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Tuesday, May 18, 2010

Electricity

A DC circuit, or direct current circuit, is a path for which a flow of charge passes.  It usually consists of a source and a load.  The source provides energy for the charged flow to carry to the load (a light bulb or motor, etc.) and back to the negative end of the source.  It also needs to be in a closed path to be a circuit.

This light bulb is in series.  The current in this is pretty simple.  The flow of charge exits the positive end of the source, the battery, and makes its way to the light bulb.  It then goes back to the negative end of the battery.  Since the bulb is in series, the current will stay the same throughout the whole thing.  Since there is only one bulb, the voltage will be the current in amperes multiplied by the resistance of the bulb in ohms.

These bulbs are in parallel.  This is different from series because the current acts different.  There is more amperes of current in the bulb closest to the battery than the one above.  Also unlike series the voltage between the two bulbs remains the same.  This is because the potential difference for light bulbs in parallel are the same.  This is a very interesting circuit.

This circuit is a complex of two light bulbs ins parallel and one in series.  As you can see in the picture above, the bulb in series is brighter than the ones in parallel, which have the same brightness.  This is because the voltage of the bulb in series is greater than the ones in parallel.  The bulbs in parallel however still have the same voltage.  The current through this kind of circuit is strange in that each bulb will have its own separate current.

Sunday, May 2, 2010

waves and optics reflection

What I learned about waves and optics is basically what a wave is.  A wave is a disturbance that moves through a medium either longitudinal or transversal style.  I also learned the different parts of a wave like the wavelength(distance between two crests) and the amplitude(height of the crest).  What I learned about optics is concave and convex lenses.  I had no idea how those worked and how they made things far and near.
What I found difficult about what we studied was just about everything.  Everything we studied gave me trouble this unit but I found with some hard work, and getting help, I was able to at least loosely grasp the concepts.  If I had to choose one thing that was the hardest I would say lenses because it took me a long time to understand how the lenses affected light that passed through them.
My problem solving skills in this unit definitely became more profound.  I realized that if I kept working on a problem and applied what I had learned in class, many problems I found hard would become simple.  This was a major help to my problem solving skills.

Sunday, April 25, 2010

The picture above is an example of Projectile Motion at an Angle.  As I rolled off this kicker, many things occurred because of projectile motion.  First off, The reason this is at an angle is because the kicker is at an angle with the ground as opposed to just rolling off an edge.  So right as my board is exiting the ramp, it has an initial velocity and x and y components to go with it.  The horizontal component of the ramp will be the same throughout my board's projectile motion.  The x-component does not change.  The vertical component, however, changes.  Right as the board exits the ramp the y-component is the initial velocity times the sin of theta.  During the actual projectile motion of the board in the air the equation for the y-component of the velocity is this:
$v_{y}=v_{oy}+gt$

Once the board reaches the highest point in its motion, the vertical component of the velocity is always 0.       It is about this point that is depicted in the picture above.  Since the place where I began my projectile motion is above the point where I will land, there are points right before I land where the y-component of my velocity will be more than the original y-component of the board's initial velocity.  The picture above is a appropriate picture to show projectile motion at an angle.

Tuesday, March 23, 2010

Einstein Quote Reflection

The Einstein quote I chose to write my essay about is "It's not that I'm so smart , it's just that I stay with problems longer ." The reason I chose this was because I think it's true and it makes me feel better about my problem solving skills. When I'm facing a difficult math or Physics problem, I can think of this quote and realize that if I just stick with the problem I can get it. That's what I think this quote means. It shows that people who get hard problems aren't necessarily "smart", but just have a good attitude and stick with their problems. This is why I think it's good to be in Honors classes because they challenge my by giving my hard problems. This quote by Einstein is a great one.

Wednesday, March 10, 2010

Speed Skating

Here is the link to our glogster for our project on Olympic Longtrack Speed Skating:Speed Skating Glog
The glogster shows the collaboration of all the team members in team 4 because we all worked on it. Enjoy

http://en.wikipedia.org/wiki/Long_track_speed_skating
http://www.zoo-m.com/flickr-storm/
http://www.fuzilogik.com/index.php/Sports-Library/Speed-Skating/Speed-Skating-Rules.html

Sunday, February 21, 2010

Energy

What I learned about energy is that it is a quantity that can produce change. The amount of energy in something is always unchanged but can be moved around and transfered. There are many mechanisms for storing energy but the ones we learned about were elastic, kinetic, and gravitational potential. Elastic is the kind in a spring or a rubber band. Kinetic is the kind of energy when an object is moving, and gravitational potential is the energy when an object can fall from a certain height. Energy can be transfered between these types by either working, heating, or using electromagnetic radiation. We only did problems with working but I still learned much about energy.

What I have found most difficult about what I have studied is differentiating between what kind of energy is present. Sometimes it's hard to tell if an object has potential or kinetic, and its even harder to put it into an equation. Overall though I was able to conquer this problem and understand what equations to use when what energy is present.

My problem solving skill after this unit have increased because it was a real challenge to deal with a whole new type of problem. These didn't use FBD's for the most part which threw me off and confused me after last unit. However, using my problem solving skills I believe that I have overcome this making those skills better.

One connection I can make between what we have studied and the real world is when a person is skateboarding on a vert ramp. When someone rides these giant ramps you cannot push for fear of falling off which means that the only way you get speed is through energy. When you drop in to the ramp, your potential energy of being around 10 feet in the air is converted into kinetic energy as you gain speed down the ramp. This kinetic energy is then converted back into potential energy when you exit the other side of the ramp and stop in the air. But then you come back down converting the energy back into kinetic. This process continues and shows how what we have learned can be applied to real life. Responding to comments: One suggestion I got in my comments was to talk about how energy is dissipated on a vert ramp. However, this does not happen as one might think. You would think that once you dropped into the ramp your speed would slowly decrease until you could not launch off the side of the ramp anymore and slow to a stop. However, this is not the case. When riding on a vert ramp skaters keep the same speed and even through a process called pumping which is very difficult to do. Skaters use perfect timing and pressure on the board to make them gain speed on a ramp and not have to have a dissipating energy.

Wednesday, February 3, 2010

Physics Skateboard Correction!

Sadly, in a new study ,I have been proven wrong about my skateboarding physics project. Because I used the friction of a sliding wheel instead of a rolling one I must correct myself. However, we have not previously studied the friction of a rotating wheel so if you find yourself amazed at the awesomeness that awaits you in my correction physics glog don't be surprised. Click on the link below to see the astonishing study.

skateboard correction


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# Conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$

I was wondering if there is some description known for the conjugacy classes of $$\mathrm{SL}_2(\mathbb{Z})=\{A\in \mathrm{GL}_2(\mathbb{Z})|\;\;|\det(A)|=1\}.$$ I was not able to find anything about this. Most references only give solutions for $$\mathrm{SL}_2(\mathbb{R})$$.

• There is plenty of literature about conjugacy within ${\rm GL}(2,\mathbb{Q}),$ which is essentially the theory of the rational canonical form (in a special case). – Geoff Robinson Apr 13 '16 at 21:40
• – Steve Huntsman Apr 14 '16 at 12:20

One can proceed as follows for $$SL_2(\mathbb{Z})$$.

1. First, the trace is a conjugacy invariant.
2. For trace $$0$$ there are two conjugacy classes represented by $$\pmatrix{0 & 1 \\ -1 & 0}$$ and $$\pmatrix{0 & -1 \\ 1 & 0}$$. These representatives can be thought of as $$90^\circ$$ and $$270^{\circ}$$ degree rotations of a lattice generated by the corners of a square centered on the origin.
3. For trace $$1$$ and $$-1$$ there are two conjugacy classes each, represented by the matrices $$M=\pmatrix{1 & -1 \\ 1 & 0}, M^2=\pmatrix{0 & 1 \\ -1 & -1}, M^4=\pmatrix{-1 & 1 \\ -1 & 0}, M^5 = \pmatrix{0 & -1 \\ 1 & 1}$$ These representatives can be thought of as $$60^\circ$$, $$120^\circ$$, $$240^\circ$$, and $$300^\circ$$ degree rotations of a lattice generated by the vertices of a regular hexagon centered at the origin.
4. For trace $$2$$ there is a $$\mathbb{Z}$$-indexed family of conjugacy classes, represented by $$\pmatrix{1 & n \\ 0 & 1}$$; these are all "shear" transformations except for the identity. For trace $$-2$$ there is a similar $$\mathbb{Z}$$-indexed family of conjugacy classes represented by $$\pmatrix{-1 & n \\ 0 & -1}$$.
5. In general, for nonzero trace the conjugacy classes come in opposite pairs, represented by a matrix $$M$$ with trace $$t>0$$ and an opposite representative $$-M$$ with trace $$-t<0$$.
6. For trace of absolute value $$> 2$$, there is one conjugacy class for each word of the form $$R^{j_1} L^{k_1} R^{j_2} L^{k_2} \cdots R^{j_I} L^{k_I}$$ up to cyclic conjugacy, where $$I \ge 1$$ and all the exponents are positive integers. A matrix representing this form is obtained from the above word by making the replacements $$R=\pmatrix{1 & 1 \\ 0 & 1}, \quad L=\pmatrix{1 & 0 \\ 1 & 1}$$ These are all "hyperbolic" transformations, having an independent pair of real eigenvectors. The slope of the expanding eigenvector is a quadratic irrational, and hence has eventually repeating continued fraction expansion. The cyclic sequence $$(j_1,k_1,j_2,k_2,\ldots,j_I,k_I)$$ can be thought of as the fundamental repeating portion of the continued fraction expansion of the slope of the expanding eigenvector, or, better, as an appropriate power of the fundamental repeating portion where the power is equal to the exponent of the given matrix.

Number theorists will tell you that the number of conjugacy classes of each trace $$t>2$$ is closely related to the class number of the number field generated by $$\sqrt{t^2-4}$$.

