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# Into Math Grade 3 Module 8 Lesson 4 Answer Key Solve Two-Step Problems

We included HMH Into Math Grade 3 Answer Key PDF Module 8 Lesson 4 Solve Two-Step Problems to make students experts in learning maths.

## HMH Into Math Grade 3 Module 8 Lesson 4 Answer Key Solve Two-Step Problems

I Can write equations using the four operations with an unknown to solve two-step problems.

Step It Out

1. Mr. Cole paints airplanes. He has 6 gallons of paint in a sprayer. He also has 7 equal¬sized buckets of paint. The total amount of paint he has is 41 gallons. How much paint is in each bucket?

The paint in each bucket is 5 gallons.

Explanation:
Let the paint in each bucket be P. As he has 6 gallons of paint in a sprayer and also has 7 equal¬sized buckets of paint. So the equation will be
6+7P = 41,
7P = 41-6,
7P = 35,
P = 35÷7
P = 5.
So the paint in each bucket is 5 gallons.

Look at the information in the problem and plan steps to solve.
A. Circle the numbers you know.
B. Complete the bar model to show how many gallons of paint Mr. Cole has. Use p for the unknown number.

P = 35 gallons.

Explanation:
The equation is 6+P = 41 where P is
P = 41-6
= 35 gallons.

C. Step 1: How can you use an equation to find the amount of paint in all the buckets?

The amount of paint in all the buckets is 35 gallons.

Explanation:
The equation to find the amount of paint in all the buckets
6+P = 41,
P = 41-6,
= 35 gallons.

Use a ___ equation to find the amount of paint in all the buckets. Solve for p.

P = 35 gallons.

Explanation:
Here, we have used subtraction to find the amount of paint in all the buckets. The equation is 41-6 = P, where P is 35 gallons.

Turn and Talk How can you use p to find the amount of paint that is in each bucket?

We can use p to find the amount of paint that is in each bucket by using subtraction.

Step It Out

D. What numbers do you know? What do the numbers represent? ___________

As 7 equal-sized buckets have 35 gallons total in the buckets.

E. Complete the bar model to show how many gallons of paint Mr. Cole has in each bucket. Use g for the unknown number.

P = 35 gallons.

Explanation:
Here, we have used g for the unknown numbers. So Mr. Cole has 7×5 which is 35 gallons.

You can use your solution from Step 1 in order to find how much paint is in each bucket for Step 2.
F. Step 2: How can you use an equation to find the amount of paint in each bucket?
Now use a ___ equation to find the amount of paint in each bucket. Solve for g.

The amount of paint in each bucket is 7 buckets.

Explanation:
The equation to find the amount of paint in each bucket is 35÷7 which is 5 buckets.

G. What operations can you use to check the reasonableness of your answers? ________________

The operations that we have used are division, subtraction.

Turn and Talk What is another strategy you could use to solve the problem?

2. Brie is painting a fence. The fence has 7 sections that are eaçh 9 feet long and one 17-foot-long section. Use steps and equations to find the total length of the fence.

A. Think about the information you know.

The total length of the fence is 80 feet.

Explanation:
Given that Brie is painting a fence and the fence has 7 sections that are eaçh 9 feet long which is 7×9 = 63 feet long and one 17-foot-long section. So the total length of the fence is 63+17 which is 80 feet.

Step 1: Draw a bar model and write a multiplication equation to find the total length of the 7 same-length sections of fence. Use s for the unknown number. Then solve.

7×9 = 63.

Explanation:
The multiplication equation is 7×9 = s, where s is 7×9 =63.

Think: What do I need to find? What information do I need to use’ What steps and equations can I use to find the answer?

The equation is 7×9 =63.

Explanation:
Here, we have found the total length of the fence and the equation is 7×9 =63.

Step 2: Draw a bar model and write an addition equation to find the total length of the fence. Use f for the unknown number. Then solve.

f is 80.

Explanation:
As length of 7 sections is 63 and length of extra section is 17. So f is 80.

Turn and Talk How can you use division and subtraction to check your answer?

Here, when we add the difference to the smaller number and if the sum equals to the larger number then our answer is correct.

Check Understanding Math Board

Question 1.
Mike buys 7 packs of erasers. Each pack has 4 erasers. He also has 5 loose erasers. How many erasers does he have now? Write two equations with letters for the unknowns. Solve.
The two equations are 7×4 = e where e is 28 erasers and 28-5 = l where l is 23 erasers.

Explanation:
Given that Mike bought 7 packs of erasers and each pack has 4 erasers which is 7×4 = e where e is 28 erasers. As he looses 5 erasers, he have 28-5 = l where l is 23 erasers.

Model with Mathematics Write two equations with letters for the unknowns. Solve. You can use a visual model to help.

Question 2.
Lanni checks out 6 sets of books from the library. Each set has the same number of books. She reads 7 of the books, and she has 11 books left to read. How many books are in each set?

There will be 3 books in each set.

Explanation:
Given that Lanni checks out 6 sets of books from the library and each set has the same number of books. As she reads 7 of the books, and she has 11 books left to read. So the equation will be 11+7 = b where b is 18 books. As there is a set of 6 books, so the equation will be 18÷6 = s where s is 3 books. So there will be 3 books in each set.

Question 3.
Sal cuts a piece of yarn for a project into 3 equal-length pieces. Then he cuts 5 inches off one of the pieces, leaving 4 inches of yarn. What was the length of the original piece of yarn?
The length of the original piece of yarn is 27 inches.

Explanation:
Given that Sal cuts a piece of yarn for a project into 3 equal-length pieces and then he cuts 5 inches off one of the pieces, leaving 4 inches of yarn. So the equation will be 4+5 = p where p is 9 in. The length of the original piece of yarn is 3×9 which is 27 inches.

Question 4.
At the lake, Jansen can rent a canoe for $5 each hour plus a one time fee of$7. He rents a canoe for 3 hours. How much does it cost Jansen to rent the canoe?
The equation will be 3×5 = r where r is $15. The equation will be 15+7= c where c is$22.

Explanation:
Given that Jansen can rent a canoe for $5 each hour plus a one time fee of$7 and he rents a canoe for 3 hours. So the equation will be 3×5 = r where r is $15. And the cost of Jansen to rent the canoe, so the equation will be 15+7= c where c is$22.

Model with Mathematics Write two equations with letters for the unknowns. Solve. You can use a visual model to help.

Question 5.
Jason has $30 to buy books. His aunt gives him$10 more. If each book that he wants costs $8, how many books can Jason buy? Answer: Jason bought 5 books. Explanation: Given that Jason has$30 to buy books and his aunt gives him $10 more, so now Jason will have$30+$10 = t where t is$40. As each book that he wants costs $8, so the number of books can Jason bought is$40÷$8 = b where b is 5 books. Question 6. Beth pays a total of$20 to rent skates and go ice skating for 2 hours. If the cost for the skate rental is $6, what is the cost for each hour of ice skating? Answer: The cost for each hour of ice skating is$7.

Explanation:
Given that Beth pays a total of $20 to rent skates and go ice skating for 2 hours and if the cost for the skate rental is$6, so the equation will be $20-$6 = s where s is $14. As Beth goes ice skating for 2 hours, so the equation will be 14÷2 = r where r is 7. So the cost for each hour of ice skating is$7.

Question 7.
Sean scores 21 points during a basketball game. One of his baskets is a three-point shot. He scores the rest of his points with two-point baskets. How many two-point baskets does Sean score during the game?
The number of two-point baskets does Sean score during the game is 9.

Explanation:
Given that Sean scores 21 points during a basketball game and one of his baskets is a three-point shot, so the equation will be 21-3 = p  where p is 18. As he scores the rest of his points with two-point baskets, so the two -point baskets does Sean score during the game is 18÷2 = b where b is 9.

Question 8.
Open Ended Write a two-step word problem that can be solved using two equations with different operations. Then write the two equations and solve the problem.

Model with Mathematics Write two equations with letters for the unknowns. Solve. You can use a visual model to help.

Question 9.
Mr. Ed’s school bus route has 8 stops. The same number of students is waiting at each stop. So far, he has picked up 20 students. There are 12 students waiting to be picked up. How many students are at each bus stop?
There are 4 students at each bus stop.

Explanation:
Given that Mr. Ed’s school bus route has 8 stops and the same number of students is waiting at each stop. So far, he has picked up 20 students and there are 12 students waiting to be picked up. So the equation will be 20+12 = b where b is 32. As there are 8 stops, so the equation will be 32÷8 = 4. So there are 4 students at each bus stop.

Question 10.
Xavier watches an old TV show by streaming it on a computer. The show had 5 seasons. Xavier has watched 7 episodes. He has 43 more episodes to watch. If each season has the same number of episodes, how many episodes does each season have?
The number of episodes does each season had is 5 episodes.

Explaantion:
As there are 5 seasons and Xavier has watched 7 episodes and he 43 more episodes to watch. So the equation is 43+7 = t where t is 50 episodes. As the show had 5 seasons, so the equation will be 50÷5 = e where e is 5. So the number of episodes does each season had is 5 episodes.

Question 11.
Jamie’s plant grows 3 inches each week for three weeks. During the fourth week, it grows 5 inches. How many inches does Jamie’s plant grow over the four weeks?

The number of inches does Jamie’s plant grow over the four weeks is 14 inches.

Explanation:
Given that Jamie’s plant grows 3 inches each week for three weeks and during the fourth week, it grows 5 inches. So the number of inches does Jamie’s plant grow over the four weeks is, the equation is 3×3 = p where p is 9. As it grows 5 inches, so the equation will be 9+5 = g where g is 14 inches.

Question 12.
Natasha buys 2 bunches of tulips. Each bunch has the same number of tulips. She puts the tulips in a vase with 5 roses. There are now 17 flowers in the vase. How many tulips are in each bunch?


