More about Equation Solving
For more about equation solving please refer to another notebook of mine: Intelligence.
There are so many methods and techniques to solve an equation. Here we will review only some of them.
There are many important equations in physics.
The are many methods to solve an ODE,
The idea of Green/s function is very simple. To solve a general solution of equation
where \(f(x)\) is the source and some given boundary conditions. To save ink we define
which takes a function \(y(x)\) to \(f(x)\), i.e.,
Now we define the Green’s function to be the solution of equation (2) but replacing the source with delta function \(\delta (x-z)\)
Why do we define this function? The solution to equation (1) is given by
To verify this conclusion we plug it into the LHS of equation (1)
in which we used one of the properties of Dirac delta distribution
Also note that delta function is even, i.e., \(\delta(-x) = \delta(x)\).
So all we need to do to find the solution to a standard second differential equation
is do the following.
Find the general form of Green’s function (GF) for operator for operator \(\hat L_x\).
Apply boundary condition (BC) to GF. This might be the most tricky part of this method. Any ways, for a BC of the form \(y(a)=0=y(b)\), we can just choose it to vanish at a and b. Otherwise we can move this step to the end when no intuition is coming to our mind.
Continuity at \(n-2\) order of derivatives at point \(x=z\), that is
Discontinuity of the first order derivative at \(x=z\), i.e.,
This condition comes from the fact that the integral of Dirac delta distribution is Heaviside step function.
Solve the coefficients to get the GF.
The solution to an inhomogeneous ODE \(y(x)=f(x)\) is given immediately by
If we haven’t done step 2 we know would have some unkown coefficients which can be determined by the BC.
So we are bound to find Green’s function. Solving a nonhonogeneous equation with delta as source is as easy as solving homogeneous equations.
We do this by demonstrating an example differential equation. The problem we are going to solve is
with boundary condition
For simplicity we define
First of all we find the GF associated with
We just follow the steps.
The general solution to
is given by
Continuity at \(x=z\) for the 0th order derivatives,
which is exactly
Discontinuity condition at 1st order derivatives,
which is
Now we combine ((4)) and ((5)) to eliminate two degrees of freedom. For example, we can solve out \(A_1\) and \(B_1\) as a function of all other coefficients. Here we have
Write down the form solution using \(y(x) = \int G(x,z) f(z) dz\). Then we still have two unknown free coefficients \(A_2\) and \(B_2\), which in fact is to be determined by the BC equation (3).
A second order ODE,
Wronskian of this is
where \(y_1\) and \(y_2\) are linearly independent solutions, i.e., \(c_1 y_1 + c_2 y_2=0\) is only satisfied when \(c_1=c_2=0\). Wronskian is NOT zero if they are linearly independent.
Singularities of an ODE is are defined when \(p(x)\) or \(q(x)\) or both of them have singular points. For example, Legendre equation
has three singular points which are \(z=\pm 1, \infty\) while \(z=0\) is an ordinary point.
Series expansion of the solution can be as simple as
which converges in a radius \(R\) where \(R\) is the distance from \(z=0\) to the nearest singular point of our ODE.
Frobenius series of the solution
The next task is to find the indicial equation.
If the roots are not differing by an integer, we just plug the two solutions to \(\sigma\) in and find two solutions independently.
If the roots differ by an integer, on the other side, we can only plug in the larger root and find one solution. As for the second solution, we need some other techniques, such as Wronskian method and derivative method.
Wronskian method requires two expression of Wronskian, which are
and
From the first expression, we have
However, we don’t know \(W(z)\) at this point. We should apply the second expression of Wronskian,
where the constant \(C\) can be set to 1 as one wish.
TO DO
The derivative method is on my to do list.
For equation that take the following form,
where \(y\equiv y(x)\), we can write down the solutions immediately,
in which \(\mathscr {Z}_p\) is the solution to Bessel equation, i.e., is one kind of Bessel function with index \(p\).
