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My Math Forum > Math need help to stop getting lost with maths formulas

 Math General Math Forum - For general math related discussion and news

 April 10th, 2015, 12:32 PM #1 Newbie   Joined: Apr 2015 From: USA Posts: 1 Thanks: 0 need help to stop getting lost with maths formulas A common problem I have is i am reading a book. It is getting interesting; then all of a sudden I see a long streak of formulas and then I get lost. I understand summation, differentiation but sometimes just seeing long forumlas with intimidating symbols loses me. I need a resource, book or other text that builds this stuff gradually. I can understand high school algebra up to grade 12. I get lost when reading scientific and mathematical papers. For example, the Page distribution is given by: \!f(k; \lambda)= \Pr(X{=}k)= \frac{\lambda^k e^{-\lambda}}{k!}, http://depts.washington.edu/wmatkins...ics/Eqn006.gif Just like when I was learning french, it took me 10 years of wasted efforts until I found Pierre Capretz french in action and I was able to master the intermediates of the language in 6 months vs 10 years. I am looking for a similar resource for maths équations and formula. anyone can help suggest?
 April 20th, 2015, 06:58 AM #2 Member   Joined: Jan 2014 Posts: 86 Thanks: 4
 April 20th, 2015, 09:00 AM #3 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions If your studies are anything like mine, you'll find that sometimes you will have no choice but to deal with what you have, rather than to go to another text to find a different description for the same thing. In which case you will have no choice but to deal with the mathematics as they are already presented to you. If such a case ever occurs, try the following on some scrap paper: 1. Do the equations come with a diagram? If so, draw it yourself. Try to compare the diagram with the equation presented to you. Do the peaks, troughs and general shape make sense? 2. Try putting some numbers in and see what they evaluate to. For example, for the Page distribution, try setting lambda = 0.5 and k = 1. What does it give? Then set k=2, then 3, then 4... what happens to the result as you change k? Formulae make more sense when you relate them back to numbers. Try doing the same thing with a different lambda. Are the values totally different? 3. If there isn't a diagram, use the numbers in part 2 above to make your own plot. When lambda = 0.5, what does the Page distribution look like as you vary k? What about the case where k=1 and lambda changes? In each case, is the curve weird-looking with discontinuities or discrete points or is it nice and smooth? 4. If you're given a sequence of operations (like the link provided) then write down by each line what is being done. For example, to go from line 1 to line 2, the person expanded the brackets on the left-hand side. To go from line 2 to line 3, the person added $\displaystyle k_{on}[ES][S]$ to both sides... etc. Write all the steps down so it is clear what is being done at each step. It also helps spot mistakes in the text if you follow the steps yourself on paper! 5. When you write out a formula for the first time, write a little list of what those parameters represent and their units. For example, let's say I'm given this: $\displaystyle F = ma + v \frac{dm}{dt}$ underneath it I'll write F = Force (N) m = mass (kg) a = acceleration (m/s^2) v = velocity (m/s) That way, if I'm unsure what something represents in a calculation further down the page, I can come back up to this bit and look up what those parameters mean. It is also good practise to make sure every parameter is labelled so it is clear what it means and what units it has. 6. Subscripts can be used to label algebraic quantities that are similar but not the same. For example, the price for three different objects might be $\displaystyle P_1$, $\displaystyle P_2$ and $\displaystyle P_3$, rather than a, b and c. Try to stick with conventions in the text you are reading however, because those conventions might crop up in other places in a similar way and it will make it easier to understand those other texts if you encounter them a second time. I hope these tips help!

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## Basic College Mathematics (9th Edition)

In order to write the fraction $\frac{3}{8}$ as a percent, we can set $\frac{3}{8}=\frac{p}{100}$ and solve for p (where $\frac{p}{100}=p$%). We can solve by using cross products. $3\times100=8\times p$ $300=8p$ Divide both sides by 8. $p=37.5$ Therefore, $\frac{3}{8}=\frac{37.5}{100}$=37.5%.

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×

# Amazing $$e^{x}$$ function

HOPE YOU LIKE IT !!!

Note by <NAME>
3 years, 4 months ago

## Comments

Sort by:

Top Newest

Try differentiating with y :P

- 3 years, 4 months ago

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Nice. But since I am not that familiar with calculus, I don't get what this means. Can you....er....explain?

- 3 years, 4 months ago

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but you rated at level 2 in calculus . how?

