text:
# Tag Info

16

The answer is Explanation: Here's a picture with all the numbers:

15

Looks like So the number is

13

12

In the second subtraction, the number being subtracted is three digits, and the first digit isn't 1. So it must be at least 200, and the divisor is at least 21. Since the first digit of the divisor isn't 2, the divisor must be at least 30. In the first subtraction, the number being subtracted is at least twice the divisor, but is only two digits. So the ...

9

9

Kind of cheating, but... 1,736 | ( 1736) x 5,843 | ( 5843) ----------- | _,___ | ( 5208) __,___ | ( 69440) _,___,___ | ( 1388800) + _,___,___ | ( 8680000) ----------- | __,___,___ | (10143448) Answers the question in its literal form. ;-) Also, note that with lots of zeroes in the multiplicands this works: ...

8

You are looking for least percentage, without concerning with the Difficulty of the puzzle. In that case, the percentage can be made arbitrarily small. Here is an example with 31 digits removed: 999)*********(******* *** *** *** *** *** 0 Here the format AAA)BBB(CCC means AAA is dividing BBB to get the ...

7

A possibility:

7

Start at the number 1. You should now be able to find how to fill in the missing value.

7

Since this does not have the no-computers tag I have used a computer to I did it in a relatively similar way to a manual approach The Python code: Running that code (sub-second):

6

I think I found the answer. Step one to find the divisor. I started with Now this number start making sense in my mind. Here's how Second step of division. Third step of division. Fourth/Final step of division Here is my actual work Answer

5

Answer: Reasoning: . . . . EDIT: Formatting

5

Answer Explanation First I will replace the $X$'s with different lowercase letters, so its easier to refer to them. F a 8 b c +---------------- A d C | e f B D g h i j k m J ------- G n g h o p H ------- q r i j s t u E ------- The ...

4

Solution: Dividing the puzzle up into sections: Some reasoning (more to come when I'm back home with a scanner): Assuming no leading zeroes: The rest is trivial-ish:

4

The "useless digits" of hints 3 and 4 are The columns are really labelled The missing digits are The rows are I confess I'm not sure I understand hint 1; perhaps it's

3

Since: Then: So the number is:

3

Edit: Changed the reasoning based on Jaap Scherphuis's comment as it seems more natural. I think the answer is Reasoning Examples

3

3

Doorknob's _,___ | ( 1000) x _,___ | ( 1000) ----------- _,___,___ | (1000000) ----------- 1,___,___ | (1000000) is not uniquely solvable; how about e.g. 1,234 x 1,000 ----------- 1,234,000 ----------- 1,234,000 On the other hand, isn't the following uniquely solvable? _,___ x _,___ ---------- _,___ +_,___,...

2

I ran a computer program and I was wondering why it didn't find a solution. It's because some numbers presented as 4-digit number are in fact 3-digit numbers. So I ran it again and it found 2350 solutions! 20 solutions have XXBDXXXX mod AXC = 0. (The problem didn't say it had to be 0) Here's all 20 solutions with mod 0: 124 * 80800 = 10019200 (A,B,C,D,E,F,...

2

2

I get Because

1

The answer is 1/3 because(3/9)/(1/3)=1 and 1/(1/3)=3 3/(1/3)=9 With the same calculations we can obtain the rest of the numbers 1, 3, 9 in the pyramid.

1

It's probably a typo Given that there are only 2 examples to work with, it must be a very simple formula. Usually the formula uses each number once. I wrote a computer search for all ways to combine the 4 numbers around the circle with operations +, -, *, / in a way that it produces the value in the circle for the left and right case. The only formulas ...

1

After @SteveV's progress: Then, given the language tag, we now Giving Meaning (approximately)

1

Ok so this has puzzled me for a while. I have a partial answer based on some assumptions. This test is designed for primary school children, looking at the rest of the paper everything is basic so this puzzle should be the same. I think we are supposed to ignore the bottom horizontal line and instead picture the shape in 8 equal slices. I think the two ...

1


text:
# Narrowband Spikes Observed during the 2013 November 7 Flare by M. Karlicky et al.

Narrowband dm-spikes belong to the most interesting fine structures of solar radio bursts that are closely connected to primary flare energy-release processes (Krueger 1979) and observed in some cases near the Type III burst starting frequency. They occur in clouds of narrowband bursts with a typical duration less than 100 ms, frequency relative bandwidth 1-3 %, and brightness temperature up to $10^{15}$ K.

For them several radio emission models were suggested. Earlier models suggested emission by the electron cyclotron maser mechanism for $Y=\omega_{pe} / \omega_{ce} < 1$, where $\omega_{pe}$ is the electron plasma frequency, $\omega_{ce}$ is the electron cyclotron frequency (Melrose & Dulk 1982). In this model, emission occurs near the harmonics of the electron cyclotron frequency. However, non-integer ratios between spike bands were observed by Krucker & Benz (1994). Therefore, a model based on emission at Bernstein mode frequencies was proposed by Willes & Robinson (1996). Moreover, the model of emission at upper-hybrid frequencies was suggested  for $Y > 1$ by Stepanov et al. (2001). In this model, emission regions are separated in height, and thus they emit at separate frequencies.

We investigate the 2013 November 7 spike event showing unique very narrow bands with non-integer frequency ratios. In interpretation of this event we consider two models: (1) with the emission from one emission source at Bernstein mode frequencies and (2) with the emission from separated emission sources at upper-hybrid frequencies.

Observation

We study the 2013 November 7 spike event  where spikes were clustered in unique very narrow bands (Figure 1). This spike event belongs to 53 spike events that were registered by Ondřejov radiospectrographs in the range 0.8-2 GHz during 1992-2020 years. The frequencies of its four narrow bands are 1003, 1276, 1572, and 1877 MHz, giving ratios between adjacent bands as 1.275, 1.23, and 1.19. Figure 2 shows the radio flux evolution of these bands and their cross-correlations.

Figure 1 – Observed spike event on 2013 November 7. (a) Broadband cloud of spikes followed by unique four very narrow bands of spikes. (b) Detail of the radio spectrum in 1 – 2 GHz range with 1 s duration. The arrow shows the band frequency variation for which the form is synchronized with the similar forms in all four bands. The spectrum is not smoothed and thus shows bins in the record.

Results

We computed growth rates for both our models. For the Bernstein model (Figure 3), we found the corresponding Bernstein modes cross regions of positive growth rates. Moreover, these modes have a non-integer frequency ratios, and their positions fit well with the observed band frequencies. The parameters in this model correspond to the plasma density $9 \times 10^{9}$ cm$^{-3}$ and magnetic field strength 113 G. In the upper-hybrid model, we also estimated growth rates  (Figure 4). The corresponding plasma density and magnetic field strength are in the range $1.1 – 2.9 \times 10^{10}$ cm$^{-3}$ and  119-111 G, respectively.

Figure 2 – (a) Evolution of radio flux at 1877 MHz (black line), at 1572 MHz (red line) +400 au, at 1276 MHz (blue line) +800 au, and at 1003 MHz (orange line) +1200 au, starting at 12:26:55 UT and lasting 1 s. (b) Cross-correlations of the radio flux profiles on 1877 and 1572 MHz (black line), 1572 and 1276 MHz (red line), and 1276 and 1003 MHz (blue line).

Figure 3 – Growth rates of the Bernstein modes as a function of the frequency and perpendicular wavenumber for parameters $\omega_{pe}/\omega_{ce} = 2.7$, $f_{pe} = 855$ MHz, $v_{tb}/c = 0.02$, $v_{t}/c = 0.25$, $n_{e}/n_{h} = 10$. Frequency corresponds to the radio emission on Bernstein mode frequencies. Green lines: dispersion modes, $s$ is the gyro-harmonic number of each branch. White dashed horizontal line denotes the plasma frequency.

