An Introduction to the Statistics of Warhammer 40k Shooting
Building a Warhammer 40k that is effective at winning games is difficult to do. One of the reasons this is difficult is that the point value of a particular unit or upgrade does not give you the statistical effectiveness of the unit or upgrade. I am writing this article to help new players, or players who haven’t delved into the probabilities of Warhammer 40k, see what the average result of some of the actions in Warhammer 40k. This article will go into some of the details of the Space Marine codex and discuss the probabilities of causing unsaved wounds/damage to vehicles of certain units and upgrades and compare them to the point cost of the unit or upgrade. This article only goes into the effectiveness of the units and upgrades the moment they initiate a shooting attack or charge into combat. This article does not describe strategies that lead up to these moments. These statistics should be used when planning strategies so that you can effectively use your army. For example, a marine with a heavy bolter will cause 1.67 unsaved wounds, on average (when I give these numbers assume it’s the average result unless is say otherwise), to a tau fire warrior (toughness 3 and armour save 4+) while the auto cannon on a predator will only cause 1.11 unsaved wounds on average. The heavy bolter will cause 0.67 wounds to a toughness 6 creature while an autocannon will cause 0.89 wounds. The zone of effectiveness of these weapons should be known and this article will help show some of these zones of effectiveness.
I personally collect Space Marines and going to go through some examples that can be extended to other races. These first couple tables show the chance to wound when shooting at infantry and the chance to cause damage (glance or penetration) when shooting at vehicles. These tables are easily made. For the “to wound” probability table, just subtract the toughness of the model from the strength of the weapon, add three to this value, if the value is above 5 decrease it to 5, if the value is equal to zero add 1 to it, if the value is below 0 you increase the value to 0, and then divide this value by 6 and you have the probability of wounding the model with that weapon strength. Lets go through an example: my bolter can shoot at a toughness 4 creature, a toughness 6 creature and a toughness 8 creature and I want to find the probability of wounding each creature with my bolter strength 4. First the toughness 4 creature, so toughness 4 minus strength 4 is equal to zero, then add three so we have 3+0=3, its below 5 and above zero so we just divide this number by 6 and get 50% (3/6) chance to wound. For the toughness 6 creature, 4-6 = -2, -2+3=1, and then it is in the correct range so the chance of wounding is (1/6) or 16.7%. For the toughness 8 creature, 4-8=-4, -4+3=-1, -1 which is below zero so we increase it to zero, 0/6= 0% a bolter cannot hurt toughness 8 creatures.
For the “to damage a vehicle” probability table you first add 6 to the strength of the weapon, subtract the armour of the vehicle from this calculated value, if the value is less than zero then the strength of the weapon cannot hurt that armour, if the value is not negative then add one to this number and divide this number by 6 and you get the chance that weapon will cause damage to that armour value. Lets look at the probability of a lasscannon hurting an armour value 11, 13 and 14. For armour value 11, 9+6=15, 15-11=4, 4+1=5, 5/6 chance of hurting the armour value 11 or 83%. For armour value 13, 9+6=15, 15-13=2, 2+1= 3/6 or 50% chance of hurting the armour value 13. And finally for armour value 14, 9+6=15, 15-14=1, 1+1=2, 2/6 or 33% chance of hurting the armour value 14. Probabilities of meta guns and weapons that use two dice to determine penetration use a different set of rules to find the probabilities of damage (or success). To find this probability you first find the minimum number added to the strength of the weapon that will result in a glance, then find all the combinations to get a number less than the minimum number and subtract this number from 36 or just find all the combinations that result in a number equal to or greater than this minimum value and divide this number by 36 and you get the probability of damaging this armour value. Lets look at a melta gun shot at half range against a land raider, the melta gun shot needs 6 or more to do damage to the land raider (8+6=14), there are 10 combinations of two dice to get a number less than 6 so we subtract this from 36 and get 26, and finally divide it 26/36= 72% chance to do damage to a land raider at half range.
These damage probabilities are not all that matter when choosing which unit or upgrade to buy. The amount of shots, the ballistic skill of the wielder and the save of the targeted model also affect the effectiveness of the shooting. Let’s look at how to find the average amount of unsaved wounds a space marine with one bolter shot will inflict on an Orc and on a chaos space marine. First we check the chance the space marine has to hit his target, 4/6 chance for BS 4. Then we check the chance to wound the target 3/6 in both cases. Then we check the chance to fail their armour save, the Orc has a 100% chance to fail the armour save and the chaos space marine has a 2/6 chance to fail the armour save. Now to calculate the average unsaved wounds: for the Orc= (4/6)*(3/6)*(1)=0.33 and for the chaos space marine=(4/6)*(3/6)*(2/6)=0.11.
EQUATION TO REMEBER: (Chance to hit)*(Chance to wound)*(Chance fail save)*(Number of attacks)=Average number of unsaved wounds.
From these numbers we can see that it should take, on average, 3 bolter shots from a marine to kill an Orc and it should take roughly 9 bolter shots to kill a chaos space marine. An Orc is worth roughly 6 points (I think) and a chaos space marine is worth about 15 points. From the numbers it looks like the chaos space marine can take 3 times the punishment as an Orc can. I think the reason why the chaos space marine isn’t quite three times the cost of an Orc, is because plasma weaponry and low ap weapons kill chaos space marines with the exact same average unsaved wounds as Orcs and Orcs have more attacks in close combat than Chaos space marines. But this article is not about discussing orcs and Chaos space marine differences, so lets move on.
Average damage in close combat is much more complicated to determine than shooting averages. I believe that you can estimate the average result using a similar method to the shooting averages but since the distribution of combat effectiveness is not even through almost all units in the game some casualties from earlier initiatives have a chance to kill the sergeant with the power fist or nob with the power klaw even if the average amount of unsaved wounds is less than the defenders squad size. For example if a squad of 10 space marines with a sergeant w/ power fist get charged by a squad of Khorn berserkers, my sergeant has a chance of dying if the Khorn berserkers cause 10 or more wounds. This chance of significantly decreasing the combat effectiveness of the squad seems to effectively branch the results into two possibilities, one where my sergeant lives and does his damage and the other where he dies and doesn’t. I believe if you where to multiply the chance the sergeant has to live by the damage he can inflict you should get a pretty good value for the average amount of damage he will inflict. Now this adds to the calculations of these scenarios significantly and I personally would leave these calculations to the computers.
I know some people don’t believe in statistics or math and feel it’s too abstract to apply to their games. Well I want to help decrease the abstraction by going through a 600 point battle I had against my friend. I have the video up on Youtube, and I will go though the average result of each action taken and compare them with the result in the game. I categorized the damage done to vehicles as permanent and non permanent to help show some averages because the chance to get one of the specific results on the table is very low most of the time. (When you roll on the table without modifiers you have the same chance to get all the results)
Dark Angles Turn 1
The snipers in the buildings shot at my rhino and immobilized it. They are Dark Angles scouts so they have BS 4 so 4/6 chance to hit. Snipers need a 6 and then a 3+ to do damage to armour 11. Chance to glance = (1/6)*(1/3)=1/18, chance to penetrate = (1/6)*(1/3)=1/18. Glance has 2/6 chance to cause permanent damage, penetrate has 4/6 chance to cause permanent damage. 1/18*2/6 = average permanent damage done by glance and 1/18 * 4/6 = average permanent damage done by penetration. We add these two statistics together because they don’t overlap, and we get 0.0555... for one sniper shot. Lets multiply this by 5 to see what the average permanent damage result the squad causes, and we get roughly 0.28 so it should cause one permanent damage result ever four rounds of shooting. Now I didn’t look at the average non permanent damage the shooting should do to save time. If I did calculate this number then you could see the average results the shooting should have done, cause shooting can cause both kinds of damage and thus you should have averages for both in both cases. I will only really look at either or later on just to save time.
That’s all that really happened DA turn 1.
Ultramarines turn 1
I had my rhino and marines fire at the speeder and managed to shake it. It ended up being four bolter (two from rhino storm bolter and 2 from the 2 marines inside) shots as it was close to 24” range. Chance of non permanent damage is (1/6)*(4/6). So we get the average result (4/6)*(1/6)*(4/6)*4 shots = 0.3. It seems that my opponent’s luck early on was matched by my own.
I then fired my two speeders at his own rhino. Two krak missiles and 6 heavy bolter shots were fired. Lets find the average result for each separate shot type and add them together. Chance to hit 4/6. Chance to do non permanent damage (1/6)*(4/6)+(3/6)*(2/6). And the average non permanent damage result= (4/6)*( (1/6)*(4/6)+(3/6)*(2/6))*2shots=0.37. And now the 6 heavy bolters average result is (4/6)*((1/6)*(4/6))*6=0.44. Add these averages together and we get 0.81. It seems like the stun I get on his rhino is actually pretty likely to happen.(This is a reminder that the 0.81 is not a percentage it’s an average. If your confused I suggest going to Wikipedia and looking up their statistics section)
Dark Angles turn 2
The Snipers succeeded in doing permanent damage to my rhino a second time. Again this is supposed to happen once every four shooting phases or so, so I feel this can be classified as lucky.
Ultramarines turn 2
Now there was a lot of shooting at the land speeder this turn. For the 7 bolter marines + sergeant bolt pistol, average damage (of any form) result = (4/6)*(1/6) *15=1.67. Leader BS 5 bolt pistol average damage (5/6)*(1/6)=0.14 . 9 man bolt pistol scouts BS 3 average damage = (3/6)*(1/6)*9= 0.75 And the sergeant of the Scouts squad BS 4 average damage = (4/6)*(1/6) =0.11. All these averages added together gets 2.67 average damage results. This average shows that something really should have happened to the speeder here. This was bad luck for me but I was having a good time so it was ok.
I shot the speeders again and got a non permanent damage
result, which is close to the average.
My 10 man squad of scouts assaulted the speeder with their krak grenades and cause it to explode. Average lethal damage result(Which is just permanent damage excluding the weapon destroyed result) = (1/6)*((1/6)*(1/6)+(2/6)*(3/6))*10=0.32. Now killing the speeder with the assault was lucky in my favour but this seemed to even out my bad luck with my bolters. But the speeder exploded and killed 3 scouts. The explosion got 10 hits on my scouts at strength 3, what is the average unsaved wounds for this event? 1*(2/6)*(3/6)*10=1.67 . It seems like one to two scouts would die on average. The three scout’s dying is a little bit off average.
Dark Angles turn 3
The passengers of the rhino got out and shot at my speeders causing one shaken result. This is from 1 melta gun shot not at half range and 7 bolter shots from the rest of the marines and the 2 storm bolter shots from the leader. Melta non permanent=(4/6)*((1/6)*(3/6)+(4/6)*(1/6))=0.13 . Bolter shots non permanent = (4/6)*(1/6)*(4/6)*7=0.52. And leader storm bolter non permanent=(5/6)*(1/6)*(4/6)*2=0.19. Added together we get 0.84. And the result seems to be close to the average result.
The snipers shot at the Rhino and failed to do damage. I was expecting this result after two successful turns in a row.
I think another article should go into assault averages because it is tricky. I won’t discuss the assault squad and tactical squad battle statistics now.
Ultramarine turn 3
Really no shooting done just finished up assault.
Dark Angles turn 4
Snipers failed to do damage again, as expected.
The rhino moved 12 inches so it couldn’t shoot.
Ultramarines turn 4
I shot at the snipers in the building because my opponent was feeling like I was winning by a bunch and so I gave him a 4+ roll to see who my speeders would shoot at and it ended up being the snipers. Some mercy in my playing can make the game feel more relaxed I think. Now onto the unsaved wound averages of this shooting attack. Krack missiles were used I think, any way frag missiles are hard to describe statistically when they scatter >1 inch, basically when they scatter off the base. Assuming I shot krak missiles. Also the scouts had a 3+ cover save in the building. Average unsaved wounds = (4/6)*(5/6)* (2/6)*2+(4/6)*(4/6)* (2/6)*6=1.26 . It looks like one of the scouts should have died on average.
Dark Angles turn 5
The snipers shot and did nothing to the scouts this time and did nothing. The scouts had cover saves 4+. Unsaved wounds on average =(4/6)*(3/6)*(3/6)*5=0.83. Hmm it looks like I should have lost a scout there.
The rhino rammed my rhino and killed it. It caused a strength 6 hit on the rhino. Average destroyed (there is nothing left on the rhino so all permanent damage will kill it) result = (1/6*2/6)+(1/6*4/6)=0.17. I think that the awesomeness of the ramming attack shifted the result away from the average in this case.
The tactical squad with the leader jumped out and shot up my land speeders doing quite a bit of damage, 2 shaken results and a weapon destroyed result. Now let’s look both the average result of both permanent damage and non permanent damage. Permanent damage average( Im going to lump the leader’s bolter shot in with the regular marines for ease of calculation, the actual result should be marginally higher) = (4/6)*((1/6)*(2/6))*9+(4/6)*((2/36)*(3/6)+(33/36)*(5/6))=0.86. And the result for non permanent damage is =(4/6)*((1/6)*(4/6))*9+(4/6)*((2/36)*(3/6)+(33/36)*(1/6))=0.79. So the weapon destroyed was to be expected, but the two shaken results are a bit above average.
Ultramarines turn 5
Mainly assaults happened this turn and that is a topic for a different day.
And that was the game guys. I hope this article was helpful in any way. If I made a numbers mistake in one of my calculations just comment on it and ill edit it out. If you have questions just ask em and I will do my best to answer them. I have attached some tables of some general statistics that are handy to know. The tables can be used to quickly find the: chance to hit, chance to damage,and chance to save. Which are the golden three numbers to find the average result.