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# Best configuration for 400W resistive heating device

Suppose I wanted to heat a large aluminum disk with joule heating. I would use one of its flat faces, which has a radius of about a foot (30cm, SA = 0.28 sq m) and enough resistors to dissipate 400W of energy.

If I use 12V across a circuit:

P = V^2/R

P = 320W

V = 12V

R = 0.45 ohms

and the current for such a circuit would be I = V/R = 12/0.45 =~ 27 A (which seems very high)

Is this correct? Is this kind of set up "safe"?

--

And if you are interested:

If you were to build this circuit, how would you arrange resistors of any R value while:

1. Distributing the heat across a large (1ft radius) surface
2. Remaining under 100W per resistor
3. Maintaining R=.45 ohm total circuit resistance as closely as possible

Here is one attempt of mine, using 10 parallel 4ohm resistors (.4 ohm circuit resistance):

simulate this circuit – Schematic created using CircuitLab

If you think it is safe/worthwhile, I am open to setups with higher voltage, but for some reason 12v seems to be the standard.

Also if I wanted to use PWM (DC) to control the output of this circuit, what would be the cheapest and most efficient way to set that up with Arduino?

My disk is about 10.8 kg with specific heat capacity (Al) = 0.9 J/gC

So I chose 320W because this would allow me to heat by 10 deg C every 5 min:

0.9*10800*10/320 =~ 300 sec = 5 min

• It'll work. Main safety issues come from the heat and the high current. If you're paranoid about safety or have difficulty finding 30A rated fuses, switches, etc, you could wire pairs of resistors as separate 6A circuits. Aug 7 '15 at 11:00

Personally, I would (if possible) have a groove milled into the disk and use a custom tubular heater that runs from the mains. I'd specify the heater at about 1.5x the power I think is required.

Tubular heaters are like the elements in a domestic oven- swaged with powdered ceramic inside- and are very long-lived and robust.

Easily controlled using an SSR- 500W would only be a few amperes.

• This would be a nice solution, but probably more expensive than I have funding for. They wouldn't happen to fabricate these for $100 or less would they? – Ryan Aug 7 '15 at 4:32 • I've been looking around for hours and the resistors are one of the cheapest and easiest to assemble, but at the cost of safety I suppose. Another solution I'm looking at is this: protovision.com/2011/09/25/milled-reprap-heated-bed-pcb But I don't have access to a CNC mill at the moment. – Ryan Aug 7 '15 at 4:40 • What country are you in? Electric skillet looks like a goodish fit. 10 Watt ceramic resistors would also be easy enough and no great safety hazard and moderate cost. | Also possible with some insulating head scratching needed is NiChrome wire of suitable rating. This can be obtained in a range of thicknesses to suit current and power needed. Aug 7 '15 at 11:54 • Custom calrod elements were not expensive last time I bought them, should be well under$100 for a couple. Aug 7 '15 at 12:23
• Ok thanks for the help everyone. I will try out a prototype with just a couple resistors and see how it goes, but since you mention the custom heating element, I may switch plans if I don't like the resistors. (btw I am in the US)
– Ryan
Aug 7 '15 at 21:17

What Spehro says re mains encapsulated element BUT it's called an electric skillet (electric frypan in NZ) - obtained at garage sales / yard sales (what country are you in) for a few dollars or somebody's attic or dumpster for free. Or Amazon / Walmart new from about $30. Typically 1000 Watt. Some are 1500 Watt, maybe more. PWM to suit. Heat distribution evenness may be an issue but a copper disk on pan base should help. Get friendly machine shop to remove sides OR use a hacksaw, file and effort if necessary. This gives you a metal heated base with sealed embedded element able to meet UL & other consumer safety specs. Amazon selection Amazon cheapie -$12.99!!! but at 7" square may not easily spread heat evenly enough.

____________________________

Walmart $low to$ high

• @Ryan But it still gets zero votes :-). Low or mo cost, robust, waterproof, available, minimal effort, ... :-). Aug 8 '15 at 14:57

format: plain


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# How do you solve by completing the square for y=3x^2+24x+8?

$y = 3 \left({x}^{2} + 8 x + 16\right) - 48 + 8$
$y = 3 {\left(x + 4\right)}^{2} - 40$
${\left(x + 4\right)}^{2} = \frac{40}{3} \to \left(x + 4\right) = \pm \frac{2 \sqrt{10}}{\sqrt{3}}$=$\pm \frac{2 \sqrt{30}}{3}$
$x = - \frac{24}{6} \pm \frac{2 \sqrt{30}}{18} = - 6 \pm \frac{\sqrt{30}}{9}$


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# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020407

## Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems

 1 Babeş-Bolyai University Cluj-Napoca, Str. Kogalniceanu 3, 400084 Cluj-Napoca, Romania 2 Romanian Institute of Science and Technology, 400022 Cluj-Napoca, Romania and, Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, 400320 Cluj-Napoca, Romania 3 Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany

* Corresponding author: A. Viorel

Received  May 2020 Revised  September 2020 Published  December 2020

The present work deals with the numerical long-time integration of damped Hamiltonian systems. The method that we analyze combines a specific Strang splitting, that separates linear dissipative effects from conservative ones, with an energy-preserving averaged vector field (AVF) integrator for the Hamiltonian subproblem. This construction faithfully reproduces the energy-dissipation structure of the continuous model, its equilibrium points and its natural Lyapunov function. As a consequence of these structural similarities, both the convergence to equilibrium and, more interestingly, the energy decay rate of the continuous dynamical system are recovered at a discrete level. The possibility of replacing the implicit AVF integrator by an explicit Störmer-Verlet one is also discussed, while numerical experiments illustrate and support the theoretical findings.

Citation: Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020407
##### References:

show all references

##### References:
The (eventual) exponential decay of the potential energy (a) as well as state-spaces trajectories (b) for decreasing step sizes $h = 1 \text{ and } 0.1$, both with identical initial conditions $u_0 = 0.01, v_0 = 0$. The double-well potential is depicted in c)
Capturing an energy plateau: a) the AVF Splitting algorithm (black) compared to the trapezoidal rule (blue) for $h = 1$ (and benchmark Runge-Kutta (red)); b) AVF Splitting algorithm (black) compared to the conformal symplectic algorithm (28) (green) for $h = 0.1$. The contour lines of the nonconvex potential (33) are depicted in c)
Erroneous total energy oscillations: a) AVF Splitting algorithm (black) compared to the trapezoidal rule (blue) for $h = 0.1$ (and benchmark Runge-Kutta (red)); b) AVF Splitting algorithm (black) compared to the conformal symplectic algorithm (28) (green) for $h = 0.01$. The contour lines of the Rosenbrock potential are depicted in c)
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2019 Impact Factor: 1.338


text:
#### Volume 13, issue 1 (2013)

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Derived $A_{\infty}$–algebras in an operadic context

### Muriel Livernet, Constanze Roitzheim and Sarah Whitehouse

Algebraic & Geometric Topology 13 (2013) 409–440
##### Abstract

Derived ${A}_{\infty }$–algebras were developed recently by Sagave. Their advantage over classical ${A}_{\infty }$–algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived ${A}_{\infty }$–algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad $d\mathsc{A}s$ encoding bidgas, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad ${A}_{\infty }$ as a resolution of the operad $\mathsc{A}s$ encoding associative algebras. We further show that Sagave’s definition of morphisms agrees with the infinity-morphisms of $d{A}_{\infty }$–algebras arising from operadic machinery. We also study the operadic homology of derived ${A}_{\infty }$–algebras.

##### Keywords
operads, $A_{\infty}$–algebras, Koszul duality
##### Mathematical Subject Classification 2010
Primary: 16E45, 18D50
Secondary: 18G55, 18G10
##### Publication
Received: 8 June 2012
Revised: 10 September 2012
Accepted: 4 October 2012
Published: 6 March 2013
##### Authors
 Muriel Livernet Université Paris 13 Sorbonne Paris Cité CNRS, UMR 7539 99 avenue Jean-Baptiste Clément 93430 Villetaneuse France http://www.math.univ-paris13.fr/~livernet/ Constanze Roitzheim School of Mathematics, Statistics and Actuarial Science University of Kent Cornwallis Canterbury CT2 7NF UK http://www.kent.ac.uk/smsas/personal/csrr/ Sarah Whitehouse School of Mathematics and Statistics University of Sheffield Hicks Building Sheffield S3 7RH UK http://www.sarah-whitehouse.staff.shef.ac.uk/


text:
Remark 68.19.3 (Variant for complete discrete valuation rings). In Lemmas 68.19.1 and 68.19.2 it suffices to consider complete discrete valuation rings. To be precise in Lemma 68.19.1 we can replace condition (3) by the following condition: Given any commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{-->}[ru] & Y }$

where $A$ is a complete discrete valuation ring with fraction field $K$ there exists at most one dotted arrow making the diagram commute. Namely, given any diagram as in Lemma 68.19.1 (3) the completion $A^\wedge$ is a discrete valuation ring (More on Algebra, Lemma 15.43.5) and the uniqueness of the arrow $\mathop{\mathrm{Spec}}(A^\wedge ) \to X$ implies the uniqueness of the arrow $\mathop{\mathrm{Spec}}(A) \to X$ for example by Properties of Spaces, Proposition 65.17.1. Similarly in Lemma 68.19.2 we can replace condition (3) by the following condition: Given any commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Y }$

where $A$ is a complete discrete valuation ring with fraction field $K$ there exists an extension $A \subset A'$ of complete discrete valuation rings inducing a fraction field extension $K \subset K'$ such that there exists a unique arrow $\mathop{\mathrm{Spec}}(A') \to X$ making the diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K) \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A') \ar[r] \ar[rru] & \mathop{\mathrm{Spec}}(A) \ar[r] & Y }$

commute. Namely, given any diagram as in Lemma 68.19.2 part (3) the existence of any commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(L) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K) \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(B) \ar[r] \ar[rru] & \mathop{\mathrm{Spec}}(A) \ar[r] & Y }$

for any extension $A \subset B$ of discrete valuation rings will imply there exists an arrow $\mathop{\mathrm{Spec}}(A) \to X$ fitting into the diagram. This was shown in Morphisms of Spaces, Lemma 66.41.4. In fact, it follows from these considerations that it suffices to look for dotted arrows in diagrams for any class of discrete valuation rings such that, given any discrete valuation ring, there is an extension of it that is in the class. For example, we could take complete discrete valuation rings with algebraically closed residue field.