Figure 4 – Growth rate of the upper-hybrid waves vs. the upper-hybrid to electron cyclotron frequency ratio; $s$ denotes the gyro-harmonic number of the most unstable wave.

Conclusions

We found that the Bernstein mode model better explains the observed spikes than the upper-hybrid model, at least in the studied event. It fits very well with the observed frequencies for reasonable plasma parameters and produces well the observed band frequency difference of 300 MHz. Moreover, it is not very probable that four separated regions give these ratios in the upper-hybrid model. Another argument against upper-hybrid model is that it requests Alfvén speed $\sim 200.000$ km s$^{-1}$ to obtain $\leq 0.01$ s cross-correlation time lag between frequency bands (Figure 2b) when we assume typical density scale height in the corona.

Based on the recent paper

Karlický, M., Benáček, J., & Rybák, J. 2021, ApJ, 910, 108.doi:10.3847/1538-4357/abe62b

References

Krucker, S. & Benz, A. O.

Krueger, A. 1979, Geophysics and Astrophysics Monographs, Dordrecht: Reidel, 1979

Melrose, D. B. & Dulk, G. A. 1982, ApJ, 259, 844. doi:10.1086/160219

Stepanov, A. V., Kliem, B., Krüger, A., et al.

Willes, A. J. & Robinson, P. A. 1996, ApJ, 467, 465. doi:10.1086/177620

*Full list of authors: Marian Karlický, Jan Benáček and Ján Rybák

format: plain


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# How do you solve 2x^2-4x-3=0 by completing the square?

Jul 11, 2016

$x = 1 \pm \sqrt{\frac{5}{2}} .$

Taking, $\sqrt{\frac{5}{2}} \cong 1.581$, we get the soln. set $= \left\{2.581 , - 0.581\right\} .$

#### Explanation:

Given that, $2 {x}^{2} - 4 x - 3 = 0.$

$\therefore 2 {x}^{2} - 4 x = 3.$

To complete the square on $L . H . S .$, we need
Last Term $= {\left(M i \mathrm{dl} . T e r m\right)}^{2} / \left(4 \times F i r s t T e r m\right) = {\left(- 4 x\right)}^{2} / \left(4 \cdot 2 {x}^{2}\right) = \frac{16 {x}^{2}}{8 {x}^{2}} = 2$

Adding $2$ on both sides,

$2 {x}^{2} - 4 x + 2 = 3 + 2$

$\therefore 2 \left({x}^{2} - 2 x + 1\right) = 5$

$\therefore 2 {\left(x - 1\right)}^{2} = 5$

$\therefore {\left(x - 1\right)}^{2} = \frac{5}{2}$

$\therefore \left(x - 1\right) = \pm \sqrt{\frac{5}{2}}$

$\therefore x = 1 \pm \sqrt{\frac{5}{2}} ,$ is the desired soln.

Taking, $\sqrt{\frac{5}{2}} \cong 1.581$, we get the soln. set $= \left\{2.581 , - 0.581\right\} .$


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# Write the set A={3,6,9, ...∞} in set builder notation

Vijay
63 Points
2 months ago
Answer –  { x : x = 3y , y $\epsilon$ N and N $\geqslant$ 1  }    where N is Natural number.

Explaination -
3 = 3 x 1
6 = 3 x2
9 = 3 x 3 …........ so on . Therefore  x = 3y ( 3 times of y )

Thanks


text:
## Journal of Integral Equations and Applications

### Boundary integral equations for the transmission eigenvalue problem for Maxwell's equations

#### Abstract

In this paper, we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We formulate the transmission eigenvalue problem as an equivalent homogeneous system of the boundary integral equation and, assuming that the contrast is constant near the boundary of the support of the inhomogeneity, we prove that the operator associated with this system is Fredholm of index zero and depends analytically on the wave number. Then we show the existence of wave numbers that are not transmission eigenvalues which by an application of the analytic Fredholm theory implies that the set of transmission eigenvalues is discrete with positive infinity as the only accumulation point.

#### Article information

Source
J. Integral Equations Applications, Volume 27, Number 3 (2015), 375-406.

Dates
First available in Project Euclid: 17 December 2015

https://projecteuclid.org/euclid.jiea/1450388941

Digital Object Identifier
doi:10.1216/JIE-2015-27-3-375

Mathematical Reviews number (MathSciNet)
MR3435806

Zentralblatt MATH identifier
1332.35247

#### Citation

Cakoni, Fioralba; Haddar, Houssem; Meng, Shixu. Boundary integral equations for the transmission eigenvalue problem for Maxwell's equations. J. Integral Equations Applications 27 (2015), no. 3, 375--406. doi:10.1216/JIE-2015-27-3-375. https://projecteuclid.org/euclid.jiea/1450388941

#### References

• A.S. Bonnet-BenDhia, L. Chesnel and H. Haddar, On the use of $t$-coercivity to study the interior transmission eigenvalue problem, C.R. Acad. Sci. 340 (2011).
• F. Cakoni and D. Colton, A qualitative approach to inverse scattering theory, Springer, Berlin, 2014.
• F. Cakoni, D. Colton and P. Monk, The linear sampling method in inverse electromagnetic scattering, CBMS-NSF 80, SIAM Publications, 2011.
• F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inv. Prob. 26 (2010), 074004.
• F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal. 42 (2010), 237–255.
• F. Cakoni and H. Haddar, A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media, Inv. Prob. Imaging 1 (2007), 443–456.
• ––––, Transmission eigenvalues in inverse scattering theory, Inv. Prob. Appl., Inside Out 60, MSRI Publications, 2012.
• ––––, Transmission eigenvalues, Inv. Prob. 29 (2013), 100201.
• F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. J. Comp. Sci. Math. 3 (2010), 142–167.
• L. Chesnel, Interior transmission eigenvalue problem for Maxwell's equations: The $T$-coercivity as an alternative approach, Inv. Prob. 28 (2012), 065005.
• D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, 3rd edition, Springer, New York, 2013.
• D. Colton and S. Meng, Spectral properties of the exterior transmission eigenvalue problem, Inv. Prob. 30 (2014), 105010.
• A. Cossonnière, Transmission eigenvalues in electromagnetic scattering, Ph.D. thesis, CERFACS, Toulouse, France, 2011.
• A. Cossonnière and H. Haddar, The electromagnetic interior transmission problem for regions with cavities, SIAM J. Math. Anal. 43 (2011), 1698–1715.
• ––––, Surface integral formulation of the interior transmission problem, J. Integral Equations Appl. 25 (2013), 1123–1138.
• H. Haddar, The interior transmission problem for anisotropic Maxwell's equations and its applications to the inverse problem, Math. Meth. Appl. Sci. 27 (2004), 2111–2129.
• G. Hsiao and W. Wendland, Boundary integral equations, Springer, Berlin, 2010.
• A. Kirsch, An integral equation for Maxwell's equations in a layered medium with an application to the factorization method, Journal of Integral Equations Appl. 19 (2007), 333–357.
• A. Kirsch, On the existence of transmission eigenvalues, Inv. Prob. Imaging 2 (2009), 155–172.
• A. Kirsch and F. Hettlich, The mathematical theory of time-harmonic Maxwell's equations, 190, Springer, Berlin 2015.
• A. Kleefeld, A numerical method to compute interior transmission eigenvalues, Inv. Prob. 29 (2013), 104012.
• R. Kress, Linear integral equations, 3rd ed., Springer Verlag, New York, 2014.
• E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media, SIAM J. Math. Anal. 44 (2012), 1165–1174.
• ––––, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem, Inv. Prob. 29 (2013), 104003.
• W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
• J.C. Nédélec, Acoustic and electromagnetic equations. Integral representations for harmonic problems, Springer-Verlag, New York, 2001.
• L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal. 40 (2008), 738–753.
• L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inv. Prob. 29 (2013), 104001.
• J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operator, SIAM J. Math. Anal. 44 (2012), 341–354.