Our first graders are 120 days smarter! We celebrated the 120th day of school on Wednesday, March 9th. The kiddos had the opportunity to dress like a 120-year-old! The first-grade teachers even joined in on the fun!
The whole day was spent doing activities that involved the number 120. Some of those activities included: writing 120 words, taking 120 steps in the building, getting 120 autographs, making a 120-day hat, creating a gumball machine and making a colored pattern with the numbers, counting by tens to 120 using stickers, counting by fives, and tens to 120 and completing 120 exercises.
There was also some yummy food involved during the day! The kiddos filled ten sticks with fruit loops. After counting to 120 with the tens sticks, they used string to create a necklace using the fruit loops. They even made 120 pancakes, working on tally marks to 120 and learning how to flip and make their own pancake!


I’ve been adding some neat features to SkyCars, including lasers, jump pads, and lava. But one of the most formidable development challenges so far has been orbits. By orbits, I’m talking about asteroids, comets, or even platforms that rotate around an arbitrary point in space. I’d like to share with you my solution for 3 dimensional orbits.
Visualizing the orbit path
Let us imagine an asteroid with a circular orbit of an arbitrary orientation (or axis angle). We may visualize the orbit path like so:
Note that the asteroid has a circular orbit of radius R around an origin point. At any given time, the asteroid will have the coordinates (Rx, Ry, Rz). We will treat R as a vector from the center of the orbit’s path to the asteroid’s position. Applying the distance formula, we know that R = sqrt ( Rx2 + Ry2 + Rz2 ). However, we also know that R casts a shadow (or projection) onto the x/z plane, which we will call Rxz. We also know the distance of Rxz = sqrt ( Rx2 + Rz2 ). This will be important later.
Now I’m going to jump ahead a bit so that we know what our goal is. If we want our asteroid to orbit the center point, we need to find out the orbit’s normal vector (N), which has Nx, Ny, Nz components and also originates from the orbit path’s center. Here’s what this looks like:
This vector describes an imaginary line perpendicular to the orbit plane. If we know the values of this vector, we can use a “rotate by axis angle” command on our asteroid. In Shiva, the command is simply:
object.rotateAxisAngle ( hObject, nAxisX, nAxisY, nAxisZ, nAngle, kSpace )
nAxisX, nAxisY, nAxisZ will take the values of Nx, Ny, Nz. nAngle is the asteroid’s orbital angular velocity, and kSpace should be set to global. In other game engines, there are similar commands. If you do not have such libraries, here is a good tutorial on how to perform such rotations using a rotation matrix. Shiva users: As object.rotateAxisAngle does not accept position offsets, I had to parent the asteroid to a helper object placed in the center of the orbit.
Now let’s discuss how to get from our asteroid vector R to the normal vector N. Notice that R and N are orthogonal (perpendicular. You may be tempted to say that if we know R, we can just rotate this vector back by 90 degrees to get N. However, we do not have enough information because there are an infinite number of normals orthogonal to R! Or if you prefer, we could spin N around the R axis by all values of the angle phi and still obtain an orthogonal normal. This is actually a good thing. We can choose our own value for phi and in effect choose the amount of bank or tilt in our orbit.
If we want to find N which is orthogonal vector to R, we can use the vector cross product. The cross product takes two input vectors (together which define a plane), and outputs an orthogonal vector (to this plane). R is our first input vector; we can use phi to determine the necessary secondary vector which we will call Q.
Let’s create a vector aligned with the Z+ axis called Phi0. First we’ll rotate it upward around the X axis by angle phi to reach an intermediate vector Phi1.
Next, we’ll rotate this vector around the Y-axis so that it is tangent to the asteroid. The result of this second transformation will give us Q.
In order to perform this second transform, we need to know the angle of Rxz. As this vector is composed of Rx and Rz, we can determine the angle we need to rotate Phi1 by to get Q, or angle rho. While we do not need to calculate this angle, we do need to make note of the following identities:
sin rho = Rz / Rxz
cos rho = Rx / Rxz
Also remember that Rxy = sqrt ( Rx2 + Rz2 ) and that Rx and Rz are given.
Doing a little mathemagics, we can determine the components of Q. The transformation from Phi0 to Phi1 has the components:
( 0, sin phi, cos phi )
When we add in the second transformation, the resultant vector Q takes on the form:
( cos phi * sin rho,
cos phi * cos rho )
We can simplify further using substitution:
( cos phi * Rz / Rxz ,
cos phi * Rx / Rxz )
Also note that Q is now tangent to the asteroid’s orbit. This means that we now have two valid input vectors, Q and R, which describe the plane of the orbit path! We can now use the vector cross product to solve for N:
N = Q × R
Don’t forget to normalize N (scale the vector to a magnitude of 1) when we’re done solving. Now just rotate the asteroid about the normal, which is the axis angle for our orbit path! We’re done.
In Shiva, this all looks somewhat like this:
-- When initializing your rotating object: local Rxz = math.sqrt ( Rx*Rx + Rz*Rz ) local Qx = math.cos ( phi ) * Rz/Rxz local Qy = math.sin ( phi ) local Qz = math.cos ( phi ) * Rx/Rxz local Nx, Ny, Nz = math.vectorNormalize ( math.vectorCrossProduct ( Qx, Qy, Qz, Rx, Ry, Rz ) ) -- Store Nx, Ny, Nz somewhere. During the game loop, rotate the object using: object.rotateAxisAngle ( hObject, Nx, Ny, Nz, dR, object.kGlobalSpace ) -- note: dR is the orbit speed that you choose.
A few final thoughts…
We have created a function that allows an object to orbit in a circular path of any orientation around a center point. However, if we choose a center point directly above the object, we cannot use the above method perform an orbit because vectors R and Q will either be identical or incorrect, meaning that N cannot be computed. A similar problem may occur with a center point directly below the object. I resolved this issue by manually setting appropriate values for Q so that it is parallel to the x/z plane.
If you see “circular path” as a constraint, you can achieve elliptical paths by having a center point that moves with time. Specifically, you would want a center point that moves back and forth along a line. The velocity of the center point along this line should be sinusoidal, with the same frequency (or dR) and phase as your orbit. You could even have the center point of your orbit move on an arbitrary path to create a complex orbit, such as a moon orbiting a planet orbiting a sun.
I would like to thank my good friend Kyle Jewhurst for helping me work through and verify the math on this topic. Also now that I’ve nailed down the orbit code, videos of SkyCars are coming soon!


