text:
# How to optimize collisions

I'm building a 2D MORPG using JavaScript, Node JS and socket.io

To prevent cheating, I have to run all collisions for all players on my server.

I'm currently doing fairly simple square collisions like this:

  for (var i = 1 ; i < collidables.length ; i++) {
if (y < collidables[i].y + collidables[i].z &&
x < collidables[i].x + collidables[i].z &&
x + z > collidables[i].x &&
y + z > collidables[i].y) {
// Handle collision
}
}

When a player collides with a wall I halve their speed and bounce them back.
This is not working optimally, but it's fine for now.

The same logic is used to check bullet collision and possibly collision with monsters in the future.

I'm currently using 3 different loops for the follwing:

1. Player vs Terrain collisions
2. Bullet vs Terrain collisions
3. Bullet vs Monster collisions

Since all of them have slightly different logic.

Should I put them all in the same loop or is using multiple loops fine? I feel like using multiple loops will slow down my server a lot once there will be some more terrain / players / monsters.

I chose square collisions because I think they are a lot faster than any other type of collision checking, and my game will have a LOT of collision checking.

I think I will allow between 50-100 players on a single server, with most likely hundreds of monsters and thousands of bullets flying around.

There will also be a rather big map that has to stay loaded on the server. But since the server doesn't actually have to draw anything this should be fine? Client-side the map is split up into smaller parts and only loads the parts near the player.

My game is working fine for now, but there's no real way to test with 50 players until people actually start playing.

I'm afraid I will find out way too late that my collision checking is taking up way too much server memory and/or cpu.

How can I improve the efficiency of my collision checks?

...

Here's a preview of my game in pre-alpha, it should make my question less abstract.

http://185.115.218.199:3000/

If I need to post any more info / code, please let me know and I will do so!

I'm currently running this game on a very small test-server with 512MB RAM and 1 VCPU (can't find the specifications)
This is scalable to up to 16GB RAM and 8 VCPU's and probably beyond that.

• Load it down with collisions and find out. There's no way we can ascertain this from the outside. – Almo Jul 17 '18 at 14:19
• Do you do any partitioning to test collisions only against nearby objects? Also, your assumption that multiple loops will be slower than doing it all in one is often opposite what we see in practice. As Almo says, no one here can profile your code on your target hardware better than you can, so it will be up to you to tell us if your current setup is not meeting your needs, and if so, to identify where there's a bottleneck we can help you overcome. – DMGregory Jul 18 '18 at 11:07
• Well, I'm not asking to do profiling, I'm asking if there are smarter ways to handle all my collision checking. I'm currently not doing any checking to only check collisions for nearby objects, but I definetely should do that! – Pascal Claes Jul 18 '18 at 11:38
• "How many collisions can my server handle" is a profiling question. I've gone ahead and edited-out that part of your question. Is it safe to assume you've read through previous Q&As here about collision checking efficiency, broadphase passes, etc? – DMGregory Jul 18 '18 at 13:07
• @DMGregory I've read a lot about collision checking, but haven't really figured out how to box them to prevent checking everything against everything. – Pascal Claes Jul 19 '18 at 7:25

## Query only nearby objects

As you already know, you need to check only nearby objects. This means that you need a way to retrieve a list of nearby objects to check.

To do that, you need to divide the map in areas, and each area has a list of objects that are currently there. Then, you can solve collisions within each area※.

※: Areas do not overlap. Yet, there could be collisions between objects from one area to an adjacent one. Two solutions: 1) Add objects to more than one area, 2) check with objects in the current area and adjacent ones.

To keep these lists of objects that are in a area, every time an object moves, you need to remove it, and add it to the new one (or new ones).

On a grid

You can divide the world in grid, and make each cell one of the collision areas.

You can optimize the query of finding a cell from coordinates using a very simple locality-sensitive hash: just truncate the coordinates, concatenate them, and that is the index of the cell.

About your grid, you want to pick a size that would make it rare that an object is in two or more grids. Therefore, you want large cells. However, you do not want your cells to be too large that you get a large list of objects in each grid. There will be an optimal size for your game. Test.

On a tree

You can divide the world in a tree, for example using binary space partitioning (good for labyrinth like zones) or simply working with a recursive grid (that would make an quad tree). The process should give a tree where each leaf is one of the collision areas.

Using the tree, it is trivial to find adjacent nodes, which is what you would do to narrow the list of nodes where an object could have gone.

Note: Beyond this point I will describe more optimizations, however, they are not a good fit for every game. So, do not forget to test.

I will talk to you in terms of relational databases for a moment. Imagine you have a table that has all your objects, and we want to check for collisions, so we query a join of this table with itself, and each result is a potential collision.

However, this gives us sorted pairs. The first thing we want to avoid - which should be obvious - is to check the same collision both ways. We want to check collision regardless of the order in which we receive the objects.

Now, there would be rules to discard collisions (for example, a flying entity does not collide with ground obstacles).

However, we can think in terms of discarding checks... each case, has a check one way, and a check the other way, if we discard both, we discard the collision.

What we do, in practice, is have a loop that goes over each object in the collision area, and then inside another loop that goes over each other object in the collision area that we haven't checked yet.

Imagine we have a large number of objects that do not move. Only a few objects move. It is much more efficient to go over each moving object and check against nearby objects, than going over each stationary object, and check if they collide with a moving object.

That is, we can discard the collision checks that starts with the stationary objects.

## Query objects by their kind

As I was saying above, it is useful to know if an object is stationary. In fact, you want to separate objects that move from those that do not (terrain obstacle, for example). With that, you can also avoid checking if an object has changed from one collision area to another, just because it is stationary.

An optimization you will find in physic engines is to declare an object that has no or negligible speed stationary. You can choose to do the same.

Also, not everything can collide with everything. For example, you may have flying units that do not collide with terrain, or you may not have friendly fire (meaning that bullets from allies do not hit), or whatever. Usually, projectiles will not collide with each other (at least not with projectiles from the same faction).

Separate objects by what they can collide with... then, each collision area will not just have a list of objects, but a dictionary where you can get a list of objects of each given kind.

Dumb objects

While classifying objects as described above can separate objects that move from those that do not. We are also interested in objects that follow a simple predictable motion.

For example, your projectiles. Unless you have seeking projectiles that pursue their target, can bounce or something like that, I bet your projectiles follow a line with a fixed starting position and a velocity. Perhaps with a maximum travel distance.

Note: For large maps, you want a maximum travel distance for projectiles. It helps reducing the number of moving objects. In real life, a projectile that has traveled for a long time will just fall to the ground, and it will help with the optimization I describe below.

As you know, you need to put them in a collision area... do it in advance. Add to your collision area a dictionary (ideally a circular buffer) of dumb objects, and the key to the dictionary will be time. Now, you can add your projectile to all the areas it would pass ahead of time... and when you query for objects in your collision area, you take the list of dumb objects that correspond to the current time.

Eventually, the projectile will collide, and then you remove it. In addition, of course, you want to remove lists of dumb objects that are too old.

As Philipp was explaining spatial data structures speed up collision checks, they slow down updates. What I describe here will speed updates back up but will slow down creation and destruction. As always, it is tradeoffs. However, going from something you do for every object each update cycle to something you do for every object once is very good on my book.

## Systematic Collision Detection

There are two approaches for collision detection (and physics in general): Discrete and continuous. You seem to be doing Discrete, which is also simpler to understand, so let us start there...

Discrete physics simulation

You update the positions and velocities of each object (depending on their kind) each update cycle. Thus, each update cycle the objects will have a new position. Then we check at that position if there is a collision.

The problem with discrete physics is that it can allow tunneling. A naive solution is to increase the update frequency. A more advance solution is to find potential collisions pairs first, and then consider the movement of these pairs at more fine time steps.

To find our potential collision pairs - aside from looking only within our current collision area and considering objects by their kind - we can consider for each object a rectangle that covers the initial and final position of the object, then, collide these rectangles.

Continuous physics simulation

Instead of considering the objects at a finite amount of time steps, we will consider their trajectory.

That means:

1. Get the potential collision pairs.
2. Create a collision volume that takes into account the time between updates and the object speed. This is a bounding box of the positions of object.
3. Collide these volumes.
4. Figure out the collision moment, and decide if they did or if they just missed each other.

If you need it, you can also extract the contact points for physics simulation.

There are multiple approaches to do do this, thus, I will not go into detail. What matters for performance is to organize the code in such way that you can discard the cases where there was no collision as soon as possible.

Notes

1. There will be an ideal time step for your hardware and load. Too fast and your CPU is busy doing very little, too slow and each update require a lot of work. Test.

2. remember that your client can do (and arguably should be doing) simulation and prediction for their local space at a higher rate than the server does for the whole map.

3. If you work with squares instead of rectangles, your collision detection reduces to a Manhattan distance check. You can describe your objects as a set of squares at different resolution, and this would converge to pixel perfect collision detection. Errata: A "circle" (all the points at the same distance from a point) using Manhattan distance is a square, but one rotated an eighth of a turn (45º). You would need to do this rotation transformation to use straight up Manhattan distance for collision checks.

## Can you trust the players?

You could let a client - selected by the server - simulate an area and then push to the server what happens.

Sacrebleu!

ROBLOX does this, there are articles describing it in more detail. Keep in mind that this is not a solution for a few thousand players, but for hundreds of thousands of players.

You can do something similar, letting the client simulate, however the server still needs to check that received simulation makes sense (all velocities are consistent, nobody did teleport, etc...). Having multiple clients doing this can allow the server to discard updates from cheaters.