• Not the square root of the trace, but rather $\sqrt{t^2-4}$, where $t$ is the trace. – David E Speyer Apr 14 '16 at 2:52
• @LeeMosher Nice answer. Do you have reference for this? For the case 6, you probably have to say that the conjugacy class is of the form $\pm R^{j_1}L^{k_1}\cdots R^{j_I}L^{k_I}$. Indeed, if the trace is negative, you cannot be conjugate to a product of $R,L$. – Jérémy Blanc Mar 11 '17 at 6:28
• I don't have a specific reference. I learned it over a long process of ingestion, maybe starting with my first book on continued fractions as a kid, and continuing with my education in mapping class groups ($PSL_2(\mathbb{Z})$ is the mapping class group of the torus). The answer of @IgorRivin answer gives some references. – Lee Mosher Mar 11 '17 at 13:57

This is the subject of Gauss' reduction theory, as discussed in Karpenkov's book (among many other places). In this 2007 paper, Karpenkov also extends the method to $SL(n, \mathbb{Z}).$

The conjugacy classes of elements of ${\rm SL}(2,\mathbb{Z})$ with given trace are counted in:

S. Chowla, J. Cowles and M. Cowles: On the number of conjugacy classes in SL(2,Z). Journal of Number Theory 12(1980), Issue 3, Pages 372-377.

• Extraordinary: the paper which has Chowla as an author, was also communicated by Chowla! – Lucia Apr 13 '16 at 22:29

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You are viewing an older version of this Concept. Go to the latest version.

# Multiplication of Rational Numbers

## Multiply fractions: multiply straight across

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Multiplication of Real Numbers

Jacob received tips of $4.00 each from three of his paper route customers. How much did he receive in total? ### Watch This ### Guidance Multiplication of two integers with the same signs produces a positive result and multiplication of two integers with unlike signs results in a negative answer. These rules can be applied to the multiplication of all real numbers. To multiply fractions, you multiply the numerators and then you multiply the denominators. The product of the numerators over the product of the denominators is the answer to the problem. Sometimes the answer can be expressed as an equivalent fraction. The rules for multiplying integers also apply to multiplying decimals. The sum of the number of digits after the decimal points determines the placement of the decimal point in the answer. #### Example A Sam spent$2.00 for a bottle of chocolate milk at the school cafeteria every school day. At the end of the week, how does this affect his net worth?

Solution: The result of is –10. The product of a positive integer and a negative integer is always negative.

#### Example B

What is ?

Solution: The result of is +6. The product of two negative integers is always positive.

#### Example C

i)

ii)

iii)

Solution: Remember, there are three simple steps to follow to multiply fractions:

1. Multiply the numerators of the fractions
2. Multiply the denominators of the fractions.
3. Simplify the fraction if necessary.

i)

ii) Express the mixed number as an improper fraction.

iii) Express the mixed numbers as improper fractions.

#### Example D

Solution: Multiply the numbers as you would whole numbers. To place the decimal point in the answer, count the number of digits after the decimal points in the problem. There are two digits after the decimal point in 14.65 and one digit after the decimal point in 2.7. This is a total of three digits after the decimal points. From the right of the answer, count three places to the left and insert the decimal point.

Jacob received tips of $4.00 each from three of his paper route customers. How much did he receive in total? The result of is +12. The product of two positive integers is always positive. ### Vocabulary Denominator The denominator of a fraction is the number on the bottom that indicates the total number of equal parts in the whole or the group. has denominator 8. Fraction A fraction is any rational number that is not an integer. Improper Fraction An improper fraction is a fraction in which the numerator is larger than the denominator. is an improper fraction. Integer All natural numbers, their opposites, and zero are integers. A number in the list ..., –3, –2, –1, 0, 1, 2, 3... Mixed Number A mixed number is a number made up of a whole number and a fraction such as . Numerator The numerator of a fraction is the number on top that is the number of equal parts being considered in the whole or the group. has numerator 5. ### Guided Practice Multiply the following fractions: 1. 2. 3. Determine the answer to Answers: 1. Multiply the numerators. Multiply the denominators. Simplify the fraction. The answer can be written as or . 2. Write the two mixed numbers as improper fractions. Multiply the denominator and the whole number. Add the numerator to this product. Write the answer over the denominator. Follow the steps for multiplying fractions. Simplify the fraction if necessary. 3. Multiply the numbers as you would whole numbers. Remember the rule for multiplying integers. When you multiply two integers that have the same sign, the product will always be positive. There are three digits after the decimal point in 135.697 and two digits after the decimal point in 34.32. Beginning at the right of the product, count five places to the left and insert the decimal point. ### Practice Multiply. Match each given phrase with the correct multiplication statement. Then, determine each product. 1. take away six groups of 3 balls 2. net worth after losing seven$5 bills
3. take away nine sets of 8 forks
4. take away four sets of four plates
5. receive eight groups of 4 glasses
6. buy seven sets of 12 placemats

Use the rules that you have learned for multiplying real numbers to answer the following problems.

### Vocabulary Language: English

Associative Property

Associative Property

The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).
Commutative Property

Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example $a+b=b+a \text{ and\,} (a)(b)=(b)(a)$.
Denominator

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.
distributive property

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.
fraction

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
improper fraction

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.
Integer

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Mixed Number

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.
multiplicative identity property

multiplicative identity property

The product of any number and one is the number itself.
Numerator

Numerator

The numerator is the number above the fraction bar in a fraction.


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# American Institute of Mathematical Sciences

July  2006, 6(4): 651-666. doi: 10.3934/dcdsb.2006.6.651

## Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow

 1 Department of Statistics, University of Michigan, 439 West Hall, 1085 S. University Ave., Ann Arbor, MI 48109-1107, United States 2 Department of Mathematics, University of Tennessee, 121 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996-1300, United States

Received  February 2005 Revised  September 2005 Published  April 2006

We derive a large deviation principle for a stochastic Navier-Stokes equation for the vorticity of a two-dimensional fluid when the magnitude of the random term tends to zero. The key is the verification of the exponential tightness for the stochastic vorticity.
Citation: Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651
 [1] Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221 [2] Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67 [3] Hugo Beirão da Veiga. Navier-Stokes equations: Some questions related to the direction of the vorticity. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 203-213. doi: 10.3934/dcdss.2019014 [4] Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234 [5] Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073 [6] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [7] Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997 [8] Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 [9] Guangrong Wu, Ping Zhang. The zero diffusion limit of 2-D Navier-Stokes equations with $L^1$ initial vorticity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 631-638. doi: 10.3934/dcds.1999.5.631 [10] Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048 [11] Hongyong Cui, Mirelson M. Freitas, José A. Langa. Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1297-1324. doi: 10.3934/dcdsb.2018152 [12] I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191 [13] Tao Wang, Huijiang Zhao, Qingyang Zou. One-dimensional compressible Navier-Stokes equations with large density oscillation. Kinetic & Related Models, 2013, 6 (3) : 649-670. doi: 10.3934/krm.2013.6.649 [14] Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521 [15] Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991 [16] Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-26. doi: 10.3934/dcdsb.2019172 [17] Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697 [18] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [19] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [20] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

2018 Impact Factor: 1.008


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# Difference between revisions of "2016 AMC 8 Problems/Problem 4"

When Cheenu was a boy he could run $15$ miles in $3$ hours and $30$ minutes. As an old man he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy?

$\textbf{(A) }6\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad \textbf{(E) }30$

## Solution

When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes $= 3\times60 + 30$ minutes $= 210$ minutes, thus running $\frac{210}{15} = 14$ minutes per mile. When he is an old man, he can walk $10$ miles in $4$ hours $= 4 \times 60$ minutes $= 240$ minutes, thus walking $\frac{240}{10} = 24$ minutes per mile. Therefore it takes him $\boxed{\textbf{(B)}\ 10}$ minutes longer to walk a mile now compared to when he was a boy.

 2016 AMC 8 (Problems • Answer Key • Resources) Preceded byProblem 3 Followed byProblem 5 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AJHSME/AMC 8 Problems and Solutions


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# To write the polynomial 5x^4+8x^5-2x+3 in standard form, which term is the leading term? a. the constant

###### Question:

To write the polynomial 5x^4+8x^5-2x+3 in standard form, which term is the leading term?
a. the constant term
b. the quintic term
c. the quartic term
d. the linear term

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### What was Dan's average speed during the 800-meter run?A. 6.7 m/sB. 7.6 m/sC. 5.7 m/sD. 8.4 m/s

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### What would happen to the rock cycle if erosion did not occur

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### Tim has a scale model of an airplane which says it is in a 1: 15 ratio what shoes that mean?

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### Which part of speech is the underlined word? we saw the french ambassador at the museum. a. adverb b. verb c. noun d. adjective

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### Giving BrainlyestHow are the small intestines and large intestines similar?They are located above the stomach and help distribute

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### Asquare membrane with four piezoresistors is diagramed below. resistors r1 and r4 are located at the

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### What is the best way to correct the comma splice in the following sentence?Mariah enjoys kayaking, Stephen prefers water skiing.

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### List three inportant facts about why jamestown is signifcant in u. s. history

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### Que pasaría si usaramos coma para separar los elementos de la enumeración de la oración: 'los grandes

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### Lifting a piano into a truck is an example of doing negative work. A. True B. False

Lifting a piano into a truck is an example of doing negative work. A. True B. False...


text: Sometimes, it's a good idea to open a copy of a document instead of the document itself. This way, you can avoid making any accidental changes to the original. You can open a copy of a document from the Windows 98 Desktop without having to use the Save As command. Just right-click the file and choose New. A copy of the document (with a generic name) opens, and when you press Ctrl+S to save it, you're prompted to name it. Make sure that you give this document a new name, such as Copy of Document1 so that you do not overwrite the original document.