A Pendulum With A Uniformly Chaning String Length
As an example, let’s consider the case of length changing pendulum,
Notice that l is a function of time and
Then the equation can be rewritten as
Comparing with the general form, we have one of the possible solutions
This solution should be
Airy Equatioin
Time-independent Schrödinger equation with a simple potential,
Comparing it with general form, we should set
So the two possible solutions are
The general solution is
Gauss’ equation has the form
which has a solution of the hypergeometric function form
The interesting part about this equation is that its Papperitz symbol is
in which the first three columns are the singularities at points \(0,1,\infty\) while the last column just points out that the argument of this equation is \(z\).
This means, in some sense, the solution to any equation with three singularities can be directly written down by comparing the equation with Gauss’ equation. If you care, the actual steps are changing variables, rewriting the equation into Gauss’ equation form, writing down the solutions.
For differential equation, whenever the highest derivative is multiplied by a small parameter, try this. But generally, the formalism is the following.
First of all, we use Hilbert space \(\mathscr L[a,b;w]\) which means the space is defined on \([a,b]\) with a weight \(w\), i.e.,
Quantum Mechanics Books
Notice that this is very different from the notation we used in most QM books.
What is the catch? Try to write down \(\braket{x}{u}\). It’s not that different because one can alway go back to the QM notation anyway.
With the help of Hilbert space, one can alway write down the vector form of some operators. Suppose we have an equation
where \(\hat L=\hat I + \hat M\). So the solution is simply
However, it’s not a solution until we find the inverse. A most general approach is the Neumann series method. We require that
where \(\gamma\in (0,1)\) and should be independent of \(x\).
As long as this is satisfied, the equation can be solved using Neumann series, which is an iteration method with
As an example, we can solve this equation
We define \(\hat M = \ket{t}\bra{\lambda}\) and check the convergence condition for \(\lambda\).
Step one is always checking condition of convergence.
Step two is to write down the series and zeroth order. Then we reach the key point. The iteration relation is
One can write down \(u_1\) imediately
Keep on going.
For the same example,
where \(\hat L=\hat I + \hat M\), we can solve it using vector space if operator is linear.
Suppose we have a \(\hat M=\ket{a}\bra{b}\), the equation, in some Hilbert space, is
Multiplying through by \(\bra{b}\), we have
which reduces to a linear equation. We only need to solve out \(\braket{b}{u}\) then plug it back into the original equation.
© 2017, Lei Ma. | Created with Sphinx and . | On GitHub | Neutrino Notebook Statistical Mechanics Notebook | Index | Page Source


Wonder Woman
----Princess Diana of Themyscira
Wonder Woman is a fictional superheroine, appearing in American comic books published by DC Comics. The character is a founding member of the Justice League. The character first appeared in All Star Comics #8 in October 1941 with her first feature in Sensation Comics #1, January 1942. The Wonder Woman title has been published by DC Comics almost continuously except for a brief hiatus in 1986. In her homeland, the island nation of Themyscira, her official title is Princess Diana of Themyscira, Daughter of Hippolyta. She has no father; she was created out of clay and brought to life by the gods of Olympus. When blending into the society outside of her homeland, she sometimes adopts her civilian identity Diana Prince.