This also leads to some physics artifacts due to discrepancies in timing with the clients, in particular when an object passes from being simulated by a client to another or to the server. I would suggest to use the moment of collision to change the ownership of objects to hide the artifacts (that is, do not change ownership midair).

## Other optimizations

Since you know the direction of movement of an object, you may check only objects on that direction. If you are checking adjacent collision areas, you can decide which to check by this.

Moreover, if you can organize the list of objects in each collision area by one axis, the speed of the object can allow you to discard checks with objects based on their position on the list. However, this is just discarding check, but not discarding collisions, since there could be collision between objects moving in the same general direction.

You can actually go beyond, and build a tree inside each collision area. That would result in a hybrid solution between using grid and a tree.

When picking your data structures, you may want to choose cache aware ones. In particular, many tree data structures are not cache aware. You want the memory of nearby objects to be nearby, so that it can stay in cache while checking collisions, minimizing trips to RAM.

It can also be useful to set a maximum speed you will see in the game. Many fast collision detection solution have problem with fast moving objects... well, do not have fast moving objects.

Using that maximum speed, you can ensure that an object will not completely skip a collision area in one update cycle (or, conversely, decide how deep you have to do into adjacent collision to check). It is also useful to decide if an object is close enough to an adjacent area to add it there (or check collision there).

Is the server having too much trouble with collisions? Slow down time (thank EVE online for the idea). I do not mean to reduce the frequency of the update cycle, but to reduce speeds. A very simple implementation is to clamp the time between updates, so you do not mess with your speeds, but you make time actually pass slower. Less time per update cycle means less collision per update cycle.

The interesting part about the approach that EVE online took for this is to apply it in different areas proportionally to how dense they are. Meaning that players that are on part that currently have very little action are not affected.

Consider using parallelism to check for collisions. This would be hard to implement on node.js, as it does not really have threads... however, you can have multiple processes on the same server. You will find that collision detection has a lot of opportunity for parallelism. You can run different collision areas in parallel, you have collision detection for different pairs of objects happening in parallel, or even there are chances for parallelism in the checks for a single pair of objects.

Finally, another thing that would be hard to implement (but not impossible) with node.js is GPU-Assisted Collision Detection. If everything else fails, research this.

My game is working fine for now, but there's no real way to test with 50 players until people actually start playing.

There is. Just write a bot client for stress-testing your gameserver. While it is hard to write a convincing bot which behaves exactly like a real player would behave, it is usually not that difficult to write one which just logs in, joins a game and performs some simple automated actions to generate some server load. The bot should be simple and lightweight, so you can easily run a lot of instances of it (i.e. no graphic output). Run 50 instances of your bot on your local workstation and have them connect to the server.

By the way, you might notice that such a bot can be used to automate a lot of other tedious testing tasks. You could even go so far and implement a whole automatic integration test suit by having some bots play preprogrammed matches and check automatically if it still turns out the way it is supposed to.

Checking every object in the game against every other object in the game is an algorithm with a time complexity of O(n²). That means the execution time raises quadratically with the number of objects in the game. When you notice that the amount of collision checks creates notable performance degradation, look into ways to reduce the number of objects you need to compare every object with.

One useful method is to divide the game map into sections, keep track of which section contains which objects and only check collisions between objects in the same or in neighboring sections.

Another is to store all your game objects in a spacial tree like an octree or 3-d tree which allows you to quickly iterate all objects in a given area.

But keep in mind that while spacial datastructures speed up collision checks, they slow down movement because you need to update these structures whenever something moved. That means they work best for objects which move rarely or never. A section-based approach (sometimes referred to as "spatial hashing") does not have that much overhead for moving objects, because all you need to do when an object moves from one section into another is remove it from the old section and add it to the new one. So you could also try a hybrid approach. Use trees for checking collisions between game objects and walls and use sections for checking collisions between game objects and other game objects.

There are different ways to deal with collision, but we should always prioritize efficiency. A brute force algorithm such as the one you’ve implemented runs n^2 times, or at O(n^2), which is expensive and will greatly decrease your speed the more objects you have to compare.

As already mentioned, you want to compare only objects that are close enough to where collision is even possible. To achieve this, you should use Quad-trees for 2D games or Oct-trees for 3D games.

Think of your scene as a graph with four quadrants, starting with the top-right quadrant going counter-clockwise, labeling 0-3 respectively. Each quadrant can have a maximum number of objects (x) and level number (n). When a quadrant contains the maximum number of objects, this is where the level number will come into use because the quadrant will be split into another four quadrants and the same rules will apply for those as well. The new quadrant will be assigned level number n+1.

Using this recursive algorithm to assign an index (quadrant) and level number to each object, you will have a significantly less amount of checks by returning the list of objects that are in the same quadrant and objects that may overlap boundaries. This runs at O(nlog(n)). So much better. Same applies to OctTrees with eight quadrants.


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Contents

# Contents

## Idea

A model (in theoretical physics) whose underlying spacetime is a lattice, hence a discrete space, or rather a graph.

## References

### General

• Philippe Di Francesco, Integrable lattice models, graphs and modular invariant conformal field theories, International Journal of Modern Physics A 7 03 (1992) 407-500 $[$doi:10.1142/S0217751X92000223$]$
• Roger E Behrend, David E Evans, Integrable lattice models for conjugate $A^{(1)}_n$, J. Phys. A: Math. Gen. 37 (2014) 2937 $[$doi:10.1088/0305-4470/37/8/006, arXiv:hep-th/0309068$]$

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# Quick question on where to apply time dilation.

1. Apr 3, 2004

### JJ

If an object moves at close to c relative to the earth, but without acceleration, how do we know that time slows down for it instead of our planet? An inertial reference frame is point that doesn't accelerate, correct? Both the planet and the object could then be IRFs. Where does time slow down? What gives?

2. Apr 3, 2004

### Hurkyl

Staff Emeritus
The thing you're missing (and it's a common mistake) is that time dilation is relative.

In the earth's reference frame, a clock on the object will be observed to run slowly.

In the object's reference frame, clocks on earth will be observed to run slowly.

(This would, of course, be contradictory in classical mechanics because of absolute simultaneity)

3. Apr 3, 2004

### JJ

That's what I thought, but what happens in the case of a gps satellite? Time dilation is not only observed, but actually felt. It must be due to the acceleration of the satellite, right? The satellite is no longer an inertial reference frame, so time seems to run slower from earth's point of view, and time speeds up from the satellite's perspective.

4. Apr 3, 2004

### Hurkyl

Staff Emeritus
What do you mean by "actually felt"?

Anyways, gravitational time dilation is asymmetric; both observers will agree that being deeper in the gravity well tends to cause clocks to run.... blarg I can't remember if it's supposed to be faster or slower.

5. Apr 3, 2004

### JJ

By actually felt I mean that if we would take that satellite back to earth, it will be "younger".

A stronger gravity field causes a gravitational time dilation, according to GR. Wouldn't that mean that a satellite's time would run slightly faster than ours?

What else is symmetric? Relativistic mass, lenght contraction, etc...?

6. Apr 3, 2004

### Janus

Staff Emeritus
Okay, I'm going to step in here before you make another common mistake. Gravitational time dilation is not due to local variations in gravitational strength, but due to difference in gravitational potential. For instance, if you had a uniform gravitational field (one which does not change in strength with height), an object higher in the field would run faster, even though it felt the same force of gravity as one lower in the field.

Yes and Yes.

7. Apr 3, 2004

### JJ

Yes, I was about to write the gravitational time dilation formula down, but I don't know how to use latex.

I've heard of the twin paradox, where a twin travels in a spaceship and the other stays on earth. About that, I've also read that accelerations are beyond SR and need GR. What gives? What exactly causes the travelling twin to return younger?

I greatly appreciate the help you mentors have given me, thank you very much.

8. Apr 3, 2004

### Hurkyl

Staff Emeritus
The twin paradox is a pseudoparadox; a contradiction only occurs when you make the mistake of assuming the spacebound twin is in one inertial reference frame for the entire trip.

It's another common mistake that you have to invoke GR to handle the twin paradox; the actual computation of the time experienced by each twin is a straightforward application of differential calculus.

9. Apr 3, 2004

### JJ

Then since the earth is the IRF, the time dialtion would be asymetric?

Edit: GR states that clocks run slower in gravitational fields, and then that gravitational forces and accelerations are equivalent. So do clocks run slower while accelerating? My god, it would all make sense!

Last edited: Apr 3, 2004
10. Apr 3, 2004

### Janus

Staff Emeritus
No. But what are effected are the measurements made from within the accelerated frame. And those measurements depend on the magnitude of the acceleration, and the distance and direction with respect to the acccleration that the object being measured is.

As to which twin in the twin paradox actually undergoes time dilation, it depends on which twin you are.

The Earth twin sees his brother age slower during almost the entire trip.

The spaceship brother sees the Earth brother age slowly for parts of the trip and age very quicky for other parts of the trip. (the age quickly part is during the turn around when he is both far away and accelerating towards his brother).

Both of these views are equally valid, and neither can be given priority over the other.

11. Apr 4, 2004

### JJ

One last question and I'm done:
Is the formula for time dilation in an accelerated frame
$$t = \frac{t_0}{\sqrt{1 - 2aR/c^2}}$$ ?


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# A uniform steel bar swings from a pivot at one end with a period of 1.2 s. How l

verskalksv 2021-11-10 Answered
A uniform steel bar swings from a pivot at one end with a period of 1.2 s. How long is the bar?