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# Synopsis: Deep Freezing Molecules

Researchers cooled trapped molecules well below $1\phantom{\rule{2.22198pt}{0ex}}\text{mK}$—a record temperature for molecules that have not been assembled from pre-cooled atoms.

Atom cooling is a mature field, and scientists hope that bringing molecules to ultracold temperatures, which is more challenging than cooling atoms, will lead to advances in precision measurement, ultracold chemistry, and quantum computing. Researchers have previously cooled molecules below $1\phantom{\rule{2.22198pt}{0ex}}\text{mK}$ indirectly, by merging two pre-cooled atomic gases into a gas of two-atom molecules. But this method only works for a limited set of molecular species. Now two teams of researchers, using different techniques, have cooled large numbers of trapped molecules down to $400\phantom{\rule{2.22198pt}{0ex}}𝜇\text{K}$—a temperature nearly 10 times lower than previous direct-cooling experiments.

Gerhard Rempe and Martin Zeppenfeld of the Max Planck Institute for Quantum Optics, Germany, and their colleagues used the so-called Sisyphus technique with formaldehyde molecules. In their experiment, molecules repeatedly climb the potential “hills” at the edges of an electric field trap, losing kinetic energy each time they do. The team cooled 300,000 molecules, at least 30 times the previous record from indirect cooling.

David DeMille of Yale University and his colleagues cooled strontium monofluoride in a modified magneto-optical trap (MOT). One common difficulty in cooling molecules is that they can easily make transitions to quantum states associated with rotation and vibration that are dark, that is, not accessible to the cooling lasers. The Yale team solved this problem for rotational states by rapidly oscillating the MOT’s magnetic field and also periodically reversing the polarization of the MOT laser. These synchronized oscillations periodically turn dark rotational states into accessible bright states and vice versa. Dark vibrational states were instead addressed with additional lasers, a technique the team developed previously.

This research is published in Physical Review Letters.

–David Ehrenstein

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text:
# Recommendations for what to do when k exceeds 0.5 in the loo package?

My “base” model is the Poisson model I described in this thread.

Saying that my base model is already complex is a fair description, that is why I called “base” instead of “simple.” The reasons why I started with that model are a bit complex and hopefully not relevant for this discussion. That said, I actually want to compare this model with simpler and more complex versions.

A follow-up question, is this plot a good reason to think that this base model is not really bad?

If I understand this correctly, loo indicates that my second model is better than the base model. That said, I still have some problems with k.

> print(loo_2)
Computed from 6000 by 1795 log-likelihood matrix

Estimate    SE
elpd_loo  -9079.3  84.5
p_loo      1346.5  48.3
looic     18158.7 169.0

Pareto k diagnostic values:
Count  Pct
(-Inf, 0.5]   (good)     721   40.2%
(0.5, 0.7]   (ok)       676   37.7%
(1, Inf)   (very bad)  78    4.3%
See help('pareto-k-diagnostic') for details.
>
> print(compare(loo_1, loo_2), digits = 3)
elpd_diff        se
105.062    22.444

Thanks, I will carefully read this Saturday morning, and probably come back with more questions after that.

Looking forward to loo 2.0. Do you know when are you going to release it?

Thanks a lot for your amazing work and all the help!

Oh that problematic one…

Here https://rawgit.com/avehtari/modelselection_tutorial/master/roaches.html is also an example of bad Poisson model with high k’s and p_loo>n, but where switching to negbin helped. Poisson is often problematic as the variance is same as mean. Negbin is not necessarily better if common dispersion parameter is used, but it’s worth testing as it’s so simple to switch.

Can you tell the values of I and T? Based on the other thread T=5, but I couldn’t find value for I.
This would help to understand whether khats are big due to weak prior or model misspecification.

The model in the thread you linked is simple, but can be flexible if I is large and priors are vague,

The posterior predictive checking can give very good fits also if the model is too flexible (too many parameters with weak prior). Could you plot also LOO-PIT (implemented in bayesplot)?

It seems both of your models have weak priors and thus observations are highly influential. This can happen also with well specified flexible models (I have experience with GPs and GMRFs) and it’s the weakness of importance sampling that it fails (and waic would fail too)

The comparison indicates that the second model is better, but having that many high k’s is problematic.

This might also be useful for you (just don’t use waic)
E. C. Merkle, D. Furr, S. Rabe-Hesketh Bayesian model assessment: Use of conditional vs marginal likelihoods https://arxiv.org/abs/1802.04452

Soonish…

2 Likes

T=5, and I=359

First time that I try to do this, not sure if i did it right:

# Extract pointwise log-likelihood and compute LOO
log_lik_1 <- extract_log_lik(fit)
loo_1 <- loo(log_lik_1)
print(loo_1)

# marginal predictive check using LOO probability integral transform
psis <- psislw(-log_lik_1, cores = 32)
ppc_loo_pit(bsdata$y, as.matrix(y_sim), lw = psis$lw_smooth)

Could you point me to an explanation for how to read this figure?

This is my first time trying to fit a negative binomial model with Stan. In my Poisson model the model and generated quantities blocks are:

model {
sigma_a ~ normal(0, 1);
sigma_b ~ normal(0, 1);
a_std ~ normal(0,1);
b_std ~ normal(0,1);
gamma ~normal(0,1);
beta ~ normal(0,1);
y ~ poisson_log(log(n) + x_beta + gamma[index_time] + a[practice] + b[practice] .* time);   // likelihood
}

generated quantities {
//vector[I * T] y_sim;
vector[I * T] log_lik;
vector[I * T] y_hat = log(n) + x_beta + gamma[index_time] + a[practice] + b[practice] .* time;
// for (it in 1:I * T) {
//  y_sim[it] = poisson_log_rng(y_hat[it]);
// }
for(it in 1:I*T){
log_lik[it] = poisson_log_lpmf(y[it] | log(n[it]) + x_beta[it] + gamma[index_time[it]] + a[practice[it]] + b[practice[it]] .* time[it]);
}
}

After reading this example, i changed these blocks to:

model {
phi ~ cauchy(0, 3);
sigma_a ~ normal(0, 1);
sigma_b ~ normal(0, 1);
a_std ~ normal(0,1);
b_std ~ normal(0,1);
gamma ~normal(0,1);
y ~ neg_binomial_2_log(log(n) + gamma[index_time] + a[practice] + b[practice] .* time, phi);   // likelihood
}

generated quantities {
vector[I * T] y_sim;
vector[I * T] log_lik;
vector[I * T] y_hat = log(n) + gamma[index_time] + a[practice] + b[practice] .* time;
for (it in 1:I * T) {
y_sim[it] = neg_binomial_2_log_rng(y_hat[it], phi);
}
for(it in 1:I*T){
log_lik[it] = neg_binomial_2_log_lpmf(y[it] | log(n[it]) + gamma[index_time[it]] + a[practice[it]] + b[practice[it]] .* time[it], phi);
}
}

where real<lower=0> phi;

Alas, when I try to fit this model I get a bunch of errors:

[1] "Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:"
[2] "Exception: neg_binomial_2_log_lpmf: Precision parameter is 0, but must be > 0!  (in 'model68685a6b6cb4_stan_68685cba1da' at line 36)"
[3] "If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,"
[4] "but if this warning occurs often then your model may be either severely ill-conditioned or misspecified."
[5] "Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:"
[6] "Exception: neg_binomial_2_log_lpmf: Precision parameter is 0, but must be > 0!  (in 'model68685a6b6cb4_stan_68685cba1da' at line 36)"
[7] "If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,"
[8] "but if this warning occurs often then your model may be either severely ill-conditioned or misspecified."
[9] "Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:"
[10] "Exception: neg_binomial_2_log_lpmf: Precision parameter is 0, but must be > 0!  (in 'model68685a6b6cb4_stan_68685cba1da' at line 36)"
[11] "If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,"
[12] "but if this warning occurs often then your model may be either severely ill-conditioned or misspecified."
[13] "Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:"
[14] "Exception: neg_binomial_2_log_lpmf: Precision parameter is 0, but must be > 0!  (in 'model68685a6b6cb4_stan_68685cba1da' at line 36)"
[15] "If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,"
[16] "but if this warning occurs often then your model may be either severely ill-conditioned or misspecified."
[17] "Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:"
[18] "Exception: neg_binomial_2_log_lpmf: Precision parameter is 0, but must be > 0!  (in 'model68685a6b6cb4_stan_68685cba1da' at line 36)"
[19] "If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,"
[20] "but if this warning occurs often then your model may be either severely ill-conditioned or misspecified."
[21] "Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:"
[22] "Exception: neg_binomial_2_log_lpmf: Precision parameter is 0, but must be > 0!  (in 'model68685a6b6cb4_stan_68685cba1da' at line 36)"
[23] "If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,"
[24] "but if this warning occurs often then your model may be either severely ill-conditioned or misspecified."
[25] "Error in sampler$call_sampler(args_list[[i]]) : " [26] " Exception: neg_binomial_2_log_rng: Random number that came from gamma distribution is 9.9818e+009, but must be less than 1.07374e+009 (in 'model68685a6b6cb4_stan_68685cba1da' at line 45)" [1] "error occurred during calling the sampler; sampling not done"  Oh it’s the old version. For the new better version see Fig 9 in https://arxiv.org/abs/1709.01449 and code in https://github.com/jgabry/bayes-vis-paper. I will explain this old figure here. It’s a qq-plot comparing uniform distribution and cdf-values (PIT-values). There is excess of LOO-PIT values which are close to zero and close to 1, which means that the left-out data is more dispersed than what you predict which means you have a problem! Since you model has so many parameters it’s possible that PPC does not reveal this same problem (and it seems you case is great case-study example of this PPC vs. LOO-PIT behavior). Based on this figure I would try negbin instead of Poisson. Variance of negbin (with neg_binomial_2 parameterization) is mu + mu^2/phi. Cauchy prior for phi let’s phi to get too close to 0, and constraint real<lower=0> phi is underflowing to zero. You need to have a zero avoiding prior or as a quick test you can try <lower=0.1> (but the zero avoiding prior would be more elegant) Thanks! Out of curiosity, will the newer version of the plot make it to cran anytime soon? I did. If I make this change and I run 4 chains, things don’t work: some chains had errors; consider specifying chains = 1 to debughere are whatever error messages were returned [[1]] Stan model 'stan-72e478f44dfb' does not contain samples. [[2]] Stan model 'stan-72e478f44dfb' does not contain samples.  My guess is that this is the same problem we are discussing on this thread. @bbbales2 thoughts? Things get even weirder. I tried to fit this with only one chain, and everything runs smoothly. My guess is that this has something to do with the starting values when you do 4 chains vs only 1 chain. So, given that this works with only one chain, I thought I should share some output: Computed from 1000 by 1795 log-likelihood matrix Estimate SE elpd_loo -8716.7 46.2 p_loo 553.4 21.2 looic 17433.4 92.3 Pareto k diagnostic values: Count Pct (-Inf, 0.5] (good) 1290 71.9% (0.5, 0.7] (ok) 388 21.6% (0.7, 1] (bad) 106 5.9% (1, Inf) (very bad) 11 0.6% See help('pareto-k-diagnostic') for details.  @avehtari if I understood you correctly, this figure shows that the negative binomial is doing a much better job than the Poisson model. Is that correct? Could you show me the more elegant solution? @jonah knows about that Did you try different random seeds for four chains? I think you need initialize chains more carefully, or do some scaling of the parameters, Now some random initial values are way beyond anything sensible. Yes! Also Pareto k’s are much better, and elpd_loo is much better, so all diagnostics tell negbin is much better than Poisson. I recommend also plotting the marginal posterior of phi. I have previously used inverse-gamma prior, and you could try that, too, but to give a well justified recommendation would require some thinking or googling. I’ll get back to this in a few days unless you are able to find something before that. 1 Like Sorry, I’m new to this… I have no clue how to initialize chains more carefully, nor how to try different seeds for the four chains, nor what you have in mind when you say that I should do some scaling of the parameters. Could you point me in the right direction? The code in that repo only uses the most recent CRAN version ofbayesplot. In that case, the function is ppc_dens_overlay 2 Likes see ?stan and option seed. Try runinng stan(...., seed=1337) with different seed values. see ?stan and option init. The default is to randomly generate initial values between -2 and 2 on the unconstrained support. If you transform -2 and 2 from unconstrained to constrained space, are these initial values sensible? If you are unlucky initial values are all 2, and if n=1 and x=1, then log(n) + x*2 + 2 + exp(2)*2 + epx(2)*2*4 is about 79 which when exponentiated to Poisson parameter is about 2e34, which is a lot. You probably need to initialize log(a_std) and log(b_std) with much smaller range than [-2, 2] or scale them in Stan code by dividing them with a big number. For example, try dividing ‘a’ and ‘b’ by 100. Then maximum Poisson parameter value (with n=1 and x=1) is about 311, which should work fine. This is something we should add in wiki / manual (if it’s not already there). Sorry I didn’t realize earlier this is the likely problem in the other thread, 1 Like I have been corresponding with avehtari who suggested that I post to this site. I too, have been getting the k>0.5 problem too often to ignore. So I began to look for alternatives. Here is my best shot so far. 1. Noting that elpd = 1/mean(Importance Ratio) , I plotted the log of the importance ratios and saw that the histogram looked very much like a translated gamma distribution. 2. Using the script below I fit a loggamma distribution to the importance ratios. Using the formula for the mean of the (transated) loggamma distribution, I then calculated elpd_tlg. 3. The script also calculates the elpd_loo and elpd_raw. I ran the script with my model on 200 datasets comparing the three calculations. To test the sample variance I ran the stan model with 5 different seeds to get a rough estimate of the sample standard deviation. Here are the conclusions. 4. The sample sd for elpd_loo and elpd_tlg tend to be noticeably smaller than the sd of elpd_raw 5. The sample sd for elpd_tlg tends be marginally smaller that sd elpd_loo 6. The parameters that I get for the loggamma distribution are such that the mean exists for all 200 x 5 samples. There is no analog of the k>0.5 situation. 7. The standard deviations (even for elpd_loo) I get are much smaller than the standard deviation output by the loo package. I have a writeup and a spreadsheed with all the details. My next post will contain the script # function to fit loggamma distribution by the method of moments # reference - p227 of Loss Distributions - Hogg and Klugman 1984 fit_loggamma=function(x){ minlogx=min(log(x)) tlogx=log(x)-minlogx m1=mean(tlogx) # translate to make all numbers positive m2=mean(tlogx^2) alpha=m1^2/(m2-m1^2) lambda=m1/(m2-m1^2) mean_tlg=1/(1-1/lambda)^alpha*exp(minlogx) #use exp(minlogx) to untranslate output=list(mean_tlg = mean_tlg, alpha_tlg = alpha, lambda_tlg = lambda) return(output) } # function to get elpd by three different formulas elpd_calc=function(loglik){ # calc #1 with the “loo” package loo_psis <- loo(loglik) # calculate importance ratios impratio=matrix(0,dim(loglik)[1],dim(loglik)[2]) for (i in 1:dim(loglik)[2]){ impratio[,i]=1/exp(loglik[,i]) } # calculate raw elpd prloo_raw=rep(0,dim(loglik)[2]) for (i in 1:dim(loglik)[2]){ prloo_raw[i]=1/mean(impratio[,i]) } elpd_raw=sum(log(prloo_raw)) # calculate elpd_tlg prloo_tlg=rep(0,dim(loglik)[2]) lambda_tlg=NULL for(i in 1:dim(loglik)[2]){ tlg=fit_loggamma(impratio[,i]) prloo_tlg[i]=1/tlg$mean_tlg
lambda_tlg=c(lambda_tlg,tlg$lambda_tlg) } elpd_tlg=sum(log(prloo_tlg)) output=list(elpd_psis = round(loo_psis$elpd_loo,3),
elpd_raw = round(elpd_raw,3),
elpd_tlg = round(elpd_tlg,3),
se_elpd_psis = round(loo_psis$se_elpd_loo,3), max_k_psis = round(max(loo_psis$pareto_k),3),
min_lam_tlg = round(min(lambda_tlg),3))
return(output)
}

This would have been good as a separate thread (now it might get lost in here)

This is ok when true, but this will be true only for certain models and likelihoods and thus useful only in special cases.

PSIS paper [1507.02646] Pareto Smoothed Importance Sampling has also example where smaller variance is achieved if a distribution can be fit to all importance ratios instead of just the M largest ones.

I didn’t see that in your script. Could you post that part, too, so that I can check where is the difference.

1 Like

It would be nice to know when “this” is true. For example in my immediate problem my model statement is of the form y ~ normal(mu, sigma) where the mu and sigma are derived from complex combinations of other parameters.

I put the “se_elpd_psis = round(loo_psis\$se_elpd_loo,3)” in the output of the “elpd_calc” function in my script above.

Here is a sample calculation that I am referring to. I ran my model with different seeds to get a ballpark estimate of the sample variability.

stan Seed 12345 23456 34567 45678 56789
elpd_psis 49.811 49.892 50.323 48.922 49.498
se_elpd_psis = 5.824
Sample Standard deviation of elpd_psis = 0.520

I was expecting that the values from the two calculations of the standard deviation to be similar. What am I missing?

I have a spreadsheet with 2000 such calculations that I would be happy to post if I could figure out how to do it. The above calculation is representative of what I got in the 2000 such calculations.

Yes it would, and the problem is that it’s not trivial (or at least not trivial for me) for a general model.

My guess (before seeing your comment on the model) was that with normal and with certain models for mu and sigma, it can be close even if not true, but there the conditions for mu and sigma are non-trivial. You were lucky that it works in your problem.

These are completely two different things. loo package returns se describing the uncertainty about the future data. What you compute here is the Monte Carlo error estimate for the uncertainty due to finite draws in MCMC. See Section 5.1 in [1507.04544] Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC

I would like to be a little better than “lucky.” To aid my understanding of this, could you point me to some examples with different distributions of the importance ratio?

I looked at the paper. Your pointing me to Section 5.1 is very helpful. One of the things you said there was “One can also compute Monte Carlo standard errors arising from the finite number of simulation draws using the formula from Gelman et al. (2013) which uses both between and within-chain information and is implemented in Stan.”

Could you point me to the formula in BDA3 (Chapter 7?) and also how I might capture it in Stan?

This Monte Carlo error is usually negligible compared to se computed by equation (23) in that paper, and in most cases you can ignore it. loo 2.0 estimates that, too, by combining N_eff computation described in Stan manual (more up to date than BDA3 (it’s annoying we can’t update the paper which has been published with a better reference)) and Monte Carlo error for importance sampling as described in [1507.02646] Pareto Smoothed Importance Sampling and Monte Carlo theory, methods and examples

When will loo 2.0 be up on Cran? The version that I can get is 1.1.

This was discussed in this same thread before… Before summer. Meanwhile you can get it from github.

OK - Thanks for your help.


text:
# Design of a calibrated sensor for remotely monitoring and managing electrical load consumption

Rounak Goswami

Indian Institute of Engineering Science and Technology, Shibpur

Guided by:

K.C. James Raju

Professor, Department of Physics, University of Hyderabad

## Abstract

Keywords: energy management systems, isolation transformers, energy metering, energyDSM, energy efficiency, true RMS

## Abbreviations

List of Abbreviations
 CT Current Transformer PT Potential Transformer BLE Bluetooth Low Energy FHMM Factorial Hidden Markov Model algorithm DMM Digital MultiMeter MSO Mixed Signal Oscilloscope PCB Printed Circuit Board MMM Monitor-Measure-Manage SWG Standard Wire Gauge

## Background

Energy conservation and efficient use of energy and resources are required for the sustainable growth and development of the human society. The economic growth of India or any country for that matter has largely been associated with energy consumption. A major chunk of the industrial, commercial and domestic requirements is in the form of electrical energy i.e, irrigation, transportation & freight, utility, HVAC, industrial applications. Hence, focus on the management of energy, and its demand and supply, is the need of the hour. Understanding various components of energy demand is very much pertinent to understand the load patterns and predict the electrical system performance for the end loads and optimize energy usage.