- 3 years, 4 months ago

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Hey, if this means d/dx of e^x is e^x, then fine- I got it. But, I thought there was some hidden joke in it. Well, I don't solve too many calculus problems because I know only very basic differentiation, bit of definite integrals and limits. I have to work on it. Getting a level two rating is not too tough. ;)

- 3 years, 4 months ago

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There is no hidden meaning

- 3 years, 4 months ago

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use ln.... loge(e^x)..go shoo away

- 3 years, 3 months ago

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good job !!

- 3 years, 3 months ago

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OH...:)

- 3 years, 4 months ago

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# Long Path Problems

@inproceedings{Horn1994LongPP,
title={Long Path Problems},
author={<NAME> and <NAME> and <NAME>},
booktitle={PPSN},
year={1994}
}
• Published in PPSN 9 October 1994
• Mathematics, Computer Science
We demonstrate the interesting, counter-intuitive result that simple paths to the global optimum can be so long that climbing the path is intractable. This means that a unimodal search space, which consists of a single hill and in which each point in the space is on a simple path to the global optimum, can be difficult for a hillclimber to optimize. Various types of hillclimbing algorithms will make constant progress toward the global optimum on such long path problems. They will continuously…
156 Citations

## Topics from this paper

Genetic Algorithm Difficulty and the Modality of Fitness Landscapes
• Mathematics, Computer Science
FOGA
• 1994
The function f mdG seems to be a powerful new tool for generalizing deception and relating hillclimbers (and Hamming space) to GAs and crossover and allows us to create functions, such as the minimum distance function fmdG, with k isolated global optima and multiple local optima attractive to both crossover and hillClimbers.
Genetic Algorithms, Problem Difficulty, and the Modality of Fitness Landscapes
This thesis addresses the appropriateness of Sewall Wright's tness landscape to the study of evolutionary computation, demonstrating that the metaphor can help us understand and predict the performance of the authors' algorithms, including both hillclimbers and recombinative GAs.
Genetic Algorithm Di culty and the Modality ofFitness
We assume that the modality (i.e., number of local optima) of a tness landscape is related to the diiculty of nding the best point on that landscape by evolutionary computation (e.g., hillclimbers
Analysis of recombinative algorithms on a non-separable building-block problem
• <NAME>
• Mathematics, Computer Science
FOGA
• 2000
An upper bound on the expected time for a recombinative algorithm to solve a nonseparable building-block problem by proving the existence of a path to the solution and calculating the time for each step on this path is given.
The analysis of evolutionary algorithms on sorting and shortest paths problems
• Computer Science, Mathematics
J. Math. Model. Algorithms
• 2004
This work analyzes simple EAs on well-known problems, namely sorting and shortest paths, and finds that sorting is the maximization of “sortedness” which is measured by one of several well- known measures of presortedness.
Heuristic algorithms and learning techniques: applications to the graph coloring problem
Taking the well-known graph coloring problem as an experimental framework, several new heuristics are developed that integrate certain learning mechanisms so as to render the search process more “self-aware”, including an algorithm that is able to record its trajectory and to interpret its own evolution.
Fitness Landscapes Based on Sorting and Shortest Paths Problems
• Computer Science
PPSN
• 2002
Fitness landscapes based on important computer science problems as sorting and shortest paths problems are investigated here and it cannot be expected that evolutionary algorithms outperform the well-known problem specific algorithms on these simple problems.
Some Combinatorial Landscapes on which a Genetic Algorithm Outperforms Other Stochastic Iterative Methods
• Computer Science
Evolutionary Computing, AISB Workshop
• 1995
This work shows one class of landscape arising from a class of real-world problems which fits the bill, and introduces a general class of landscapes which also display this ‘GA advantage’.
On the Effect of Connectedness for Biobjective Multiple and Long Path Problems
• Computer Science
LION
• 2011
It is argued that connectedness is not the single property to study for the design of multiobjective local search algorithms, and opens new discussions on a proper definition of multi objective fitness landscapes.
Stepping stones and hidden haystacks: when a genetic algorithm defeats a hillclimber
• <NAME>
• Computer Science, Mathematics
Proceedings of 1997 IEEE International Conference on Evolutionary Computation (ICEC '97)
• 1997
It is argued in particular that a specific notion of hillclimbing behaviour can with certain merits, and with certain qualifications, be included in this collection of descriptive tools with which to assess potential amenability to evolutionary search.