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Tuesday, August 10, 2010

Maven, Spring, Hibernate, Struts2, same old same old.. and what? Scala!

I recently had a project using the standard J2EE technology stack, but with an added twist of using Scala (in addition to Java). It was relatively smooth, with a few gotchas, which I will note here.

Lack of IDE support

I enjoy using vi much more than any of the Big Three IDEs for Java, so it wasn't as much of an issue for me, but my teammate was using IDEA, and he was having all sorts of problems (one reason is because one of the Scala plugins for IDEA seems to be completely busted as of this writing). I did try researching the plugins for other IDEs, and I found the Eclipse plugin to be the most sensible (unsurprising, as most of the developers of Scala seem to be using Eclipse).

If you have Scala code that depends on Java code, it's not much of a problem (as you can just compile the Java first, then compile the Scala). On the flip side, if both Scala and Java code depend on each other, you'll have trouble in the form of red squiggly underlines in Netbeans and IDEA that you simply cannot turn off.

Luckily though...

Maven support works beautifully

You won't have any trouble building your project from the command line with Maven - even if there are you have Scala and Java code with interdependencies. Here's what I added to my POM, updated for Scala 2.8.0, to make the whole thing work:

<plugin>
<groupId>org.scala-tools</groupId>
<artifactId>maven-scala-plugin</artifactId>
<version>2.12</version>
<configuration>
<args>
<arg>-deprecation</arg>
<arg>-unchecked</arg>
</args>
</configuration>
<executions>
<execution>
<id>compile</id>
<goals>
<goal>compile</goal>
</goals>
<phase>compile</phase>
</execution>
<execution>
<id>test-compile</id>
<goals>
<goal>testCompile</goal>
</goals>
<phase>test-compile</phase>
</execution>
<execution>
<id>process-resources</id>
<phase>process-resources</phase>
<goals>
<goal>compile</goal>
</goals>
</execution>
<execution>
<id>process-test-resources</id>
<phase>process-test-resources</phase>
<goals>
<goal>testCompile</goal>
</goals>
</execution>
</executions>
</plugin>

If you put your sources in src/main/scala, IDEs should be able to pick them up without a problem.

No more generating getters and setters

In Java, you'd navigate through something like 5 menu options so the IDE could generate 20 lines of boilerplate to make your class a "JavaBean". The nice folks who wrote Scala understand your pain and offer the @BeanProperty annotation:

import scala.reflect.BeanProperty
import java.io.Serializable
import java.lang.{Long => JLong}
import java.lang.{Integer => JInt}
import javax.persistence._

@Entity
@Table(name = "reviews")
class Review extends Serializable {
@BeanProperty
@Id
@GeneratedValue(strategy = GenerationType.AUTO)
var id: JLong = _

@BeanProperty var rating: JInt = _
@BeanProperty var content: String = _
}

That's a standard Hibernate entity, written in Scala (Hibernate is none the wiser). The @BeanProperty annotation generates the getX and setX methods for you, so that Hibernate will be happy (the regular x and x_= Scala methods exist also).

Hibernate really only wants the Java classes, dammit!

Notice in the above example that Hibernate will barf if you try to make your id a scala.Long instead of java.lang.Long. The same holds for other types like Double and Float (Integer is Int in Scala, but I made the import alias just to be explicit).

Hibernate also really wants your collections to be Java collections instead of Scala collections. You can still work with Scala collections, but you'll have to import implicit conversions for them:

import scala.collection.JavaConversions._

For the performance wary, all that does is wrap the Java collections so you can interact with them in a more idiomatic way (it won't copy the collection).

Struts2

Struts seems to work quite well with Scala. Stacktraces will even show the specific line (in the Scala source) where the error occurred. The only minor gotcha I had was that when your Action classes extend ActionSupport, the SUCCESS, ERROR, etc constants are not in scope. You'll have to import them from Action directly:

import com.opensymphony.xwork2.Action._

Conclusion

Opinions on the project varied on our 2 person team. I was able to crank out much more code at a faster pace, but my coworker struggled a lot with the IDE support. On the flip side, integrating Scala into a Java project was a heck of a lot smoother than I thought.


text:
# Non negative eigenvalues for a certain definition of hermitian matrix

If a matrix A is defined by $(A)_{ij}=(X_i)^*X_j$, show that eigenvalues of A are non negative.

I was able to show that A is hermitian and an eigenvalue $\lambda$ can be written for an eigenvector Y as: $\lambda=(Y)^*(AY)/((Y)^*Y)$.

However I am unable to show that this is non negative.

Thanks for the help.

Hint: Note that $$Y^*AY = \left(\sum_{i=1}^n Y_i X_i\right)^*\left(\sum_{i=1}^n Y_i X_i\right)$$
Let's $x_i = (X_i)^*$, then $A = xx^*$. Now if $y$ is eigenvector of $A$ we have $$Ay = xx^*y = (x^*y)x = \lambda y,$$ so $y = \alpha x$ for some number $\alpha$, which implies $\lambda = x^*x = \sum x_i^*x_i = \sum |x_i|^2 > 0$.


text:
1. ## constructing a line

Given: a line and a point

Construct a line through the point perpendicular to the given line

2. ## Re: constructing a line

What's the problem? Have you constructed the line yet?

3. ## Re: constructing a line

No, I would like to see how to do it.


text:
# PUFM 1.4 Subtraction

Photo by Martin Thomas via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

When adding, we combine two addends to get a sum. For subtraction we are given the sum and one addend and must find the “missing addend”.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?”

Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse.

By Denise Gaskins

# Game: Target Number (or 24)

[Photo by stevendepolo.]

Math concepts: addition, subtraction, multiplication, division, powers and roots, factorial, mental math, multi-step thinking
Number of players: any number
Equipment: deck of math cards, pencils and scratch paper, timer (optional)

# Do Your Students Understand Division?

[I couldn’t find a good picture illustrating “division.” Niner came to my rescue and took this photo of her breakfast.]

I found an interesting question at Mathematics Education Research Blog. In the spirit of Liping Ma’s Knowing and Teaching Elementary Mathematics, Finnish researchers gave this problem to high school students and pre-service teachers:

• We know that:
$498 \div 6 = 83$.
How could you use this relationship (without using long-division) to discover the answer to:
$491\div6=?$
[No calculators allowed!]

The Finnish researchers concluded that “division seems not to be fully understood.” No surprise there! Check out the pdf report for detailed analysis.

By Denise Gaskins

# Contig Game: Master Your Math Facts

[Photo by Photo Mojo.]

Yahtzee and other board games provide a modicum of math fact practice. But for intensive, thought-provoking math drill, I can’t think of any game that would beat Contig.

Math concepts: addition, subtraction, multiplication, division, order of operations, mental math
Number of players: 2 – 4
Equipment: Contig game board, three 6-sided dice, pencil and scratch paper for keeping score, and bingo chips or wide-tip markers to mark game squares

# 30+ Things to Do with a Hundred Chart

[Photo by geishaboy500.]

Are you looking for creative ways to help your children study math? Even without a workbook or teacher’s manual, your kids can learn a lot about numbers. Just spend an afternoon playing around with a hundred chart (also called a hundred board or hundred grid).

Here are a few ideas to get you started…

# Hit Me! (A Math Game)

Photo by paparutzi.

Math concepts: addition, subtraction, negative numbers, mental math, absolute value
Number of players: any number
Equipment: math cards (two decks may be needed for a large group)

# How to Teach Math to a Struggling Student

photo by MC Quinn via flickr (CC BY 2.0)

Paraphrased from a homeschool math discussion forum:

Help! My daughter struggles with arithmetic. I guess she is like me: just not a math person. She is an outstanding reader. When we do word problems, she usually has no trouble. She’s a whiz at strategy games and beats her dad at chess every time. But numbers — yikes! When we play Yahtzee, she gets lost trying to add up her score. The simple basics of adding and subtracting confuse her.