## Statement of the problems

Conventional electrical consumption monitoring systems usually rely on power monitoring & metering equipment such as a smart meter or a conventional metering device. But this type of monitoring of end load consumption is done by a lump from a specific metering point. There are some smart power consumption monitoring systems, but they are either cost-intensive or the sensors placed at each load consume a sizable power defeating the purpose of energy savings or they could not be remotely/centrally monitored & managed. Using wired/BLE/Ethernet as the transmission medium, the results populated on the server, which in turn are disaggregated/analyzed for the electricity consumption on each load using FHMM or similar algorithms for the user to observe the consumption of all or each of the loads. The parameters of each individual end load in this method heavily depends on the execution of the algorithm.

However, obtaining enough information for energy monitoring is limited by the number of measuring equipment that can be installed. Cost and privacy issues are involved. The status of devices is difficult to infer from limited information using the technology called non-intrusive appliance load monitoring, while it can be easily managed by intrusive techniques. However, because of the limited information in estimating the operational status of the devices, a lot of efforts to improve the estimation accuracy are in progress.

## Objectives of the Research

The problems with all the previously stated models of energy monitoring are many, poor demand visibility, addressing large power factor variations, unpredictable power quality to name a few. A slight aberration in consumption could impact the overall grid. The objective is to bridge the existing gap between this connected consumption and disconnected information, through an intelligent interconnected network of sensors which feed data into the system and allows real-time monitoring of appliance states. At the very fundamental level of this monitoring system stands a calibrated sensor which will be hooked to every connected electrical load for M-M-M purpose.

Sensing and measuring can be achieved by multiple techniques such as shunts, current transformers (CT), hall-effect sensors, Rogowski coils and magnetic field based transducers. Out of these, shunt and CT methods are found to be more reliant for obtaining accurate and reliable measurements for end use. The CT sensor method has been chosen for isolating the electrical load from the conditioning and digital microcontroller circuit.

I worked on the calibration of CT sensors with toroidal core. Proper calibration of the sensor is required to ensure reliable and reproducible measurements for predictable end use. An attempt has been made to identify the best-suited material, shape for the core which serves the purpose, the proper wire gauge and optimum turns ratio of the primary to secondary windings to address a load current ranging from possibly 1mA up to 1A. Comparisons have been made between the performance of different toroidal cores made with different core materials. Performance for a given core material with different number of turns on primary for a fixed load current, and with fixed number of turns on primary while addressing varying load currents have also been studied.

The basic purpose of this sensor is to sense the state of the load, measure the current consumption and remotely control the AC loads if necessary.

## Scope

The system when realized will have a lot of scope. It will not only be able to monitor the end loads, but it will also measure and manage the loads at the same time. This CT sensor which forms the fundamental building block of this management system can be further refined and improvised with additional circuitry to measure electrical parameters like voltage, power factor (lead/lag), phase, frequency, active power, reactive power, surge/spike and harmonics associated with the load.

## LITERATURE REVIEW

Studies have been done on the various types of loads namely, continuous loads, discrete loads, highly inductive loads, non-linear loads,loads which demand a very high starting current etc. A paper by Kurmi, Sharma, Singh[1] introduces the various possible loads in power systems.

Thorough studies have also been done on the various existing methods of current detection and sensing and energy monitoring systems. An application note by Yang Zhen [2] discusses the concepts and fundamentals of various current sensing circuits. Forghani-Zadeh and Rincón-Mora [3], in their paper, have given a comparative study of the various methods of current sensing. A paper by Sung-Yong Son[4] gives an alternative algorithm based approach to management systems in smart homes and also gives an idea of the challenges faced in realizing such a system. Soft Ferrite, A User's Guide by Magnetic Materials Producers Association gives us an insight into the soft ferrites, the material which has been used as the transformer core in our case.

## Information

Shunt resistor technique works on the simple principle that voltage drop across a resistor is proportional to the current going through it. The parasitic inductance present in the shunt affects high precision current measurement. They also introduce additional resistance in the source circuit and also limits high current sensing applications due to large power dissipation.

Rogowski coil is based on the principle of Faraday’s law of induction and the output voltage Vout of the Rogowski coil is given by integrating the current Ic to be measured. The Rogowski coil has a low sensitivity which can be compensated by either adding more turns or using an integrator with a higher gain.

Hall effect sensors are devices based on the Hall-effect, i.e, they are activated by an external magnetic field. Signal conditioning circuitry like an amplifier stage and temperature compensation are necessary to make the output usable for most applications.

## Summary

The act of measurement of current ranges from picoamps to tens of thousands of amperes. The selection of a current sensing method depends on requirements such as magnitude, accuracy, bandwidth, robustness, cost, isolation or size. The current value may be directly displayed by an instrument, or converted to digital form for use by a monitoring or control system. The added benefit of isolation and non -intrusive monitoring along with high precision makes the CT technique best suited for our purpose.

## Transformer

Theory of operation of a real transformer

A transformer basically operates on the principle of mutual induction. Flux is present in the primary coil of a transformer due to excitation from some source. Not all the flux produced in the primary coil passes through the secondary coil - some of the flux lines leave the iron core and pass through the air instead. The portion of the flux that goes through one of the transformer coils, but not the other one is called leakage flux and the flux that links both the windings is called the mutual flux. The ratio of the primary voltage caused by the mutual flux to the secondary voltage caused by the mutual flux is equal to the turns ratio of the transformer. A current is drawn in its primary circuit, even when the secondary circuit is open circuited. This current is the current required to produce flux in a real ferromagnetic core. This no-load current can be divided into two parts, magnetization current and core-loss current.

Transformer Core

The core of the transformer transfers energy from the primary winding to the secondary winding through magnetic coupling. As a matter of fact, the core selection is the first designing step and after that, all other building details are defined: the number of turns and the conductor element (that can be copper foil, solid wire, litz wire), the isolation, the output connections. The common transformer cores used in various applications are:

• Solid Iron/Steel

• Laminated Silicon alloy/silicon steel

• Amorphous and Nanocrystalline Laminations

• Soft Ferrites

• Powdered cores (distributed air gap materials)

Out of the above, soft ferrites are the cheapest and they can operate over a wide range of frequencies. They are ceramic, homogenous material composed of various oxides with iron oxide as the main constituent. They are ferromagnetic in nature. They have saturation flux density in the range of 0.3 to 0.5 Tesla and relative permeability of the order of about 1000 . Soft ferrites have low eddy current and hysteresis loss owing to high electrical resistivity and small hysteresis area over a wide range of frequencies. Soft ferrites are available in many shapes such as ring core, E Core, I core etc. Nanocrystalline cores have desirable magnetic properties like very high saturation flux density, high magnetic permeability and low coercivity.

Losses in a transformer

1.  Copper (I2R) Losses: Copper losses are the resistive heating losses in the primary and secondary windings of the transformer. They are proportional to the square of the current in the windings.

2.  Eddy Current Losses: Eddy current losses are resistive heating losses in the core of the transformer. They are proportional to the square of the voltage applied to the transformer.

3.   Hysteresis Losses: Hysteresis losses are associated with the rearrangement of the magnetic domains in the core during each half-cycle. They are a complex, nonlinear function of the voltage applied to the transformer.

4.   Leakage flux: The fluxes which escape the core and pass through only one of the transformer windings are leakage fluxes. These escaped fluxes produce a self-inductance in the primary and secondary coils, and the effects of this inductance must be accounted for.

Instrumentation transformers

Instrumentation transformers are of two types:

Potential Transformers -Their primary is connected in parallel with the main circuit. Moreover, potential transformers are used as "sourcing elements".

Current Transformers -Their primary is connected in series with the main circuit and they are mostly used for metering and control purposes.

For the purpose of current detection, we have used a current transformer. The current transformer has two components, namely core and windings. The ideal current transformer may be defined as one in which any primary current is reproduced in the secondary circuit in the exact ratio and phase relationship.

Ip=Primary Current

Is=Secondary Current

Np=Number of turns in the primary

Ns=Number of turns in the secondary

H=Magnetic field strength (Amps/metre)

B=Magnetic flux density (Tesla)

le=Effective mean length of flux path

Ae=Effective cross-sectional area

For an ideal current transformer: INp = Is Ns $\frac{{I}_{p}}{{I}_{s}}=\frac{{N}_{s}}{{N}_{p}}$

Therefore the ratio of primary and secondary winding currents equal to the turns ratio.

In an actual transformer, the windings have resistance and reactance and also the transformer has magnetizing and loss component of current to maintain the flux. Therefore, in an actual transformer the ratio of current is not equal to the turns ratio and also there is a phase difference between the primary current and the secondary currents reflected back on the primary side and consequently, we have ratio error and phase angle error.

The following are the basic relations from magnetism:

μ = $\frac{B}{H}$   [Definition of Relative Permeability]

H = $\frac{{N_{p}}I_{p}}{l_{e}}$       [Ampere's Circuital Law]

Erms = 4.44φfNsErms= 4.44BfNsAe [Faraday's Law of Electromagnetic Induction]

For a given toroidal transformer core, the maximum value of H is Hs which corresponds to the saturation flux density Bsat. So the maximum value of the NpIp product is given by

(NpIp)max = Hs×le

Knowing the value of maximum primary current Ip max we can get the value of maximum number of primary turns Np max.