## References

SHOWING 1-10 OF 17 REFERENCES
Reverse HillclimbingGenetic Algorithms and the Busy Beaver Problem
• Computer Science
ICGA
• 1993
This paper introduces a new analysis tool called reverse hillclimbing, and demonstrates how it can be used to evaluate the performance of a genetic algorithm, using the {\it Busy Beaver problem}, an interesting problem of theoretical importance in its own right.
Relative Building-Block Fitness and the Building Block Hypothesis
• Mathematics, Computer Science
FOGA
• 1992
A class of fitness landscapes (the “Royal Road” functions) that are designed to investigate the ability of the GA to produce fitter and fitter partial solutions by combining building blocks are described and some unexpected experimental results concerning the GA's performance on simple instances of these landscapes are presented.
When will a Genetic Algorithm Outperform Hill Climbing
• Computer Science
NIPS
• 1993
An "idealized" genetic algorithm (IGA) is analyzed that is significantly faster than RMHC and that gives a lower bound for GA speed.
How Genetic Algorithms Really Work I.mutation and Hillclimbing
In this paper mutation and hillclimbing are analyzed with the help of representative binary functions and a simple asexual evolutionary algorithm. An optimal mutation rate is computed and a good
How Genetic Algorithms Really Work: Mutation and Hillclimbing
The invention relates to the production of washing powders of stabilized or enhanced appearance which contain a fluorescent whitening agent of the formula or of the formula wherein R1 is hydrogen or
Adaptation in natural and artificial systems
Names of founding work in the area of Adaptation and modiication, which aims to mimic biological optimization, and some (Non-GA) branches of AI.
Genetic Algorithms in Search Optimization and Machine Learning
This book brings together the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields.
Theory of Error-correcting Codes
The field of channel coding started with <NAME>'s 1948 landmark paper. Fifty years of efforts and invention have finally produced coding schemes that closely approach Shannon's channel
Difference-preserving codes
• Mathematics, Computer Science
IEEE Trans. Inf. Theory
• 1974
This paper discusses the application of DP codes to pattern recognition and classification problems and presents a construction of efficient DP codes whose information content is asymptotically of the order of theoretical upper bounds.
Making genetic algorithms y: a lesson from the Wright brothers
• Advanced Technology for Developers. 2 February
• 1993


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# 101b: Swissmetro MNL, Biogeme Style¶

This example is a mode choice model built using the Swissmetro example dataset. We will use a style for writing the utility functions that is similar to the style used in Biogeme. First we create the DB and Model objects, as usual:

d = larch.DB.Example('SWISSMETRO')
m = larch.Model(d)

We can attach a title to the model. The title does not affect the calculations as all; it is merely used in various output report styles.

m.title = "swissmetro example 01b (simple logit)"

Unlike Biogeme, the usual way of using Larch does not fill the main namespace with all the parameters and data column references as distinct objects. Instead, we can use two master classes to fill those roles.

from larch.roles import P,X  # Parameters, Data

All of our parameter references can be written as instances of the P (roles.ParameterRef) class, and all of our data column references can be written as instances of the X (roles.DataRef) class.

The swissmetro dataset, as with all Biogeme data, is only in co format. Which is great, because it lets us ignore the ca format and just write out the utility functions directly.

m.utility[1] = ( P.ASC_TRAIN
+ P.Time * X.TRAIN_TT
+ P.Cost * X("TRAIN_CO*(GA==0)") )
m.utility[2] = ( P.Time * X.SM_TT
+ P.Cost * X("SM_CO*(GA==0)") )
m.utility[3] = ( P.ASC_CAR
+ P.Time * X.CAR_TT
+ P.Cost * X("CAR_CO") )

Note that when the data field is too complex to be expressed as a single python identifier (variable name), we can write it as a quoted string instead.

We can estimate the models and check the results match up with those given by Biogeme:

>>> m.maximize_loglike()
messages: Optimization terminated successfully ...
>>> m.loglike()
-5331.252...
>>> m['Time'].value
-0.01277...
>>> m['Cost'].value
-0.01083...
>>> m['ASC_TRAIN'].value
-0.701...
>>> m['ASC_CAR'].value
-0.1546...