Since I find math difficult myself, it’s hard for me to know what she needs. What’s missing to make it click for her? She used to think math was fun and tested well above grade level, but I listened to some well-meaning advice and totally changed the way we were schooling. I switched from using workbooks and games to using Saxon math, and she got extremely frustrated. Now she hates math.

# Subtracting Mixed Numbers: A Cry for Help

Photo by powerbooktrance.

Paraphrased from a homeschool math discussion forum:

Help me teach fractions! My son can do long subtraction problems that involve borrowing, and he can handle basic fraction math, but problems like $9 - 5 \frac{2}{5}$ give him a brain freeze. To me, this is an easy problem, but he can’t grasp the concept of borrowing from the whole number. It is even worse when the math book moves on to $10 \frac{1}{4} - 2 \frac{3}{7}$ .

Several homeschooling parents replied to this question, offering advice about various fraction manipulatives that might be used to demonstrate the concept. I am not sure that manipulatives are needed or helpful in this case. The boy seems to have the basic concept of subtraction down, but he gets flustered and is unsure of what to do in the more complicated mixed-number problems.

The mother says, “To me, this is an easy problem” — and that itself is one source of trouble. Too often, we adults (homeschoolers and classroom teachers alike) don’t appreciate how very complicated an operation we are asking our students to perform. A mixed-number calculation like this is an intricate dance that can seem overwhelming to a beginner.

I will go through the calculation one bite at a time, so you can see just how much a student must remember. As you read through the steps, pay attention to your own emotional reaction. Are you starting to feel a bit of brain freeze, too?

Afterward, we’ll discuss how to make the problem simpler…

# Nim Revisited

Photo by Sister72.

Dave at MathNotations offers another version of Nim that will give your students something to think about:

[1,2]-3-[4,5]-6-[7,8]…21 Helping Children Devise and Understand Winning Strategies

Get monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.

# Euclid’s Game on a Hundred Chart

Math concepts: subtraction within 100, number patterns, mental math
Number of players: 2 or 3
Equipment: printed hundred chart (also called a hundred board), and highlighter or translucent disks to mark numbers — or use this online hundred chart

## Set Up

Place the hundred chart and highlighter where all players can reach them.

## How to Play

• Allow the youngest player choice of moving first or second; in future games, allow the loser of the last game to choose.
• The first player chooses a number from 1 to 100 and marks that square on the hundred chart.
• The second player chooses and marks any other number.
• On each succeeding turn, the player subtracts any two marked numbers to find and mark a difference that has not yet been taken.
• Play alternates until no more numbers can be marked.

# Game: Hundred Chart Nim

Photo by Håkan Dahlström via flickr.

Math concepts: addition and subtraction within 100, logical strategy
Number of players: 2 or 3
Equipment: printed hundred chart (also called a hundred board) and beans, pennies, or other tokens with which to mark numbers — or use this online hundred chart

## Set Up

Place the hundred chart and a small pile of tokens where both players can reach them.

# Ben Franklin Math: Elementary Problem Solving 3rd Grade

The ability to solve word problems ranks high on any math teacher’s list of goals. How can I teach my students to solve math problems? I must help them develop the ability to translate “real world” situations into mathematical language.

In two previous posts, I introduced the problem-solving tools algebra and bar diagrams. These tools help our students organize the information in a word problem and translate it into a mathematical calculation.

## Working Math Problems with Poor Richard

This time I will demonstrate these problem-solving tools in action with a series of 3rd-grade problems based on the Singapore Primary Math series, level 3A. For your reading pleasure, I have translated the problems into the universe of a well-written biography of Ben Franklin, Poor Richard by James Daugherty.

# Number Bonds, Number Rainbows

Many of us use the idea of number bonds with our young students. A number bond is a mental picture of the relationship between a number and the parts that combine to make it.

Now we have a new, colorful way to show these relationships, thanks to Maria at Homeschool Math Blog. If you teach math to young children, check this out:

# Penguin Math: Elementary Problem Solving 2nd Grade

The ability to solve word problems ranks high on any math teacher’s list of goals. How can I teach my students to reason their way through math problems? I must help my students develop the ability to translate “real world” situations into mathematical language.

In a previous post, I analyzed two problem-solving tools we can teach our students: algebra and bar diagrams. These tools help our students organize the information in a word problem and translate it into a mathematical calculation.

Now I want to demonstrate these problem-solving tools in action with a series of 2nd grade problems, based on the Singapore Primary Math series, level 2A. For your reading pleasure, I have translated the problems into the universe of one of our family’s favorite read-aloud books, Mr. Popper’s Penguins.

# 7 Things to Do with a Hundred Chart

This post has been revised to incorporate all the suggestions in the comments below, plus many more activities. Please update your bookmarks:

Or continue reading the original article…

# Egyptian Math in Hieroglyphs

Egyptians wrote in hieroglyphs, a type of picture writing, and in hieratics, which were like a cursive form of hieroglyphs.

Hieroglyphs came first. They were carved in the stone walls of temples and tombs, written on monuments, and used to decorate furniture. But they were a nuisance for scribes, who simplified the pictures and slurred some lines together when they wrote in ink on paper-like papyrus. This hieratic writing — like some people’s cursive today — can be hard to read, so we are only using hieroglyphic numbers on this blog.

Download this page from my old newsletter, and try your hand at translating some Egyptian hieroglyphs:

Then try writing some hieroglyphic calculations of your own.

Edited to add: The answers to these puzzles (and more) are now posted here.

## To Be Continued…

Read all the posts from the September/October 1998 issue of my Mathematical Adventures of Alexandria Jones newsletter.

Get monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.

# Number Bonds = Better Understanding

[Rescued from my old blog.]

A number bond is a mental picture of the relationship between a number and the parts that combine to make it. The concept of number bonds is very basic, an important foundation for understanding how numbers work. A whole thing is made up of parts. If you know the parts, you can put them together (add) to find the whole. If you know the whole and one of the parts, you take away the part you know (subtract) to find the other part.

Number bonds let children see the inverse relationship between addition and subtraction. Subtraction is not a totally different thing from addition; they are mirror images. To subtract means to figure out how much more you would have to add to get the whole thing.

# Negative Numbers for Young Students

[Rescued from my old blog.]

Would you like to introduce your students to negative numbers before they study them in pre-algebra? With a whimsical number line, negative numbers are easy for children to understand.

Get a sheet of poster board, and paint a tree with roots — or a boat on the ocean, with water and fish below and bright sky above. Use big brushes and thick poster paint, so you are not tempted to put in too much detail. A thick, permanent marker works well to draw in your number line, with zero at ground (or sea) level and the negative numbers down below.

# Order of Operations

[Rescued from my old blog.]

Marjorie in AZ asked a terrific question on the (now defunct) AHFH Math forum:

“…I have always been taught that the order of operations (Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction) means that you work a problem in that order. All parenthesis first, then all exponents, then all multiplication from left to right, then all division from left to right, etc. …”

Many people are confused with order of operations, and it is often poorly taught. I’m afraid that Marjorie has fallen victim to a poor teacher — or at least, to a teacher who didn’t fully understand math. Rather than thinking of a strict “PEMDAS” progression, think of a series of stair steps, with the inverse operations being on the same level.

# The Game That Is Worth 1,000 Worksheets

[Rescued from my old blog. Image via Wikipedia.]

Math concepts: greater-than/less-than, addition, subtraction, multiplication, division, fractions, negative numbers, absolute value, and multi-step problem solving.