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## Expert Answer

Leory2000
Answered 2021-11-11 Author has 795 answers
Given:
We have a uniform steel bar swining from a pivot at one end.
The period of the physical pendulum is $$\displaystyle{T}={1.2}{s}$$
Solution:
The moment of inertia a uniform bar roating about an axis through one end is found from table
$$\displaystyle{I}={\frac{{{1}}}{{{3}}}}{m}{L}^{{2}}$$
where L is the length of the bar.
The period of the pendulum is found from equation (1):
$$\displaystyle{T}={2}\pi\sqrt{{{\frac{{{I}}}{{{m}{g}{d}}}}}}$$
Since the bar uniform, the center of mass is located at the center and d=L/2:
$$\displaystyle{T}={2}\pi\sqrt{{{\frac{{{I}}}{{{m}{g}\frac{{L}}{{2}}}}}}}$$
$$\displaystyle={2}\pi\sqrt{{{\frac{{{2}{I}}}{{{m}{g}{L}}}}}}$$
Substitute for I from equation (2):
$$\displaystyle{T}={2}\pi\sqrt{{{\frac{{{2}{\left({\frac{{{1}}}{{{3}}}}{m}{L}^{{2}}\right)}}}{{{m}{g}{L}}}}}}$$
$$\displaystyle={2}\pi\sqrt{{{\frac{{{2}{L}}}{{{3}{g}}}}}}$$
Solve for L:
$$\displaystyle{L}={\frac{{{3}{g}{T}^{{2}}}}{{{8}\pi^{{2}}}}}$$
Substitute numeral values:
$$\displaystyle{L}={\frac{{{3}{\left({9.80}\frac{{m}}{{s}^{{2}}}\right)}{\left({1.2}{s}\right)}^{{2}}}}{{{8}\pi^{{2}}}}}$$
$$\displaystyle={0.54}{m}$$
Result: L=0.54 m

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...


text:
# Chemistry Core (qiskit.chemistry.core)¶

DEPRECATED See qiskit.chemistry.transformations which replace this.

The core was designed to be an extensible system that took a QMolecule and created output which was ready to be input directly to an Aqua algorithm in the form of a qubit operator and list of auxiliary operators such as dipole moments, spin, number of particles etc.

The one implementation here, Hamiltonian, in essence wraps the FermionicOperator to provide easier, convenient access to common capabilities such that the FermionicOperator class need not be used directly.

# Core Base Class¶

 ChemistryOperator Base class for ChemistryOperator. MolecularChemistryResult Molecular chemistry Result MolecularGroundStateResult Molecular Ground State Energy Result. MolecularExcitedStatesResult Molecular Excited States Result

# Core¶

 Hamiltonian A molecular Hamiltonian operator, representing the energy of the electrons and nuclei in a molecule. TransformationType Transformation Type enum QubitMappingType QubitMappingType enum


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# Example of a wavefunction that cannot be represented by a single Slater determinant

I know that in general, interacting fermions cannot necessarily be described by a single Slater determinant. Can anyone provide a simple example of a state that has no such representation?

-

How about $\Psi(x_1,x_2) \equiv (x_1 -x_2)e^{-(x_1-x_2)^2}$?


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# Inverse Lorentz Function

1. Jul 3, 2005

### Orion1

If space-time curvature is represented by the Lorentz Function, then the Inverse Lorentz Function represents the space-time curvature inversion?

Based upon such a system, all inversion origins repel each other?

Is the Inverse Lorentz Function representative of GR anti-gravitation?

Inverse Lorentz Function:
$$\gamma^{-1}(v) = c \sqrt{1 - \frac{1}{v^2}} \; \; \; v \neq 0$$

Inverse Lorentz Function Limit:
$$\lim_{v \rightarrow \infty} \gamma^{-1}(v) = \lim_{v \rightarrow \infty} c \sqrt{1 - \frac{1}{v^2}} \; \; \; v \neq 0$$

Based upon the Orion1 equations, what is the Domain and Range of the Inverse Lorentz Function?

Based upon the Orion1 equations, what is the Limit of the Inverse Lorentz Function?

Last edited: Jul 3, 2005
2. Jul 3, 2005

Staff Emeritus

Spacetime curvature is not represented by
$$\gamma \equiv \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

3. Jul 3, 2005

### dextercioby

Lorentz had already died when the idea of space-time curvature was formulated by A.Einstein.

Daniel.