The value of Bsat restricts the maximum value of Erms that can be obtained at the CT secondary.

It is to be noted that distributed capacitance and leakage inductance in a CT restricts the turns ratio from being too high. But if the turns ratio is very low then the output may be unstable because of distortion.

## Burden Resistance

The secondary of a CT should not be kept open. So a burden resistor is kept across the secondary terminals of the CT. It determines the value of secondary current Is. It helps to avoid the loading effect at the CT secondary and also addresses the no load and full load connected condition. Burden Resistor Rb is also necessary if we want to connect multiple CTs in a cascaded fashion. Let VCT = Erms be the emf obatained at the transformer secondary. Then the secondary current Is is given by

Is = $\frac{V_{CT}}{R_{b}}$

Now, Is = $\frac{I_{s}N_{p}}{N_{s}}$

Therefore, Rb = $\frac{V_{CT}N_{s}}{I_{p}N_{p}}$

Corresponding to the value of Np max we can obtain the value of Rb min.

## B-H Curve

The B-H curve is a plot of magnetic flux density (B) v/s magnetic field strength (H) for a particular material. This is an important curve for selecting the material of the transformer. It helps us to compare the magnetic properties of one material over another while selecting a material for a specific application.

The following two graphs show the B-H characteristics for two different materials with the highlighted zone showing the desired region of operation for linearity.

F39
P11
B-H curve (from material datasheet)

The zone of operation on the B-H curve should be as linear as possible for our purpose since we want to replicate the primary current as a proportionally calibrated emf on the secondary side.

Calculation of Imin and Imax:

From the fig 1.a) it can be seen that as per the highlighted zone, Hmin =30A/m and Hmax =120A/m.

Now for F39 core , le = 30.14 mm. Hence for a fixed number of turns in the primary Np =19, we have Imin = $\frac{H_{min}l_{e}}{N_{p}}$Imin = $\frac{30 \times 30.14 \times 10^{-3}}{19}$ = 0.047 A = 47 mA

Imax = $\frac{H_{max}l_{e}}{N_{p}}$Imax = $\frac{120\times30.14\times10^{-3}}{19}$  =0.190 A = 190 mA

The above calculation gives the range of current can be addressed in a linear relation using the F39 core.

## Zero Crossing Detector and Snubber

A zero crossing detector detects the point of zero crossing of the voltage. This mechanism inside the optocoupler ensures that the gate of the power triac is triggered only when the supply voltage crosses zero so that the load is not exposed to sudden high voltage when turned on.

The following waveforms clearly depict how a triac driver only triggers at the point of zero crossing and also stops at the point of zero crossing.

Waveforms for triggering of Triac gate using Optoisolator Triac Driver

Since the current is not in phase with the voltage in case of capacitive and inductive loads, hence the semiconductor device which is acting as the electronic switch is exposed to high commutating dv/dt, which may be even higher than the static dv/dt rating of the device.Hence, the snubber circuit, which is basically an arc suppression circuit is put across the device to damp out the voltage spikes or high dv/dt which may result in false triggering of the device or may even cause electrical damage.

Waveform for depicting voltage across power triac while driving highly inductive loads

## Filter Circuits

Noise can be a very disturbing factor in signal detection and measurement, especially if the signal is itself very weak. The filter circuits are those which eliminate high-frequency noise and only allow the signal to pass. These are called low pass filters and can also have the ability to filter harmonics if specially designed. The main problem with passive filter circuits is that the signal may get attenuated when passing through it.

## Basic Block Diagram

The following figure shows the sensor circuit integrated into the total energy management system.

Block Diagram of the sensor circuit with the system

## Main Components

The main components in the circuit and their purpose are as described below:

Intelligent Power Supply

It is a power supply module for providing power to the microcontroller and signal conditioning circuit with·the following properties:

‣Highly accurate and low quiescent power consumption.

‣Delivers as much power as is demanded by the load.

‣Low footprint.

‣Low derating of power.

‣Wirelessly manageable sleep/wake up modes to avoid unnecessary power consumption.

Failsafe Mechanism and Snubber

It is for the purpose of protection. It protects the control circuit from any short circuit or overloading of the AC load and also protects the sensor circuit and the ac load from any switching or lightning-induced surges and spikes in the line causing electrical damage. It consists of components like fuse, MOVs etc. Snubber is for the dv/dt and di/dt protection of the semiconductor device.It can be any arc suppression circuit such as an R-C series circuit.

Signal Conditioning Circuit

It is the instrumentation and feedback circuitry which is directly fed with the CT sensor output. It has the following properties:

‣It has the capability to control multiple sensors (max 8 in this case).

‣It has high noise rejection capabilities.

‣It has a provision for configurable gain.

‣It consumes low power and has a minimal footprint.

‣It is wirelessly managed.

Wireless Microcontroller

It is the like the brain of the management system and it can have duplex communication with the remote server. It has the following properties:

‣It follows low power communication protocols.

‣It is a processor with offline computation and managing capabilities.

‣It has low quiescent power consumption and minimal footprint.

Triac

For the purpose of switching the load, we require a semiconductor switch. An SCR provides only unidirectional conduction controlled by a third terminal called the gate. A Diac is a bidirectional switch which can conduct in both directions above a certain threshold voltage but with no controlling third terminal.

The features of bidirectional conduction in diac and third controlling terminal in thyristor has been combined into a single semiconductor device called triac. It means TRIode for Alternating Current. It is a three terminal semiconductor junction device. It works in two active quadrants Q1 and Q3. The two main terminals are MT1 and MT2 and gate(G) is the controlling third terminal. There are four possible combinations:

Possible combinations for triggering of Triac
 Applied Voltage Gate triggering Current Quadrant of Operation State of Conduction MT2+ G+ Q1 ON MT2+ G- Q1 OFF MT2- G- Q3 ON MT2- G+ Q3 Off

The triac can be thought of as two back to back thyristors. It can drive ac loads. It is to be noted that in the above table the datasheet refers to MT2 and not MT1. This is because the triac is like a solid state switch and load will be connected to T2 only and not T1.

The graph below clearly explains the schemes for the above possible operations:

Triac V-I characteristics

In the circuit, we have used BT 136.

Optoisolator

Optocoupler is a device which contains infrared LED and some photodetector such as photodiode, darlington pair, phototransistor, triac etc. It acts like a solid state relay. It acts as an interface between digital control circuit and AC loads usually operating at different voltage levels (LV /HV). The gate of optoisolator internally works on an optical principle. Low voltage digital logic (DC) of microcontroller can be used as an input to the optoisolator to provide gate triggering pulse as an output to the power triac to drive and control the state of high voltage AC loads. The extremely high isolation voltage also protects the low voltage logic circuits (LHS) from surges and spikes arising out of load/power line (RHS). It also eliminates traditional methods of providing electrical isolation like the use of relay controlled contact or isolation transformers and can be used for fast electronic control of the load along with minimum size. Its output terminals also have very high dv/dt withstanding capability. It also has an internal zero crossing detector circuitry implemented within the chip which ensures that the power triac gate is triggered only when the supply voltage crosses zero so that the AC load does not experience any abruptly high voltage during on/off operation.

In the circuit, we have used MOC 3043.

Toroidal Current Transformer

Structure of the core has been chosen as toroidal because such cores provide a continuous path for the flux lines to pass without any air gap. So the reluctance of the core is very low with minimum leakage flux. Four different core materials for the toroidal core of CT have been used. Those are F39, P11, T38 and Nanocrystalline. For the purpose of winding SWG 26 enamelled copper wire was used.

Some important parameters related to toroidal core transformers are listed below:

✶C1, C2 → Geometric Core constants

✶A→ Outer Diameter of the toroid [A=2×r2]

✶B→ Inner Diameter of the toroid [B=2×r1]

✶C→ Height of the toroid [C=h]

✶le → Effective length

✶Ae → Effective area

✶AL →Inductance factor

✶L → Inductance

Toroidal Core

The following relations mathematically relate the above parameters:

C1 = $\frac{2 \pi }{h {\log_e (\frac{r_{2}}{r_{1}}) }}$

C2 = $\frac{2 \pi (\frac{1}{r_{1}}-\frac{1}{r_{2}})}{h^2 \log_e^3 (\frac{r_{2}}{r_{1}})}$

le = $\frac{C_{1} ^ 2}{C_{2}}$

Ae = $\frac{C_{1}}{{C_2}}$

AL = $\frac{\mu A_{e}}{l_{e}}$

L = AL×N2

## Methodology

• The circuit design was designed keeping the EMI/EMC considerations by using the DIP TraCe software which gives a file in xyz format. Then it was first converted into a gerber file from which the g code was generated. The circuit was printed on PCB using a CNC PCB milling machine rather than chemical processing (etching) for prototyping purpose. It was printed on single layer FR-4 glass epoxy laminated board.
PCB Design
Sensor Circuit
Sensor Circuit on PCB
• Respective components were then placed on the board and soldered at an appropriate soldering temperature, 300ºC. Thermal paste was used wherever necessary.
• Considering the limitations of Np and Ns, winding of transformer was done manually with bare hands carefully so that the enamel is not removed. At the ends, the enamel was removed to get the copper contact at the terminals. Tight winding is done ensuring that there is minimum air gap left between toroid and the wire. Elecreical isolation was maintained between primary and secondary windings using sleeve and the transformer was then placed in the circuit.
• The observations were taken under the following conditions for each of the four cores F39, P11, T38 and Nanocrystalline:

★Keeping Np constant and varying Ip.

★Keeping Ip constant and varying Np.