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text:
## Abstract and Applied Analysis

### On Certain Subclass of Harmonic Starlike Functions

A. Y. Lashin

#### Abstract

Coefficient conditions, distortion bounds, extreme points, convolution, convex combinations, and neighborhoods for a new class of harmonic univalent functions in the open unit disc are investigated. Further, a class preserving integral operator and connections with various previously known results are briefly discussed.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 467929, 7 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412605989

Digital Object Identifier
doi:10.1155/2014/467929

Mathematical Reviews number (MathSciNet)
MR3198195

Zentralblatt MATH identifier
1317.30016

#### Citation

Lashin, A. Y. On Certain Subclass of Harmonic Starlike Functions. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 467929, 7 pages. doi:10.1155/2014/467929. https://projecteuclid.org/euclid.aaa/1412605989

#### References

• J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae: Series A I. Mathematica, vol. 9, pp. 3–25, 1984.
• J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 470–477, 1999.
• H. Silverman and E. M. Silvia, “Subclasses of harmonic univalent functions,” The New Zealand Journal of Mathematics, vol. 28, no. 2, pp. 275–284, 1999.
• H. Silverman, “Harmonic univalent functions with negative coefficients,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 283–289, 1998.
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text:
Tracking Buildings 2020

Cooling

More efforts needed

Tracking progress 2020

Energy demand for space cooling has more than tripled since 1990, making it the fastest-growing end use in buildings. Space cooling was responsible for emissions of about 1 GtCO2 and nearly 8.5% of total final electricity consumption in 2019. While highly efficient AC units are currently available, most consumers purchase ones that are two to three times less efficient. To put cooling on track with the SDS, energy efficiency standards need to be implemented to improve AC energy performance more than 50% by 2030. Together with improved building design, increased renewables integration and smart controls, this measure would cut space cooling energy use and emissions and limit the power capacity additions required to meet peak electricity demand.
Next Lighting


text:
• J. Stat. Distrib. App. Pub Date : 2020-09-07
Mohammad A. Aljarrah; Felix Famoye; Carl Lee

A new generalized asymmetric logistic distribution is defined. In some cases, existing three parameter distributions provide poor fit to heavy tailed data sets. The proposed new distribution consists of only three parameters and is shown to fit a much wider range of heavy left and right tailed data when compared with various existing distributions. The new generalized distribution has logistic, maximum

更新日期：2020-09-08
• J. Stat. Distrib. App. Pub Date : 2020-09-05
Jose H. Guardiola

Today, data mining and gene expressions are at the forefront of modern data analysis. Here we introduce a novel probability distribution that is applicable in these fields. This paper develops the proposed spherical-Dirichlet distribution designed to fit vectors located at the positive orthant of the hypersphere, as it is often the case for data in these fields, avoiding unnecessary probability mass

更新日期：2020-09-07
• J. Stat. Distrib. App. Pub Date : 2020-08-28
Marianne Müller

To compare fit statistics for the Rasch model based on estimates of unconditional or conditional response probabilities. Using person estimates to calculate fit statistics can lead to problems because the person estimates are biased. Conditional response probabilities given the total person score could be used instead. Data sets are simulated which fit the Rasch model. Type I error rates are calculated

更新日期：2020-08-29
• J. Stat. Distrib. App. Pub Date : 2020-08-17

At this juncture when mixed models are heavily employed in applications ranging from clinical research to business analytics, the purpose of this article is to extend the exact distributional result of Wald (Ann. Math. Stat. 18: 586–589, 1947) to handle models involving a number of variance components.Due to the unavailability of exact distributional results for underlying statistics, currently available

更新日期：2020-08-18
• J. Stat. Distrib. App. Pub Date : 2020-08-01
Indranil Ghosh

Discrete analogue of a continuous distribution (especially in the univariate domain) is not new in the literature. The work of discretizing continuous distributions begun with the paper by Nakagawa and Osaki (1975) to the best of the knowledge of the author. Since then several authors proposed discrete analogues of known continuous models. In this paper, we propose and study a discrete analogue of

更新日期：2020-08-01
• J. Stat. Distrib. App. Pub Date : 2020-07-23
Cornelis J. Potgieter

The density deconvolution problem is considered for random variables assumed to belong to the generalized skew-symmetric (GSS) family of distributions. The approach is semiparametric in that the symmetric component of the GSS distribution is assumed known, and the skewing function capturing deviation from the symmetric component is estimated using a deconvolution kernel approach. This requires the

更新日期：2020-07-23
• J. Stat. Distrib. App. Pub Date : 2019-11-05
Oscar Alberto Quijano Xacur

We introduce a new distribution with support on (0,1) called unifed. It can be used as the response distribution for a GLM and it is suitable for data aggregation. We make a comparison to the beta regression. A link to an R package for working with the unifed is provided.

更新日期：2019-11-05
• J. Stat. Distrib. App. Pub Date : 2015-11-26
Lipika Ghosh,Jiancheng Jiang,Yanqing Sun,Haibo Zhou

In this paper we use Cox's regression model to fit failure time data with continuous informative auxiliary variables in the presence of a validation subsample. We first estimate the induced relative risk function by kernel smoothing based on the validation subsample, and then improve the estimation by utilizing the information on the incomplete observations from non-validation subsample and the auxiliary

更新日期：2019-11-01
• J. Stat. Distrib. App. Pub Date : 2014-01-01
Li Qi,Yanqing Sun

In this paper, we investigate several well known approaches for missing data and their relationships for the parametric probability regression model Pβ (Y|X) when outcome of interest Y is subject to missingness. We explore the relationships between the mean score method, the inverse probability weighting (IPW) method and the augmented inverse probability weighted (AIPW) method with some interesting

更新日期：2019-11-01
• J. Stat. Distrib. App. Pub Date : 2019-08-23
Fiaz Ahmad Bhatti; G. G. Hamedani; Mustafa C. Korkmaz; Gauss M. Cordeiro; Haitham M. Yousof; Munir Ahmad

In this paper, a flexible family of distributions with unimodel, bimodal, increasing, increasing and decreasing, inverted bathtub and modified bathtub hazard rate called Burr III-Marshal Olkin-G (BIIIMO-G) family is developed on the basis of the T-X family technique. The density function of the BIIIMO-G family is arc, exponential, left- skewed, right-skewed and symmetrical shaped. Descriptive measures

更新日期：2019-08-23
• J. Stat. Distrib. App. Pub Date : 2019-08-20
Roger W. Barnard; Chamila Perera; James G. Surles; A. Alexandre Trindade

Motivated by an engineering pullout test applied to a steel strip embedded in earth, we show how the resulting linearly decreasing force leads naturally to a new distribution, if the force under constant stress is modeled via a three-parameter Weibull. We term this the LDSWeibull distribution, and show that inference on the parameters of the underlying Weibull can be made upon collection of data from

更新日期：2019-08-20
• J. Stat. Distrib. App. Pub Date : 2019-07-24
Saman Muthukumarana; David Martell; Ram Tiwari

We present a novel Bayesian approach to random effects meta analysis of binary data with excessive zeros in two-arm trials. We discuss the development of likelihood accounting for excessive zeros, the prior, and the posterior distributions of parameters of interest. Dirichlet process prior is used to account for the heterogeneity among studies. A zero inflated binomial model with excessive zero parameters

更新日期：2019-07-24
• J. Stat. Distrib. App. Pub Date : 2019-06-12
Wolf-Dieter Richter

The class of (p1,…,pk)-spherical probability laws and a method of simulating random vectors following such distributions are introduced using a new stochastic vector representation. A dynamic geometric disintegration method and a corresponding geometric measure representation are used for generalizing the classical χ2-, t- and F-distributions. Comparing the principles of specialization and marginalization

更新日期：2019-06-12
• J. Stat. Distrib. App. Pub Date : 2019-06-11
Mei-Ling Ting Lee; G. A. Whitmore

Many systems experience gradual degradation while simultaneously being exposed to a stream of random shocks of varying magnitudes that eventually cause failure when a shock exceeds the residual strength of the system. In this paper, we present a family of stochastic processes, called shock-degradation processes, that describe this failure mechanism. In our failure model, system strength follows a geometric

更新日期：2019-06-11
• J. Stat. Distrib. App. Pub Date : 2019-06-08
Maria C.S. Lima; Gauss M. Cordeiro; Edwin M.M. Ortega; Abraão D.C. Nascimento

Various applications in natural science require models more accurate than well-known distributions. In this context, several generators of distributions have been recently proposed. We introduce a new four-parameter extended normal (EN) distribution, which can provide better fits than the skew-normal and beta normal distributions as proved empirically in two applications to real data. We present Monte

更新日期：2019-06-08
• J. Stat. Distrib. App. Pub Date : 2019-06-06
Wolf-Dieter Richter

Stochastic representations of star-shaped distributed random vectors having heavy or light tail density generating function g are studied for increasing dimensions along with corresponding geometric measure representations. Intervals are considered where star radius variables take values with high probability, and the derivation of values of distribution functions of g-robust statistics is proved to

更新日期：2019-06-06
• J. Stat. Distrib. App. Pub Date : 2019-06-06
Li-Pang Chen; Grace Y. Yi; Qihuang Zhang; Wenqing He