Have you and your children been struggling to learn the math facts? The game of Math Card War is worth more than a thousand math drill worksheets, letting you build your children’s calculating speed in a no-stress, no-test way.

# Kids Do the Craziest Things

[Rescued from my old blog.]

My youngest daughter wanted to do Singapore math today. Miquon Red is her main math text this quarter, but we add a bit of Singapore Primary Math 1B whenever she’s in the mood. We turned to the lesson on subtracting with numbers in the 30-somethings. The first problem was pretty easy for her:

30 – 7 = []

I reminded her that she already knows 10 – 7. She agreed, “10 take away 7 is 3.” Then her eyes lit up. “So it’s 23! Because there are two tens left.”


text:
American Institute of Mathematical Sciences

November 2018, 12(4): 773-804. doi: 10.3934/amc.2018046

Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases

 1 Institute of Communications and Navigation, German Aerospace Center (DLR), D-82234 Oberpfaffenhofen, Germany 2 Institute for Communications Engineering, Technical University of Munich (TUM), D-80290 Munich, Germany

Received  February 2018 Published  September 2018

Fund Project: A. Wachter-Zeh's work was supported by the Technical University of Munich|Institute for Advanced Study, funded by the German Excellence Initiative and European Union Seventh Framework Programme under Grant Agreement No. 291763 and the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) unter Grant No. WA3907/1-1

An interpolation-based decoding scheme for $L$-interleaved subspace codes is presented. The scheme can be used as a (not necessarily polynomial-time) list decoder as well as a polynomial-time probabilistic unique decoder. Both interpretations allow to decode interleaved subspace codes beyond half the minimum subspace distance. Both schemes can decode $\gamma$ insertions and $\delta$ deletions up to $\gamma +L\delta \leq L({{n}_{t}}-k)$, where ${{n}_{t}}$ is the dimension of the transmitted subspace and $k$ is the number of data symbols from the field ${{\mathbb{F}}_{{{q}^{m}}}}$. Further, a complementary decoding approach is presented which corrects $\gamma$ insertions and $\delta$ deletions up to $L\gamma +\delta \leq L({{n}_{t}}-k)$. Both schemes use properties of minimal Gröebner bases for the interpolation module that allow predicting the worst-case list size right after the interpolation step. An efficient procedure for constructing the required minimal Gröebner basis using the general Kötter interpolation is presented. A computationally- and memory-efficient root-finding algorithm for the probabilistic unique decoder is proposed. The overall complexity of the decoding algorithm is at most $\mathcal{O}\left( {{L}^{2}}n_{r}^{2} \right)$ operations in ${{\mathbb{F}}_{{{q}^{m}}}}$ where ${{n}_{r}}$ is the dimension of the received subspace and $L$ is the interleaving order. The analysis as well as the efficient algorithms can also be applied for accelerating the decoding of interleaved Gabidulin codes.

Citation: Hannes Bartz, Antonia Wachter-Zeh. Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Advances in Mathematics of Communications, 2018, 12 (4) : 773-804. doi: 10.3934/amc.2018046
References:
 [1] W. W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, 3, American Mathematical Soc., 1994. doi: 10.1090/gsm/003. [2] C. Bachoc, F. Vallentin and A. Passuello, Bounds for Projective Codes from Semidefinite Programming, Adv. Math. Commun., 7 (2013), 127-145. doi: 10.3934/amc.2013.7.127. [3] H. Bartz, Algebraic Decoding of Subspace and Rank-Metric Codes, PhD thesis, Technical University of Munich, 2017. [4] H. Bartz, M. Meier and V. Sidorenko, Improved Syndrome Decoding of Interleaved Subspace Codes, in 11th International ITG Conference on Systems, Communications and Coding 2017 (SCC), Hamburg, Germany, 2017. [5] H. Bartz and V. Sidorenko, List and probabilistic unique decoding of folded subspace codes in IEEE International Symposium on Information Theory (ISIT), 2015. doi: 10.1109/ISIT.2015.7282407. [6] H. Bartz and V. Sidorenko, On list-decoding schemes for punctured reed-solomon, Gabidulin and subspace codes in XV International Symposium "Problems of Redundancy in Information and Control Systems", 2016. doi: 10.1109/RED.2016.7779321. [7] H. Bartz and A. Wachter-Zeh, Efficient interpolation-based decoding of interleaved subspace and Gabidulin codes, in 52nd Annual Allerton Conference on Communication, Control, and Computing, 2014, 1349-1356. doi: 10.1109/ALLERTON.2014.7028612. [8] D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, vol. 3, Springer, 1992. doi: 10.1007/978-1-4757-2181-2. [9] P. Delsarte, Bilinear forms over a finite field with applications to coding theory, J. Combin. Theory, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8. [10] T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inf. Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376. [11] T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inf. Theory, 57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232. [12] T. Etzion, E. Gorla, A. Ravagnani and A. Wachter-Zeh, Optimal Ferrers diagram rank-metric codes, IEEE Trans. Inform. Theory, 62 (2016), 1616-1630. doi: 10.1109/TIT.2016.2522971. [13] E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 3-16. [14] M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inf. Theory, 56 (2010), 3207-3216. doi: 10.1109/TIT.2010.2048447. [15] V. Guruswami and C. Xing, List decoding Reed-Solomon, algebraic-geometric, and Gabidulin subcodes up to the singleton bound, STOC'13??Proceedings of the 2013 ACM Symposium on Theory of Computing, 843-852, ACM, New York, 2013. doi: 10.1145/2488608.2488715. [16] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013. [17] A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science, vol. 5393 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2008, 31-42. doi: 10.1007/978-3-540-89994-5_4. [18] R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449. [19] M. Kuijper and A. Trautmann, Gröbner bases for linearized polynomials, URL http://arXiv.org/abs/1406.4600. [20] W. Li, V. Sidorenko and D. Silva, On transform-domain error and erasure correction by Gabidulin codes, Des. Codes Cryptogr., 73 (2014), 571-586. doi: 10.1007/s10623-014-9954-4. [21] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1997. [22] P. Loidreau and R. Overbeck, Decoding rank errors beyond the error correcting capability, in Int. Workshop Alg. Combin. Coding Theory (ACCT), 2006,186-190. [23] H. Mahdavifar and A. Vardy, Algebraic list-decoding on the operator channel, in IEEE Int. Symp. Inf. Theory (ISIT), 2010, 1193-1197. doi: 10.1109/ISIT.2010.5513656. [24] H. Mahdavifar and A. Vardy, List-Decoding of Subspace Codes and Rank-Metric Codes up to Singleton Bound, in IEEE Int. Symp. Inf. Theory (ISIT), 2012, 1488-1492. [25] J. N. Nielsen, List Decoding of Algebraic Codes, PhD thesis, 2013. [26] ∅. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584. doi: 10.1090/S0002-9947-1933-1501703-0. [27] S. Puchinger, J. S. R. Nielsen, W. Li and V. Sidorenko, Row reduction applied to decoding of rank metric and subspace codes, Designs, Codes and Cryptography, 82 (2017), 389-409, URL http://arXiv.org/abs/1510.04728v2. doi: 10.1007/s10623-016-0257-9. [28] N. Raviv and A. Wachter-Zeh, Some Gabidulin codes cannot be list decoded efficiently at any radius, IEEE Trans. Inform. Theory, 62 (2016), 1605-1615. doi: 10.1109/TIT.2016.2532343. [29] R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inf. Theory, 37 (1991), 328-336. doi: 10.1109/18.75248. [30] V. R. Sidorenko and M. Bossert, Decoding interleaved Gabidulin codes and multisequence linearized shift-register synthesis, in IEEE Int. Symp. Inf. Theory (ISIT), 2010, 1148-1152. doi: 10.1109/ISIT.2010.5513676. [31] V. R. Sidorenko, L. Jiang and M. Bossert, Skew-feedback shift-register synthesis and decoding interleaved Gabidulin codes, IEEE Trans. Inf. Theory, 57 (2011), 621-632. doi: 10.1109/TIT.2010.2096032. [32] D. Silva, Error Control for Network Coding, PhD thesis, University of Toronto, Toronto, Canada, 2009. [33] D. Silva, F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291. [34] V. Skachek, Recursive code construction for random networks, IEEE Trans. Inf. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163. [35] A. L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - A new concept in the area of network coding in IEEE Information Theory Workshop 2019 (ITW 2012), 2010. doi: 10.1109/CIG.2010.5592788. [36] A.-L. Trautmann and M. Kuijper, Gabidulin decoding via minimal bases of linearized polynomial modules, preprint, arXiv: 1408.2303. [37] A.-L. Trautmann, N. Silberstein and J. Rosenthal, List decoding of lifted Gabidulin codes via the Plücker embedding, in Int. Workshop Coding Cryptogr. (WCC), 2013. [38] A. Wachter-Zeh, Bounds on list decoding of rank-metric codes, IEEE Trans. Inf. Theory, 59 (2013), 7268-7277. doi: 10.1109/TIT.2013.2274653. [39] A. Wachter-Zeh and A. Zeh, List and unique error-erasure decoding of interleaved Gabidulin codes with interpolation techniques, Des. Codes Cryptogr., 73 (2014), 547-570. doi: 10.1007/s10623-014-9953-5. [40] B. Wang, R. J. McEliece and K. Watanabe, Kötter interpolation over free modules, in Proc. 43rd Annu. Allerton Conf. Comm., Control, and Comp., 2005. [41] S. Xia and F. Fu, Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172. doi: 10.1007/s10623-008-9221-7. [42] H. Xie, Z. Yan and B. W. Suter, General linearized polynomial interpolation and its applications, in IEEE Int. Symp. Network Coding (Netcod), 2011, 1-4. doi: 10.1109/ISNETCOD.2011.5978942.