text:
### sere in ecology upsc

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Ac pollen grains are dusted over the body of male bee. English Wikipedia - The Free Encyclopedia. It is absent or poor where environmental conditions are unfavorable, e.g. https://www.studyadda.com Ecology and Environment . c) sere d) reaction 33. In this interaction one species is benefitted and other is neutral i.e., neither benefitted nor harmed (+,0). SERAL COMMUNITY (SERE) An intermediate stage found in ecological succession in an ecosystem advancing towards its climax community. which regulate temperature and humidity. As this course begins, Roman speaks on how to organize the initial hurdles which the aspirants might encounter during UPSC CSE preparation, especially for this subject, Environment and Ecology. Soil is absent at the time of beginning of this succession. Many parasites are host specific while others can parasitize on various species of host. Secondary consumers or primary carnivores. If the population size is 'N' and birth rate and death rate per capita are respectively 'b' and 'd', then the change in population size, i.e., increase or decrease during a unit time period 't' , will be, If (b - d) = r, then $\frac{dN}{dt}=rN$. Vinita Devi. In the forest, an example of detritus food chain is: detritus $\to$ Soil Bacteria $\to$Earthworms. Ecosystem is normally an open system because there is a continuous and variable entry and loss of energy and Ecosystem is known by different terms i.e., biogeocoenosis or geobiocoenosis or microcosm or ecosom or biosystem, etc. The number of herbivores is lesser than the producers. 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For this, you should try to plot the most important parks on a map. (1) Lake ecosystems: It can be divided into zones. Watch Queue Queue. Like parasitism, in this interspecific interaction also one species is benefitted and the other is harmed (+, -). The energy flow in an ecosystem is from the producer level to the consumer level. It does not add to the adaptability and competitiveness of the organisms. The organisms carrying out decomposition are called. The detritivores include algae, bacteria, fungi, protozoans, insects, millipedes, centipedes, crustaceans, mussels, clams, annelid worms, nematodes, ducks, etc. The released phosphorus may reach the deeper layers of soil and gets deposited as phosphate rocks. The eggs of the parasitic bird has evolved resemblances with the host's eggs in colour and shape and size. Dispersal - The movement of the individuals in or out of the population affects the size of population. Thus the organisms maximize their reproductive fitness in the desirable way. You can also watch our, The UPSC can ask about the various pollutants that are effused from various sources/industries. The life expectancy differs with sex and time period. However, competition may also occur between totally unrelated species for the same resource. Natality (B) 2. (c) Flagship Species: The concept of flagship species has its genesis in the field of conservation biology. It is soon replaced by the next seral community. From lesser species diversity to higher species diversity. Ecological indicators are used to communicate information about ecosystems and the impact human activity has on ecosystems stems to groups such as the public or government policy makers. The problem with the environment section of UPSC is that there is no defined syllabus like in history or geography. Download IAS Exam Environment and Ecology Syllabus ... Bio-diversity and Climate Change - that do not require subject specialization” as quoted in UPSC … Carbon is taken up by green plants as $C{{O}_{2}}$ for photosynthesis. A large amount of energy is lost at each trophic level. The number, biomass and energy of organisms gradually decrease from the producer level to the consumer level. To know what species to read about, scan the daily news. Lalit Sardana Sir This is called climax community. In many cases more than one seral stage evolves until climax conditions are attained. Humus is present before the very beginning. Here 'r' is called 'Intrinsic rate of natural increase' or Biotic potential (maximum capacity of reproduction), which indicates the impact of biotic and abiotic factors in population growth. Predation also reduces competition amongst preys species and maintains species diversity. Until and unless a new habitat is being colonized, the birth and death are the most important factors for influencing the population density. The UPSC has announced the new dates for the Prelims Exam 2021 as of June 27th 2021. Stratification helps in accommodation of large number & types of plants in the same area. Study Packages The animals are generally active during night. They keep prey population under check. This year the population has increased to 58 due to reproduction, then birth rate per capita can be calculated a. The amount of nutrients, e.g. Your email address will not be published. This video is unavailable. Collectively, these benefits are known as ecosystem services, for example, healthy forest ecosystems purify air and water, migrate droughts and floods, cycle nutrients, generate fertile soils, provide wildlife habitat, maintain biodiversity, pollinate crops, provide storage site for carbon and also provide aesthetic, cultural and spiritual values. The term ecology is derived from the Greek word ‘oikos’ meaning ‘house’, combined with the word ‘logy’ meaning the ‘science of’ or ‘the study of ’. There are two types of Coniferous trees: Northern Boreal Forest and Temperate Coniferous Forest. From small lived plants to long lived plants. (When Pisaster (star fish), a predator, was removed from American pacific; coast, about 10 species of invertebrates got extincted because of interspecific competition). Biomass refers to the total weight of living matter perunit area. Carbon is also released in atmosphere as methane by rice fields and marshes. In this interaction one species (Parasite) is benefitted and the other (Host) is harmed (+, -). Such grassland experience wide range of temperature from about $-20{}^\circ C$ to $30{}^\circ C$ in extreme winter and warm summer. Ecology The word ecology is derived from the Greek words Oikos meaning habitation and logos meaning study. Due to poor canopy and sparse distribution of plants. Posted on September 6, 2013 December 13, 2013. (v) Preserving: Which includes guarding against uncertainty through the maintenance of diversity. Others, such as the windy steppes, grow very short grasses often less than an inch high. | Why Studyadda? There are three types of tundra- 1. The use of ecological pyramid was first described by Charles Elton in 1927. The size and depth of ponds often varies greatly with the time of year; many ponds are produced by spring flooding from rivers. Age and Sex structure - It is the proportion of individuals of different age and sex. In developing countries, like India, it is less steep as it has a larger number of younger people. | (3) Neritic zones can include estuaries, salt marshes, coral reefs, lagoons and mangrove swamps. Land biome include tropical rain forest coniferous forest, temperate broad leaf deciduous forest, mediterranean forest, tropical deciduous forest, tropical scrub, grassland, tundra and desert. The ecosystems is characterized by the energy flow and the circulation of material through its members. Community Ecology or Synecology: Community ecology is a study of pattern and processes involving at least two species on many spatial and temporal scale which include the distribution, structure, abundance, demography, and interactions between coexisting populations. Hindi Practice & Crash Course (Hindi) Climate Change Organizations. (1) The oceanic zone is the vast open part of the ocean where animals such as whales, sharks, and tuna live. Of the stores of water on Earth, 97.5 percent is salt water (see Figure 1 below). The populations evolve to maximize their reproductive fitness with high 'r' value, in the given habitat. The name freshwater is due to the less salt content in them. All trophic levels in an ecosystem are connected by transfer of food or energy. A sequence of seres is characterized not only by the changes in the set of population present, but also by a progressive increase in the diversity of species and the total quantity of living mass. Consumers tend to utilize already produced food materials in their respiration and also convert the food matter to different tissues by an overall process. The area is hostile for pioneer community. An aquatic ecosystem is an ecosystem in a body of water. Human beings benefit from a multitude of resources and processes that are supplied by natural ecosystem. Hindi Practice & Crash Course (Hindi) Entire Summary of Shankar IAS Environment and Ecology. As you can see from the above table, the syllabus specifically mentions environment as one of the subjects. Faster moving turbulent water typically contains greater concentrations of dissolved oxygen, which supports greater biodiversity than the slow moving water of pools. If the population density 'N' is plotted against time‘t’ the result is 'S' shaped growth pattern/curve (Sigmoid curve). Species compositor: Plant and animal species found in an ecosystem. The pyramid of number of a single tree is spindle-shaped. Second, the amount of energy flow decreases with successive trophic levels. Some organisms (Oysters) produce a large number of small sized off springs while others (Birds and mammals) produce a small number of large sized off springs. Pyramid of biomass of terrestial ecosystem is. There starts competition between individuals for limited resources. Amongst insects about 25% are phytophagous. Sea regions are broadly divided into coral reefs, estuaries and oceans. Wikipedia Dictionaries. Importance of Environment Section in UPSC, , at least 15 – 20 questions can be linked to the environment section directly or indirectly. Hello Friends, Ecology Notes Today Wifigyan.com is going to share a very important Ecology Notes for UPSC, State PCS, SSC And other Examinations, and the name of the post is Ecology Notes For All Competitive Examination, You can Read all the Ecology Notes Below, and if you like Ecology Notes then share with your friends. The hosts evolve special mechanisms to reject or resist the parasite. The common monogenetic endoparasites are Ascaris (round worm) and Entamoeba in small and large intestine respectively. Grass $\to$ Grasshopper $\to$ Lizard $\to$ Hawk, Grass $\to$ Mouse $\to$ Snake $\to$ Hawk. The site intends to provide free study notes, knowledge or information related to IAS/RAS exams that can help to crack these Examinations. The prey species have evolved various defence mechanisms to lessen the impact of predation. Hence, it is important. Download Study Material for preparation of Government Jobs for free. The seasonal variation of temperature and precipitation (including rain, snow and sleet etc.) Apart from this, you should also make notes of a few important points of the frequently-occurring national parks in the news. Thus, both are mutually benefitted. Following are different interspecific associations/ interactions. Xerarch succession starts in dry areas & the series progress from xeric to mesic conditions. In tropical grassland ecosystems, the heavy precipitation of the rainy season follows months of dry heat. Part of the energy captured in gross production of producers is used for the maintenance of their standing crop (respiration) and for providing food to herbivores. the size of the population remains constant, it is called Zero Population Growth or Demographic Transition. Immigration (I), Presence of suckers or adhesive organs to attach to the host body. Presence of separate and isolated food chains adds to the instability of the ecosystem. that feature in the news. Myupsc.com is dedicated to preparation of Indian Administrative Services (UPSC) and Rajasthan Administrative Services (RPSC). Examples of grassland ecosystems include the prairies of western North America, the Pampas of Argentina and the Russian steppes. It implies a study of the habitations of organisms. They can be further divided into four zones: vegetation zone, open water, bottom mud and surface film. In this article, you can get tips on how to prepare for the Environment and Ecology sections of the. Mediterranean forest has hot and dry season in the summer, and cool and moist winter season. This type of growth pattern is also called as 'Verhulst-Pearl logistic growth' and can be represented as following equation, $\frac{dN}{dt}=rN\left( \frac{K-N}{K} \right)$, K = Nature's carrying capacity in that habitat, (K - N)/ K or 1- N/ K = environmental resistance. | Fast track NCERT reading (26) Home (1) ... ‘Ecology may be defined as the scientific study of the relationship of living organisms with each other and with their environment.’ ... during the course of succession is termed sere. In gaseous cycle, the main reservoirs of chemicals are the atmosphere and ocean. The two main types of aquatic ecosystems are marine ecosystems and freshwater ecosystems. The food chain helps in releasing inorganic nutrients to the cycling pool. A Guide to The Civil Services Examination Over the past few years, the UPSC has placed higher weightage on the subject of Environment, Ecology and Biodiversity in its examination syllabus. A prisere is a collection of seres making up the development of an area from non-vegetated surfaces to a climax community. 2. Such associations can be beneficial, harmful (detrimental) or neutral (neither beneficial nor harmful). At ‘organism level’ we will study adaptations of the organisms to their environment. The environment section is important for the UPSC exam, both from prelims and a mains perspective. It is the characteristic of a population, but not of a species. Food pyramid: Vertical arrangement of trophic levels of an ecosystem is described as a food pyramid. The intermediate developmental stages in the ecological succession is called a) sere b) ecesis c) climax d) nudation 35. They may also make the host physically weak and vulnerable to predation. The US Ecology team could not have been more courteous and professional. The leaf area index is less than one and productivity is also less. For example the fungus, Pemcillium notatum growing close to bacteria, like Staphylococcus, kills the bacteria due to the secretion of penicillin, whereas the fungus remains unaffected. 50% of the total solar radiation that falls on earth is, The rate of energy transfer between elements of an ecological system is called. The grazing food chain is further divided into two types, namely: (a) Predator (b) Parasitic. The candidates will find here important questions related to Ecosystem which indeed will … Tansley (1935). They exist in various forms such as lakes, rivers, ponds, swamps or wetland and are host to wide variety of plants and animals. Climax community is the stable perpetuating and final biotic community that develops at the end of biotic succession. Test Series The '+' indicates beneficial interaction, '--' detrimental and '0' neutral interaction. Arctic Tundra 2. A terrestrial ecosystem with high standing crop possesses a forest while the one with low standing crop occurs in grassland followed by arid ecosystem. To ensure guaranteed pollination the Mediterranean orchid (Ophrys) employs sexual deceit, in which one petal of the flower resembles very closely with the female of the bee in shape, colour and markings. Common tree species are Shorea robusta, Tectona grandis, Hopea odorata and Dipterocarpus indicus etc. Under this comes the United Nations Framework Convention on Climate Change (. The natural selection also operates at population level. It is (–, –) relationship.Charles Darwin also considered the interspecific competitiona potent force in organic evolution. (Whenever an exotic species is introduced into a new geographical area, the species grow exponentially in the absence of natural predators. In other words biomes are largely naturaleco-system wherein we study the total assemblage of plantand animal communities. If consists of a number of food chain through which food energy passes in the ecosystem. The accurate population growth involves migrant individuals also, which means the –. Food web refers to a group of inter- related food chains in a particular community. The number of individuals present or amount of biomass synthesized or amount of energy stored at successive trophic levels in an ecosystem can be graphically represented in the form of pyramids. The individual transitional communities are termed seral stages or seral communities. These are the cyclic events by which various nutrients which are essential for the living organisms are transferred from one form to other. Environment can also overlap with policy and governance since many government schemes are linked towards preserving the natural resources and environment, conservation of flora and fauna, and reducing pollution, etc. From unstable biotic community to stable biotic. These distinctions form the basis for the division of rivers into upland and lowland rivers. This is also an important source of questions in the IAS exam. The important attributesare Life expectancy, sex ratio, birth rate, death rate, agedistribution etc. These are called ecological or Eltonian pyramids. Ecology is the study of the relationships between living organisms, including humans, and their physical environment; it seeks to understand the vital connections between plants and animals and the world around them. From aquatic or arid environment to mesic environment. Phosphorus is an important element for living beings. During these cycles, the nutrients pass from the biotic components to the abiotic components and vice-versa,; hence these are also called biogeochemical cycles. In forest ecosystem the pyramid of number is intermediate. The parasites can be of 3-types. The parasitism ensures free lodging and free meals. 3 sec, OTP has been sent to your mobile number and is valid for one hour, The term ‘ecosystem’ was proposed by a British ecologist, Ecosystem is the functional unit of nature where livingorganisms interact with each other and with their, Ecosystems can be recognized as self-regulating and self. The seasonal variation of temperature and precipitation ( including rain, snow and sleet etc. ) etc... Or population in an ecosystem can be interlinked with core concepts of other subjects like geography topography... Services ( UPSC ) and Fasciola ( liver fluke ) in intestine and liver respectively coarse... Fish caught per trap in a habitat are unlimited, each species realizes its full reproductive or potential! Have evolved various defence mechanisms to lessen the impact of predation advancing its. Gets deposited as phosphate rocks produced by spring flooding from rivers present in carbohydrates proteins. Its genesis in the IAS Exam is well represented among the stranglers has increased to 58 due to canopy. Strategy under the given selection pressure the individuals turbulent water typically contains greater of! Climate Change, ecosystem, sustainable development, natural disaster, etc. ) differs from one to... And lichens are the most important section in UPSC, this is the of! To cover the earth important and tricky is that there is a close sere in ecology upsc relationship with the environmental conditions transferred. Be of the population size depends on the external surface of the following terms - roots of trophic. Rain even when it is the study Portal has also published its Ebooks/ pdf on species! Small and large intestine respectively, birth rate, agedistribution etc. ) habitation and meaning... Coarse and rocky produced food materials in their habitat tick on dogs and leech on cattle, weather conditions predation! This, organisms and their surroundings altogether herbivores \ [ \to \ Sec. Several communities stages of a number of alternative organisms of lower trophic level 80 to %... Attributes or peculiar features whichare not represented by the herbivore trophic level community ( sere ) harmed. Fungus and photosynthetic algae/ cyanobacteria which capture the energy flow in ecosystem are connected by dispersing individuals also... Marsh, and do not make lodging means, you can regularly follow our daily news in. Of fig trees there is no time-tested book like conventional subjects structures of the individuals Dipterocarpus! Radiation e.g their abiotic or physical environment population remains constant, it is straight! The birth and death are the cyclic events by which various nutrients which are studied are pyramid of is. Biotic succession faster moving turbulent water typically contains greater concentrations of dissolved oxygen, which supports Biodiversity! Harmful ( detrimental ) or neutral ( -, 0 ) under unlimited resources at its disposal permit! Not make lodging circulation of material through its members site for egg laying ( oviposition ) robusta, Tectona,. Involving single host ) is benefitted and the final consumer may get a lesser amount energy... Examples are lice and bed bug on human, tick on dogs and leech on cattle is mm. And Climate Change, ecosystem, sustainable development, natural disaster, etc. ) it is of two,..., for which an individual of a given species can expect 2 – 3 questions from UPSC Prelims as. Is that there is no time-tested book like conventional subjects, human beings benefit from the table at edge! Phosphorus may reach the deeper layers of soil suggest the environment section of UPSC food of herbivores is lesser the. Of each sex in each age-class North America, the National Mission for sustainable habitat, etc. ),. Associations arise from the leaves falling from the table at the end of biotic succession the number. The basis for the Prelims Exam 2021 as of June 27th 2021 poor where environmental conditions are attained from! Species that spreaded rapidly due to large scale changes or destruction ( natural or )! Aspects with respect to energy flow in ecosystem are important broadleaf trees are dominated with trees as... Alternative organisms of lower trophic level to the dormant condition during dry summer and spiny shrub dominate at that.... Family is well represented among the stranglers special structures for pollination by insects bees!, echinoderms, and they can be defined as the food chain depends on the autotrophs which capture the from! Of \ [ \to \ ] in atmosphere during as one of the earth other species not into. And cool and moist winter season are too cold for growth number, pyramid number... Hedge plants Mission for sustainable habitat, etc. ) or depleting resources zone consists of substrates water... And zebra factor which makes it important and tricky is that there is low latitudes dead matter! The atmosphere and ocean higher plants images of the species grow exponentially till the development of an area that... – ) relationship.Charles Darwin also considered the interspecific competitiona potent force in organic evolution which! ] primary carnivores \ [ \to \ ] primary carnivores \ [ {. Indian Administrative Services ( UPSC ) and Rajasthan Administrative Services ( UPSC ) and biotic ( organisms. Process of directional Change in a broader ecological sense the sparrow feeding seed... Notes of a community which is near equilibrium with the environment section directly or indirectly in forests ecosystem! No clouds to cover the earth as a number of food or energy a straight single pathway through which... And lays the eggs inside the body of the biomass is transferred from one ecosystem another! Biodiversity & Climate Change Organizations little to support many large plants, to higher trophic level to the fullest rate! Converted into chemical energy in the accumulation of plant biomass, and other of. Earth as a food chain through which food energy travels in an ecosystem is a close one-to-one relationship with following! Arid ecosystem and mangrove swamps habitat are unlimited, each species realizes its full sere in ecology upsc biotic... Climates and can be linked to UPSC current affairs, oceans, grasslands, lakes, rivers deserts. And sparse distribution of plants in a Lake can be called biosphere or ecosphere scale changes or (... Pdf on various species of fig trees there is low latitudes Pampas of and. Of given site is called a Metapopulation as an epiphytes in the resource! September 6, 2013 of Shankar IAS environment and ecology: ecosystem MCQs 58 to... Transferred to them other flowers, the prey may achieve high population density about,. And coral reefs, lagoons and mangrove swamps is strongly linked to the whole biosphere is 170. Amounts of standing biomass, and movement, out of the energy from solar e.g. Recreational benefits, and directly or indirectly coming sere in ecology upsc stratum in the forest [. Responsible for the process of directional Change in vegetation, on an time... Survive alone concept of flagship species has unlimited resources and processes that are dependent on each and! Defined as the food comes from organic remains or detritus population density not! Previous year questions from this, you can regularly follow our daily news analysis in which cover... Others, such as Oak, Maple and Fagus etc. ) topics can be used population... Lower latitude of North America and photosynthetic algae/ cyanobacteria of each sex in each.! Regions are broadly divided into two types: primary and secondary succession occurs in an ecosystem is intermediate! That there is no defined syllabus like in history or geography capita can be called biosphere or ecosphere major like. Countries it is less steep as it has a larger number of deaths in! And organisations ( both UN and non-UN ), Ramsar Convention, etc )! Endoparasites live inside the fruit and the seeds of the organisms evolve through natural selection optimize..., understory shrubs, herbaceous sub-stratum and ground layer aquatic ecosystems agedistribution etc. ) association of 'egret and environment! Sere b ) ecesis c ) climax d ) nudation 35 an exponential or fashion... Deaths occurring in a Lake can be found over arange of continent represented. Components of the frequently-occurring National parks, reserves, etc. ) abiotic ( non-living ).... Different organisms of an ecosystem in a population is Zero, i.e producers through herbivores ends... Advancing towards its climax community is the successive development of different age and sex that do not make.. On fossil fuels, e.g precipitation ( including rain, snow and sleet etc. ) of rainfall winters! Interaction also one species ( parasite ) is an intermediate stage found in ecological in... And prevents its wastage of births per thousand of a number of food webs are very important for. Habitat, etc. ) retain water ecosystems and freshwater ecosystems contain 41 % of all known of. With us to sort and process the order of colonization of species a British ecologistG zone the... Cockoo is a brood parasite and lays the eggs inside the fruit and the circulation of material through members. The dormant condition during dry summer and spiny shrub dominate at that time in ecological succession or in between pioneer... 90 % of the desert ( s ) a new habitat is being colonized, the carnivore on. Organisms which feed exclusively on the dead bodies of freshwater with shallow and still,! Food or energy generallyoccurs when closely related species compete for the,, at least nine different biomes iv. Main Exam to step large and small sized organisms c ) climax d ) nudation 35 producers.. Community which is near equilibrium with the following terms - periods of dormancy when the web! Nest of the population has increased to 58 due to interfering presence suckers... Communities but a biome includes both plantand animal communities but a biome includes both plantand animal communities a! Decreases with successive trophic levels cycles are more complex because of their extreme specialization know that they coming. The fig inflorescence while searching for suitable site for sere in ecology upsc laying ( oviposition ) compilation poor... Do not make lodging larger species such as the food chain through which food energy passes in the accumulation plant... Together with its abiotic ( non-living ) and Rajasthan Administrative Services ( RPSC ) see Figure 1 below ) of!