>>> print(m.report('txt', sigfigs=3))
=========================================================================================...
swissmetro example 01b (simple logit)
=========================================================================================...
Model Parameter Estimates
-----------------------------------------------------------------------------------------...
Parameter       InitValue       FinalValue      StdError        t-Stat          NullValue
ASC_TRAIN        0.0            -0.701           0.0549         -12.8            0.0
Time             0.0            -0.0128          0.000569       -22.5            0.0
Cost             0.0            -0.0108          0.000518       -20.9            0.0
ASC_CAR          0.0            -0.155           0.0432         -3.58            0.0
=========================================================================================...
Model Estimation Statistics
-----------------------------------------------------------------------------------------...
Log Likelihood at Convergence           -5331.25
Log Likelihood at Null Parameters       -6964.66
-----------------------------------------------------------------------------------------...
Rho Squared w.r.t. Null Parameters      0.235
=========================================================================================...

Tip

m = larch.Model.Example('101b')


text:
Prereqs: this other post introducing the determinant as a tensor. And it wouldn’t hurt to know a bit about permutations and the symmetric group, but it’s not at all required

## Last time…

…we saw that the determinant can be thought of a special kind of tensor: a multilinear function $\phi:V\times \dots \times V\to \mathbb{F}$ from a product of vector spaces $V$ to their common base field $\mathbb{F}$ (often $\mathbb{R}$ or $\mathbb{C}$). More concretely, the determinant eats up vectors (in the form of columns of a matrix) and spits out a scalar which measures the oriented volume of the parallelepiped spanned by those vectors. It’s much easier to say in pictures:

We talked about how multilinearity captures the scaling nature of the determinant: the total volume scales linearly as we vary one side length and leave the others fixed. Multilinearity is the signature property of all tensors. But the determinant has another property, one that a generic tensor doesn’t: antisymmetry. Antisymmetry, and its partner symmetry, are crucial properties to a world of fascinating tensors. Generic tensors don’t have any relationships between their input slots–if you swap two positions in the input, you don’t know anything about the output of the tensor. But symmetric and antisymmetric (also called alternating tensors) describe special cases where permuting the inputs results in a predictable output: for symmetric tensors, the output is unchanged, and for antisymmetric tensors, the output changes sign according to the permutation.

What does this really mean? To start, let’s think about inner products and the determinant, excellent examples of symmetric and antisymmetric tensors. The moral will be that for these special tensors, they are measuring some form of geometric quantity that remains invariant when we change our perspective. Swapping vector positions shouldn’t change the total volume or the angle between two vectors!

## Symmetric tensors: an example through inner products

An inner product $\langle \cdot , \cdot \rangle$ over $\mathbb{R}$ is a positive definite symmetric 2-tensor. Translation? (note: this definition only works for $\mathbb{R}$ and not $\mathbb{C}$. Complex inner products are conjugate linear in one of the slots and thus not multilinear)

• 2-tensor: Tensors are multilinear functions, and the number indicates how many vectors the tensor eats up. So, a 2-tensor is a function with linearity in each component. 2-tensors arise so commonly that they have their own name–bilinear forms. A great deal of linear algebra is dedicated to classifying them, and their roles in almost every major area of mathematics and applied mathematics is irreplaceable–they allow us to measure error in machine learning, proper time in special relativity, and correlations in signal theory.
• Symmetric: This means that if we swap the two inputs, the output is unchanged. In other words, the inner product between two vectors shouldn’t depend on their order. In the case of the dot product…
• Positive definite: Inner products give rise to norms by $\langle v, v \rangle = ||v||^2$, allowing us to measure the size of a vector $v$. Positive definite means that the size is always positive unless $v = 0$.

See what symmetric did here? Saying that the inner product was symmetric gave us some geometric information: it told us that it didn’t matter the order in which we calculated our angle. Inner products measure “similarity”, and symmetric meant that $u$ is as similar to $v$ as $v$ is to $u$. In other words, the inner product is left invariant under swapping the input vectors.

## Antisymmetric tensors: an example through determinants

We spent a ton of time in the last post discussing why the determinant was tensorial: when we independently scaled our input vectors (thought of as the columns of a matrix), the determinant scaled linearly for each input vector. But we didn’t address the intriguing sign in front of the determinant. When we perform a single swap of our input vectors, the volume stays the same, but we multiply by $-1$. And since any permutation of inputs can be rewritten as a series of swaps, we can calculate the determinant for all permutations of given input vectors! If you’ve taken abstract algebra, this is captured by the statement that all elements in the symmetric group can be decomposed into products of 2-cycles, and the sign $\pm 1$ in front of the determinant is determined by whether the product has an even ($+1$) or odd ($-1$) number of 2-cycles. This property is called antisymmetry, and these tensors are sometimes called alternating tensors.