• Ip has been varied by varying the wattage of the bulbs connected as the load while Np has been varied from 1 to Np max. For the case of constant Np, it has been kept close to Np max (say for Np max =22, Np has been kept as 20). For the case of constant Ip, it has been kept close to Ip max (say for addressing current up to 1A, Ip has been kept as 0.868 A which is the current drawn by two 100W bulbs).
• Input is the line current applied to transformer primary and output is the emf taken across the CT secondary using high precision Scientific 4½ DMM incorporating True RMS using integral function and Tektronix 2002B 16 channel 70MHz MSO with noise filter.

## Comparative Graphs from collected data

• 1
• 2
• 3
• 4
Np v/s Erms for a constant load of 200W
• 1
• 2
• 3
• 4
Ip v/s Erms for constant turns ratio

From the above two graphs it is clear that higher values of emf can be achieved at the CT secondary using T38 core but the results, although less, are more linear for F39 and P11 cores. F39 and P11 cores give almost similar outputs though the P11 output is lower than the F39 output. This is possibly because of the lower value of initial permeability in the case Np or Ip . Nanocrystalline gives a lower value of emf due to a lesser number of turns in the primary and secondary.

## Waveforms obtained from MSO

P11 Core

Fig 8.b) is the sampled version of Fig 8.a) The waveform obtained is not sinusoidal and there is a lagging tendency at the leading edges in both positive and negative cycles which is probably due to lower initial permeability value of P11.

T38 Core

From Fig 9.a) and Fig 9.b) it is clear that the waveform of the latter one is more close to a sinusoid. This is probably due to the highly linear B-H characteristics of T38 at lower values of magnetizing strength H i.e., at lower values of load current.

From Fig 10 it is quite clear that the waveform obtained for the F39 core is nowhere close to sinusoidal and has lots of induced harmonics due to noise.

Nanocrystalline Core

Fig 11.a)is the closest to sinusoidal. This is because of the near perfect magnetic properties of Nanocrystalline which helps to replicate the variation in secondary emf in the same way as the variation in primary current.

Some noise introduced from the line in the primary winding was getting amplified in the secondary. The cores which have lower coercive field strength HC and lower remanent flux density Br are more susceptible to noise. This can be easily understood from the differences in the readings obtained from the DMM and the MSO. Most of the readings were higher in MSO as compared to DMM. The DMM has a fixed internal noise filter which rejects the noise to give the readings while the MSO has an adjustable noise filter which can be varied from 6 kHz to 70 MHz. If the filter frequency is set at higher levels, then the displayed values are higher because the noise also contributes to the output obtained.

The best material out of the four tested as toroidal core of CT is definitely Nanocrystalline, as expected, due to its high permeability which offers low reluctance to flux path, high Hs which allows a more linear range of operation, higher Bsat due to which higher values of emf can be observed in the secondary, low Hc and Br which leads to lower power loss in the hysteresis cycle. It is to be noted that choosing a higher SWG allows a higher number of turns to be accommodated due to its thinner diameter but it also decreases the current handling capability of the wire. Keeping both the factors,i.e, current carrying capacity and number of turns required, SWG 26 was found to be the best suited for our purpose.

## CONCLUSION

The selection of the core material depends on the application and the desired range of current detection. Although the very basic primary sensor circuit with four different toroidal core materials was realized and a comparative study was completed, further work is needed to find the best-suited material and optimum turns ratio, which will give a calibrated output emf against any primary load current.

## REFERENCES

Web Resouces & Documents referred to for studying the components, datasheets and specifications:

F39, P11 datasheets: https://www.mmgca.com/catalogue/MMG-Neosid.pdf

Nanocrystalline Core datasheet: https://www.mouser.com/pdfdocs/VACChokesandCoresDatasheet.pdf

BT 136 datasheet: https://www.promelec.ru/pdf/BT136-600-NXP.pdf

Optoisolator Triac Driver datasheet: http://www.mouser.com/ds/2/149/fairchild%20semiconductor_moc3031m-544476.pdf

Information about ferrite and toroid parameters: https://product.tdk.com/info/en/catalog/datasheets/ferrite_summary_en.pdf

Application note about Zero Voltage Crossing Optically Isolated Triac Driver: https://www.onsemi.com/pub/Collateral/AN-3004.pdf.pdf

Papers referred for literature review:

[1] Kurmi, V.P., Sharma, D, Singh, D.K. (2016). Analysis of the behaviour of various electrical loads with variable voltage and variable frequency, International Journal of Science, Engineering and Technology Research (IJSETR),Volume 5, Issue 5, May 2016

[2] Yang Zhen, Microchip Technology Inc., Current Sensing Circuit Concepts and Fundamentals, Microchip, Application Note AN-1332

[3] Forghani-zadeh, Hassan P., Rincón-Mora, Gabriel A. (2005). Current-Sensing Techniques for DC-DC Converters, Georgia Tech Analog Consortium, School of Electrical and Computer Engineering, Georgia Institute of Technology

[4] Sung-Yong Son (2015). Home Electricity Consumption Monitoring Enhancement Using Smart Device Status Information, International Journal of Smart Home, Vol. 9, No. 10, (2015), pp. 189-196

## ACKNOWLEDGEMENTS

With great pleasure, I express my gratitude to my project guide Dr K.C. James Raju, Professor, Centre for Advanced Studies in Electronics Science and Technology (CASEST), School of Physics, University of Hyderabad for his guidance.

I offer my sincere thanks to Mr G. Krishna Prasad and his team Conservision Technologies at TIDE Centre, University of Hyderabad for their constant support, motivation and guidance in developing the calibrated monitor-measure-manage sensor.

I also want to thank my teammates for this project Mr Sadanand Powar and Mr Ninad Nikalje without the help of whom this work could not have been completed.

I express my heartful thanks to my professor, Dr Abhijit Chakrabarty, Department of Electrical Engineering, Indian Institute of Engineering Science and Technolgy, Shibpur for his recommendation letter.

I am extremely grateful to the Indian Academy of Sciences for providing me with the fellowship and such a great platform to do a fruitful summer project.

The AuthorCafe platform was extremely user-friendly and helped me in compiling my report.

Last but not the least I would like to thank my parents and friends who have always been very supportive, without the help of whom this project would not have been possible.

#### References

• https://www.onsemi.com/pub/Collateral/AN-3004.pdf.pdf

#### Source

• Fig 1 a: http://www.mmgca.com/catalogue/MMG-Neosid.pdf
• Fig 1 b: http://www.mmgca.com/catalogue/MMG-Neosid.pdf
• Fig 2: https://www.fairchildsemi.com/application-notes/AN/AN-3004.pdf
• Fig 3:   https://www.fairchildsemi.com/application-notes/AN/AN-3004.pdf
• Fig 5: https://electronicspost.com/sketch-the-v-i-characteristics-of-a-triac-describe-some-of-its-important-applications/
• Fig 6: http://g3ynh.info/zdocs/magnetics/mag-core.html
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Written, reviewed, revised, proofed and published with


text:
Let $\alpha\in[0,9].$ Is the set of real numbers such that the average of the digits of each of these numbers $=\alpha,$ countable or uncountable?

Let $$\alpha \in [0,9]\subset\mathbb{R}.$$ Is the following set countable or uncountable:

The set of all (nonnegative) real numbers which, when written in decimal expansion form

$$k_1\ldots k_m\cdot k_{m+1}\ldots$$

where the $$k_i$$ are digits between $$0$$ and $$9$$, satisfy

$$\lim_{n\to\infty}\left( \frac{\sum_{j=1}^n k_j}{n} \right) = \alpha ?$$

• Comments are not for extended discussion; this conversation has been moved to chat. Mar 13 at 14:07

4 Answers

1. There are uncountably many numbers with $$\alpha = 0$$.

Let $$\{a_n\}$$ be a strictly increasing sequence of natural numbers. For each sequence, we'll build a number with $$\alpha=0$$. Since there are uncountably many such sequences (in particular, the set of sequences s.t. $$a_{n+1} = a_n + 1$$ or $$a_{n+1} = a_n + 2$$ is uncountable), we can generate uncountably many numbers.

Let $$s_n = \sum_{i=1}^n a_i$$. Let $$x$$ be the number such that its $$i$$'th digit $$x_i = 1$$ iff $$i = s_n$$ for some $$n$$, and $$x_i=0$$ otherwise.

Then $$\alpha(x) = 0$$. It suffices to show that $$\lim \frac n {s_n} = 0$$ (it's the average up to $$s_n$$'th digit). It follows from the fact that $$a_n \ge n$$ and therefore $$s_n \ge \frac {n^2} 2$$.

1. There are uncountably many numbers for any $$\alpha \in (0, 1)$$. Handling $$\alpha \ge 1$$ is trivial (just add $$\lfloor\alpha \rfloor$$ to all digits).

Consider digits between $$s_{n-1}$$ and $$s_{n}$$, There are $$a_n$$ such digits. Let $$c_n \in \mathbb N$$ s.t. $$\frac {c_n} {a_n}$$ is the closest to $$\alpha$$. Let $${c_n}$$ digits between $$s_{n-1}$$ and $$s_n$$ be $$1$$ and the rest of the digits be $$0$$, then the average of these digits is $$\frac {c_n} {a_n}$$.

Since $$a_n$$ strictly increases, $$\lim \frac {c_n} {a_n} = \alpha$$. Therefore, for any $$\epsilon$$ there exists $$n_0$$ s.t. $$\frac {c_n} {a_n} \in [\alpha - \epsilon, \alpha + \epsilon]$$ for $$n \ge n_0$$. Then the average up to $$s_n$$ is \begin{align*} avg_{s_n} &= \frac {c_1 + \cdots + c_n} {s_n}\\ &= \frac {c_1 + \cdots + c_{n_0}} {s_n} + \frac {c_{n_0 + 1} + \cdots + c_{n}} {s_n} \\ &= \frac {c_1 + \cdots + c_{n_0}} {s_n} - \alpha \frac {s_{n_0}} {s_n} + \frac {\alpha s_{n_0} + c_{n_0 + 1} + \cdots + c_{n}} {s_n} \\ &\to 0 + 0 + \ell \in [\alpha - \epsilon, \alpha + \epsilon], \end{align*} since $$\alpha s_{n_0} + c_{n_0 + 1} + \cdots + c_{n} \ge \alpha s_{n_0} + a_{n_0 + 1} (\alpha - \epsilon) + \cdots + a_{n}(\alpha - \epsilon) \ge \alpha s_n - \epsilon s_n$$ (the same for $$\alpha+\epsilon$$). Since this holds for arbitrary $$\epsilon$$, $$\lim avg_n = \alpha$$.