show all references

References:
 [1] W. W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, 3, American Mathematical Soc., 1994. doi: 10.1090/gsm/003. [2] C. Bachoc, F. Vallentin and A. Passuello, Bounds for Projective Codes from Semidefinite Programming, Adv. Math. Commun., 7 (2013), 127-145. doi: 10.3934/amc.2013.7.127. [3] H. Bartz, Algebraic Decoding of Subspace and Rank-Metric Codes, PhD thesis, Technical University of Munich, 2017. [4] H. Bartz, M. Meier and V. Sidorenko, Improved Syndrome Decoding of Interleaved Subspace Codes, in 11th International ITG Conference on Systems, Communications and Coding 2017 (SCC), Hamburg, Germany, 2017. [5] H. Bartz and V. Sidorenko, List and probabilistic unique decoding of folded subspace codes in IEEE International Symposium on Information Theory (ISIT), 2015. doi: 10.1109/ISIT.2015.7282407. [6] H. Bartz and V. Sidorenko, On list-decoding schemes for punctured reed-solomon, Gabidulin and subspace codes in XV International Symposium "Problems of Redundancy in Information and Control Systems", 2016. doi: 10.1109/RED.2016.7779321. [7] H. Bartz and A. Wachter-Zeh, Efficient interpolation-based decoding of interleaved subspace and Gabidulin codes, in 52nd Annual Allerton Conference on Communication, Control, and Computing, 2014, 1349-1356. doi: 10.1109/ALLERTON.2014.7028612. [8] D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, vol. 3, Springer, 1992. doi: 10.1007/978-1-4757-2181-2. [9] P. Delsarte, Bilinear forms over a finite field with applications to coding theory, J. Combin. Theory, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8. [10] T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inf. Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376. [11] T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inf. Theory, 57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232. [12] T. Etzion, E. Gorla, A. Ravagnani and A. Wachter-Zeh, Optimal Ferrers diagram rank-metric codes, IEEE Trans. Inform. Theory, 62 (2016), 1616-1630. doi: 10.1109/TIT.2016.2522971. [13] E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 3-16. [14] M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inf. Theory, 56 (2010), 3207-3216. doi: 10.1109/TIT.2010.2048447. [15] V. Guruswami and C. Xing, List decoding Reed-Solomon, algebraic-geometric, and Gabidulin subcodes up to the singleton bound, STOC'13??Proceedings of the 2013 ACM Symposium on Theory of Computing, 843-852, ACM, New York, 2013. doi: 10.1145/2488608.2488715. [16] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013. [17] A. Kohnert and S. 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Skachek, Recursive code construction for random networks, IEEE Trans. Inf. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163. [35] A. L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - A new concept in the area of network coding in IEEE Information Theory Workshop 2019 (ITW 2012), 2010. doi: 10.1109/CIG.2010.5592788. [36] A.-L. Trautmann and M. Kuijper, Gabidulin decoding via minimal bases of linearized polynomial modules, preprint, arXiv: 1408.2303. [37] A.-L. Trautmann, N. Silberstein and J. Rosenthal, List decoding of lifted Gabidulin codes via the Plücker embedding, in Int. Workshop Coding Cryptogr. (WCC), 2013. [38] A. Wachter-Zeh, Bounds on list decoding of rank-metric codes, IEEE Trans. Inf. Theory, 59 (2013), 7268-7277. doi: 10.1109/TIT.2013.2274653. [39] A. Wachter-Zeh and A. Zeh, List and unique error-erasure decoding of interleaved Gabidulin codes with interpolation techniques, Des. Codes Cryptogr., 73 (2014), 547-570. doi: 10.1007/s10623-014-9953-5. [40] B. Wang, R. J. McEliece and K. Watanabe, Kötter interpolation over free modules, in Proc. 43rd Annu. Allerton Conf. Comm., Control, and Comp., 2005. [41] S. Xia and F. Fu, Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172. doi: 10.1007/s10623-008-9221-7. [42] H. Xie, Z. Yan and B. W. Suter, General linearized polynomial interpolation and its applications, in IEEE Int. Symp. Network Coding (Netcod), 2011, 1-4. doi: 10.1109/ISNETCOD.2011.5978942.
Decoding region for the Kötter-Kschischang [18] decoder $(L = 1)$ and for list decoding a homogeneous interleaved KK subspace code with $(L = 4)$. Both codes have minimum subspace distance ${{d}_{s}} = 8$
Decoding region for Kötter-Kschischang [18] codes $(L = 1)$ and for probabilistic unique decoding of $(L = 4)$-interleaved KK subspace codes. Both codes have minimum subspace distance ${{d}_{s}} = 8$. The decoding region for insertions increases with the interleaving order $L$
Simulation results and upper bounds on $P_f$ of the probabilistic-unique decoder with parameters $m = {{n}_{t}} = 6, k = 2, L = 2$
Multiplications vs. the number of insertions for $L = 4$
Number of multiplications for root-finding step for different interleaving orders L. The complexity of the recursive GE is independent of $\gamma$
Memory requirements of the root-finding step for ${{n}_{t}} = 80, k = 60$
Simulation results of the probabilistic-unique decoder with parameters $m = {{n}_{t}} = 6, k = 2$ and $L = 2$
 $\gamma + L\delta$ $\gamma$ $\delta$ Transmissions Observed dec. failures Simulated $P_{f}$ 8 8 0 $1.2\cdot10^{6}$ 18749 $1.56\cdot10^{-2}$ 6 1 $4.0\cdot10^{5}$ 6202 $1.55\cdot10^{-2}$ 7 7 0 $4.4\cdot10^{6}$ 1025 $2.33\cdot10^{-4}$ 5 1 $6.0\cdot10^{5}$ 144 $2.40\cdot10^{-4}$ 6 6 0 $9.0\cdot10^{6}$ 21 $2.33\cdot10^{-6}$ 4 1 $3.4\cdot10^{6}$ 17 $5.00\cdot10^{-6}$ 5 5 0 $3.2\cdot10^{6}$ 750 $2.33\cdot10^{-4}$ 3 1 $8.0\cdot10^{5}$ 184 $2.30\cdot10^{-4}$ 4 4 0 $4.4\cdot10^{6}$ 21 $4.77\cdot10^{-6}$ 2 1 $1.6\cdot10^{6}$ 0 $0$
 $\gamma + L\delta$ $\gamma$ $\delta$ Transmissions Observed dec. failures Simulated $P_{f}$ 8 8 0 $1.2\cdot10^{6}$ 18749 $1.56\cdot10^{-2}$ 6 1 $4.0\cdot10^{5}$ 6202 $1.55\cdot10^{-2}$ 7 7 0 $4.4\cdot10^{6}$ 1025 $2.33\cdot10^{-4}$ 5 1 $6.0\cdot10^{5}$ 144 $2.40\cdot10^{-4}$ 6 6 0 $9.0\cdot10^{6}$ 21 $2.33\cdot10^{-6}$ 4 1 $3.4\cdot10^{6}$ 17 $5.00\cdot10^{-6}$ 5 5 0 $3.2\cdot10^{6}$ 750 $2.33\cdot10^{-4}$ 3 1 $8.0\cdot10^{5}$ 184 $2.30\cdot10^{-4}$ 4 4 0 $4.4\cdot10^{6}$ 21 $4.77\cdot10^{-6}$ 2 1 $1.6\cdot10^{6}$ 0 $0$
Comparison of different decoding schemes for interleaved KK subspace codes
 Decoding scheme Decoding region Op. in ${{\mathbb{F}}_{{{q}^{m}}}}$ Li-Sidorenko-Silva [20,31] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( Ln_{t}^{2} \right) \phantom{{}^{2}}$ Wachter-Zeh-Zeh [39] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{2} \right)$ Guruswami-Xing [15] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{6}}n_{t}^{2} \right)$ Bartz-Meier-Sidorenko [4] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{3} \right)$ This contribution $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{4}}n_{t}^{2} \right)$
 Decoding scheme Decoding region Op. in ${{\mathbb{F}}_{{{q}^{m}}}}$ Li-Sidorenko-Silva [20,31] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( Ln_{t}^{2} \right) \phantom{{}^{2}}$ Wachter-Zeh-Zeh [39] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{2} \right)$ Guruswami-Xing [15] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{6}}n_{t}^{2} \right)$ Bartz-Meier-Sidorenko [4] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{3} \right)$ This contribution $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{4}}n_{t}^{2} \right)$
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text:
# The Lives of the Most Eminent English Poets/Volume 1/Denham