But what does this mean geometrically? The sign $\pm 1$ detects the orientation of the volume: if it agrees with the right-handed coordinate system it lives in, then it is $+1$, and if it points in the opposite direction, then it is $-1$. As with much geometry, this will be much more clear in a nice picture in 2D. Here, we can define orientation as “where the right hand rule applied to our input vectors points” (we’re basically doing a cross product of our inputs). Look at what happens when we flip our input vectors!

So the antisymmetry property told us how orientation changes when we swap inputs: if we swap any two inputs, the volume stays the same, but the sign encoding orientation flips!

A warning–this actually doesn’t make any sense for higher dimensional volumes! The cross product is special and really only works in dimension 3 with vectors as input. In fact, mathematicians use the determinant itself to define orientation for more general volumes: first, take a “standard” ordered set of vectors, like the standard basis vectors $(\hat{x}, \hat{y}, \hat{z})$ and define orientation by checking the sign of the determinant of the matrix mapping to a new coordinate system $(v_1, v_2, v_3)$. If the sign is $+1$, the orientations agree, and if the sign is $-1$, they disagree. This basic idea even allows us to generalize to shapes that aren’t just flat space called manifolds. These are fundamental objects to the world around us, and they’re absolutely fascinating. You’ve been working with circles and spheres your whole life, and I’m sure you’ve run into the torus the last time you craved a glazed donut. You’ve probably even run into a few möbius bands too. Some of these manifolds, like the sphere or the torus, have meaningful orientations, meaning you can define an “inside” and “outside”, and others, like the mobius band, do not. The wiki for the möbius band has some delightful animations explaining why they cannot be oriented. But ultimately, the reason they cannot comes down to definitions to construct manifolds and properties of the determinant. If you want to learn more about this sort of thing, check out differential geometry–it’s a beautiful field of mathematics.

## Wrap up

We’ve finally unpacked the loaded statement from last post: the determinant is an alternating $k$-tensor $\text{det}: V\times \dots \times V \to \mathbb{F}$ from $n$ copies of a vector space to the base field $\mathbb{F}$, usually $\mathbb{F} = \mathbb{R}, \mathbb{C}$. The tensor part tells us the determinant is multilinear and so scales linearly as we scalar multiply and add in each separate component. The alternating part tells us that it’s an antisymmetric function, so when we swap two inputs, we should multiply by $-1$. Geometrically, the tensor part captures how volume scales when we scale individual sides of a hyperparallelepiped (just a parallelogram in $n$ dimensions), and the alternating part tells us whether its orientation agrees ($+1$) or disagrees ($-1$) with the right-handed coordinate system it lives in. Along the way, we ran into another fascinating tensor–the special case of a real inner product, which has our familiar dot product as an even more special case. This tensor was symmetric, and so when we swapped inputs, the output was unchanged.

It’s worth mentioning that the symmetric and antisymmetric properties are very deep. Like, the fundamental particles of the universe deep. For instance, symmetry and antisymmetry allow us to distinguish bosons, like the light particles photons, from fermions, like electrons: the wavefunctions for bosons are symmetric, and the wavefunctions for fermions are antisymmetric. This seemingly innocent distinction has profound physical consequences like the Pauli exclusion principle, which forces electrons to live in different states. This in turn is essential for all of chemistry and life as we know it–that electrons cannot occupy the same state forces them to arrange in special configurations within atoms, and these electron configurations ultimately govern the chemical properties of matter. If that wasn’t enough, other flavors of symmetry, like rotation symmetry and charge symmetry, are central to the standard model of physics, which classifies all fundamental particles of the universe that we understand. But that’s another story.

Food: Mother’s day eggs benedict, with a twist: chipotle-lime hollandaise with garlic, coriander, turmeric, and mustard, sous vide eggs, asparagus, bacon, and some darn good english muffins made by Grandma. Yes, I am a basic brunch boi.


