pattern matching – Extract the powers of exponentials in a list

I have a very complicated and long expression (thousands of terms) that boils down to a form of

expr = a1 Exp[b1] + a2 Exp[b2] + a3 Exp[b3]

Note that all the expressions a and b are huge symbolic expressions, but that this expression is developed in this way.

Now I would like to have a list with b1, b2, b3.
All I can do is

Collection[expr, E^_]

but that does not give me a list.

GOP rejected Obama's executive power, but accepted Trump's position.

GOP rejected Obama's executive power, but accepted Trump's position.

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  1. GOP rejected Obama's executive power, but accepted Trump's position.

    WASHINGTON (AP) President Barack Obama surprised Republicans by bypassing Congress and, relying on what he called his pen and phone, used executive powers to implement his program, protecting millions of young immigrants from deportation.More


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Aggressive Geometry – Variety of commuting matrix $[A,B]= 0 $ – can we geometrically explain the divisibility of $ F_ q $ number of points by the high powers of $ q $?

$ DeclareMathOperator Comm {Comm} DeclareMathOperator Id {Id} $Consider the variety $ Comm $ commuting matrices $[A,B]= $ 0 on a field K $. It is very studied and interesting for various reasons.

We have evidence free action of $ K ^ 2 $: $ A to A + a Id $, $ B to B + b Id $ above.
Hence the number of $ F_q $ points is divisible by $ q ^ 2 $. However actually
it is divisible by a higher power of $ q $: for $ 2 times2 $ dies by $ q ^ 3 $, for $ 3 times3 $ by $ q ^ $ 5, etc., so:

Question Are there any free action of $ K ^ 3 $ sure $ Comm $, or any other geometric explanation of the divisibility above?

A similar superior divisibility seems to be true for the triples, $ n $-tuples of switching matrices, so we have similar questions.


Note 1
Number of $ F_q $ points of $ Comm $ W. Feit, N. Fine, Pairs of commuting matrices on a finite field, 1960.
For each size of matrix $ n $ it is given by polynomial in $ q $ with effective coefficients.

For example, for $ 2 times2 $ the dice, it's $ q ^ 3 (q ^ 3 + q ^ 2-q) $.

For $ 3 times3 $, he is $ q ^ 5 (q ^ 7 + q ^ 2 (q ^ 2-1) (q ^ 3-1) / (q-1) + (q ^ 2-1) (q ^ 3-1)) $

For general $ n $we have a sum of money $ n $, with the summand corresponding to the score $ 1 ^ {b_1} 2 ^ {b_2} cdots $ being $[n]_q! / prod_i [b_i]_q! $.


Note 2
In general, count the equivalence even at $ K ^ n $ does not imply
algebraic equivalence as discussed here: MO300946, MO301249.
Although, in this particular case, there may be a geometric reason.


Note 3
If my grades are correct, triple trips count for $ n = $ 2 is $ q ^ 4 (q ^ 4 + q ^ 3 + q ^ 2-q-1) $, and for quadruples is $ q ^ 5 (q ^ 5 + q ^ 4 + q ^ 3-q-1) $.
There should be good generation functions for shuttles:
MO271752, MO272045.

We can also observe $ (N-1) $ is divisible by $ (q-1) $since we have a free action of $ K ^ * $ sure $ Comm setminus0 $ (exclude null matrices)

Analogous to Euler's guess on the sum of powers

And if instead of considering $$ a_1 ^ 3 + a_2 ^ 3 + a_3 ^ 3 = b_1 ^ 3 \
a_1 ^ 4 + a_2 ^ 4 + a_3 ^ 4 + a_4 ^ 4 = b_1 ^ 4 \
vdots $$

we have rather considered
$$ a_1 ^ 3 + a_2 ^ 3 + a_3 ^ 3 = b_1 ^ 3 + b_2 ^ 3 \
a_1 ^ 4 + a_2 ^ 4 + a_3 ^ 4 + a_4 ^ 4 = b_1 ^ 4 + b_2 ^ 4 + b_3 ^ 4 \
vdots $$
My feeling is that it will have an infinity of solutions.

powers – How can I motivate my players' characters to explore when they are teleported?

If the players are having fun, is this a problem? I think exploration is part of the game and I love doing it, but your players may not want it. Be sure to discuss the group's expectations regarding the type of elements that the campaign should contain.

It is possible that once you report it, they can solve the problem themselves. If they agree, they would like to explore the game, but they do not feel that they have a reason to do so, read on.

Basically, the problem here is that players do not see the interest of do not teleport. It's easier, faster and probably safer. You must find a reason for them to explore. Below, I gave some of the things I would use to do that.

Reward them for it

The easiest way to encourage anyone to do anything is to make them do it. This can be in elements, xp or opportunities. You must inform your players that it is clearly advantageous to explore the world in which they live.