Technological advances associated with data acquisition are leading to the production of complex structured data sets. The recent development on classification with multiclass responses makes it possible to incorporate the dependence structure of predictors. The available methods, however, are hindered by the restrictive requirements. Those methods basically assume a common network structure for predictors

更新日期：2019-06-06
• J. Stat. Distrib. App. Pub Date : 2019-04-15
Johannes T. Ferreira; Andriëtte Bekker

The eigenvalue distributions from a complex noncentral Wishart matrix S=XHX has been the subject of interest in various real world applications, where X is assumed to be complex matrix variate normally distributed with nonzero mean M and covariance Σ. This paper focuses on a weighted analytical representation of S to alleviate the restriction of normality; thereby allowing the choice of X to be complex

更新日期：2019-04-15
• J. Stat. Distrib. App. Pub Date : 2019-04-08
Yu Shi; Zizhao Zhang; Weng Kee Wong

When a model-based approach is appropriate, an optimal design can guide how to collect data judiciously for making reliable inference at minimal cost. However, finding optimal designs for a statistical model with several possibly interacting factors can be both theoretically and computationally challenging, and this issue is rarely discussed in the literature. We propose nature-inspired metaheuristic

更新日期：2019-04-08
• J. Stat. Distrib. App. Pub Date : 2019-03-08
Mark Huber; Nevena Marić

A multivariate symmetric Bernoulli distribution has marginals that are uniform over the pair {0,1}. Consider the problem of sampling from this distribution given a prescribed correlation between each pair of variables. Not all correlation structures can be attained. Here we completely characterize the admissible correlation vectors as those given by convex combinations of simpler distributions. This

更新日期：2019-03-08
• J. Stat. Distrib. App. Pub Date : 2019-03-07
Klaus Müller; Wolf-Dieter Richter

We introduce rank-k-continuous axis-aligned p-generalized elliptically contoured distributions and study their properties such as stochastic representations, moments, and density-like representations. Applying the Kolmogorov existence theorem, we prove the existence of random processes having axis-aligned p-generalized elliptically contoured finite dimensional distributions with arbitrary location

更新日期：2019-03-07
• J. Stat. Distrib. App. Pub Date : 2018-12-29
Maria A Veretennikova,Alla Sikorskii,Michael J Boivin

The objective of this study was to test statistical features from the electroencephalogram (EEG) recordings as predictors of neurodevelopment and cognition of Ugandan children after coma due to cerebral malaria. The increments of the frequency bands of EEG time series were modeled as Student processes; the parameters of these Student processes were estimated and used along with clinical and demographic

更新日期：2018-12-29
• J. Stat. Distrib. App. Pub Date : 2018-12-05
Emrah Altun; Haitham M. Yousof; G.G. Hamedani

In this paper, a new extension of the generalized half-normal distribution is introduced and studied. We assess the performance of the maximum likelihood estimators of the parameters of the new distribution via simulation study. The flexibility of the new model is illustrated by means of four real data sets. A new log-location regression model based on the new distribution is also introduced and studied

更新日期：2018-12-05
• J. Stat. Distrib. App. Pub Date : 2018-12-04
Alex Dytso; Ronit Bustin; H. Vincent Poor; Shlomo Shamai

The family of Generalized Gaussian (GG) distributions has received considerable attention from the engineering community, due to the flexible parametric form of its probability density function, in modeling many physical phenomena. However, very little is known about the analytical properties of this family of distributions, and the aim of this work is to fill this gap. Roughly, this work consists

更新日期：2018-12-04
• J. Stat. Distrib. App. Pub Date : 2018-11-03
Zubair Ahmad; M. Elgarhy; G. G. Hamedani

We propose a new family of univariate distributions generated from the Weibull random variable, called a new Weibull-X family of distributions. Two special sub-models of the proposed family are presented and the shapes of density and hazard functions are investigated. General expressions for some statistical properties are discussed. For the new family, three useful characterizations based on truncated

更新日期：2018-11-03
• J. Stat. Distrib. App. Pub Date : 2018-08-13

We propose a five parameter transmuted geometric quadratic hazard rate (TG-QHR) distribution derived from mixture of quadratic hazard rate (QHR), geometric and transmuted distributions via the application of transmuted geometric-G (TG-G) family of Afify et al.(Pak J Statist 32(2), 139-160, 2016). Some of its structural properties are studied. Moments, incomplete moments, inequality measures, residual

更新日期：2018-08-13
• J. Stat. Distrib. App. Pub Date : 2018-07-18
Mei Ling Huang,Christine Nguyen

Quantile regression estimates conditional quantiles and has wide applications in the real world. Estimating high conditional quantiles is an important problem. The regular quantile regression (QR) method often designs a linear or non-linear model, then estimates the coefficients to obtain the estimated conditional quantiles. This approach may be restricted by the linear model setting. To overcome this

更新日期：2018-07-18
• J. Stat. Distrib. App. Pub Date : 2018-01-18
Frantisek Duris; Juraj Gazdarica; Iveta Gazdaricova; Lucia Strieskova; Jaroslav Budis; Jan Turna; Tomas Szemes

Ratio distribution is a probability distribution representing the ratio of two random variables, each usually having a known distribution. Currently, there are results when the random variables in the ratio follow (not necessarily the same) Gaussian, Cauchy, binomial or uniform distributions. In this paper we consider a case, where the random variables in the ratio are joint binomial components of

更新日期：2018-01-18
• J. Stat. Distrib. App. Pub Date : 2018-01-08
Muhammad Zubair; Muhammad H. Tahir; Gauss M. Cordeiro; Ayman Alzaatreh; Edwin M. M. Ortega

We propose and study a new compounded model to extend the half-Cauchy and power-Cauchy distributions, which offers more flexibility in modeling lifetime data. The proposed model is analytically tractable and can be used effectively to analyze censored and uncensored data sets. Its density function can have various shapes such as reversed-J and right-skewed. It can accommodate different hazard shapes

更新日期：2018-01-08
• J. Stat. Distrib. App. Pub Date : 2017-11-22
Mahmoud Aldeni; Carl Lee; Felix Famoye

In this paper, the class of T-R {generalized lambda} families of distributions based on the quantile of generalized lambda distribution has been proposed using the T-R{Y} framework. In the development of the T-R{Y} framework, the support of Y and T must be the same. It is typical that the random variable Y has one type of support and T is restricted to the same support. Taking Y to be a generalized

更新日期：2017-11-22
• J. Stat. Distrib. App. Pub Date : 2017-11-15
Hoben Thomas; Thomas P. Hettmansperger

Risk ratios are distribution function tail ratios and are widely used in health disparities research. Let A and D denote advantaged and disadvantaged populations with cdfs F A (x) and F D (x) respectively, F A (x)≤F D (x). Consider a selection setting where those selected have x>c a critical value. Scanlan observed in empirical data that as c is lowered the failure ratio F R(c)=F D (c)/F A (c) and

更新日期：2017-11-15
• J. Stat. Distrib. App. Pub Date : 2017-11-15
Anastasios N. Arapis; Frosso S. Makri; Zaharias M. Psillakis

We consider a sequence of n, n≥3, zero (0) - one (1) Markov-dependent trials. We focus on k-tuples of 1s; i.e. runs of 1s of length at least equal to a fixed integer number k, 1≤k≤n. The statistics denoting the number of k-tuples of 1s, the number of 1s in them and the distance between the first and the last k-tuple of 1s in the sequence, are defined. The work provides, in a closed form, the exact

更新日期：2017-11-15
• J. Stat. Distrib. App. Pub Date : 2017-11-01
Matthew Matheson; Alvaro Muñoz; Christopher Cox

The generalized gamma (GG) distribution is a widely used, flexible tool for parametric survival analysis. Many alternatives and extensions to this family have been proposed. This paper characterizes the flexibility of the GG by the quartile ratio relationship, log(Q2/Q1)/log(Q3/Q2), and compares the GG on this basis with two other three-parameter distributions and four parent distributions of four

更新日期：2017-11-01
• J. Stat. Distrib. App. Pub Date : 2017-11-01
Peter Congdon

Generalized Poisson regression is commonly applied to overdispersed count data, and focused on modelling the conditional mean of the response. However, conditional mean regression models may be sensitive to response outliers and provide no information on other conditional distribution features of the response. We consider instead a hierarchical approach to quantile regression of overdispersed count

更新日期：2017-11-01
• J. Stat. Distrib. App. Pub Date : 2017-11-01
Gauss M. Cordeiro; Antonio E. Gomes; Cibele Q. da-Silva; Edwin M. M. Ortega

For any continuous baseline G distribution, Zografos and Balakrishnan (Statistical Methodology 6:344–362, 2009) introduced the gamma-generated family of distributions with an extra shape parameter. Based on this family, we define a new four-parameter extension of the Burr III distribution. It can have decreasing, unimodal and decreasing-increasing-decreasing hazard rate function. We provide a comprehensive

更新日期：2017-11-01
• J. Stat. Distrib. App. Pub Date : 2017-10-30
Grace Y. Yi; Wenqing He

Case-control studies are important and useful methods for studying health outcomes and many methods have been developed for analyzing case-control data. Those methods, however, are vulnerable to mismeasurement of variables; biased results are often produced if such a feature is ignored. In this paper, we develop an inference method for handling case-control data with interacting misclassified covariates

更新日期：2017-10-30
• J. Stat. Distrib. App. Pub Date : 2017-10-16
Kimberly F. Sellers; Andrew W. Swift; Kimberly S. Weems

Following publication of the original article (Sellers et al., 2017), the authors reported that the typesetters had misinterpreted some of the edits included in their proof corrections, namely instances of “sp” to denote that an extra space was required. The original article has been corrected. Sellers, K.F., Swift, A.W., Weems, K.S.: A flexible distribution class for count data. J. Stat. Distrib.