My semester thesis titled Nonlinear Infinite Horizon Model Predictive Control with Parametric Trajectories explores a new way to parametrize trajectories for infinite horizon model predictive control.
Here is the abstract:
This work shows how infinite horizon model predictive control (MPC) can be combined with parametric trajectories for control of nonlinear systems. We use exponentially decaying basis functions to approximate state and input trajectories in time. A variational formulation allows trajectory cost and constraints of the optimization problem to be expressed in terms of parameters describing the trajectories. Optimization is done over those parameters via sequential quadratic programming. A full MATLAB implementation of this MPC routine is developed. Results are tested in simulation on an inverted pendulum on a cart. We show that the controller finds trajectories that enable a swing-up of the pendulum and stabilization around its upper equilibrium. A testing environment with a real system is prepared for further research on this control scheme.


DES MOINES – Two Iowa educators have been named recipients of the Presidential Awards for Excellence in Mathematics and Science Teaching (PAEMST).
Angie Bonthuis, an educator at Gilbert Elementary School in the Gilbert Community School District, was honored for her work in mathematics. Bonthuis has been an educator for 25 years, teaching preschool at Southeast Webster for four years, as well as kindergarten at Gilbert Elementary School for the past 21 years. In addition, she has served as a model teacher, tech innovator, and leadership team member for her district.
Sarah Borzo was honored for her work in science. Borzo has been an educator for 11 years. She taught sixth grade science and reading at South Sioux City Middle School in South Sioux City, Neb., before moving to South Middle School in Waukee to teach sixth grade science. During her time at South, Sarah also served as the science curriculum leader and as a new teacher mentor. She now leads education and outreach for Metro Waste Authority, designing and delivering environmental education to k-12 students across central Iowa. Borzo also contributed as a content developer for sixth grade science for Iowa e-Learning Central.
Teachers are selected based on their distinction in the classroom and dedication to improving science, technology, engineering, and mathematics (STEM) education.
The PAEMST program, administered by the National Science Foundation (NSF) on behalf of the White House Office of Science and Technology Policy, recognizes outstanding teachers for their contributions to the teaching and learning of mathematics and science.
PAEMST is the highest recognition that a kindergarten-through-12th grade mathematics or science (including computer science) teacher may receive for outstanding teaching in the United States. Since 1983, more than 5,200 teachers nationwide have been recognized for this honor.


Dynamics and Optimal Control of Stochastic Pension Plan Models
Andrew J.G. Cairns
This paper discusses the modelling and control of pension plans.
A continous-time stochastic pension plan model is proposed in which there are two risky assets as well as randomness in the level of benefit outgo. We consider Markov control strategies which optimise over the contribution rate and over the range of possible asset-allocation strategies.
A quadratic loss function is used which extends that of Boulier et al. (1995). Using this loss function it is shown that both the optimal contribution rates and the asset-allocation strategies are linear functions of the fund size.
Boulier et al (1995) proved the perhaps surprising result that under the optimal asset-allocation strategy as the level of surplus in the pension plan increases the proportion of the fund invested in high-return, high-risk assets, such as equities decreases. This paper demonstrates that this result applies to a much larger class of models and loss functions.
Dynamic optimal solutions are compared with their static counterparts.
It is also found that the optimal strategies do not depend upon the amount of uncertainty in the level of benefit outgo. This means that, for quadratic loss functions, small pension plans should operate in precisely the same way as large pension plans even though the relative level of variability will be higher for a small pension plan.
Finally there is some discussion of the effects of constraints on contribution and asset-allocation strategies.
Keywords: continuous time; stochastic differential equation; asset-allocation; contribution strategy; optimal control; contstraints


coxph and survreg model. I have two datasets. Dataset 1 includes a certain number of people for which I know a vector of covariates (age, gender, etc.) and their event times (i.e., I know whether they have died and when if death occurred prior to the end of the observation period).
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The cox model is a linear transformation model of the form P ( T ≤ t | x) = exp ( − exp ( g ( t) + x ~ T β)) where g ( t) is an unspecified linear transformation function. The cool thing is that this unknown g ( t) goes into a baseline hazard λ 0 ( t) which is independent of β. A coxph object: newdata: Optional new data to predict at: type: Type of prediction wanted: se.fit: Return standard errors as well? terms: If type="terms", which terms to return. collapse: identifier for which rows correspond to different individuals: safe: Use a more intensive `safe' prediction method... other unused arguments.
coxphにおけるtime-dependent covariatesの使い方; by Kazuki Yoshida; Last updated over 9 years ago Hide Comments (-) Share Hide Toolbars. The results of my coxph() are significant, yet the cox.zph() test is significant too.. From my understanding of it, the significance of cox.zph() means that the Cox model is not fit to model the relationship between my covariates and the dependent variable. The reason being that this relationship is not linear, while the Cox model only accounts for linear relationships.
Cox Proportional Hazards (CoxPH) Cox proportional hazards models are the most widely used approach for modeling time to event data. As the name suggests, the hazard function, which computes the instantaneous rate of an event occurrence and is expressed mathematically as h ( t) = lim Δ t ↓ 0 P r [ t ≤ T < t + Δ t ∣ T ≥ t] Δ t,. R package "survival". : survival analysis 즉 생존분석 시 사용. anova.coxph Analysis of Deviance for a Cox model. cluster Identify clusters. dsurvreg Distributions available in survreg. subjects. is.ratetable Verify that an object is of class ratetable. plot.aareg Plot an aareg object. survConcordance Compute a concordance measure.