Some technicalities

1. For the limit to exist, we also need to consider all $$n' \in [s_{n-1}, s_n]$$. It suffices to guarantee that $$a_n = o(s_{n-1})$$, and I gave an example of such a family of sequences above.
2. For $$\alpha \in (0,1)$$, I don't actually guarantee that the generated numbers are distinct. We can guarantee this with some adjustments, but it's simpler to do the following: generate one number, and then change its $$s_n$$'th digits to be either $$0$$ or $$1$$ (all combinations). As shown in the case $$\alpha = 0$$, it doesn't change the limit of the average.
• I was about to post almost the same thing. I started with sequences of positive integers, which is obviously uncountable, took partial sums (an invertible operation) to get uncountably many strictly increasing sequences, and then took partial sums again, as you do, to get the thinning out sequence $s_n$. Your way of showing that the strictly increasing sequences are uncountable is very nice too. Mar 12 at 16:57
• When you wrote, "Let $s_n = \sum_{i=1}^n a_n$." Did you mean: Let $s_n = \sum_{i=1}^n a_i$ ? Mar 12 at 18:39
• Also, is $\{a_n\}$ a strictly increasing sequence of integers or real numbers? Mar 12 at 21:52
• Yes, it should be $\sum_i a_i$. $a_i$ are natural numbers. Mar 12 at 22:20

Let $$\ \alpha\in [0,9].$$

Proof outline

1. There exists a sequence of integers $$\ k_1,\ k_2,\ldots\$$ such that $$\ \displaystyle\lim_{n\to\infty}\left( \frac{\sum_{j=1}^{n} k_j}{n} \right) = \alpha.\quad$$
2. Create a new sequence by changing every member of the above sequence of the form $$\ \large{k_{2^j}}\$$ i.e. changing $$\ k_1\to \ {k_1}',\quad k_2\to\ {k_2}',\quad k_4\to\ {k_4}',\quad k_8\to\ {k_8}',\quad k_{16}\to\ {k_{16}}',\$$ etc, and in our new sequence we keep other members the same as before: $$\ {k_3}' = k_3,\ {k_5}' = k_5,\ {k_6}' = k_6,\$$ etc. Then the limit as in $$(1)$$ but for our new sequence, is also equal to $$\ \alpha,\$$ i.e. we have: $$\ \displaystyle\lim_{n\to\infty}\left( \frac{\sum_{j=1}^{n} {k_j}'}{n} \right) = \alpha.$$
3. This gives us a countably infinite collection $$\ (\ k_1,\ k_2,\ k_4,\ k_8,\ k_{16},\ \ldots\ )$$ of members of the above sequence- and we can change each of these terms to what we want without affecting the sum of the series. Now for each $$\ j\in\mathbb{N},\$$ map $$\ \large{k_{(2^j)}}\to 0\$$ and $$\ \large{k_{(2^j)'}}\to 1.\$$ This now resembles Cantor's Diagonalisation argument with sequences containing only $$\ 0's\$$ and $$\ 1's\$$ as entries, which is famously uncountable, and so the set of all real number sequences satisfying $$\ \displaystyle\lim_{n\to\infty}\left( \frac{\sum_{j=1}^{n} k_j}{n} \right) = \alpha\quad$$ is uncountable.

Funnily enough, I have found Step 2 easy to prove, but couldn't come up with an easy proof for Step 1.

Proof of Step 2:

Limit of altered ("new") sequence

\begin{align} =\displaystyle\lim_{n\to\infty}\left(\frac{\displaystyle\sum_{j=1}^{n} {k_j}'}{n}\right)\\ \\ =\displaystyle\lim_{n\to\infty}\left(\frac{\displaystyle\sum_{j=1}^{n} k_j + \displaystyle\sum_{j=0}^{\lfloor \log_2(n) \rfloor} ({k_{2^j}}'-k_{2^j}) }{n}\right)\\ \\ =\displaystyle\lim_{n\to\infty}\left(\frac{\displaystyle\sum_{j=1}^{n} k_j}{n}\right) + \displaystyle\lim_{n\to\infty}\left(\frac{\displaystyle\sum_{j=0}^{\lfloor \log_2(n) \rfloor} ({k_{2^j}}'-k_{2^j}) }{n}\right)\\ \\ =\alpha + \displaystyle\lim_{n\to\infty}\left(\frac{\displaystyle\sum_{j=0}^{\lfloor \log_2(n) \rfloor} ({k_{2^j}}'-k_{2^j}) }{n}\right),\\ \\ \text{and this final limit term equals zero, because it's absolute value is}\\ \\ \leq\ \displaystyle\lim_{n\to\infty}\left(\frac{\displaystyle\sum_{j=0}^{\lfloor \log_2(n) \rfloor} \vert{k_{2^j}}'-k_{2^j}\vert }{n}\right)\ \leq\ \displaystyle\lim_{n\to\infty}\left(\frac{\displaystyle\sum_{j=0}^{\lfloor \log_2(n) \rfloor} 9 }{n}\right)\\ \\ \leq\ \displaystyle\lim_{n\to\infty}\left(\frac{ 9\log_2(n) +1 }{n}\right) = 0.\\ \end{align}

If you are willing to allow a little heavy machinery, one can show that the set is uncountable for every $$\alpha$$, even just restricting to the set of numbers of the form $$0.x_1 x_2 x_3 \dots$$. In fact the argument below will give more precise information than just uncountability.

Setup. Rather than work with the averages of the digits $$\frac{1}{n} \sum_{j=1}^{k} x_j$$ directly, we will instead work with the frequencies of occurrences of specific digits, and make the connection at the end. If $$x = 0.x_1 x_2 \dots$$ and $$r \in \{0,1,2,\dots, 9\}$$, let us define $$D_x(r) = \lim_{n \to \infty} \frac{1}{n} \#\{1 \leq j \leq n : x_j = r\}$$ whenever the limit exists.

Black box theorem. Let $$p = (p_0, p_1, \dots, p_9)$$ be a probability vector over the set $$\{0,1,2,\dots, 9\}$$ (that is, all $$p_i \geq 0$$ and $$\sum p_i = 1$$). Then the Hausdorff dimension of the set $$\{x \in [0,1] : D_x(r) \text{ exists and is equal to } p_r \text{ for each } r \in \{0,1,2,\dots,9\} \}$$ is given by $$\frac{H(p)}{\log 10}$$, where $$H(p) = \sum_{i=1}^{10} -p_i \log p_i$$ is the Shannon entropy of the vector $$p$$. The relevant fact about $$H$$ that we'll need later is that $$H(p) > 0$$ whenever at least two of the $$p_i$$ are $$>0$$.

Essentially this theorem specifies the "size" (in the sense of Hausdorff dimension) of the set of numbers that have specified frequencies of digits in their base 10 expansions. It was proved originally by Patrick Billingsley, I think. I can't find an exact reference right now but I'd be happy to look a little more if you are curious. The proof uses a little ergodic theory and a little geometric measure theory (hence the aforementioned "heavy machinery").

Connection. Fix $$\alpha \in [0,9]$$. There are two cases, depending on if $$\alpha \in \mathbb{Z}$$ or not. If $$\alpha \in \mathbb{Z}$$, let $$x = 0.x_1 x_2 \dots$$ be any number satisfying $$D_x(\alpha - 1) = D_x(\alpha + 1) = 1/2$$. Then $$\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{n} x_j = \lim_{n \to \infty} \sum_{r=0}^9 r \cdot \frac{1}{n} \#\{1 \leq j \leq n : x_j = r \} = \sum_{r=0}^9 r \cdot D_x(r) = (\alpha - 1) \cdot \frac12 + (\alpha + 1) \cdot \frac12 = \alpha.$$ By Billingsley's theorem, the set of $$x$$ satisfying this has strictly positive Hausdorff dimension, which implies that it is uncountable.

Finally, consider the case where $$\alpha$$ is not an integer. Then write $$\alpha = A + t$$ where $$A \in \mathbb{Z}$$ and $$t \in (0,1)$$. Then we let $$x = 0.x_1x_2 \dots$$ be any number satisfying $$D_x(A) = 1-t$$, $$D_x(A+1) = t$$. A similar calculation as above will then show that any such $$x$$ satisfies $$\frac{1}{n} \sum_{j=1}^{n} x_j = A+t = \alpha$$, and again applying Billingsley's theorem shows that the set of such $$x$$ has positive Hausdorff dimension and is therefore uncountable.

• Can you please provide the source of the Black Box Theorem? Jul 7 at 22:27
• – Adam
Jul 8 at 18:15

Given $$\alpha\in[0,9]$$, for $$n\in\Bbb N$$ let $$x_n=\lfloor n\alpha\rfloor-\lfloor (n-1)\alpha\rfloor.$$ Note that $$x_n$$ is a non-negative integer and $$\le 9$$, so can be used as a decimal digit. Let $$A\subseteq \Bbb N$$ be an arbitrary set (so there are continuum-many choices for $$A$$). Then consider the number $$X_{\alpha,A}=0.d_1d_2d_3\ldots$$ that has (no integer part and) as $$n$$th decimal digit $$d_n=\begin{cases}1&n=2^k,k\in A\\0&n=2^k,k\notin A\\x_{n-k}&2^k Then one verifies quickly that the average digit is $$\alpha$$.