更新日期：2017-10-16
• J. Stat. Distrib. App. Pub Date : 2017-10-15
John P. Nolan

Erlang renewal models, also called chi-squared models, provide a tractable model for genetic recombination that exhibits positive interference. Closed form expressions for multilocus probabilities are derived for the crossover process when it is a renewal process with the distance between crossovers modeled by a Erlang distribution. These expressions yield explicit formulas for the map functions, coincidence

更新日期：2017-10-15
• J. Stat. Distrib. App. Pub Date : 2017-10-02

Poisson-Tweedie mixtures are the Poisson mixtures for which the mixing measure is generated by those members of the family of Tweedie distributions whose support is non-negative. This class of non-negative integer-valued distributions is comprised of Neyman type A, back-shifted negative binomial, compound Poisson-negative binomial, discrete stable and exponentially tilted discrete stable laws. For

更新日期：2017-10-02
• J. Stat. Distrib. App. Pub Date : 2017-09-29
Melissa A. Bingham; Marissa L. Scray

Although there have been fairly recent advances regarding inference for three-dimensional rotation data, there are still many areas of interest yet to be explored. One such area involves comparing the rotational symmetry of 3-D rotations. In this paper, nonparametric inference is used to test if F 1=F 2, where F i is the degree of rotational symmetry of distribution i, through a permutation test. The

更新日期：2017-09-29
• J. Stat. Distrib. App. Pub Date : 2017-09-26
Kimberly F. Sellers; Andrew W. Swift; Kimberly S. Weems

The Poisson, geometric and Bernoulli distributions are special cases of a flexible count distribution, namely the Conway-Maxwell-Poisson (CMP) distribution – a two-parameter generalization of the Poisson distribution that can accommodate data over- or under-dispersion. This work further generalizes the ideas of the CMP distribution by considering sums of CMP random variables to establish a flexible

更新日期：2017-09-26
• J. Stat. Distrib. App. Pub Date : 2017-09-20
Wolf-Dieter Richter

First, likelihood ratio statistics for checking the hypothesis of equal variances of two-dimensional Gaussian vectors are derived both under the standard $\left (\sigma ^{2}_{1},\sigma ^{2}_{2},\varrho \right)$ -parametrization and under the geometric (a,b,α)-parametrization where a 2 and b 2 are the variances of the principle components and α is an angle of rotation. Then, the likelihood ratio statistics

更新日期：2017-09-20
• J. Stat. Distrib. App. Pub Date : 2017-09-15
Jean-François Plante

Rank correlation is invariant to bijective marginal transformations, but it is not immune to confounding. Assuming a categorical confounding variable is observed, the author proposes weighted coefficients of correlation for continuous variables developed within a larger framework based on copulas. While the weighting is clear under the assumption that the dependence is the same within each group implied

更新日期：2017-09-15
• J. Stat. Distrib. App. Pub Date : 2017-09-15
Lukun Zheng; Jiancheng Jiang

The maximum likelihood estimator (MLE) of Gini-Simpson’s diversity index (GS) is widely used but suffers from large bias when the number of species is large or infinite. We propose a new estimator of the GS index and show its unbiasedness. Asymptotic normality of the proposed estimator is established when the number of species in the population is finite and known, finite but unknown, and infinite

更新日期：2017-09-15
• J. Stat. Distrib. App. Pub Date : 2017-09-01
Martín Egozcue; Jiang Wu; Ričardas Zitikis

In Punta del Este, a resort town in Uruguay, real-estate property is in demand by both domestic and foreign buyers. There are several stages of selling residential units: before, during, and after the actual construction. Different pricing strategies are used at every stage. Our goal in this paper is to derive, under various scenarios of practical relevance, optimal strategies for setting prices within

更新日期：2017-09-01
• J. Stat. Distrib. App. Pub Date : 2017-08-31

A logistic regression model is a specialized model for product-binomial data. When a proper, noninformative prior is placed on the unrestricted model for the product-binomial model, the hypothesis H 0 of a logistic regression model holding can then be assessed by comparing the concentration of the posterior distribution about H 0 with the concentration of the prior about H 0. This comparison is effected

更新日期：2017-08-31
• J. Stat. Distrib. App. Pub Date : 2017-08-15
Sher B. Chhetri; Alfred A. Akinsete; Gokarna Aryal; Hongwei Long

In this work, a new five-parameter Kumaraswamy transmuted Pareto (KwTP) distribution is introduced and studied. We discuss various mathematical and statistical properties of the distribution including obtaining expressions for the moments, quantiles, mean deviations, skewness, kurtosis, reliability and order statistics. The estimation of the model parameters is performed by the method of maximum likelihood

更新日期：2017-08-15
• J. Stat. Distrib. App. Pub Date : 2017-08-01
Judex Hyppolite

The economic researcher is sometimes confronted with panel datasets that come from a population made of a finite number of subpopulations. Within each subpopulation the individuals may also be heterogenous according to some unobserved characteristics. A good understanding of the behavior of the observed individuals may then require the ability to identify the groups to which they belong and to study

更新日期：2017-08-01
• J. Stat. Distrib. App. Pub Date : 2017-07-13
Subrata Chakraborty; S. H. Ong

In this paper a new generalization of the hyper-Poisson distribution is proposed using the Mittag-Leffler function. The hyper-Poisson, displaced Poisson, Poisson and geometric distributions among others are seen as particular cases. This Mittag-Leffler function distribution (MLFD) belongs to the generalized hypergeometric and generalized power series families and also arises as weighted Poisson distributions

更新日期：2017-07-13
• J. Stat. Distrib. App. Pub Date : 2017-07-05
Sugata Ghosh; Subhajit Dutta; Marc G. Genton

We construct a d-dimensional discrete multivariate distribution for which any proper subset of its components belongs to a specific family of distributions. However, the joint d-dimensional distribution fails to belong to that family and in other words, it is ‘inconsistent’ with the distribution of these subsets. We also address preservation of this ‘inconsistency’ property for the symmetric Binomial

更新日期：2017-07-05
• J. Stat. Distrib. App. Pub Date : 2017-07-04
Morad Alizadeh; S. M. T. K. MirMostafee; Edwin M. M. Ortega; Thiago G. Ramires; Gauss M. Cordeiro

We introduce and study general mathematical properties of a new generator of continuous distributions with three extra parameters called the odd log-logistic logarithmic generated family of distributions. We present some special models and investigate the asymptotes and shapes. The new density function can be expressed as a linear combination of exponentiated densities based on the same baseline distribution

更新日期：2017-07-04
• J. Stat. Distrib. App. Pub Date : 2017-07-03
Gwo Dong Lin

We consider univariate distributions with finite moments of all positive orders. The moment problem is to determine whether or not a given distribution is uniquely determined by the sequence of its moments. There is a huge literature on this classical topic. In this survey, we will focus only on the recent developments on the checkable moment-(in)determinacy criteria including Cramér’s condition, Carleman’s

更新日期：2017-07-03
• J. Stat. Distrib. App. Pub Date : 2017-05-03
Mei Ling Huang; Christine Nguyen

For extreme events, estimation of high conditional quantiles for heavy tailed distributions is an important problem. Quantile regression is a useful method in this field with many applications. Quantile regression uses an L 1-loss function, and an optimal solution by means of linear programming. In this paper, we propose a weighted quantile regression method. Monte Carlo simulations are performed to

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• J. Stat. Distrib. App. Pub Date : 2017-04-07
Habtamu K Benecha,Brian Neelon,Kimon Divaris,John S Preisser

Mixture distributions provide flexibility in modeling data collected from populations having unexplained heterogeneity. While interpretations of regression parameters from traditional finite mixture models are specific to unobserved subpopulations or latent classes, investigators are often interested in making inferences about the marginal mean of a count variable in the overall population. Recently

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Alessandro Selvitella

The Simpson’s Paradox is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, it has a deep statistical significance. In this paper, we discuss basic examples in arithmetic, geometry, linear algebra, statistics,

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• J. Stat. Distrib. App. Pub Date : 2017-03-21
Wolf-Dieter Richter; Kay Schicker

In low dimensions, the relatively easily implementable acceptance-rejection method for generating polyhedral convex contoured uniform distributions is compared to more sophisticated particular methods from the literature, and applied to drug combination studies. Based upon a stochastic representation, the method is extended to the general class of polyhedral convex contoured distributions of known

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We study general mathematical properties of a new class of continuous distributions with three extra shape parameters called the exponentiated Marshal-Olkin family of distributions. Further, we present some special models of the new class and investigate the shapes and derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions and probability weighted moments

更新日期：2016-11-05
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Home > Reverse Engineering > Dongles and Nyberg-Rueppel signature scheme

## Dongles and Nyberg-Rueppel signature scheme

“Dongle me” by cyclops is, as name suggest, a crackme that requires a hardware dongle, or a software emulator. These two technical problems, combined with an uncommon authentication scheme, make it an interesting target to analyse.

First, we need to learn how our target detects and communicates with the dongle. In this case, since the executable is small and not packed/protected/obfuscated, a quick glance at imports section is sufficient to get to the dongle discovery procedure — HidD_* APIs from hid.dll give it away instantly.

Pseudocode:

#define VENDOR_ID  0x04d8
#define PRODUCT_ID 0x003f

device_t g_device = NULL;

bool find_hid_dongle(int vendor_id, int product_id){

device_t dev;

while(dev = next_hid_device()){

if(dev->vid == vendor_id && dev->pid == product_id){
g_device = dev;
return TRUE;
}
}

return FALSE;
}

And a high level view of authentication:

bool authenticate(){
char buffer[N];

if(!find_hid_dongle(VENDOR_ID, PRODUCT_ID)){
return FALSE;
}

if(is_valid(buffer)){
return TRUE;
}

return FALSE;
}

To make it even clearer: HID devices installed in the OS are enumerated one by one. Dongle is recognized by specific values of variables exposed by every HID: product_id and vendor_id. If found, device is opened with CreateFile and read with ReadFile (standard windows APIs). Finally, read data is validated. Before creating a dongle, or an emulator, we need to get familiar with HID.