LITTLE MIX
Little Mix are a British girl group formed in 2011 during the eighth series of the British version of The X Factor. The group consists of Jade Thirlwall, Perrie Edwards and Leigh-Anne Pinnock; former member Jesy Nelson left the group in December 2020 due to mental health issues. The first group to win The X Factor, they signed with Simon Cowell's record label, Syco Music. They released a cover of Damien Rice's "Cannonball" as their winner's single, which debuted atop the UK Singles Chart upon release. The group are known for their strong vocals and signature harmonies, as well as their representative songs about female empowerment and unity.[1]
Being one of the most successful acts to emerge from The X Factor, the group has since earned four more UK number-one singles: "Wings" in 2012, "Black Magic" in 2015, "Shout Out to My Ex" in 2016 and "Sweet Melody" in 2021. Each of their albums, DNA (2012), Salute (2013), Get Weird (2015), Glory Days (2016), LM5 (2018) and Confetti (2020), have peaked in the top five of the UK Albums Chart, with five of the albums certified platinum to 3× platinum by BPI. Notably, their fourth album, Glory Days, became the longest-reigning girl group number one album since the Spice Girls' debut album 20 years earlier, and the highest first week UK album sales for a girl band since 1997.[2] Throughout their career, the group have sold over 60 million records worldwide, making them one of the best-selling girl groups of all time.[3][4][5]
Winning the Brit Award for Song of the Year with "Shout Out to My Ex" in 2017 and the Brit Award for British Video of the Year for "Woman Like Me" in 2019, they have also received six MTV Europe Music Awards, two Teen Choice Awards, five Global Awards and nine Brit Award nominations. Get Weird Tour was the biggest-selling UK arena tour of 2016.[6] As of March 2021, Little Mix have five platinum certified albums and twenty-four certified singles in the UK.[7] In the media, Little Mix has appeared on Debrett's 2017 list of the most influential people in the UK,[8] Forbes' 30 Under 30 list in 2018,[9] on the Sunday Times Young Rich List for four consecutive years, from 2017 to 2020.[10]Read more
Little Mix have recorded material for six studio albums as well as non-album singles.Here is a list of all their recorded albums and songs
Leigh-Anne Pinnock was born on 4 October 1991,[1] and was raised in High Wycombe, Buckinghamshire.[2] Pinnock is of Barbadian and Jamaican ancestry.[3] She is the daughter of Deborah Thornhill, a teacher at the Harlington Community School, and John L. Pinnock, a mechanic, both whom filed for divorce in 2009. She is the youngest of three children and growing up, she looked up to her two older sisters Sian-Louise Pinnock and Sairah Pinnock. Other than her sisters, she also spent most of her time with her grandfather in Jamaica. Pinnock primarily attended Sir William Ramsay School. Then, she enrolled at the Sylvia Young Theatre School for a couple of years when she was eleven. Before joining Little Mix, she worked as a waitress in Pizza Hut
Jade Amelia Thirlwall (born 26 December 1992) is an English singer, songwriter, and a member of the British girl group Little Mix. The group was formed on the eighth series of The X Factor in 2011 and became the first group to win the competition and have since sold over 60 million albums and singles worldwide, making them one of the best-selling girl groups. In 2020 she was nominated at the Ethnicity Awards and won the Equality Award.[1][2]
Jessica Louise Nelson[1] was born on 14 June 1991.[2] She was raised in Romford, East London to parents John Nelson, a businessman, and Janice White, a Police Community Service Officer. Nelson released six albums with the group; DNA (2012), Salute (2013), Get Weird (2015), Glory Days (2016), LM5 (2018) and Confetti (2020).[14] In December 2020, she announced her departure from the group due to prolonged issues with her mental health.
Perrie Louise Edwards[1] was born on 10 July 1993,[2] and raised in the Whiteleas area of South Shields, Tyne and Wear.[3] She is the daughter of Deborah "Debbie" Duffy and Alexander Edwards, who are both singers. Her parents divorced when she was young and have both remarried. She also has an older brother named Jonnie Edwards and a paternal half-sister named Caitlin Edwards. Edwards attended Radipole Primary School in Weymouth, Dorset before moving back to South Shields.[4] She attended St. Peter and Paul RC Primary School, and Mortimer Community College in South Shields,[4] and later transferred to Newcastle College, where she received a BTEC in Performing Arts.[4] Edwards lived in Hamilton, New Zealand for two years as a teenager.
AUTHOR:RABECCA NZAU
Back to top


Donec id elit non mi porta gravida at eget metus.
My email address Jan. 12, 2011 Read 5 books. Borrowed 3 books. Returned 1 book Novice Amateur Senior Enthusiast Expert Leader Veteran Master


Method CallApiMethodAsync
| Improve this Doc View SourceCallApiMethodAsync<TResponse, TResult>(ApiMethod, Func<TResponse, TResult>, CancellationToken, IEnumerable<JsonConverter>)
Asynchronously invokes a method on the Tumblr API, and performs a projection on the response before returning the result.