OF Sir JOHN DENHAM very little is known but what is related of him by Wood, or by himself.

He was born at Dublin in 1615; the only son of Sir John Denham, of Little Horsely in Essex, then chief baron of the Exchequer in Ireland, and of Eleanor, daughter of Sir Garret More, baron of Mellefont.

Two years afterwards, his father, being made one of the barons of the Exchequer in England, brought him away from his native country, and educated him in London.

In 1631 he was sent to Oxford, where he was considered "as a dreaming young man, given more to dice and cards than study;" and therefore gave no prognosticks of his future eminence; nor was suspected to conceal, under sluggishness and laxity, a genius born to improve the literature of his country.

When he was, three years afterwards, removed to Lincoln's Inn, he prosecuted the common law with sufficient appearance of application; yet did not lose his propensity to cards and dice; but was very often plundered by gamesters.

Being severely reproved for this folly, he professed, and perhaps believed, himself reclaimed; and, to testify the sincerity of his repentance, wrote and published "An Essay upon Gaming."

He seems to have divided his studies between law and poetry; for, in 1636, he translated the second book of the Æneid.

Two years after, his father died; and then, notwithstanding his resolutions and professions, he returned again to the vice of gaming, and lost several thousand pounds that had been left him. In 1642, he published "The Sophy." This seems to have given him his first hold of the publick attention; for Waller remarked, "that he broke out like the Irish rebellion three score thousand strong when nobody was aware, or in the least suspected it;" an observation which could have had no propriety, had his poetical abilities been known before.

He was after that pricked for sheriff of Surrey, and made governor of Farnham Castle for the king; but he soon resigned that charge, and retreated to Oxford, where, in 1643, he published "Cooper's Hill."

This poem had such reputation as to excite the common artifice by which envy degrades excellence. A report was spread, that the performance was not his own, but that he had bought it of a vicar for forty pounds. The same attempt was made to rob Addison of his Cato, and Pope of his Essay on Criticism.

In 1647, the distresses of the royal family required him to engage in more dangerous employments. He was entrusted by the queen with a message to the king; and, by whatever means, so far softened the ferocity of Hugh Peters, that by his intercession admission was procured. Of the king's condescension he has given an account in the dedication of his works.

He was afterwards employed in carrying on the king's correspondence; and, as he says, discharged this office with great safety to the royalists: and being accidentally discovered by the adverse party's knowledge of Mr. Cowley's hand, he escaped happily both for himself and his friends.

He was yet engaged in a greater undertaking. In April 1648, he conveyed James the duke of York from London into France, and delivered him there to the queen and prince of Wales. This year he published his translation of "Cato Major."

He now resided in France, as one of the followers of the exiled king; and, to divert the melancholy of their condition, was sometimes enjoined by his master to write occasional verses; one of which amusements was probably his ode or song upon the Embassy to Poland, by which he and lord Crofts procured a contribution of ten thousand pounds from the Scotch, that wandered over that kingdom. Poland was at that time very much frequented by itinerant traders, who, in a country of very little commerce and of great extent, where every man resided on his own estate, contributed very much to the accommodation of life, by bringing to every man's house those little necessaries which it was very inconvenient to want, and very troublesome to fetch. I have formerly read, without much reflection, of the multitude of Scotchmen that travelled with their wares in Poland; and that their numbers were not small, the success of this negociation gives sufficient evidence.

About this time, what estate the war and the gamesters had left him was sold, by order of the parliament; and when, in 1652, he returned to England, he was entertained by the earl of Pembroke.

Of the next years of his life there is no account. At the Restoration he obtained that which many missed, the reward of his loyalty; being made surveyor of the king's buildings, and dignified with the order of the Bath. He seems now to have learned some attention to money; for Wood says, that he got by this place seven thousand pounds.

After the Restoration he wrote the poem on Prudence and Justice, and perhaps some of his other pieces: and as he appears, whenever any serious question comes before him, to have been a man of piety, he consecrated his poetical powers to religion, and made a metrical version of the psalms of David. In this attempt he has failed; but in sacred poetry who has succeeded?

It might be hoped that the favour of his master and esteem of the publick would now make him happy. But human felicity is short and uncertain; a second marriage brought upon him so much disquiet, as for a time disordered his understanding; and Butler lampooned him for his lunacy. I know not whether the malignant lines were then made publick, nor what provocation incited Butler to do that which no provocation can excuse.

His frenzy lasted not long[1]; and he seems to have regained his full force of mind; for he wrote afterwards his excellent poem upon the death of Cowley, whom he was not long to survive; for on the 19th of March, 1668, he was buried by his side.

DENHAM is deservedly considered as one of the fathers of English poetry. "Denham and Waller," says Prior, "improved our versification, and Dryden perfected it." He has given specimens of various composition, descriptive, ludicrous, didactick, and sublime.

He appears to have had, in common with almost all mankind, the ambition of being upon proper occasions a merry fellow, and in common with most of them to have been by nature, or by early habits, debarred from it. Nothing is less exhilarating than the ludicrousness of Denham: he does not fail for want of efforts: he is familiar, he is gross; but he is never merry, unless the "Speech against peace in the close Committee" be excepted. For grave burlesque, however, his imitation of Davenant shews him to have been well qualified.