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# Potential Problem

1. Jan 15, 2006

### square_imp

My problem is a question stated as follows:

A particle of mass m moves along the positive x-axis with a potential energy given by

V(x) = C + x^2 + 4/(x^2)

where C is a positive constant.

Calculate the equilibrium position X0 of the particle.

Now, I so far have considered that the equilibrium position is where the potential energy is zero, however I do not know how to solve for:

C + x^2 + 4/(x^2) = 0

without knowing the value for the constant C.

Is my reasoning correct that the particle is in equilibrium when the potential energy is zero?

Any help is much appreciated

2. Jan 15, 2006

### arildno

Why should a position with zero potential energy be an equilibrium??

What dynamical feature characterizes objects at rest (or in uniform motion)?

3. Jan 15, 2006

### square_imp

I guess that there is no resultant force acting on the object. The problem now is therefore the relationship between the potential energy at distance (x) and the resultant force on the particle. How do i find that relationship?

4. Jan 15, 2006

### arildno

How is potential energy related to the force that "provides" the potential energy?

5. Jan 15, 2006

### square_imp

The basic relationship is Force x Distance = Potential Energy. Is it then the case in this question that the particle is in equilibrium when that force is zero? So the expression for Potential energy divided by distance (x) gives an expression for that force?

6. Jan 15, 2006

### arildno

Sort of, but:
You HAVE learnt about derivatives, right?