## Human interface devices

Human Interface Devices are devices that interact with humans — take input from them and/or present them output. Good examples are mouse and a keyboard. Primary motivation for HID, was to simplify the process of installation of PC input devices. Prior to HID, custom devices were required to implement their own protocols to communicate with the user, which of course forced developers to create custom drivers to handle these protocols, and users to install these drivers. HID protocol makes things simpler. Instead of custom drivers, developers need to “describe” their protocol using HID descriptors. Descriptors are parsed and interpreted by the OS itself, allowing developers to easily create complex devices from pritmitives provided by the OS.

To implement a hardware HID dongle, we can use Teensy, or any compatible clone. I found a Teensy 1.0 clone available for few bucks, preloaded with a PS3 hack :). With USB raw hid example it’s trivial to implement a simple dongle that waits for data from our keygen, saves it in EEPROM and then sends it back to the OS, in a loop. If you want to take a look at the implementation, see here.

Creating an emulator is more complicated and requires a HID miniport driver. While it certainly is possible to create it basing only on Win DDK HID samples, it is no easy feat, if you don’t have experience in kernel development. Fortunately, there is an excellent open source project called vmulti, created by Daniel Newton, that does exactly what we need: creates virtual HID devices that can be read / written to. Adapting it to our needs is a technicality that I feel isn’t worth describing, so check the sources if you are interested in details.

Emulator will simply pass stuff received from the keygen to anyone who invokes ReadFile(). Here’s a nice screenshot of device manager with successfully installed emulator:

## Authentication

Finally, the most interesting part :). To recognize used authentication scheme, make shure to apply IDA signatures for MIRACL bignum library, otherwise you will unnecessarily spend a lot of time identifying functions. Crackme uses elliptic curve cryptography, so it’s very possible that failing to recognize MIRACL usage, would cost you digging deep into the assembly code with slim chances of making any sense out of it.

Cyclops used an elliptic curve variant of Nyber-Rueppel signature scheme. Current user’s username is “hashed” with CRC32. Then, CRC is compared to a message extracted from an ECNR signature read from the dongle. To pass authentication, we need to sign the CRC and push it to the dongle.

## Nyberg-Rueppel Signature Scheme

Let $E$ be an elliptic curve defined over $\mathbb{Z}_p$ ($p > 3$ prime) such that $E$ contains a cyclic subgroup $H$ in which the discrete logarithm problem is intractable.

Let $K=\{(E,k,P,Q): Q=k*P\}$, where $P \in H \subset E$. Points $P$ and $Q$ are public, while $k$ is secret. For $K$ defined as above, for a (secret and random) $t \in \mathbb{Z}_{|H|}$ and for a (message) $m \in \mathbb{Z}_p$, define:

$sign_K(m, t) = (s_1, s_2)$

where

$(x_1, y_1) = t*P$

$s_1 = x_1 + hash(m)$ mod $p$

$s_2 = t - k*s_1$ mod $p$

and

$verify_K(m, s_1, s_2) = true \iff z=hash(m)$

where

$(x_2, y_2) = s_2*P + s_1*Q$

$z = s_1-x_2$

This works, because $s_2*P + s_1*Q = (t - k*s_1)*P + s_1*k*P = t*P$.

Since we now know what cryptosystem we are up against, and how to generate correct signatures, let’s take a look at curve parameters used in the crackme. Since author claims it’s solvable without patching, we expect either:

1. a small curve, where ECDLP can be effectively solved
2. a weak curve, that is suspectible to a cryptographic attack (examples in [3])
3. something tricky :)

Quick inspection of parameters shows that 1) and 2) are not the case. Used curve is secp112r1, which is bad news for us, since secp* curves (their parameters) are chosen by Certicom research [2], to be optimal for cryptographic purposes. This means they provide maximal possible bit-strength security against fastest known ECDLP solving algorithms and aren’t weakened by special-case attacks.

Before using these curves, you may wonder how were their parameters chosen, after all, maybe NSA helped to pick them ;). Fortunately, these concerns were anticipated and curves were generated by a SHA-1 powered PRNG, seeded with a known value and then checked for desired properties, so no parameter could be predetermined. Since seeds and PRNG implementation are public [2], you can even repeat the process and verify chosen curves yourself.

We are left with option 3). Before doing anything complicated, I wanted to check if the ECDLP from crackme was referenced anywhere on the Internet. secp112r1 parameters ($a, b, p$) are common and won’t provide any interesting clues, but coordinates of point $P$  (or $G$ , using Certicom’s nomenclature) could, as cyclops decided to use a non-standard base point.

Searching for 9487239995A5EE76B55F9C2F098 ($x$ coord. of $P$) yields nothing of interest, but searching for 188281465057972534892223778713752 (same value, but in base 10) turns out to be a bull’s eye ;).

In 2009, Bos, Marcelo, KaiharaKleinjung, Lenstra and Montgomery solved an ECDLP over secp112r1, using 200 PS3 consoles. ECDLP instance they were solving was $Q=k*P$, where

P = (188281465057972534892223778713752, 3419875491033170827167861896082688),

Q = (1415926535897932384626433832795028, 3846759606494706724286139623885544).

Solution (which took ~6 months to find) is k=312521636014772477161767351856699. The exact same ECDLP is used in our crackme, so fortunately, all we have to do is to implement Nyber-Rueppel scheme and use $k$ above to emit correct signatures :).

See here for sources (dongle\ folder, crackme in crackme\ :)).

References

[1] Henna Pietiläinen, Elliptic curve cryptography on smart cards, page 24, http://goo.gl/Lpr4h

[2] Certicom Research, SEC 2: Recommended Elliptic Curve Domain Parameters, 2000, http://goo.gl/sJV5A

[3] Matthew Musson, Attacking the Elliptic Curve Discrete Logarithm Problem, http://goo.gl/L4Dz8


text:
# Exterior calculus: what about symmetric tensors?

1. Sep 22, 2009

### mrentropy

Hi all,

Quick question I haven't been able to find the answer to anywhere:

Can I use exterior calculus for symmetric tensors?

I'm familiar with the exterior calculus approach to things like Stokes's theorem and Gauss's law, but that's vector stuff. It seems to me the only tensors in exterior calculus are anti-symmetric tensors. This is fine. I understand the wedge product, so this makes sense.

The problem is my tensors aren't anti-symmetric, they're symmetric. I do lots of things with rank-2 symmetric tensor fields in flat space. Perfectly pedestrian things like the viscous stress tensor in a fluid, the stress in a sold, and the Maxwell stress; all of this is nonrelativistic BTW.

So is exterior calculus totally useless for what I do, or am I missing something?

Thanks,
Peter

2. Sep 22, 2009

### tiny-tim

Hi Peter!
Yup!

3. Sep 22, 2009

### lurflurf

There is an analogous theory, but exterior calculus will not itself be very helpful.

4. Sep 23, 2009

### Phrak

Are you sure? Say I only care about the elements of the symmetric tensors in an equation. Is it possible to recaste a tensor equation involving symmetric tensors into one involving skew symmetric tensors?

5. Sep 23, 2009

### jambaugh

Here is the context in which you can "appreciate" exterior and other products of vectors and tensors.

An N-dimensional vector is an element of the fundamental representation of the Lie group GL(N) of invertible linear operators acting on a vector space. (Think of the set of NxN invertible matrices). This group GL(N) is the group of automorphisms of the vector space i.e. the group of transformations preserving linear combinations.

Now if you define an abstract product $\otimes$ acting on vectors, and products of vectors and products of products of vectors etc, so that it generates an algebra then only hold it to the conditions that it

a.) be bilinear i.e. the product respects the procedure of taking linear combinations
$$X\otimes (aY + bZ) = a(X\otimes Y) + b(X\otimes Z)$$
and likewise for $(aY + bZ) \otimes X$; and

b.) that it be associative,
$$(X\otimes Y)\otimes Z \equiv X\otimes(Y\otimes Z)$$

then you have what is called the free associative product on a linear space. I know this is pretty abstract but bear with me. Here is the thing the "free product" for the algebra generated by elements of a vector space is the tensor product. What is more it is a way to generate other representations of the group GL(N) acting on the space.

Now the point of this is, the free product generates higher order objects from the vectors and these are the tensors. Also these tensors are other objects upon which the automorphism group GL(N) acts. (Since the tensor product respects linear combinations and so does GL(N) the action of GL(N) on tensor products of vectors still respects linearity.) In fact one way to define the vectors as a fundamental representation is that any finite dimensional representation of GL(N) can be found as a sub-representation (invariant subspace) of one of the tensor representations obtained by this free product. But in general these tensors are not irreducible in that the action of GL(N) on a space of tensors can be broken down into separate actions on subspaces of specific types of tensors. For example GL(N) will map anti-symmetric tensors to anti-symmetric tensors, and symmetric tensors to symmetric tensors. There is a whole big subject of representation theory for this group which delves into enumerating the irreducible parts.

OK. Here is the punch line. Any other reasonable/useful products we define on a set of objects can generally be expressed by giving the free product modulo a set of identities. This is how you define for example the "wedge product" or outer product of vectors and anti-symmetric tensors. You impose the anti-symmetry property. You can likewise impose a symmetry property defining a product yielding symmetric tensors. In particular these two products end up generating (from the vectors) classes of irreducible tensor representations. (Think of the identities as filtering out all but one irreducible part of the generally reducible tensor representations.)

There are other weirder products you can define but these two are special in that they generate uniform classes of irreducible representations. This partly because the defining identities themselves make no basis dependent references to the spaces and so are themselves unaffected by the automorphism group GL(N).