Of his more elevated occasional poems there is perhaps none that does not deserve commendation. In the verses to Fletcher, we have an image that has since been adopted:

"But whither am I stray'd? I need not raise
"Trophies to thee from other men's dispraise;
"Nor is thy fame on lesser ruins built,
"Nor need thy juster title the foul guilt
"Of eastern kings, who, to secure their reign,
"Must have their brothers, sons, and kindred slain."

After Denham, Orrery, in one of his prologues,

"Poets are sultans, if they had their will;
"For every author would his brother kill."

And Pope,

"Should such a man, too fond to rule alone,
"Bear like the Turk no brother near the throne."

But this is not the best of his little pieces: it is excelled by his poem to Fanshaw, and his elegy on Cowley.

His praise of Fanshaw's version of Guarini contains a very spritely and judicious character of a good translator:

"That servile path thou nobly dost decline,
"Of tracing word by word, and line by line.
"Those are the labour'd births of slavish brains,
"Not the effect of poetry, but pains;
"Cheap vulgar arts, whose narrowness affords
"No flight for thoughts, but poorly stick at words.
"A new and nobler way thou dost pursue,
"To make translations and translators too.
"They but preserve the ashes, thou the flame,
"True to his sense, but truer to his fame."

The excellence of these lines is greater, as the truth which they contain was not at that time generally known.

His poem on the death of Cowley was his last, and, among his shorter works, his best performance: the numbers are musical, and the thoughts are just.

"Cooper's Hill" is the work that confers upon him the rank and dignity of an original author. He seems to have been, at least among us, the author of a species of composition that may be denominated local poetry, of which the fundamental subject is some particular landscape, to be poetically described, with the addition of such embellishments as may be supplied by historical retrospection or incidental meditation.

To trace a new scheme of poetry has in itself a very high claim to praise, and its praise is yet more when it is apparently copied by Garth and Pope[2]; after whose names little will be gained by an enumeration of smaller poets, that have left scarcely a corner of the island not dignified either by rhyme, or blank verse.

"Cooper's Hill," if it be maliciously inspected, will not be found without its faults. The digressions are too long, the morality too frequent, and the sentiments sometimes such as will not bear a rigorous enquiry.

The four verses, which, since Dryden has commended them, almost every writer for a century past has imitated, are generally known:

"O could I flow like thee, and make thy stream
"My great example, as it is my theme!
"Though deep, yet clear; though gentle, yet not dull;
"Strong without rage, without o'erflowing full."

The lines are in themselves not perfect; for most of the words, thus artfully opposed, are to be understood simply on one side of the comparison, and metaphorically on the other; and if there be any language which does not express intellectual operations by material images, into that language they cannot be translated. But so much meaning is comprised in few words; the particulars of resemblance are so perspicaciously collected, and every mode of excellence separated from its adjacent fault by so nice a line of limitation; the different parts of the sentences are so accurately adjusted; and the flow of the last couplet is so smooth and sweet; that the passage, however celebrated, has not been praised above its merit. It has beauty peculiar to itself, and must be numbered among those felicities which cannot be produced at will by wit and labour, but must arise unexpectedly in some hour propitious to poetry. He appears to have been one of the first that understood the necessity of emancipating translation from the drudgery of counting lines and interpreting single words. How much this servile practice obscured the clearest and deformed the most beautiful parts of the ancient authors, may be discovered by a perusal of our earlier versions; some of them the works of men well qualified, not only by critical knowledge, but by poetical genius, who yet, by a mistaken ambition of exactness, degraded at once their originals and themselves.

Denham saw the better way, but has not pursued it with great success. His versions of Virgil are not pleasing; but they taught Dryden to please better. His poetical imitation of Tully on "Old Age" has neither the clearness of prose, nor the spriteliness of poetry.

The "strength of Denham," which Pope so emphatically mentions, is to be found in many lines and couplets, which convey much meaning in few words, and exhibit the sentiment with more weight than bulk.

On the Thames.

"Though with those streams he no resemblance hold,
"Whose foam is amber, and their gravel gold;
"His genuine and less guilty wealth t' explore,
"Search not his bottom, but survey his shore."

On Strafford.

"His wisdom such, as once it did appear
"Three kingdoms' wonder, and three kingdoms' fear.
"While single he stood forth, and seem'd, although
"Each had an army, as an equal foe,
"Such was his force of eloquence, to make
"The hearers more concern'd than he that spake:
"Each seem'd to act that part he came to see,
"And none was more a looker-on than he;
"So did he move our passions, some were known
"To wish, for the defence, the crime their own.
"Now private pity strove with publick hate,
"Reason with rage, and eloquence with fate."

On Cowley.

"To him no author was unknown,
"Yet what he wrote was all his own;
"Horace's wit, and Virgil's state,
"He did not steal, but emulate!
"And when he would like them appear,
"Their garb, but not their cloaths, did wear."

As one of Denham's principal claims to the regard of posterity arises from his improvement of our numbers, his versification ought to be considered. It will afford that pleasure which arises from the observation of a man of judgement, naturally right, forsaking bad copies by degrees, and advancing towards a better practice, as he gains more confidence in himself.

In his translation of Virgil, written when he was about twenty-one years old, may be still found the old manner of continuing the sense ungracefully from verse to verse.

"Then all those
"Who in the dark our fury did escape,
"Returning, know our borrow'd arms, and shape,
"And differing dialect: then their numbers swell
"And grow upon us; first Chorœbeus fell
${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}}$"Before Minerva's altar; next did bleed
"Just Ripheus, whom no Trojan did exceed
"In virtue, yet the gods his fate decreed.
"Then Hypanis and Dymas, wounded by
"Their friends; nor thee, Pantheus, thy piety,
"Nor consecrated mitre, from the same
"Ill fate could save; my country's funeral flame
"And Troy's cold ashes I attest, and call
"To witness for myself, that in their fall
"No foes, or death, nor danger I declin'd,
"Did and deserv'd no less, my fate to find."

From this kind of concatenated metre he afterwards refrained, and taught his followers the art of concluding their sense in couplets; which has perhaps been with rather too much constancy pursued.

This passage exhibits one of those triplets which are not infrequent in this first essay, but which is to be supposed his maturer judgement disapproved, since in his latter works he has totally forborn them.

His rhymes are such as seem found with out difficulty, by following the sense; and are for the most part as exact at least as those of other poets, though now and then the reader is shifted off with what he can get.

"O how transform'd!
"How much unlike that Hector, who return'd

And again:

"From thence a thousand lesser poets sprung
"Like pretty princes from the fall of Rome."

Sometimes the weight of rhyme is laid upon a word too feeble to sustain it:

"Troy confounded falls
"From all her glories: if it might have stood
"By any power, by this right hand it shou'd.
"—And though my outward state misfortune hath.
"Deprest thus low, it cannot reach my faith."
"—Thus, by his fraud and our own faith o'ercome,
"A feigned tear destroys us, against whom
"Tydides nor Achilles could prevail,
"Nor ten years conflict, nor a thousand sail."

He is not very careful to vary the ends of his verses: in one passage the word die rhimes three couplets in six.

Most of these petty faults are in his first productions, when he was less skilful, or at least less dextrous, in the use of words; and though they had been more frequent they could only have lessened the grace, not the strength of his composition. He is one of the writers that improved our taste, and advanced our language, and whom we ought therefore to read with gratitude, though, having done much, he left much to do.

1. In Grammont's Memoirs many circumstances are related, both of his marriage and his frenzy, very little favourable to his character.R.
2. By Garth, in his "Poem on Claremont," and by Pope, in his "Windsor Forest." H.