Declaration
public Task<TResult> CallApiMethodAsync<TResponse, TResult>(ApiMethod method, Func<TResponse, TResult> projection, CancellationToken cancellationToken, IEnumerable<JsonConverter> converters = null)
where TResponse : class where TResult : class
Parameters
| Type | Name | Description |
|---|---|---|
| ApiMethod | method | The ApiMethod to invoke. |
| System.Func<TResponse, TResult> | projection | The projection function that transforms |
| System.Threading.CancellationToken | cancellationToken | A System.Threading.CancellationToken that can be used to cancel the operation. |
| System.Collections.Generic.IEnumerable<Newtonsoft.Json.JsonConverter> | converters | An optional list of JSON converters that will be used while deserializing the response from the Tumblr API. |
Returns
| Type | Description |
|---|---|
| System.Threading.Tasks.Task<TResult> | A System.Threading.Tasks.Task<TResult> that can be used to track the operation. If the task succeeds, the System.Threading.Tasks.Task`1.Result will
carry a |
Type Parameters
| Name | Description |
|---|---|
| TResponse | The type of the response received from the API. This must be a type that can be deserialized from the response JSON. |
| TResult | The actual type that is the result of the method. |
Exceptions
| Type | Condition |
|---|---|
| System.ObjectDisposedException | The object has been disposed. |
| System.ArgumentNullException |
|
CallApiMethodAsync<TResult>(ApiMethod, CancellationToken, IEnumerable<JsonConverter>)
Asynchronously invokes a method on the Tumblr API.
Declaration
public Task<TResult> CallApiMethodAsync<TResult>(ApiMethod method, CancellationToken cancellationToken, IEnumerable<JsonConverter> converters = null)
where TResult : class
Parameters
| Type | Name | Description |
|---|---|---|
| ApiMethod | method | The ApiMethod to invoke. |
| System.Threading.CancellationToken | cancellationToken | A System.Threading.CancellationToken that can be used to cancel the operation. |
| System.Collections.Generic.IEnumerable<Newtonsoft.Json.JsonConverter> | converters | An optional list of JSON converters that will be used while deserializing the response from the Tumblr API. |
Returns
| Type | Description |
|---|---|
| System.Threading.Tasks.Task<TResult> | A System.Threading.Tasks.Task<TResult> that can be used to track the operation. If the task succeeds, the System.Threading.Tasks.Task`1.Result will
carry a |
Type Parameters
| Name | Description |
|---|---|
| TResult | The type of the response received from the API. This must be a type that can be deserialized from the response JSON. |
Exceptions
| Type | Condition |
|---|---|
| System.ObjectDisposedException | The object has been disposed. |
| System.ArgumentNullException |
|


|
Binding of Isaac - Afterbirth+ Lua Reference/Guide
|
To access the debug console, mods must be enabled. If you don't have any mods to enable, download one from Steam Workshop, or create an empty folder in your mods.
To show the console, while in a run press the grave/tilde (`~) key. It should be below Esc on most keyboards.
To hide the console, just hit Enter without typing anything.
Beware of spoilers! You can easily learn things accidentally via the debug console that you might have wanted to learn from regular gameplay!
The debug console is a feature of Isaac that's been around since Rebirth, but until AB+ it's only been enabled in developer builds. Now modders can finally unleash its power! It can be a big time saver for testing your changes during development of mods.
The debug console is not completely stable, so if you enter commands like numbers outside certain ranges, it may crash. Some of the crashes are documented below, but there are likely others.
If you're testing a specific item or item combo, it's useful to enter your commands and then exit and continue so your items are reloaded when you continue and you don't need to keep retyping the same command. Also, try the clipboard! You can paste multiple commands at once or copy the last few commands you entered using the copy command.
Some commands that are still in the game aren't listed here since they're only useful for Nicalis developers, or are no longer useful. For instance, the achievement and eggs commands unlock stuff in your game save, but saving your persistent data is disabled when mods are turned on.
Syntax
Very similar names are incredibly common among entities, so you'll likely need to type the name to figure out the number, then delete it and type the number manually. The decimal syntax lets you specify: Type.Variant.Subtype.Champion. spawn <IP_ADDRESS>
will get you a gaper(10)'s flaming variant(.2) with a blue champion type(.4).