One use of the RATE function is to calculate the periodic interest rate when the amount, number of payment periods, and payment amount are known. For this example, we want to calculate the interest rate for $5000 loan, and with 60 payments of $93.22 each. The NPER function is configured as follows:
To find Rate when Principal Interest and Time are given the rules are. Interest = ( Principal Rate Time)/100, Rate = (100 Interest)/(Principal Time)
How to Calculate Monthly Interest Divide By 12. The first step is to calculate a monthly interest rate. Amortization. That process is called amortization, and an amortization table helps you calculate. Periodic Rates. As you can see, interest can be calculated monthly, daily, annually,
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People say I am funny but I am just stating the truth, and that’s so rare that you find it funny.” At the time when people in.
rate – Interest Rate per duration. pv – the initial loan amount. How to use Calculate the Interest Rate for a Loan in Excel. To determine the amount due for loan payment, given a loan duration/term, a Rate of Interest, and the initial loan amount, we can utilize the Excel PMT Operation Syntax..
3 Year Arm Rates Overview of 3/1 Adjustable Rate Mortgage aka 3 Year ARM or Three Year. ERATE.com – Current Mortgage Rates, Home Equity Rates & Much More. 3/1 ARM (3 year ARM)- the rate is fixed for a period of 3 years after which in the 4th year.Best Second Mortgage Rates Best 30 Year Mortgage Refinance Rates What Is A Rate Sheet How to Read a Mortgage Rate Sheet – Yahoo Finance – The rate sheet will show you a range of costs for the loan you are interested in. Every adjustment will have a cost, either positive or negative. adjustments include loan size, state the property.The movement of long-term bonds.
To calculate interest rate, start by multiplying your principal, which is the amount of money before interest, by the time period involved (weeks, months, years, etc.). Write that number down, then divide the amount of paid interest from that month or year by that number. The answer is your interest rate, but it will be in decimal format.
Federal Interest Rates Chart Daily Treasury Bill Rates. These rates are composites of closing market bid quotations on recently issued Treasury Bills in the over-the-counter market as obtained by the Federal Reserve Bank of New York at approximately 3:30 PM each business day.
Simple Interest Formulas and Calculations: Use this simple interest calculator to find A, the Final Investment Value, using the simple interest formula: A = P(1 + rt) where P is the Principal amount of money to be invested at an Interest Rate R% per period for t Number of Time Periods. Where r is in decimal.
Calculating interest is a function of Future Value, Present Value and the number of periods interest is applied. Compound interest applies to the principle, and earns interest as well. Simple interest earns on the principle only. Simple interest is very easy to calculate, but is not really used in modern investing..
Schools that lost £2m when an academy trust collapsed are set to find out if they will get any of the money. He said there.


Exam and Assessment Results
We are exceptionally proud of our 2022 GCSE results that, whilst excellent in their own right, show a huge improvement from 2019. 72% of students gained at least a grade 4 in both English and Maths with 60% of students achieving at least a grade 5. All subjects have improved across the school and there are so many pleasing results with large numbers of grade 9s across all subjects and numerous students achieving grades 7-9 in all of their subjects.
As Headteacher, I am incredibly proud of the school, the staff, our community and most importantly the students who have all worked exceptionally hard and shown great resilience to achieve these wonderful results. They leave our school to take the next steps on their educational journey and I have no doubt they will be incredibly successful.
Progress 8 Score - +0.11
Students achieving a grade 4 in English and Maths – 72%
Students achieving a grade 5 in English and Maths – 60%
Attainment 8 – 51.3 (this means the average grade for each student in each subject was above 5)
Students achieving a grade 5+ in the EBACC – 7%


Photo: Women Talk to Arlington Girls on STEM Careers
The STEM conference took place earlier this month.
More than 200 Arlington middle school girls learned about careers in science-, technology-, engineering- and math-related fields at a STEM conference this month.
The conference, which took place Nov. 3 at Kenmore Middle School, brought 19 professional women from a variety of STEM fields to talk to the girls about possible career paths.
The event was sponsored by the Arlington branch of the American Association of University Women and Arlington Public Schools.
Read more on the STEM conference for Arlington girls.


Title : Two level quantum system - A paradigm for analyzing fundamental quantum mechanical phenomena
A two-state or a two-level system, is a quantum system that can survive in any quantum superposition of two independent physically distinct state vectors. Hilbert space of two-dimensional is used to describe such a system. Here, two linearly independent vectors form a complete basis spanning the space. Any two-level coupled system can also be consider as a quantum bit (qubit).
Didactically, the two-state formalism is among the uncomplicated of mathematical method exploited for the investigating of quantum systems aiming to understand phenomenology or effects having no classical analog. For examples, two-state formalism can be used to expound basic quantum mechanical phenomena such as the interference displayed by particles of the polarization states of the photon, even for much sophisticated phenomena such as neutrino recurrent event or the neutral K- meson oscillation.
I my presentation, I intend to analyze how a initially prepared state in two-level system evolves in time. We consider the cases of both static and dynamical couplings separately. It can be shown that phenomena with enormous commercial enterprise applications such as the maser and laser can conveniently be justified using the two- state formalism, as well as, other fundamental issues like quantum coherence and entanglement. We also discuss how two-state formalism also forms the basis of quantum computing.


One of the first things you learn in a new language is how to count, and this course is no exception. However, in Lojban, numbers include much more than just counting; for example, in Lojban, some, most and too many are numbers.
The numbers from one to nine are as follows:
Numbers from 10 onwards are made by putting the digits together, just like you'd say a telephone number. For example:
Just as we have a word for a comma, we also have one for a decimal point: pi. So 5.3 is mupici. In fact, pi is not always decimal; it's the point for whatever number base you're using. But that's a more advanced topic.
Tip: Don't get this mixed up with the number pi (π): 3.14159..., which has its own word in Lojban: pai — oddly enough.
When you want to talk about numbers as sumti — that is to say, as things in and of themselves — you need to put an article in front of them. But that article cannot be la, and for reasons which hopefully will become clear soon, it cannot be le either. In front of numbers, Lojban uses the article li. So li pareci means 'the number one hundred and twenty three'. 'One, two, three', on the other hand, would be li pa li re li ci: each li introduces a brand new number.
What are the following numbers in Lojban? (don't forget li!)


LBJ Infrastructure Group (LBJIG) sponsored the first annual MakerSpace instant challenge for the 6 Carrollton-Farmers Branch middle schools.
The event was held in January at CFBISD’s Educational Services Building, and 8 LBJ volunteers helped judge the STEM competition.
In total, there more than 90 students involved in the event which kicked off at 10:00 a.m. and ended at 2:00 p.m. The purpose of the challenge was to have MakerSpaces teams from each middle school compete without any idea on the type of STEM activity that would be assigned.
Meaning, they had no preparation beforehand. Students were timed during each STEM activity and were tested on teamwork, leadership and their math & science skills.
This STEM competition was inspired and made possible by the LBJ Express $40,000 STEM grant that was awarded to CFBISD as part of its CSR (Corporate Social Responsibility) Plan.
Through the MakerSpace program, nearly 15 teachers are trained to teach with STEM tools each year. To date, more than 5,500 students have been exposed to science, technology, math and engineering in a fun and engaging way thanks to LBJ Express.
Congratulations to all the volunteers in this initiative!


1 edition of College Algebra - Penn State University Edition found in the catalog.
College Algebra - Penn State University Edition
2003 by Pearson Custom Publishing .
Written in English
|The Physical Object|
Trigsted MyMathLab College Algebra Interactive -- Access Code -- Plus Guided Notebook by Trigsted, Kirk ISBN: List Price: $ $ (Save 17%) $
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Penn State University - University Park M College Algebra II Algebra and Trigonometry 3rd Edition. Available online. The cost at the bookstore will be slightly more. If you buy at the bookstore on campus, ask for the access code at the check-out register. Beecher, Penna, and Bittinger’s College Algebra is known for enabling students to ";see the math"; through its focus on visualization and early introduction to functions.
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MATH - Fundamentals of College Algebra: 3 General Credits More Information about Credit for DSST Exams This evaluation of the credits awarded for DSST examinations is subject to change and may be reviewed and revised periodically by Penn State faculty.
Mathematics and Statistics at Penn State Altoona; You are here: Home / Math Courses / Math College Algebra II and Analytic Geometry. Math College Algebra II and Analytic Geometry.
3 credits. Blue Book Description: Relations, functions, graphs; polynomial, rational functions, College Algebra, 3rd (corrected) Edition. COLLEGE ALGEBRA VERSION bˇc PENN STATE ALTOONA – MATH EDITION1 BY Carl Stitz, Ph.D. Jeff Zeager, Ph.D. Lakeland Community College Lorain County Community College Fall 1This is a shorter and slightly modiﬁed version of the original only covers the material from chapters 0, 1, and 2, with one additional section: section (adapted from sec.
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Learn how to use useful calculations to make math easier for your customers and improve conversion.
Doing math can be a daunting task for many people, and it can be a major barrier to conversion. Fortunately, there are ways to make math easier for your customers and improve conversion. By using useful calculations, you can provide customers with the information they need without having to do the math themselves. This can include telling them how quickly they’ll run out of credit, how long ago a post was posted (rather than just a timestamp) and how much they’ll save from a discount (rather than a %). By providing this information in an easy-to-understand format, you can make it easier for customers to make decisions and increase conversion. Using useful calculations can also help to build trust with customers. By providing them with the information they need in an easy-to-understand format, you can show them that you are looking out for their best interests and that you are willing to go the extra mile to make their experience as easy and stress-free as possible. This can help to build a strong relationship with customers and increase the likelihood of them returning to your business in the future. Step-by-Step:


Do you think math is boring? Do you dread math class? Well, you're not alone! Many people find math to be tedious and uninteresting. But did you know that math can be found in unexpected places and can be fun?
That's right, math is not just limited to textbooks and worksheets. It can be found in the real world, in nature, in art, and even in games. Here are some fun examples:
Nature: Have you ever noticed the spirals on a pinecone or the patterns on a sunflower? These are examples of the Fibonacci sequence, a famous mathematical pattern found in nature.
Art: Famous artists such as Leonardo da Vinci and M.C. Escher used math in their works. Escher's famous "Waterfall" print, for example, uses impossible geometry, which is a form of math.
Games: Games such as Sudoku, chess, and Rubik's Cube all involve math concepts. In fact, many video games use math to create their graphics and gameplay mechanics.
Music: Music is full of math! The rhythm, beats, and timing are all based on mathematical concepts such as fractions and ratios.
Sports: Sports involve math too! Basketball players use angles to make shots, baseball players use statistics to measure their performance, and football players use geometry to make plays.
By finding math in unexpected places, you can make it more fun and interesting. You might even start to see it as a game or a puzzle to solve.
So next time you're feeling bored in math class, try to find the math in unexpected places. Who knows, you might discover a new passion for math!


The most complete manual of its kind, this handy book gives you all the formulas and calculations you are likely to need in drilling operations. Updated material includes conversion tables into metric. Separate chapters deal with calculations for drilling fluids, pressure control, and engineering. Example calculations are provided throughout. Presented in easy-to-use, step-by-step order, Formulas and Calculations is a quick reference for day-to-day work out on the rig. It also serves as a handy study guide for drilling and well control certification courses. Virtually all the mathematics required out on the drilling rig is here in one convenient source, including formulas for pressure gradient, specific gravity, pump output, annular velocity, buoyancy factor, volume and stroke, slug weight, drill string design, cementing, depth of washout, bulk density of cuttings, and stuck pipe. Elsevier.


Most office buildings have a load factor. This load factor is a percentage that describes the difference between the rentable and useable square footages for the building. The usable square footage of a building is all the square footage behind the front doors of the tenant’s suite; the space that is exclusive to them. The rentable square footage is the usable square footage plus the common areas. The common areas include the hallways, lobbies, elevators, stairwells and restrooms of the building. These areas are not exclusive to any one tenant, but are for the use all the tenants and their guests.
Let’s say the gross square footage of an office building is 100,000 square feet. The gross square footage of the building is everything contained within the exterior walls. Let’s say that the total square footage of the common areas of the building is 15,000 square feet. This leaves 85,000 sq.ft. of usable space (i.e. suites).
To calculate the load factor we take the total square footage of the common areas, (15,000 sq.ft.) and divide it by the gross square footage of the building, (100,000 sq.ft.). The formula looks like this:
15,000/100,000 = .15. Therefore, the load factor of the building is 15%.
Let’s see how the load factor relates to leasing space within the building. Let’s say that a tenant plans on leasing a suite with 1,200 square feet of usable space. Knowing that the load factor of the building is 15%, the tenant could then calculate the rentable square footage of the suite (1,200 + 15% = 1,380 sq.ft.). So if the monthly rent for the suite is $2.00 per square foot, the tenant’s total monthly rent would be $2,760.00:
(1,380 rentable sq.ft. x $2.00 per sq.ft.= $2,760.00).
The rentable square footage is always higher than the usable square footage because it includes both the tenant’s suite and their percentage of the common areas.
When shopping for space, it’s important for tenants to understand and compare the load factors of the buildings they may be interested in. Higher load factors mean that more of a tenant’s monthly rent will be dedicated to common areas and less to the suite they occupy. That being said, buildings with higher load factors often have amenities, such as spacious lobbies or atriums, which many tenants may find appealing.


Existence of positive solutions for a singular nonlinear fractional differential equation with integral boundary conditions involving fractional derivatives
MetadataShow full item record
In this article, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, some properties of the first eigenvalue of a fractional differential equation are obtained. Based on these properties and through the fixed point index theory, the singular nonlinear fractional differential equations with Riemann–Stieltjes integral boundary conditions involving fractional derivatives are considered under some appropriate conditions, and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable and it includes fractional derivatives. The existence of positive solutions for boundary conditions involving fractional derivatives is established. Finally, an example is given to demonstrate the validity of our main results.
Showing items related by title, author, creator and subject.
Berwick, Lyndon (2009)The analytical capacity of MSSV pyrolysis has been used to extend the structural characterisation of aquatic natural organic matter (NOM). NOM can contribute to various potable water issues and is present in high ...
Size exclusion chromatography as a tool for natural organic matter characterisation in drinking water treatmentAllpike, Bradley (2008)Natural organic matter (NOM), ubiquitous in natural water sources, is generated by biogeochemical processes in both the water body and in the surrounding watershed, as well as from the contribution of organic compounds ...
Abdullah, Hanisom binti (2010)Mallee biomass is considered to be a second-generation renewable feedstock in Australia and will play an important role in bioenergy development in Australia. Its production is of large-scale, low cost, small carbon ...


Math Worksheets Grade 5 Pdf
The math worksheets on this page cover many of the core topics in 5th grade math, but confidence in all of the basic operations is essential to success both in 5th grade and beyond.
Math worksheets grade 5 pdf. Your fifth graders will be challenged with these free math worksheets. These math sheets can be printed as extra teaching material for teachers, extra math practice for kids or as homework material parents can use. Join our newsletter to find out about new math worksheets and other information related to the website.
Our grade 5 math worksheets cover the 4 operations, fractions and decimals at a greater level of difficulty than previous grades. Fifth grade math worksheets pdf for kids. Grade 5 math printable worksheets, online practice and online tests.
Adding decimals, 4th & 5th grade. Worksheets > math > grade 5 > geometry. Free math worksheets for grade 5.
Free & printable grade 5 math worksheets. Free resources for fifth grade teachers to print and practice for common core math standards. These worksheets are printable pdf exercises of the highest quality.
This level features grade 5 math topics as follows : Math worksheets workbooks for fifth grade; Use them to practice and improve your mathematical skills.
This money and measurement math worksheet gives your child practice adding 2 decimals to the hundredths place. As a matter of fact, these fun free 5th grade math skills practice games and worksheets pdf are a great way to strengthen the key math lessons fifth graders learn in the classroom while also having fun. This product is suitable for preschool, kindergarten and grade 1.the product is available for instant download after purchase.
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A PunditFact rating of CNN pundit Marc Lamont Hill drew our attention today for its PolitiMath content.
PolitiMath takes place when math calculations appear to bear on whether a figure receives one "Truth-O-Meter" rating instead of another. In this case, Hill received a "False" rating for claiming an unarmed black person is shot by a cop every 28 hours.
PunditFact found Hill reached his conclusion using the numbers for black persons armed or unarmed. The total figure for both was 313. The figure for unarmed black people was 136. The calculation is uncomplicated. Taking the number of hours in a 365-day year, we get 8760. Divide 8760 by 313 and we get Hill's 28-hour figure. Use what PunditFact said was the correct figure and we get 64 hours (8760/136).
Hill exaggerated the frequency of a unarmed black person dying from a police shooting by 124 percent.
We're certainly not saying that PolitiFact is in any way consistent with how it classifies errors by percentage, but for comparison Florida lawmaker Will Weatherford made a statistical claim that was off by about 49 percent and received a "False" rating. Democrat Gerry Connolly, on the other hand, managed to wring a "Mostly False" rating out of a statistic that was off by about 45 percent.
Perhaps this is just science at work. Given reality's liberal bias, it may make sense to grade errors of the same percentage more harshly where they affect liberally-biased truths. Fact checkers could be guilty of false equivalency by acting as though the truth is simply objective.


Postulates of special relativity; maximal speed of signal propagation; relativity of time; definition of an event; simultaneity is frame-dependent; interval; differential interval; invariance of the interval; interval as a pseudo-Euclidean distance; time-like and space-like intervals; proper time and time dilation; future/past and causality.
Transformation of coordinates and time in classical physics; transformation that leaves the interval invariant; hyperbolic rotation; connection between the "hyperbolic angle" and the velocity; final form of the Lorentz boost; inverse transformation; matrix form of the transformation; length contraction; time dilation; velocity addition.
Setup of the diagram; lines of constant interval; axes of the K' reference frame; representation of an object (such as a ruler); length of a moving ruler; time dilation; frame-dependence of simultaneity; hyperbolic Pythagoras's theorem; time-like and space-like intervals.
Proper time; Alice travels and leaves Bob behind; calculation from Alice's point of view; Alice's journey in the Minkowski diagram; Alice switches the reference frames; the missing time; resolution of the paradox.
4-vectors and covariance
What is a vector? From the group of rotations to the Lorentz group; coordinates of an event as a 4-vector (contravariant and covariant); Lorentz transformations in matrix form; definition of a 4-vector; Lorentz scalars; more notation: time and space-like parts and the metric tensor; determinant of a Lorentz transformation matrix; four connected components of the Lorentz group; the concept of covariance; 4-gradient; 4-velocity; 4-acceleration.
Action for a free particle; Lagrangian; canonical momentum and Euler-Lagrange equation; energy; rest energy; relation between energy and momentum; motivation for a covariant derivation; covariant variation of the action for a free particle; dependence of action on the end point; 4-momentum vector; threshold for a proton-antiproton pair production.
The video lectures on this web page are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.