You are already familiar with the antisymmetric (wedge) products generating the totally antisymmetric tensors but --though you may not realize it-- you are even more familiar with the totally symmetric product. This total symmetry translates to commutativity and so the totally symmetric tensors on an N dimension space equates to the set of polynomials of N variables. Identify the degree k term coefficients of a polynomial with the coefficients of a rank k totally symmetric tensor. (The variables themselves correspond to the basis elements.)

OK. Now that I've taken you through China just to get to the store up the street, I hope by the tour I have given you a feel for the context by which the following emerges. The totally symmetric correspondent of the exterior calculus is just the standard standard calculus of analytic functions on N variables. (Recall that analytic functions have power series expansion and thus you can think of them as infinite degree polynomials or infinite sums of totally symmetric tensors.)

But Wait There's More! OK the generating object in all this was the Vector Space which is basically defined by linearity, i.e. linear combinations of vectors yields vectors. It had the automorphism group, the group of general linear transformations GL(N). You can impose additional structure on the space e.g. give it a metric. I could go into indefinite metrics such as we have in Minkowski space-time but lets stick to Euclidean spaces for now. The additional structure reduces the number of transformations which preserve it so you get a smaller group of automorphism, the orthogonal group O(N). (If you allow indefinite metrics you get O(p,n) p+n=N).

I bring this up because the metric corresponds to an inner product on the space. Also given more structure you get less automorphisms and more possible products preserving them. In particular you can extend the outer product (Grassmann product) to include also an inner product term since the inner product is also invariant under the smaller group. This is the Clifford product.

In the way that outer (Grassmann) and symmetric (commutative) products naturally express the linear structure, Clifford products and Clifford algebras naturally incorporate the additional metric structure.

OK I'm done now. I know I've laid a bunch of abstract stuff which is difficult to absorb in one reading but treat it as a source for references to further study.

Ultimately I'm describing the category structure of vector spaces. You have a category of objects and the automorphism structure. You can then look at the free associative products preserving structure and how the automorphisms map under it. Then impose identities and see what you get. This is how you generate the "calculus" over this category.

6. Sep 23, 2009

### jambaugh

OK, I got a bit overzealous. Could you give an example problem to see what you'd like to "recast"?

7. Sep 23, 2009

### mrentropy

Sure: the Navier-Stokes equations, starting with writing out explicitly the viscous stress tensor in a Newtonian fluid. Actually what I'd really like is compressible N-S with MHD, but I think if I see straight up NS I could figure out how to do the rest. :)

I just stepped in the door so haven't yet had the chance to read all the detailed replies. Thanks in advance.

8. Sep 23, 2009

### jambaugh

I'm sorry I'm not quite clear on the actual mathematics problem. You're trying to express the viscous stress tensor? You're trying to find a particular one for a given solution? You're trying to solve Navier-Stokes with certain boundary conditions?

9. Sep 23, 2009

### jambaugh

See if this may help...
One can use "diadics" to represent vectors and tensors in terms of a standard basis:
$$\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$$
$$\mathbf{u}\wedge \mathbf{v} = (v_1 u_2 - u_1 v_2)\mathbf{i}\wedge\mathbf{j} + \cdots$$

Think of the outer (wedge) product of vectors as the commutator of the tensor product:
$$\mathbf{i}\wedge \mathbf{j} = \mathbf{i}\otimes\mathbf{j} - \mathbf{j}\otimes\mathbf{i}$$
Except be sure you totally antisymmetrize multiple products instead of just taking straight commutator. The commutator product is not associative but if you totally antisymmetrize completely you recover associativity.

To express symmetric tensors you can define a similar totally symmetrized product. In the typical diadic formulation the tensor product is written just as adjacency ij but here I'm including it explicitly. You can then use ij to be the totally symmetric product i.e. commuting product. Thus you can express a totally symmetric rank 2 tensor as:

$$\mathbf{S} = S_{xx} i^2 + S_{yy}j^2 + S_{zz}k^2 + S_{xy}ij + S_{xz}ik + S_{yz}jk$$

Now if you want to take say a dot product you'll need to expand this symmetric product of distinct basis vectors (but not of powers to avoid factor of two issues).
$$\mathbf{i}\cdot ij = \mathbf{i}\cdot( \mathbf{i}\otimes \mathbf{j} + \mathbf{j}\otimes \mathbf{i}) = \mathbf{j}+ 0$$

In terms of the calculus just use the normal:
$$\nabla = \partial_x \mathbf{i} + \partial_y \mathbf{j} + \partial_z\mathbf{k}$$.
But be very careful about mixing wedge and symmetric products. Expand and then apply.

If you are working on paper you may want to always put symmetrized products of basis vectors in parentheses and shorten the tensor product to just adjacency for conciseness:
$$(\mathbf{i}\mathbf{j})= \mathbf{ij} + \mathbf{ji} \equiv \mathbf{i}\otimes\mathbf{j} + \mathbf{j}\otimes\mathbf{i}$$

Another convention is to put totally antisymmetrized products in square brackets:
$$[uv] = uv - vu$$
$$[uvw] = uvw - uwv -vuw -wvu + vwu + wuv$$

Again the symmetric tensors are equivalent to the polynomials (rank = degree) on the basis treated as the variables. So expanding symmetrized products is relatively easy. You'll also be able to apply calculus in the standard fashion. But be careful to explicitly identify the derivative operators type of diadic product. e.g. straight gradient is tensor product so expand symmetric or outer products in terms of tensor products first. Wedge products applied to anti-symmetric tensors yields anti-symmetric tensors so totally antisymmetrize.

As far as applying Stokes type formulas there may be something you can do with symmetrized forms. Let me see what I can work out.

10. Sep 23, 2009

### mrentropy

You're right, and I'll definitely do that.

OK so the reason for all of this is in a nutshell is that I'm trying to figure out if it's worth learning exterior calculus, specifically for doing numerical simulations (discrete exterior calculus, aka DEC), for fluid dynamics simulations. I haven't found any that do viscous flow, just Euler flow, the difference being that in the latter case you don't have to solve a stress tensor.

(BTW I first learned Stokes's and Gauss's from the exterior calc perspective so it's not totally foreign to me.)

I've found it helpful to derive correct finite difference / finite element discretizations using DEC approaches for simple things like Poisson's equation, but nothing sophisticated like Navier-Stokes.

So it would be nice to know how to express the Stokes viscous term - and then the full Navier-Stokes - in the language of exterior calculus.

The Stokes viscous term is that the viscous stress tensor is:

$\sigma'_{\alpha \beta} = \eta (v_{\alpha;\beta} + v_{\beta;\alpha} - \frac{2}{3}g_{\alpha\beta}v^\gamma_{\ \ ;\gamma}) + \xi g_{\alpha\beta}v^\gamma_{\ \ ;\gamma}$

I've taken this from Landau & Lifgarbagez and put it in covariant form instead of Cartesian index notation, but anyway there you go. (I use semicolons to denote covariant differentiation: indices that appear following a ";" are differentiated upon.)

$v_{\alpha}$ is the fluid velocity, and $v_{\alpha;\beta}$ is its covariant derivative. Obviously $\sigma'_{\alpha \beta}$ is symmetric. Oh yes and $\eta$ and $\xi$ are constants (first and second viscosities).

From this one then forms the full stress tensor

$\Pi_{\alpha\beta} = P g_{\alpha\beta} + \rho v_\alpha v_\beta - \sigma'_{\alpha \beta}$

where $P$ is the pressure and $\rho$ is the fluid density. Then the NS equations (i.e. momentum conservation) come from taking the divergence of this - i.e. d/dt of momentum density (LHS) is equal to the divergence of the momentum flux tensor $\Pi_{\alpha \beta}$ (which is a symmetric tensor), on the RHS:

$\frac{\partial}{\partial t} \left( \rho v_\alpha \right) = - \Pi_{\alpha \gamma ;}^{\ \ \ \gamma}$

Of course that's all pretty complicated but I'd be happy for starters just to have the foggiest notion how to write the viscous stress tensor for an incompressible fluid, which pretty simple. In that case,
$\sigma'_{\alpha \beta} = \eta \left( v_{\alpha;\beta} + v_{\beta;\alpha} \right)$
since $v^\gamma_{\ \ ;\gamma} = 0$.

Anyway that's what I'm after. Hope that makes sense. Thanks so much!

11. Sep 23, 2009

### mrentropy

Oh yes and $g_{\alpha \beta}$ is the metric, but you knew that.

Maybe a simpler example to start is this: The Euler equation (setting the density $\rho$ equal to 1) can be written:

$\partial_t {\vec v} + {\vec v} \cdot \nabla {\vec v} = - \nabla P$

Nevermind how you solve for $P$ at the moment. Anyway the second term on the LHS is equivalent to setting a stress tensor equal to
$\overleftrightarrow{\Pi} = {\vec v} {\vec v}$
and then taking its divergence,
$\partial_t {\vec v} = - \nabla \cdot \overleftrightarrow{\Pi} - \nabla P$
assuming that we're incompressible (since density is constant) so $\nabla \cdot {\vec v} = 0$.
Any ideas how one would write this in the language of exterior calculus?

(There's supposed to be a double-headed arrow over the $\Pi$. I can rewrite in Cartesian index notation:
$\Pi_{ij} = v_i v_j$
$\partial_t v_i = -\nabla_k \Pi_{ki} -\nabla_i P$
Sorry to bounce back and forth in the notation.)

You can also add the pressure into the stress tensor definition but nevermind that for the moment.

P.S. jambaugh, I just now noticed your second post. This makes much more sense to me than the first (I'm sure it was valid, I just don't yet understand it). I think taking a quick look at what you wrote I might be able to figure it out. It's definitely a big help. I'll give it a shot and let you know if/what I come up with.