Syntax
s is for special room types, and d is for normal room types. The rooms will be different depending on what floor you're on, so goto d.10 will take you to a different room on stage 1 than it will on stage 3.
Syntax
Move to another floor on the same seed. You can use this to skip ahead to a later floor, go backwards to an earlier floor, or reset the current floor by entering the current floor's id.
Entering nonexistent floors can crash the game.
Syntax
Spawns the grid entity with the given ID at a random place in the room. For some reason it doesn't always do something, possibly because of failing to overwrite other grid entities or trying to spawn outside the room or something. Pits won't look right until you leave the room and come back.
Various cheats that are useful for testing. Some of the ones not listed here are obsolete, not very useful, or possibly broken. Here are the ones that are still useful.
debug 3 Complete invincibility
debug 4 +40 damage. Good with Soy Milk.
debug 7 Display damage values, similar to the Spider Mod item
debug 8 Your active item always has max charge.
debug 9 Your luck is very high, so luck-based random events like random tear effects happen more often.
debug 10 All enemies take rapid damage at all times, so rooms are usually cleared in under a second.
aliases: g
Name collisions are somewhat common among items, so you'll likely need to type the name to figure out the number, then delete it and type the number manually. For items, use "c15" to give the item by number instead of name. For trinkets numbers, use "t15" Notice how it can autocomplete if you don't type the full name. It doesn't work with some odd item names like "<3" so you'll have to figure out the collectible number and use "c15" instead.
Unfortunately you can't type pills or cards by name, so you have to figure out the number. You can give yourself pills even if that pill's effect is not in the rotation. It won't subsequently appear in rotation afterwards.
Syntax
remove *
is a special version of the command that should remove all items.
Syntax
Gives Isaac a number of totally random costumes, without having the associated items.
The same as holding R. Resets the game back to the beginning of a new run.
listcollectibles Prints a list of all items in your possession, and how many of each
Syntax
The commands above will spawn one, then fifty gapers, and then give you one MEAT! and then two hundred more MEAT!s. repeat looks back up the previous console command. The last one that wasn't a repeat command is re-entered as many times as requested. You can use this to give yourself tons of items or spawn tons of enemies. Exercise restraint! Remember that an extreme number of repetitions of some commands might crash the game or leave it hanging for a long time.
Removes all easter egg effects from the current run.
Syntax
It's much easier just to exit to the character select and type the seed the normal way. Easter egg seeds won't work. It's case-sensitive (uppercase only) and doesn't replace "i" with "1," etc.
Syntax
Starts a new run on a random seed on the given challenge number. Numbers greater than the number of challenges included in the base game will cause the game to crash.
Syntax
Gives you an item combo, taking 69 random items from the specified item pool (in this case, 0 is the treasure room item pool).
This command wasn't designed for general consumption, as macros can only be added in C++ code. Sorry! If there's enough demand we can consider adding a similar feature for modder access. It just pastes a bunch of hardcoded console commands one after another using an abbrevition that's quick to type. You can use the clipboard to similar effect.
Syntax
Set a value corresponding to a sum of the desired curses. Darkness (1) + Labyrinth (2) + Lost (4) + Unknown (8) + Cursed (16) + Maze (32) + Blind (64) = 127
The stage or reseed commands are useful after adding curse of the labyrinth to force the floor to be re-generated.
Generates a new layout for the current floor with a new floor seed. This floor will no longer match other people playing the same seed.
Syntax
Copies the last three console commands to your clipboard. Paste one or more commands with ctrl+v, or command+v on mac.
Clears the console. Useful for video recording purposes or if you want to dispose of incriminating evidence.
aliases: l
Syntax
Directly runs a line of Lua code. The first example simply prints "Hello World!" to the console, while the second prints the current HP and max HP of every vulnerable enemy in the room.
Syntax
Takes a path to a lua file and runs it immediately. Paths are relative to the folder where the game is installed.
Syntax
Takes a mod folder name and runs the main.lua for it. This can be used to reload a mod without relaunching the game.
Reports memory usage for Lua