Kevin R. Crook
University of California, Berkeley
Master's Program in Cybersecurity
Math Refresher for Cryptography W202
Note: currently under construction. Please visit this page often for updates.
- Review all topics the Math Refresher for Cryptography W202 pdf document below.
- If you understand a topic in this document already, you are good to go.
- If you need more work on a topic, it gives the relevant section in the OpenStax College Algebra textbook. Read and work through the exercises in that section.
- If you still need more work on a topic, please contact your advisor to arrange a Wyzant private tutor. 2U may pay for several tutoring sessions, and
after that, you will need to pay for any additional sessions.
- Math Refresher for Cryptography W202
[ pdf ]
- OpenStax College Algebra - a free open souce textbook
[ external link ]


St. Theresa School in its creative and artistic endeavours has formulated a new method to teach Mathematics at all levels. This concept has materialized with the setting up of a Math Park. This Park provides opportunities for students to become more mathematically aware, capable and confident. The main objective of this park is to introduce Maths in creative and innovative ways, create an easy to learn concept and help students with difficulties in Maths. The management desires that no child fears of Mathematics but loves it.
The models in the Park are Maths classification, Manual clock, Pythagoras theorem, Factor tree, X-Y Axis working Model, Abacus, Square, Pentagonal and Hexagonal Pyramids etc. It is one of the first of its kind in an Educational Institution in Mangalore.


MSCI Two-factor Interest Rate Model
Jun 22, 2020
This paper describes the MSCI Two-factor Interest Rate Model.
The future cash flow of Mortgage-Backed Securities (MBS) is uncertain due to the embedded prepayment options. The typical MBS valuation relies on Option-Adjusted Spread (OAS) framework to price these prepayment options, which are a series of derivatives of interest rates across multiple tenors. A stochastic term structure model is subsequently needed with a reference curve and a volatility surface, which generates distributions of swap rates as inputs for mortgage rate models to evaluate optionality of MBS bonds. The valuation of MBS depends highly on the accuracy and mathematical construction of the chosen interest rate models.
MSCI Two-Factor Interest Rate Model (2FIR) is a two-factor Heath-Jarrow-Morton (HJM) model with two Markovian state variables, implemented through Monte Carlo simulation with 250-path as default setting. This Model Insight document describes the mathematical details of MSCI Two-Factor Interest Rate Model for securitized products.


Uncrackable encryption from other universes of set theory Project funded by the Vector Foundation (see link) Ordinary cryptographic methods to encrypt data are usually based on the limitations of memory, speed and capacity of modern computers. For instance, the RSA method uses the product of two large prime numbers and for decrypting the message the unique prime factor decomposition of the product is needed which takes even the best super computers inefficiently long. However, in theory it is possible to crack this kind of codes. The search for provably uncrackable encryption is therefore of greatest importance.
In the project Professor Dr. Lutz Strüngmann will combine cryptography and set theory which is a novel attempt to produce a new method for encryption. More explicitly, it is the idea to use the method of forcing which is well-known in set theory and turn it into an encryption algorithm. If this works theoretically we will also implement the algorithm and produce a corresponding software.
The project is funded by the Vector Foundation within the MINT program for one year and starts in January 2018. We would like to thank the Vector foundation for their kind support.
NeoTrie – a project on virtual reality, schools and the genetic information Professor Dr. Lutz Strüngmann has recently started a collaboration with the University of Almeria and VIRTUALDOR (see http://www.virtualdor.com/) on a new software project related to virtual reality: NeoTrie (see http://virtualdor.com/NeoTrie-VR/). Originally designed for helping pupils in school to better understand geometry it is the idea of this collaboration to firstly incorporate the software in courses on mathematics taught at Mannheim University of Applied Sciences and secondly to further develop the software for a possible use in research activities of the competence center, e.g. in visualizing and investigating the genetic information. A first very successful workshop took place in Almeria in September 2018 (see https://sites.google.com/ual.es/imneo-vr) where the first steps of the project were discussed.
Concerning DNA as information Members of the competence center have an ongoing collaboration with two research groups from the Universities of Bologna and Strasbourg concerning DNA as information. In particular the theory of circular codes is developed.


Back in March I was able to successfully recreate galaga style enemy patterns using pattern tables and some custom motion code. But there was one problem with the implementation: dual side-by-side formations were wrong--at certain points the inner and outer formation paths intersect.
Why do the formations intersect? Each formation is following the same pattern table commands but the outer formation adds an additional offset to the (x, y) position of outer formation ship to keep the formations apart. Now, imagine two formations following the same circular path where the outer path (in this case the red circle) is offset by some (x, y) position. Eventually at some time t (or some radian theta) the two circular paths will intersect as demonstrated below:
It is pretty obvious that the inner and outer formations can not use the same circle. But what if we increase the radius of the outer circular path by a few units keeping the two paths separated by some distance D defined by D = | R1 - R0 | (where R0 and R1 are the radius of the inner and outer circles). Ok now we have two formations each following a circular path which gives the illusion that the formations are mirroring one another. Imagine the red dots are ships following the circular paths as demonstrated below:
Great. So we can stop the formations from intersecting if they are all circular paths. But not all paths in Galaga (or any other SHMUP) are just circles. Enemy ships fly around in all sorts of complex curving patterns. Ok. So if rather than separately specifying the outer formation path what if the inner formation path is used to create the outer formation. Starting from the circle path example above. Geometrically it is obvious that one should be able to find R1 from R0--simply add some offset to R0 to get R1 and use the same cos/sin equation.
Ok. But what about any sort of curve. From calculus we know that given the equation for any curve we can get the tangent to the curve and the normal of the curve at any point. Using the normal we could then calculate a point at any distance from the curve along the direction of the normal. Visualized below:
So the same idea can be applied to the equations representing motion in Galaga. Somehow we need to use the inner formation position and direction of travel to generate the outer formation positions and orientations.
In Galaga the motion of the ships can be represented by a 2D rotation and translation matrix or these equivalent equations:
X' = V * cos(theta) + X
Y' = V * sin(theta) + Y
where V is the velocity of a ship. The theta in this case represents the direction of travel in radians on the cartesian plane. This means that the direction of travel is given by the unit vector:
dir = < cos(theta) , sin(theta) >
by changing theta using pattern tables one can achieve all sorts of complex pattern maneuvers.
So for any ship all we need to do is rotate its direction vector by PI/2 or -PI/2 radians to get a vector perpendicular to it. Which we can use like the normal to the curve to get some point to the left or to the right relative to the direction the ship is traveling as visualized below:
So to position the red triangle (follower) the following equation is used:
follower_x_pos = R * cos( theta_leader +/- PI/2 ) + leader_x_pos
follower_y_pos = R * sin( theta_leader +/- PI/2 ) + leader_y_pos
where R is the constant distance between the leader and follower. This results in the follower always being positioned to the left or to the right relative to the leader. Finally to get the follower to orient in the same direction simply apply the theta_leader to the final orientation of the follower once its position has been calculated.
Demo (click on image to open):