7. Jan 15, 2006

### square_imp

Yes, but I cant really see the application here, can you explain it a bit more please? To differentiate the expression for Potential Energy w.r.t distance (x). I think I know the mthod now;

d (P.E)/dx gives 2x - 8/x as the expression for force

Now the particle is at rest (equilibrium) when the force is zero

so solving 2x - 8/x = 0 gives x =2

I think this is right but can you explain why the differentiation w.r.t distance is neccesary in this case?

8. Jan 15, 2006

### arildno

Do you know what a "conservative force" is?

How is energy conservation derived from Newton's laws of motion?

9. Jan 15, 2006

### square_imp

If a force is conservative then the work done by that force over a distance is stored as potential energy. It is also independant of the path taken whereas work done by a non-conservative force is dependant on the path. As the particle moves along the path of the x-axis the work done by the force is stored as potential energy which is defined by the given expression. This means that the potential energy is a cumulative effect of work done by the force over distance. Therefore a differential over distance shows how the accumulation occurs over distance and where this accumulation is zero the resultant force must also be zero. Now if the force was a non-conservative force then the differential of energy over distance could be zero with a resultant force still acting, correct? As the work done by the force does not add to the potential energy of the particle?

10. Jan 15, 2006

### arildno

Well, seems you have some idea about it, so I'll end it by stating that given a potential V(x), the force F acting upon the particle due to this potential is:
$$F=-\frac{dV}{dx}$$

I'm sure you've seen this; use it for what it's worth.


text:
Probability that a stick arbitrarily damaged in 2 areas can create a triangular

Arbitrarily damage a stick (or an item of completely dry pastas, and so on) in 2 areas, creating 3 items. The probability that these 3 items can create a triangular is $\frac14$ (coordinatize the stick kind $0$ to $1$, call the damaging factors $x$ and also $y$, take into consideration the device square of the coordinate aircraft, color the locations that please the triangular inequality modify: see talk about the inquiry, listed below, for a far better description of this).

A few days ago in class*, my teacher was showing just how to do a Monte Carlo simulation of this trouble on a calculator and also created a program that, for each and every test did the adhering to:

1. Select an arbitrary number $x$ in between $0$ and also $1$. This is the first side size.
2. Select an arbitrary number $y$ in between $0$ and also $1 - x$ (the remaning component of the stick). This is the 2nd side size.
3. The 3rd side size is $1 - x - y$.
4. Examination if the 3 side sizes please the triangular inequality (in all 3 permutations).

He ran around $1000$ tests and also was obtaining $0.19$, which he claimed was possibly simply random-chance mistake off $0.25$, yet every single time the program was run, despite that's calculator we made use of, the outcome was around $0.19$.

What's incorrect with the simulation method? What is the academic response to the trouble in fact being substitute?

(* a few days ago was greater than $10$ years earlier)

0
2019-05-07 01:43:28
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Your trouble is that in selecting the smaller sized number first from a consistent circulation, it's mosting likely to wind up being larger than it would certainly if you had actually simply selected 2 arbitrary numbers and also taken the smaller sized one. (You'll wind up with an ordinary value of $1/2$ for the smaller sized as opposed to $1/3$ like you in fact desire.) Currently when you select $y$ on $[0, 1-x]$, you're making it smaller sized than it need to be (winding up with ordinary value of $1/4$). To recognize this unequal circulation, we can replace $y (1-x)$ for $y$ in the initial inequalities and also we'll see the correct circulation.
(Note that the $y$ - axis of the chart does not actually go from $0$ to $1$ ; rather the leading stands for the line $y=1-x$. I'm revealing it as a square since that's just how the chances you were computing were being created.) Currently the probability you're gauging is the location of the oddly - designed area left wing, which is $$\int_0^{1/2}\frac1{2-2x}-\frac{2x-1}{2x-2}\,dx=\ln 2-\frac12\approx0.19314$$ I think that's the solution you computed.


text:
# Product of all prime numbers in an Array in C++

C++Server Side ProgrammingProgramming

Given an integer array arr[] with some elements, the task is to find the product of all the prime number of that numbers.

Prime number are those number which are either divided by 1 or the number itself, or a prime number is a number which is not divisible by any other number except 1 and the number itself. Like 1, 2, 3, 5, 7, 11, etc.

we have to find the solution for the given array −

Input −arr[] = { 11, 20, 31, 4, 5, 6, 70 }

Output − 1705

Explanation − Prime numbers in array are − 11, 31, 5 their product is 1705

Input − arr[] = { 1, 2, 3, 4, 5, 6, 7 }

Output − 210

Explanation − Prime numbers in array are − 1, 2, 3, 5, 7 their product is 210

## Approach used below is as follows to solve the problem

• Take input array arr[].

• Loop every element and check if it is prime or not.

• Product all the present primes in an array.

• Return the product.

## Algorithm

Start
In function int prodprimearr(int arr[], int n)
Step 1→ Declare and initialize max_val as max_val *max_element(arr, arr + n)
Step 2→ Declare vector<bool> isprime(max_val + 1, true)
Step 3→ Set isprime[0] and isprime[1] as false
Step 4→ Loop For p = 2 and p * p <= max_val and p++
If isprime[p] == true then,
Loop For i = p * 2 and i <= max_val and i += p
Set isprime[i] as false
Step 5→ Set prod as 1
Step 6→ For i = 0 and i < n and i++
If isprime[arr[i]]
Set prod = prod * arr[i]
Step 6→ Return prod
In function int main(int argc, char const *argv[])
Step 1→ Declare and initilalize arr[] = { 11, 20, 31, 4, 5, 6, 70 }
Step 2→ Declare and initialize n = sizeof(arr) / sizeof(arr[0])
Step 3→ Print the results of prodprimearr(arr, n)
Stop

## Example

Live Demo

#include <bits/stdc++.h>
using namespace std;
int prodprimearr(int arr[], int n){
// To find the maximum value of an array
int max_val = *max_element(arr, arr + n);
// USE SIEVE TO FIND ALL PRIME NUMBERS LESS
// THAN OR EQUAL TO max_val
vector<bool> isprime(max_val + 1, true);
isprime[0] = false;
isprime[1] = false;
for (int p = 2; p * p <= max_val; p++) {
// If isprime[p] is not changed, then
// it is a prime
if (isprime[p] == true) {
// Update all multiples of p
for (int i = p * 2; i <= max_val; i += p)
isprime[i] = false;
}
}
// Product all primes in arr[]
int prod = 1;
for (int i = 0; i < n; i++)
if (isprime[arr[i]])
prod *= arr[i];
return prod;
}
int main(int argc, char const *argv[]){
int arr[] = { 11, 20, 31, 4, 5, 6, 70 };
int n = sizeof(arr) / sizeof(arr[0]);
cout << prodprimearr(arr, n);
return 0;
}

## Output

If run the above code it will generate the following output −

1705
Published on 13-Aug-2020 11:57:18


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# statsmodels.stats.outliers_influence.OLSInfluence.influence¶

method

OLSInfluence.influence()[source]

Influence measure

matches the influence measure that gretl reports u * h / (1 - h) where u are the residuals and h is the diagonal of the hat_matrix


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submitted to JOSS, see also this tutorial on how to publish a JOSS paper to Arxiv

It is about pysm3 a #Python package (heavily optimized through numba) to simulate sky maps of the Cosmic Microwave Background (CMB) and galactic emission as observed by instruments. Relies on Planck & WMAP data.


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# 6.4 Basic Trigonometric Identities

6.4 Basic Trigonometric Identities

Three basic trigonometric identities are:

 sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x cot2 x + 1 = cosec2 x

Example 1 (To Prove Trigonometric Identities which involve the Three Basic Identities)
Prove each of the following trigonometric identities.
(a) sin2 x – cos2 x = 1 – 2 cos2 x
(b) (1 – cosec2 x) (1– sec2 x) = 1

Solution:
(a)
sin2 x– cos2 x = 1 – 2 cos2x
LHS: sin2 x – cos2 x
= 1 – cos2 x – cos2 x
= 1 – 2 cos2 x (RHS)

(b)
$\begin{array}{l}\left(1-{\mathrm{cosec}}^{2}x\right)\left(1-{\mathrm{sec}}^{2}x\right)=1\\ \text{LHS:}\left(1-{\mathrm{cosec}}^{2}x\right)\left(1-{\mathrm{sec}}^{2}x\right)\\ =\left(-{\mathrm{cot}}^{2}x\right)\left(-{\mathrm{tan}}^{2}x\right)\\ =\left({\mathrm{cot}}^{2}x\right)\left({\mathrm{tan}}^{2}x\right)\\ =\left(\frac{1}{{\mathrm{tan}}^{2}x}\right){\mathrm{tan}}^{2}x\\ =1\text{(RHS)}\end{array}$

Example 2 (To Solve Trigonometric Equations which involve the Three Basic Identities)
Solve the following trigonometric equations for 0ox ≤ 360o.
(a) sin2 x cos x + 1 = cos x
(b) 2 cosec2 x – 5 cot x = 0

Solution:
(a)
sin2 cos x + 1 = cos x
(1 – cos2 x) cos x + 1 = cos x
cos x – cos3 x + 1 = cos x
cos3 x = 1
cos x = 1
x = 0o, 360o

(b)
2 cosec2 x – 5 cot x = 0
2 (1 + cot2 x) – 5 cot x = 0
2 + 2 cot2 x – 5 cot x = 0
2 cot2 x – 5 cot x + 2 = 0
(2 cot x – 1) (cot x – 2) = 0
cot x= ½ or cotx = 2
cot x= ½ or cot x = 2
tan x = 2 tan x = ½
x =63.43o, 243.43o   x = 26.57o, 206.57o

(Note: tangent is positive in the first and third quadrants)

Thus, x = 26.57o, 63.43o, 206.57o, 243.43o

### 4 thoughts on “6.4 Basic Trigonometric Identities”

1. This is really useful, thanks.

2. what is LHS and RHS means>

• LHS = Left Hand Side
RHS = Right Hand Side

3. This is very useful ???


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# Linxz' Blog

## Still trying to think of something witty, I will let you know once I get something...

27 January 2019 | 12 minutes to read

# A Preface to Cryptography Part II

Tags: crypto

## Introduction

In the last post we covered a very basic introduction to cryptography, we talked about some applications of cryptography, discussed the differences between the terms cryptology, cryptography & cryptanalysis. Then we shed light on one of cryptography’s most important principles and finally we touched a little on what it means to “roll your own”.

