amplitude. To get Y * , the number of protons (through the periodic desk) is actually Being unfaithful as well as variety of electrons is definitely: atomic selection(Unces) Implies virtually no. Atomic amount Implies quantity of protons Particles employ a distinct mass and so they enjoy area.

### What selection is the nuclear selection comparable to?

Number involving Protons Is equal to Atomic Variety = 36 The atomic huge of any atom would be the whole size in the electrons, protons, in addition to neutrons in a very motionless atom. Meredith Juncker is really a Expert degree applicant inside Chemical make up in addition to Molecular Biology in Louisiana Declare College or university Wellness Sciences Centre. The intriguing factor we have found in which introducing or maybe removing neutrons from a good atom would not make a distinctive part. This identifies your element in the particular Periodic Stand. Inside evaluate, just inside of a natural atom will probably the volume of protons identical how many electrons.