Last edited: Sep 23, 2009
12. Sep 23, 2009

### Phrak

I don't believe I have an solid example of a tensor equation that doesn't appear to be antisymmetric, but is.

One candidate is Maxwell's equations. Are these antisymmetric equations in three dimensions? I haven't considered it, actually.

In any case, in 4 dimensions we have d*dAμ = -J, where the scalars found in Maxwell's equations are elements of antisymmetric tensors. A is the 1-form 4-potential. I must say, this is not the sort of thing I had in mind when asking about 'recasting', but I suppose it's worth considering.

The vacuum wave equation is another one we might consider.

d'Alembertian(Ei) = 0 and d'Alembertian(Bi) = 0 don't look antisymmetric. But in forms, they are. d*d*F = 0.

Personnally, I would be more interested in knowing whether the scalar elements of an equation involving symmetric tensors, such as those mrentropy had in mind, could be found within an equation involving antisymmetric tensors without invoking higher dimensions.

Last edited: Sep 23, 2009
13. Sep 24, 2009

### jambaugh

You don't need exterior calculus as you have it all in the general tensor calculus and you seem to be familiar with index notation. I don't think you'll find any magic bullets in the exterior calculus but --yes-- I'd say it is worth learning. Find a good book on Differential Geometry and it will cover both. In any relevant application there will be times when using the exterior calculus is helpful but it is a subset of the more general tensor calculus and you need to be able to fall back on that.

The principle arena where exterior calculus is useful in defining differential forms. The anti-symmetry keeps track of orientation nicely when one integrates.

BTW This thread has gotten me remembering a little trick of notation I worked out once upon a time. I'm trying to type it up now. The idea is to use characteristic functions to internalize the limits of integration (recall a characteristic function is defined for some set to be one for elements of the set but otherwise zero.) Applying the gradient to a characteristic function gives a nice generalization of Dirac's delta function and allows one to "internalize" the various Stokes type integral formulas as differential identities in the integrand. I'll send you a copy when I get a complete first draft.

14. Sep 24, 2009

### mrentropy

Well, yes and no. I mean after all if that were completely true, then *nobody* would need exterior calculus, right?

The beauty of differential forms (to me anyway) is that things like Stokes's and Gauss's law just "pop out".

In practical numerical terms, this means that if you want to discretize your PDE on a mesh (regular or unstructured), that things that should be conserved are, in fact, conserved.

Contrast this, say, with magnetohydrodynamic simulations of fluids where at each timestep you project the solution for the B-field onto the space of divergenceless vector fields, to maintain div B = 0. Ideally you wouldn't have to do this; your discretization would automatically conserve the div B = 0 property on time integration up to some machine error.

There's actually a fair bit of subtlety to this in that discretizing the Hodge star operator introduces all sorts of errors on a mesh, but outside of this, the DEC approach works pretty well, so far as I'm aware. In the case of an Euler fluid, the thing that corresponds to the div B = 0 problem in MHD is conserving vorticity. If I'm not mistaken this comes out of the symmetry of the momentum flux density tensor for an inviscid fluid, which I mentioned earlier.

So, my thinking is, if I understood how to apply ext calc (EC) to symmetric tensors, that I would see how to discretize things in such a way that the stuff that's supposed to be conserved is, in fact, conserved, for the systems of interest to me. I'd love to learn more EC, but I have a million other things I'd like to learn too so I wanted to know *ahead* of time if the risk/return equation balanced out in my favor.

I suspected that the answer might be that there was some magic where you could see that a symmetric rank n tensor could be built out of antisymmetric rank m tensors or somesuch.... and it appears that this might be the case. The problem for me then is to revisit the appropriate conservation laws to remind myself how they come out of the rank-2 tensors of interest to me, and then go from there. That is, keeping the underlying physics in mind.

Thanks so much for all the help!

15. Sep 24, 2009

### jambaugh

Don't confuse necessity with utility. You can also do a lot of 3-vector work using quaternions. Quaternions aren't "necessary" but may be useful in specific applications.
Give me a week and I'll show you more dramatic "popping out" of these theorems.
I understand this in general but its been quite a while since I actually worked with numerical (finite element) methods on PDE's. I'm not going to be much help with the details without doing my homework.

Let me suggest you take a look at Regge calculus in GR which I think is almost exactly what you are describing. The tools you seek may be there.

That is possible but will not be helpful. Understand that symmetric and anti-symmetric tensors are in distinct irreducible representations of the GL(N) automorphism group I mentioned. You can build symmetric tensors from anti-symmetric tensors in the same sense as you can build symmetric tensors from vectors. But a.) the constructions will be much messier in general and b.) will necessarily be stepping outside the EC you'd like to use so no advantage is to be had. Again I think you'll want to work at the more fundamental level of diadics and tensor calculus. But keep at it I could be wrong.

What I suggest is that you look within the numerical methods you know EC helps and see how the boon e.g. conservation law preservation manifests. Then look to see how to preserve that boon when you step out of EC into TC.

16. Sep 24, 2009

### mrentropy

Thanks for the help. I do remember the discussion in MTW* re Regge calculus and it does seem apropos as I recall, so thanks for the reminder - I'll check it out.

Regards,
Peter

*Misner, Thorne & Wheeler


text:
### SirRembocodina's blog

By SirRembocodina, history, 10 months ago, translation,

In this thread it is shown how to go though all the submasks of a given mask without extra space or time. I needed to do the same thing for the overmasks, but I did not find anything. Based on the formula mask ^ (~mask) = -1, my brother gave me this code:

    for (int over = (1 << n) - 1; over > 0; over = ((over - mask - 1) & ~mask) + mask) {
cout << over << " ";
}

It really works. But can I simplify it, so it could be easier remembered during the contest?

• +28

 » 10 months ago, # |   +94 for (int mask = need; mask < (1 << n); mask = (mask + 1) | need) { }
•  » » 10 months ago, # ^ |   +15 Genial, thank you.
 » 10 months ago, # |   +80
 » 10 months ago, # |   +11 you want the overmasks of $S$ (i.e.: the masks of which $S$ is submask). I think it's the same as the NOT ($\text{~}$) of submasks of $\text{~} S$. Correct me if I'm wrong.
 » 5 weeks ago, # |   +39 This post has no comment newer than 8 months ago. How did it appear under "Recent actions"?
•  » » 5 weeks ago, # ^ |   +12 Probably someone posted a comment and deleted it.
•  » » » 5 weeks ago, # ^ |   +11 Or OP made an edit.


text:
# Wireless Artnet Node

I wanted to ask for help in researching appropriate Microcontrollers to create a Wireless Artnet/sACN Node for DMX communications with film and stage lighting Fixtures. The ultimate goal for the system is to have a mesh network of Nodes that are capable of putting out realtime DMX frames at 44fps. the main reason for a mesh system is to allow a user to control lighting fixtures from an Ipad and allow them to move freely around the stage without worrying about loosing signal connection.

At the moment I've been experimenting with an ESP32 Devboard from Sparkfun and their DMX shield. I've used their example libraries (here) as well as another Arnet Library for the ESP32 (here). I'm still looking into other libraries but so far the ones I have tried all "stutter" (only send one frame of DMX every second or so), then crash after a while and I can't find the bug int the code. I run the ESP32 Module as a wireless access point and this appears to be a cause.

My main question is would it be best to use 2 Microcontrollers for the nodes say an ESP32 for the wifi mesh and Artnet aspects and then an ATmega or Cortex-M4F /ARM Cortex-M7 (teensy 3.0/teensy 4.0) or something else to output the DMX information to the fixutre? Or would it be best to stick with a single microcontroller? Also how does the Serial buffer size affect the ability to output the information of a 512 channel universe, would a 256byte buffer be too small? What other factors, hardware wise would limit its ability to perform constantly for long periods of time?

If it isn't obvious yet I'm fairly new to all of this and exploring out of interest so please be kind. ;)

• It's essentially impossible to make a wireless system for streaming a time sequence of events which is both "prompt" and "reliable". You either have to accept the risk of having to hold over old data, or you build in a buffering delay sufficient to give a few retires before you need to act on the data. Regardless of that, figure out the strategy with respect to protocol and buffering before you chose hardware, not after. – Chris Stratton Nov 1 '19 at 4:08
• Hey Chris, I definitely agree. The problem is I'm not 100% sure what I'm looking for. in regards to dmx most of the code puts a universe's 512 channels into an array, then using Serial.write(buffer, length) where buffer is the array to send the data over serial. Is the Serial buffer hardware fixed or can it be manipulated? I see conflicting information on this. thanks for the response. – Connor N Nov 1 '19 at 13:01


text:
### Home > CCAA8 > Chapter 1 Unit 1 > Lesson CCA: 1.1.3 > Problem1-25

1-25.

Freda Function has another quadratic function for you to investigate! Graph the equation $y=x^2+3$ and then answer the questions from problem 1‑23.

Look at the equation. Which way do you think the parabola will open up?

Below are the questions from problem 1-23 for you to answer.
Look carefully at your parabola when you are answering them.

• How would you describe the shape of your parabola?  For example, would you describe your parabola as opening up or down?  Do the sides of the parabola ever go straight up or down (vertically)?  Why or why not?  Is there anything else special about its shape?

• Does your parabola have any lines of symmetry?  That is, can you fold the graph of your parabola so that each side of the fold exactly matches the other?  If so, where would the fold be?  Do you think this works for all parabolas?  Why or why not?  For more information on lines of symmetry, see the Math Notes box at the end of this lesson.

• Are there any special points on your parabola?  Which points do you think are important to know?

• Are there $x$- and $y$-intercepts?  What are they?  Are there any intercepts that you expected but do not exist for your parabola?

• Is there a highest (maximum) or lowest (minimum) point on the graph of your parabola?  If so, where is it?  This point is called a vertex

Use the eTool below to help you graph the equation.
Click on the link at right for the full eTool version: CCA 1-25 HW eTool