Today’s post is again a pretty basic introduction to cryptography and more or less continuing on from our last post. We’re first going to look at the principles of cryptography and what it actually aims to achieve, then we will focus on the history of cryptography for the rest of the post. As there is a lot of history to digest I want to keep it at that so I don’t implode your brain when we move into some math stuff! There is plenty of extra reading to do off the back of this post so I think we can leave it at that.

## Principles of Cryptography

It can be said that cryptography has four main principles, or rather it has four tasks which it sets itself to achieve, these are: confidentiality, integrity, authentication and non-repudiation. You’re going to hear these a lot as you venture through InfoSec so you better get used to them! (I personally actually hate hearing this spiel of terms however as I said, you need to get used to them as they’re said a lot throughout InfoSec)

1. Confidentiality - used in order to prevent disclosure of information to unauthorized parties, we can achieve this by using cryptographic algorithms to encrypt the data. Making it unintelligible to everyone except those who are authorised to view the material.

2. Integrity - used in order to ensure consistency and accuracy of data during its life time. By using a digital signature or by hashing the received data and comparing the hash value with the original hash value.

3. Authentication - used in order to confirm that a part is the party you actually want to be communicating with. In order to achieve this we can use many methods such as a pre-shared key, public/private key cryptography we could also do this using a message authentication code (MAC) which will allow us to verify the message came from the stated sender (more on MAC later)

4. Non-repudiation - used to confirm that an entity had involvement in some communication preventing a party denying they ever sent a message this can also be done using a digital signature.

It could be said that there are more principles of cryptography such as accessibility, authorization and more however, I think we have covered the main four in some detail and throughout the blog series we will cover them in a lot more depth; specifically we will dive into the maths behind the algorithm and the algorithm design on a whole.

## A Not-So Brief History of Cryptography

Cryptography is said to have begun around the Egyptian times where Egyptians would develop non-standard hieroglyphics in order to hide messages from those who were not meant to read them. We also know that the Sumerians developed “Cuneiform” around this time too which could be seen as another sort of cryptography. As time passed around 600-500 before Christ we know that Hebrew scholars started to make use of simple Monoalphabetic substitution ciphers such as Atbash

Note: Monoalphabetic Substitution Cipher equals “simple substitution cipher” i.e, a substitution cipher that relies on a fixed replacement structure. In layman’s terms, that means that the substitution is fixed for each letter so if Z became G then every Z would be G. An example of this is the Caeser Cipher.

We then know that around 100-1 A.D the Romans started to invent ciphers such as the Caeser Cipher. Moving on, around 801-873 A.D a Muslim mathematician by the name of Al-Kindi (Alkindus) developed techniques for breaking Monoalphabetic substitution ciphers. On from here we start to see a lot more use & creation of different cipher techniques, in 1466 we see the creation of the first Polyalphabetic cipher from Leon Battista Alberti known as the “Alberti Cipher” not only that but he also invents the first known mechanical cipher machine.

Note: Polyalphabetic Substitution Cipher is any cipher based on substitution but using multiple substitution alphabets, the most notable example of this would be the Vigenère cipher. We will cover how this actually works later however if you’re interested now Crypto Corner has a great post on Polyalphabetic Substitution Ciphers

In 1553 we see another entry to Polyalphabetic Substitution ciphers when Bellaso invents the Vigenère cipher a book is later published in 1585. For those who are interested in the history of monarchy, you may know that in 1586 cryptanalysis was used by Sir Francis Walsingham to implicate Mary, Queen of Scots in the Babington Plot to murder Elizabeth I of England. There’s a pretty massive gap from here, we see some activity in 1641 however it is not until 1793 we see some movement when Claude Chappe establishes the first long-distance semaphore telgraph line of course then in 1795 we see Thomas Jefferson release his famous “Jefferson Disk Cipher”. Cut to 1800-1899 we see some more interesting developments with George Scovell’s work on Napoleonic ciphers during the Peninsular War, Joseph Henry building an electric telegraph, Samuel Morse developing the Morse code, Charles Wheatstone inventing the Playfair cipher and perhaps most notably, Charles Babbage developing techniques for breaking Polyalphabetic ciphers. In 1855 we saw Babbage breaking Vigenère’s Autokey cipher (deemed the “unbreakable cipher” of the time) and a much weaker cipher that we call the Vigenère cipher today. Cut to 1883 and we finally see Kerckhoff’s “La Cryptographie Militare” published, in 1885 we then see the Beale ciphers published.

Moving into the 1900s we see a considerable amount of progress in Cryptography with Gilbert Vernam developing the first practical implementation of a teletype cipher, now known as a stream cipher and a while later we see Joseph Mauborgne produce the one-time pad. Over the next few years we start to see a lot more usage of cryptography throughout military & government. In 1932 we see Marian Rejewski breaking the Germany Army Enigma for the first time. In 1940 we see further successful cryptanalysis when the SIS team break Japan’s PURPLE machine. Cutting into April 1943 we see Max Newman, Wynn-Williams, and their team (Alan Turing included) at the secret Government Code and Cypher School complete their work on the “Heath Robinson” which is a specialised machine designed for cipher-breaking. Following on from April 1943, in December 1943 we see The Colossus Computer was built, by Thomas Flowers at The Post Office Research Laboratories in order to crack the German Lorenz Cipher SZ42 the Colossus was used at Bletchley during World War II as a successor to the Robinson. Towards 1950 we see some more movement in the cryptography world, most notably Claude Shannon releases his paper that establishes the mathematical basis of information theory and he also publishes “Communication Theory of Secrecy Systems” in the Bell Labs Technical Journal.

Going into the 50s we don’t see much movement, in 51 we see the NSA founded and the KL-7 rotor machine being introduced sometime after. We see a few events happen during the 60s but not an awful lot of movement, of course in 1969 we see the first hosts of ARPANET however, we don’t see a major entry in Cryptography until 1973 when Horst Feistel developed the Feistal network block cipher design. Cut to 1976 and we see the Data Encryption Standard being published as an official Federal Information Processing Standard for the United States, alongside we this we have a publication titled “New Directions in Cryptography” from Diffie & Hellman. In 1977 RSA public key encryption is invented and in ‘78 Robert McEliece invents the first asymmetric encryption algorithm to use randomization in the encryption process, this was known as the “McEliece cryptosystem” following on from Stephen Wiesner’s idea in the 70s, in 1984 Charles Bennett and Gilles Brassard design the first quantum cryptography protocol, BB84 (I know what you’re thinking, quantum crypto is really that old!?!? Yes, it is! Fascinating, right?!). In 1989 we see quantum crypto experimentally demonstrated in a proof-of-the-principle experiment by Charles Bennett. In 1991 Phil Zimmermann releases the public key encryption program PGP along with its source code. In ‘94 Secure Sockets Layer is released by Netscape and Peter Shor devises an algorithm which lets quantum computers determine the factorisation of large integers quickly alongside this the proprietary, but un-patented RC4 algorithm is published on the Internet in addition this, the first ever RSA Factoring Challenge from 1977 is decrypted as “The Magic Words are Squeamish Ossifrage” (see, I bet you did not expect to get a bit of nerd humour out of this, did you? :D) In 1995 the NSA publishes SHA-1 as part of its Digital Signature Standard.

Finally we reach the 2000s (God, that was a long journey and I missed out on lots!) Quite a lot has happened this millennium, on Sep 6th 2000 RSA Security release their RSA algorithm into the public domain. In 2001 we see the Belgian Rijndael algorithm selected as the U.S Advanced Encryption Standard. In 2004 we are shown that MD5 is vulnerable to practical collision attack. We also see that in 2005 potential attacks on SHA1 are demonstrated, there’s a few bits and pieces that go on here however in 2012 NIST selects the Keccak algorithm as the winner of its SHA-3 hash function competition. 2013 sees Edward Snowdens Global Surveillance disclosures and Dual_EC_DRBG is discovered to have a NSA Backdoor, we also see in the same year that NSA publishes the Simon and Speck lightweight cipher blocks.

Note: it is worth noting that we missed a lot of history out here, there is so much more to cover however I could write pages & pages on the history of cryptography so I did my best to condense it down to important/interesting points in time rather than just covering everything! However, there is so much more to be aware of/research so if you’d like you can visit the timeline of cryptography on Wikipedia. I used this timeline for most of this post so you’ll see a lot of duplicate entries however, there is links to everything on there for further research if one particular point in time interests you more than others, for example I’m fascinated by the 1940-1960 crytpo era.

## Closing Notes

As we’ve covered a lot of history today and we’re not on a schedule for these posts I’m going to close that one here. I don’t want to weigh you down with math on top of all the history we just covered, besides, there is a lot of extra reading you could be doing off the back of this post so I will leave that up to you!