Some rewards you can use:

  • XP: I do not know how the upgrade works in this system, but I've already played in several different systems where we added an XP Direct reward for discovering new locations.
  • Unique objects: some objects can only be obtained through side quests. Side quests can only be found while exploring.
  • NPC Allies: By not exploring, they miss the opportunity to meet people. Create clearly useful NPCs that they can only meet while exploring and they will be encouraged to look for more.

Do not give them a destination

To extend the second point above, perhaps there are whole story quests or arcs that compel them to search for something and that teleporting will not help them.

I do not know how their exact powers work, but many systems say you can not teleport to a place you do not know. Give them new places and destinations regularly and they are forced to explore it for the first time. This makes their teleportation a fast travel tool more than an adventure tool.

Penalize them for not doing it

This is not my favorite approach, but you can try something if they still do not understand.

Do things happen without them? Riots, advertisements of billboards or more subtle things. Whatever the case may be, continue this story even if the players did not come. The player did not cross the city? They were not there to save this bus loaded with orphans. They did not explore this building? They missed the bandit's warning that he was going to murder the president.

Whatever your story, make it clear that they have forgotten something. Communicate them through news broadcasts or advice from their allies.

Drag them

They have a flat no? So they have neighbors? Or do they have loved ones? Anyone for whom the PCs would do any thing? Use these characters to get the PCs to explore. Maybe grandmother needs help to shop? Maybe their neighbor owes a lot of money to someone and that he needs help to do it? Heaps of classic superhero stories start with the heroes who shop every day, so encourage them to go out and do them.

Have fun

This one is more related to your specific group and the type of game that they enjoy. Why are they teleporting always? Is it because they think it's more fun or that it's easier?

Determine what kind of game they want and make sure they have a lot when they are going to explore. Maybe they are the kind of band that just like to complete missions, so give them a lot of mini-quests that they can complete while exploring. Or maybe they just like boss fights and that they can not be bothered by the preparation. Set up car chases and major action sequences that they can only join when they are at the same time.

algorithms – Batch Sizing for convolutional neural networks – powers of 2 or prime numbers?

The conventional wisdom for convolutional neural networks (CNNs) is to make the batch size a power of 2 because of the optimizations performed later in the convolutional layers. The optimizations here relate to computer optimization, in other words the acceleration.

Similar logic applies to Fast Fourier Transformation (FFT). However, the FFT is best used when it comes to a size that is a power of prime numbers (for example, $ 2 ^ M3 ^ N5 ^ P $ or $ M, N, P $ are positive integers.) I wondered if this applied to CNN as well.

Is there a series of powers $ sum_ {n = 1} ^ { infty} a_nx ^ n $ such that $ lim_ {n to infty} frac {a_ {n + 1}} {a_nx ^ n} $ diverge?

The usual examples of the power series $ sum_ {n = 1} ^ { infty} a_nx ^ n $ in the calculation texts always satisfy the fact that "$ lim_ {n to infty} frac {a_ {n + 1}} {a_n} $ converge. "Is there an example of power series such as $ lim_ {n to infty} frac {a_ {n + 1}} {a_n} $ divergent?

pr.probability – expected value of the powers of a Gaussian matrix

Let $ Z $ to be a fixed $ d times d $ matrix and leave $ G $ to be a chance $ d times d $ matrix with each entry i.i.d. N $ (0, 1) $.

Is it true that:

$$ mathrm {Tr} ( mathbb {E} _G[ (Z^T + G^T)^ell (Z + G)^{ell-k-1} Z (Z+G)^k ] ) geq 0 :, $$

or $ ell, k $ are positive integers and $ ell – k geq 1 $?

When $ d = $ 1, this reduces to:

$$
mathbb {E} _g[ (z+g)^{2ell – 1} z ] geq 0 :,
$$

which is easy to verify, since the LHS is of the form $ a_2 z ^ 2 + a_4 z ^ 4 + … + a_ {2 ell} z ^ {2 ell} $ with positive coefficients.

powers – Do quick objects to grab and move (damage) work together?

The damaging modifier on the Move an object Power (page 119) states:

Damage: Your effect can deal damage, such as a Normal Force application with damage equal to its rank. This includes targets that can be injured in seizure and remote attacks.

The text would imply that Move Object is able to "grab" a target:

This includes damaging targets to win

In addition, the DCA version of the Hero Handbook contains six examples of moving objects used to move and grab people, the Wonder Woman's Lasso of Truth being the most important.

the Fast entry Advantage (page 84) indicates:

When you hit with an unarmed attack, you can immediately perform an immediate check against that opponent as a free action (see Capture, page 196). Your unarmed attack deals its normal damage and counts as initial attack check required to seize your opponent.

The question is:

Is a remote telekinetic version of an unarmed strike considered an unarmed strike for the purpose of quick entry?

And would it be different to use Fast Grab during a close attack that required a real unarmed strike?

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