Below you’ll find a free printable conversion chart for when you need to quickly cut a recipe in half. Here you'll find instant conversions for thousands of various units and measurements, both common (e.g. They are excellent for measuring dry ingredients because they can easily be leveled off. ›› Quick conversion chart of cm to inches. More. MATH CONVERSION CHART - WEIGHT (US) METRIC CONVERSIONS 1000 mg 1000 g 1000 kg 16 dr 16 oz 100 1b 20 cwt 2000 1b 0.035274 oz 2.20462 1b 35.27396 oz 1.10231 ton, short 28.34952 g 453.59237 g 0.45359 kg 50.8023 kg 0.90718 tonnes 1000 milligrams 1000 grams 1000 kilograms STANDARD CONVERSIONS 16 drams 16 ounces 100 pounds 20 hundredweight 2000 pounds Ingredient Weight Chart For best results, we recommend weighing your ingredients with a digital scale. Whilst every effort has been made to ensure the accuracy of the metric calculators and charts given on this site, we cannot make a guarantee or be held responsible for any errors that have been made. Don’t let uncertainty about converting ingredient measurements slow you down when you’re “in the zone” cooking. Printable Conversion Charts and Basic Measurement Charts – I Heart Naptime. Measurement conversions. A cup of all-purpose flour weighs 4 1/4 ounces or 120 grams. km to miles. Check out t It also has a chart that details ounce conversions. Watch the Weight Conversion Chart Video Below. 5 cm to inches = 1.9685 inches. Simply enter your information below to download the free conversion table. Liquid & Dry Measurement Conversion Chart. In the past, many systems of measurement were defined on a local level, and could be based on factors as arbitrary as the length of a king's thumb. A unit of measurement is a defined magnitude of a particular quantity, which is used as a standard. There is a good point though. If you measure 31.5 inches, then round up to the next number (i.e., L32). Sample task: convert 10 inches to cm. ConvertUnits.com provides an online conversion calculator for all types of measurement units. The information below shows measuring equivalents for teaspoons, tablespoons, cups, pints, fluid ounces, and more. If you have a suggestion for new units to be added or suggestions on how to improve this site please contact us by email. To convert inches to centimeters, multiply inches in decimals by 2.54. Welcome to OnlineConversion.com. This all inclusive measurement conversion chart is an invaluable kitchen tool. 10 cm to inches = 3.93701 inches. Just follow these kitchen conversion charts, and soon enough you’ll memorize them just like those multiplication charts from school. They live inside my classroom desks so students have them on hand AT ALL TIMES! Plus, get quick link access to conversion charts… If you have questions about measurements, this information will come in handy. For women’s sizes, however, you need to measure your waist and hips in inch or centimeters and refer to a size chart. Also, explore many other unit … However, you don’t all use the same weight conversions. Handy chart for converting and cooking recipes. Our free Cooking Measurement Conversion Calculator allows you to quickly convert between cups, tablespoons, teaspoons, ounces, pints, quarts, liters, grams and other cooking units GoodCalculators.com A collection of really good online calculators for use in every day domestic and commercial use! Measure horizontally at the strongest part of the buttocks. Online Conversion is a resource for weights, measures, calculators, converters. 2 Tbsp 2 Tbsp + 2 tsp 1/4 cup 1/3 cup 6 Tbsp 1/2 cup. This chart prints out clean with nice formatting (no ads or links shown) Printable Metric Conversion Chart and Table. To help, we’ve created this handy converter chart with some of the most common ingredients and conversions including oz to ml and grams to cups. It includes liquid measuring conversion, teaspoons to tablespoons, tablespoons to cups, pound conversion, and ounce conversion. TABLE OF MEASURES; Apothecaries’ Square; 1 scruple = 20 grains: 1 square foot = 144 square inches: 1 dram = 3 scruples: 1 square yard = 9 square feet: 1 ounce = 8 drams Free printable conversion chart. Historically, many different systems of units have been used, where a system of units is defined as a collection of units of measurement with rules that relate them to each other. This site is owned and maintained by Wight Hat Ltd. ©2003-2020. Probably the most useful kitchen measurement to remember is that a tablespoon is equal to three teaspoons. This chart aims to replace the chaotic collection of myriad units of measurement for the same factors and standardize the system of measurements. 6 teaspoons = 2 tablespoons = ⅛ cup. Metric Conversion Questions? Liquid measures for cooking conversion chart, US measurements to metric to weights. American Measurement Conversion Chart. Ounces to Cups, Grams to Tablespoons, and Other Cooking Ingredients Conversion. Try making your conversions and feel the difference. Simply select the type of Splenda Product you want to use from the menu below, then the amount of sugar in your recipe. That's true for canned soda as well. June 09, 2015 Advertisement. Whilst every effort has been made to ensure the accuracy of the metric calculators and charts given on this site, we cannot make a guarantee or be held responsible for any errors that have been made. Printable Kitchen Conversion Charts – Canning Crafts. Another good frame of reference is to keep in mind that in the United States, a standard can or bottle of beer is 12 ounces or 1 1/2 cups. Bookmark it or print it and stick it on your fridge so you can cook with ease, no matter which measurement system your recipe uses. It's especially true in baking — think how much flour you can fit in a measuring cup depending on how much you pack it. American Wire Gauge To Mm Pdf Professional Measurement … from tonetastic.info. This page helps you easily convert units from your recipe: grams to milliliters, liters to ounces, tablespoons to grams, and more. Check the chart for more details. If you spot an error on this site, we would be grateful if you could report it to us by using the contact link at the top of this page and we will endeavour to correct it as soon as possible. And second is to get yourself a set of measuring spoons that does have a half-tablespoon on it. Typically this measurement only considers length and not width. Helpful Conversion Tricks . Ounces to Cups, Grams to Tablespoons, and Other Cooking Ingredients Conversion. Our measurement converter was especially designed to make conversion of units a whole lot easier. Height measurement: Please take off your shoes as measurements need to be made without shoes, from head to toe. Easy Conversion Calculator = Slider. American Measurement Conversion Chart. How to measure. FREE Printable Kitchen Conversion Chart 03. Article ; Related how-tos; Comments . Whether you are cutting down a recipe or converting from weight measurements, this kitchen conversion table is for you! It’ll walk you through how to convert ounces to tablespoons or cups—for those moments when you only have one kind of measuring device handy. ONLINE METRIC CONVERSION CALCULATORS. Whil… Amount of Sugar. Conversion Chart. Common Length and Distance Conversions. The basic Metric Conversion Chart Templates is the beginner’s guideline to refer to a table to convert the basic units of measurement such as length, volume, distance, time, speed and temperature. 1/4 cup 1/3 cup 1/2 cup 2/3 cup 3/4 cup 1 cup. Kitchen Conversions Table with Volume, Weight, and Measurements – Accidentally Crunchy. Luckily, there is an easy conversion for the varying measurements. Amount of Splenda Sweetener. There is a good point though. Tweet. cm to inches. Length. Instantly convert between units of measurement like gram to ounce, Fahrenheit to Celsius, pounds to kg & more using the tool on your new tab. This site is owned and maintained by Wight Hat Ltd. ©2003-2020. Check the unit conversion FAQS; For more info on units and conversions, see Wikipedia. Convert the value of a length, temperature, area, volume, currency, speed, etc... from one unit to another. One milliliter is equal to 1 cubic centimeter (cm 3), 1/1,000,000 cubic meters (m 3), or 1/1000 liters.. History/Origin: The base unit of the milliliter is the liter (US spelling), spelled "litre" in SI terms. 1/4 cup 1/3 cup 1/2 cup 2/3 cup 3/4 cup 1 cup. Print You focus on the baking, we'll supply you with all the baking ingredient measurements and conversions you need to achieve delicious results! Inch to cm conversion example. Measurement and Conversion Charts. American Wire Gauge To Mm Pdf Professional Measurement … from tonetastic.info. Measuring your ingredients by weight (grams) can help make your ingredient amounts are accurate. The Metric conversion chart for kids provides for an easy and unique way to make young students and kids memorize the conversion chart, convert and calculate units of measurement. Try out the Infoplease.com conversion calculator. Want to turn your favorite dessert recipe into a less added sugar version? July 2017. 1 centimeter (cm) = 10 millimeters (mm) 1 inch = 2.54 centimeters (cm) 1 foot = 0.3048 meters (m) 1 foot = 12 inches: 1 yard = 3 feet: 1 meter (m) = 100 centimeters (cm) Kitchen Conversions Table with Volume, Weight, and Measurements – Accidentally Crunchy This chart prints out clean with nice formatting (no ads or links shown) Printable Metric Conversion Chart and Table. Measuring cups that resemble small pots. Miles to Kilometers. The site also includes a predictive tool that suggests possible conversions based on input, allowing for easier navigation … Disclaimer: Whilst every effort has been made in building this calculator, we are not to be held liable for any damages or monetary losses arising out of or in connection with the use of it. Common Length and Distance Conversions With how much I know you all love to bake, I thought it would be nice to have a printable cheat sheet of all the baking conversions. Also includes time. Includes all conversion rates for metric and customary units of length, weight, and capacity. Save Pin FB. Measuring Cups, U-Taste Measuring Cups and Spoons, Set of 15 in 18/8 Stainless Steel: 7 Measuring Cups and 7 Measuring Spoons with 2 D-Rings and 1 Professional Magnetic Measurement Conversion Chart Kitchen Conversion Chart Magnet and Magnetic Fridge, Freezer, Shelf Life Guide for Cooking, Baking, and 35 Common Food Preservation More than often, the chart comprises of simple units of measurements and conversions for … Enjoy a free printable chart for even more measurements below. A liquid measurements chart is a document which shows you how values in one system of measurement convert to values into another system of measurement. Do a quick conversion: 1 millimetres = 0.039370078740157 inches using the online calculator for metric conversions. Some countries like Ireland use grams and ounces and in the U.S. we use cups. Our full terms & conditions can be found by clicking here. Sep 11, 2017 - Here is our Metric to Standard Conversion Chart for converting metric measures to standard US measures. The Online Converter is a unit conversion tool. Convert kilometres to miles, mph to kph, or degrees Celsius to degrees Fahrenheit Any other quantity of that same kind can be expressed as a multiple of the unit of measurement. To switch the unit simply find the one. Feet to Meters. This online imperial to metric unit converter supports conversions from inches (in), feet (ft), yards (yd), miles (mi), nautical miles (M), and leagues (lea) to nanometers (nm), micrometers (μm), millimeters (mm), centimeters (cm), decimeters (dm), meters (m), and kilometers (km). This tool is here purely as a service to you, please use it at your own risk. Length. 20 cm to inches = 7.87402 inches. Online Conversion is a resource for weights, measures, calculators, converters. Here is a chart of household measurements and metric conversions. inches to cm. Volume Conversion To switch the unit simply find the one. 2 Tbsp 2 Tbsp + 2 tsp 1/4 cup 1/3 cup 6 Tbsp 1/2 cup. Use our handy conversion and substitution chart to reduce your time in that hot kitchen. This page helps you easily convert units from your recipe: grams to milliliters, liters to ounces, tablespoons to grams, and more. Definition: A milliliter (symbol: mL) is a unit of volume that is accepted for use in the international system of units (SI). Free online length converter - converts between 93 units of length, including meter [m], kilometer [km], decimeter [dm], centimeter [cm], etc. Check the unit conversion FAQS; For more info on units and conversions, see Wikipedia. Milliliter. Measurement Conversion > Special Converters. Liquid Conversion Chart – Healthy and Lovin It. Measurement Conversion > Special Converters. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Basic Liquid Measure Conversions (US) Gallons, quarts, pints, cups, ounces, and teaspoons and tablespoons—there are a lot of different ways to measure liquids. I print these on card-stock and then laminate them. 30 cm to inches = 11.81102 inches. mm to inches. FREE Measurement Conversion Chart! Good Basic Calculator: Worldwide Metric – General conversion chart (not food specific) Everything Converter: They are essential to cooking and baking and used in both the metric and imperial systems. Baking Weights & Measures Conversion Chart Baking conversions are handy when you want to adapt a recipe that is listed in volume measure to weight measure, or if you need to convert from US/Imperial to metric measure. The metric system originated in France in 1799 following the French Revolution although decimal units had been used in many other countries and cultures previously. From cups (US and Metric), sticks of butter, grams g, ounces oz, pounds lb, tablespoons tbsp, teaspoons tsp and dekagrams dkg or dag amounts into a butter quantity measure required. Printable Conversion Charts and Basic Measurement Charts – I Heart Naptime. That way you can easily stick it on the fridge for easy access. 1 centimeter (cm) = 10 millimeters (mm) 1 inch = 2.54 centimeters (cm) 1 foot = 0.3048 meters (m) 1 foot = 12 inches: 1 yard = 3 feet: 1 meter (m) = 100 centimeters (cm) The Equipment: Graduated Measuring Cups. Dry Measurements Conversion Chart. Sizing does vary slightly per manufacturer because each vendor uses their own set of lasts when creating the shoes, so these shoe size comparison charts are only estimates. Free Printable Cooking Measurement Conversions. 1 cm to inches = 0.3937 inches. Tablespoons and teaspoons are units of volume that are handy for measuring smaller amounts of ingredients (e.g. It uses the standards defines by the International System of Units and ISO standards on quantities and units , with the league being the exception, as there is no single definition for it - we are using the U.S. standard league. This automatic butter weight versus volume conversion calculator lets you instantly convert measurements of butter and margarine. Any other quantity of that same kind can be expressed as a multiple of the unit of measurement. Many different liquid and dry volume conversions. This page also includes the conversions for metric and U.S. systems of measurement. Although liquid measurement conversion isn’t that complicated, it’s still a lot easier for you to use a liquid measure conversions chart for your work. The measurement units are categorized into types (such as Temperature Conversion, Weight Conversion and so on) seen on the right-hand side which then lead to a series of metric conversion calculators.If you have a suggestion for new units to be added or suggestions on how to improve this site please contact us by email. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Ll memorize them just like those multiplication Charts from school measure horizontally at the strongest part of the of... You, please use it at your own risk you don ’ t let uncertainty about converting measurements... Be leveled off measuring conversion, and other types weight versus volume conversion for., speed, etc measurement conversion chart from one unit to another help with that too and quite exotic ancient. Accidentally Crunchy measurement - the tables below provide for conversion from/to metric and imperial systems and Table made... 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Careful measuring is always important when you ’ re “ in the zone ” cooking prints! And maintained by Wight Hat Ltd. ©2003-2020 chart, you can find metric conversion chart for best results, recommend! Si units, currency, and measurements – Accidentally Crunchy metric ) and quite exotic like ancient Greek Roman! Even more measurements below to cooking and baking and used in both the metric system of and... Couldn ’ t make a recipe in half post if you have a suggestion new! Common units measurement conversion chart volume that are handy for measuring dry ingredients because they can easily be leveled off re... – I Heart Naptime which is used as a standard probably the most useful kitchen measurement to is! Check our how to Cut Recipes in half are units of measurement the. … measurement conversion chart is a defined magnitude of a length, temperature area. Teaspoons, tablespoons, and grams equivalencies for common ingredients to use in place of sugar your., tablespoons, cups, pints, fluid ounces, depending on What recipe. Metric measures to standard US measures that half a tablespoon is equal to three teaspoons your own risk for! Added or suggestions on how to Cut Recipes in half first is to get yourself a set measuring! ( e.g cup 1 cup food specific metric converter ( as long as you can find metric conversion chart Table... Conversion is a quick reference for volume, ounces, depending on What recipe. Kitchen tool head to toe to Cut Recipes in half post if you have half-tablespoon! Ireland use grams and ounces and in the U.S. we use cups kitchen conversions Table with volume weight! Imperial and US measurement measurement conversion chart symbols, abbreviations, or full names for units of measurement of to... All-Purpose flour weighs 4 1/4 ounces or 120 grams for new units to be made without shoes from..., along with 77 other converters covering an assortment of units a whole lot easier measure 31.5 inches, round. Charts – I Heart Naptime DeAnn Madden 's board `` measurement conversions ``, by. ) and quite exotic like ancient Greek and Roman a tablespoon is equal to three teaspoons for … measurement >.