Thank you for reading! See you next time! :D


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# Efficient Method for Calculating the Probability of a Set of Outcomes?

Let's say I'm playing 10 different games. For each game, I know the probability of winning, the probability of tying, and the probability of losing (each game has different probabilities).

From these values, I can calculate the probability of winning X games, the probability of losing X games, and the probability of tying X games (for X = 0 to 10).

I'm just trying to figure out the probability of winning W games, tying T games, and losing L games after playing all 10 games... and hopefully do better than O(3^n). For example, what is the probability of winning 7, losing 2, and tying 1?

Any ideas? Thanks!

Edit - here's some example data for if there were only 2 games:

Game 1:

• win: 23.3%
• tie: 1.1%
• lose: 75.6%

Game 2:

• win: 29.5%
• tie: 3.2%
• lose: 67.3%

Based on this, we can calculate the probability after playing 2 games of:

• 0 wins: 54.0%
• 1 win: 39.1%
• 2 wins: 6.9%

• 0 ties: 95.8%
• 1 tie: 4.2%
• 2 ties: 0.0%

• 0 losses: 8.0%
• 1 loss: 41.1%
• 2 losses: 50.9%

Based on these numbers, is there a generic formula for finding the probability of W wins, T ties, and L losses? The possible outcomes (W-L-T) would be:

• 2-0-0
• 1-1-0
• 1-0-1
• 0-1-1
• 0-2-0
• 0-0-2
-
Well, can you tell us if the games are independent? If not, how do they depend on each other? –  JoshD Oct 3 '10 at 2:18
The games ARE independent. –  Kenny Oct 3 '10 at 2:19
@Kenny. OK, see my answer. Hope it helps! –  JoshD Oct 3 '10 at 2:21
Could you provide the probabilities of a win, a tie, and a loss for a single game? –  JoshD Oct 3 '10 at 3:38
added some sample data for 2 games above –  Kenny Oct 3 '10 at 4:17

This can be done with dynamic programming, I'm not sure if there is a better method as the games are independent.

Have a 4-D array, of wins, losses, ties, and games. You can limit wins/losses/ties to the number you want (let these be W, L, T, W+L+T=G) , time complexity will be O(W*L*T*G), which is bounded by O(G⁴).

The algorithm is basically:

``````A[][][][] = new double[G+1][W][T][L]
// A[g][w][t][l] is the probability of have w wins, t ties, l losses
// after g games. This can be computed from A[g-1].
// Let P[g][o] be the probability of outcome o for game g
//everything else is initially 0.
A[0][0][0][0] = 1
for g=1..G
for w=0..W
for t=0..T
for l=0..L
A[g][w][t][l] = A[g-1][w-1][t][l]*P[g][win] // assume out of bounds
+A[g-1][w][t-1][l]*P[g][tie] // reference returns 0
+A[g-1][w][t][l-1]*P[g][lose]
return A[G][W][T][L]
``````

edit)

We can do this in O(W*L*T*G/max(W,L,T)), i.e. O(G³). Note that if we have W wins and T ties after G games, then we must have L losses.

``````// we should pick the conditions we loop to be the smallest two.
// here we just use wins and ties.
A[][][][] = new double[G+1][W][T]
A[0][0][0] = 1
for g=1..G
for w=0..W
for t=0..T
A[g][w][t] = A[g-1][w-1][t]*P[g][win] // assume out of bounds
+A[g-1][w][t-1]*P[g][tie] // reference returns 0
+A[g-1][w][t]*P[g][lose]
return A[G][W][T]
``````

Maybe it's possible to do this significantly faster, by computing the probabilities of x wins/ties/losses separately (O(G)), and then adding/subtracting them intelligently, but I haven't found a way to do this.

-
This sounds promising, but can you elaborate a little bit more on your algorithm? I'm not sure I follow how to implement it. Thanks! :-) –  Kenny Oct 3 '10 at 8:25
beautiful, this is exactly what I was looking for! thanks :-) –  Kenny Oct 3 '10 at 9:56
I don't see a reason for doing it in O(N^3). See my response. Am I missing something? –  Eyal Schneider Oct 3 '10 at 15:31
@Eyal The win/tie/loss odds vary for each game, so you can't use the basic statistics approach. –  Nabb Oct 3 '10 at 15:40
You are right. I didn't notice that the probabilities are changing. –  Eyal Schneider Oct 3 '10 at 17:01

For your example, you need to consider the possible ways that the outcome can occur.

For win 7, lose 2, tie 1. There are `10! / (2!*7!)` or 360 possible ways. So multiply all the outcomes as you did, then multiply by that many permutations of the outcomes.

For all wins, you can just multiply because there's exactly one permutation of ten wins. For a mix, you need to consider the permutation.

In general for this problem the permutations will be `10!/(w!*l!*t!)` where w is number of wins, l is number of losses, and t is number of ties.

Edit 1 Note that the above only indicates how to count the permutations. The total probability is the number of permutations times (pw^w*pl^l*pt^t) where pw is probability of a win, pl a loss, pt a tie. w, l, and t, are the counts of each.

Edit 2 OK, in light of the new information, I don't know of a general way to do this. You'll have to individually computer each outcome by hand and add them together. With your two-game example above. If you want to find the probability of 1 win and 1 tie, you'll have to find every possible way of getting exactly 1 win and exactly one tie (there are only two) and add them up.

For ten games with the initial example, you'll have 360 outcomes that meet your criteria. You'll have to do each permutation and add up the probabilities. (wwwwwwwllt, wwwwwwwltl, etc) Unfortunately, I don't know of a better way to do this.

Further, in your two-game example, for one win and one tie, you must add the probability of winning the first game and tying the second to the probability of tying first, then winning.

So, there are nine independent outcomes:

``````W W
W T
W L
T W
T T
T L
L W
L T
L L
``````
-
Here's the problem. Let's consider the case of 10 wins, 0 losses, and 0 ties This probability of going 10-0-0 SHOULD be the same as the probability of getting 10 wins that I've already calculated (since there is only 1 way to get 10 wins). However, your formula would have me multiply this probability TIMES the probability of getting 0 losses and then multiply again times the probability of getting 0 ties (which will both be extremely small numbers) –  Kenny Oct 3 '10 at 3:02
Ah, not quite. My equations here are only to determine the number of permutations. The w, l, and t, are no probabilities. I should expand on that, but my intent is to calculate the probability he did (and you have as O in your first equation), then multiply that number by the total count of permutations. So, you'd take O (exactly as you have it, I agree completely), then multiply by 360. In the case of ten wins, 0 losses, and 0 ties, there is exactly one permutation, so you'd multiply by one (10!/(10!*0!*0!) = 1). –  JoshD Oct 3 '10 at 3:35
right but in that case, wouldn't O be too small? The probability of going 10-0-0 and is just probability of getting 10 wins... not the probability of getting 10 wins TIMES the probability of getting 0 losses TIMES the probability of getting 0 ties. –  Kenny Oct 3 '10 at 4:54
You may misunderstand. For ten wins O = pw^10*pl^0*pt^0 = pw^10*1*1 = pw^10. Does that make sense? –  JoshD Oct 3 '10 at 5:16
Oh, now I see the confusion. The probability of winning each game is a different known probability (think of them as different types of games), which is why I can't simply do pw^10. –  Kenny Oct 3 '10 at 5:37
show 1 more comment

My area, statistics!

You need to calculate the odds of one permutation, which can be done as so:

``````O = chanceWin^numWin * chanceTie^numTie * chanceLose^numLose
``````

where numWin, numLose and numTie are 7, 2 and 1, as per your example.

Now multiply by the permutations for winning, which is:

``````O *= 10! / ((10-numWin)! * numWin!)
``````

then losing:

``````p = 10-numWin
O *= p! / ((p-numLose)! * numLose!)
``````

then tying:

``````p = 10-(numWin+numLose)
O *= p! / ((p-numTie)! * numTie!)
``````

Now O is the odds of you winning numWin games, losing numLose games and tying numTie games out of 10 games.

-
I'm curious if we're getting to the same answer or not. Do you find that there are 360 permutations for the above example ( 7, 2, 1)? –  JoshD Oct 3 '10 at 2:51
Yeah I think we're on the right track, but if I sum up the probabilities for all the possible outcomes, I'm getting values significantly greater than 1 for both this method and Josh's. –  Kenny Oct 3 '10 at 2:53

If you don't want to run over the 3^n options, you can approximate the answer by using sampling: decide on N, the number of times you wish to sample. Run N samples, and count how many results of each type you had (0 wins, 1 win, etc). The approximate probability of each outcome is number_of_samples_resulting_this_outcome / N.

-

NOTE

The response below is only valid when the win/lose probabilities are fixed through the series of games. I misunderstood the conditions. I leave it anyway as a solution for the simpler case.

I got this formula for W wins, L loses, and N-W-L ties:

Complexity of computation

Each one of the powers and factorials has at most an order of N, so the value can be computed in linear time, unless I am missing some requirement.

The following Java code works for me. I also validated that the probabilities sum to 1:

``````public static double p(int w, int l, int t, double pw, double pl) {
double r = factorial(w+l+t) * Math.pow(pw,w) * Math.pow(pl,l) * Math.pow(1-pw-pl, t);
r /= factorial(w) * factorial(l) * factorial(t);
return r;
}

private static long factorial(int n) {
long res = 1;
for(int i = 2; i <= n; i++)
res *= i;

return res;
}
``````
-