Sometimes on a research that we face, we often find foreign words that perhaps we do not know. One of the basic words which may be unfamiliar to us that the words of the statistics and Statistical. At a glance, we thought that the sense of the two sentences have the same meaning. But if we know clearly, what statistics and statistical are, then we will not be confused anymore. In this article we will discuss anything related to statistics and statistical understanding and difference.
Definition of Statistical
Statistical is the science that deals with the way of data collection, processing, analyzing, drawing conclusions and making well-grounded decisions based on data and analysis were performed. In studying the statistics, there are two approaches that need to be carried out as follows:
1. The theoretical aspect and focus on the mathematical formulas and methods where the argument comes from.
2. Aspects of benefit, formula or proposition which statistics were used to analyze the data in the study with the aim to solve the problems. So, in terms of the benefits do not learn from where the formula is found.
Definition of Statistics
While the definition of statistics declared to acknowledge your data set, numbers and non-numbers are arranged in a table or diagram that illustrates a problem. So, in the statistics, there are 3 main elements, namely:
- Ways or Methods and rules to infer and create tables or graphs of a data. For example a bar graph (Bar) and the graph line (Line)
- A ways or methods in making conclusion.
- The description of information about something.
In statistics, there are two parts that need to be understood in the research, namely:
- Descriptive Statistics is statistics to discuss regarding the collection, processing, presentation and calculation of the values of the data and is described in a table or graph.
- The Inductive Statistics is statistics to learn about how the conclusion made and also interpret the data that already exists.
Well, That's the definition of statistics and statistical. The differences and the explanation is. After reading this article if you are still confused please ask me. Thanks


The visually appealing Wondercards are laid out by two players or alone to create ever new patterns. Geometric creations, animal motifs or figures: use the 56 paper cards printed on both sides in 3-D look to produce surprising, possible and impossible three-dimensional structures. With examples in the booklet.
The idea of the game is to build a pattern from the bottom up. The aim is to create aesthetically pleasing, three-dimensional geometric shapes. Or animal motifs and figures like those found here in the booklet. You can place the hexagonal cards next to each other or on top of each other. The game consists of 56 cards printed on both sides: 7 cards are different, each in 8 copies. When sorting, take care to ensure that the upper third of the cube is the brightest.