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# source:NEMO/trunk/doc/latex/global/index.ist@11577

Last change on this file since 11577 was 11577, checked in by nicolasmartin, 14 months ago

New LaTeX commands \nam and \np to mention namelist content
(Partial commit to serve as a backup before other large edits)
In order to benefit of the syntax highlighting and to have a simpler syntax for
citing namelist block (\nam) and parameter (\np) with an optional variable assignment (\forcode{…}),
at this time the only viable solution I found is to require a double marker for
what it looks like the same item:

1. Marker with the real name: 'tra_adv' block or 'ln_flx' parameter
2. Marker with underscore character escaping: 'tra\_adv' block or 'ln\_flx' parameter

Despite many searches and attempts, I did not find a workaround to edit on-the-fly one or
the other marker.
In fact, the problem is on one side that the LaTeX index interprets '_' as a switch for lowering like
in math mode while on the other hand the backslash is considered for Pygments as a typo in Fortran
(red box).

For instance, \nam and \np have as of now the aforementioned 2 mandatory arguments in
the previous order (between braces) + an optional argument for \np when the parameter is defined
(between brackets at the first position):

• \nam: LaTeX code in the \nam{tra_adv}{tra\_adv} → PDF ' in the &namtra_adv (namelist X.X) ' with syntax highlighting, the hyperlink and the index entry
• \np: LaTeX code \np[=.true.]{ln_flx}{ln\_flx} → PDF ln_flux=.true. with syntax highlighting for the whole string and the entry in the 'parameters' index
File size: 124 bytes
RevLine


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tutorialpoint.org

B.Pharm Lab. Instruction Manuals

APHE Anatomy, Physiology, and Health Education

Pharmacy study material

# 3.5.3 Pharmacology I

## Lab 12 Collection of blood from orbital sinus of rats and separation of RBC (Cont'd...)

### Method (cont'd...)

Step 4: Collection of whole blood: Pour 2-3% of halothane in a closed chamber and keep the animal in the chamber till anesthesia. Now, take it out of the chamber and insert the capillary tube (previously dipped into TSC solution) into the medial canthus of the eye and direct caudally and slightly dorsally. Pressure should be applied after blood collection to prevent hematomas. Collect the blood into two tubes (previously filled with TSC solution) in equal amount (1.0-1.5 ml. in each tube). Give one or two drops of antibiotic eye drops in rat's eye to avoid bacterial infection. The animal will be in the state of arousal after 15-20 minutes of the application of the anesthetics.

Fig. Orbital Sinus of a rat. Ref:http://www.theodora.com/rodent_laboratory/images/retro_orbital_sinus.jpg

Step 5: Separation of RBC from whole blood: Add NS or PBS (1:1 ratio with blood) in tubes and centrifuge at 2350 rpm for 3-4 minutes. Discard the supernatant out of tubes. Repeat the procedures for 4-5 times to wash the RBC with NS. Collect the precipitate. Because RBC is precipitated out in the tubes for it's higher molecular weight (68000).

Step 6: Dilution of RBC: Say, $X$ ml. of RBC is available after separation. Require to prepare $Y$ percent of suspension. Add $(100-Y) \times X/Y$ ml. of normal saline in $X$ ml. of RBC to get $Y$% of suspension. Collect $X$ ml. of RBC in a test tube and add the measured amount of normal saline or phosphate buffer saline into it to get the suspension.

### Precautions

i. The volume of collected blood should be as per the guideline otherwise the animal may die due to hypovolumic shock or physiological stress.

ii. Frequent collection should be avoided to avoid the anaemia.

iii. If an animal is required to be used for long period, then the haemacrit of the animal should be monitored to avoid the complications (haemacrit of mice, rats and rabbits are 39-49%, 36-54% 30-50% respectively).

iv. Cautiously do the anesthesia to avoid the anesthesia induced death.


text:
Math and Arithmetic

# How do you write 16 over 100 as a decimal in simplest form?

Top Answer
###### Answered 2012-04-27 18:51:08

16/100 = 4/25 There are no more common factors for 4 and 25 so this is the simplest form.

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## Related Questions

### Write each percent as a decimal and as a fraction in simplest form?

To write a percent as a decimal, divide it by 100 To write a percent as a fraction, put it over 100 To reduce a fraction to its simplest form, find the GCF of the numerator and the denominator and divide them both by it. If the GCF is 1, the fraction is in its simplest form. 75% = 0.75 = 75/100 = 3/4

### What is the simplest form for 3 over 100?

It is already in its simplest form, or it can be written as the decimal 0.03.

### What is 75 over 125 as a decimal?

3 over 5 in its simplest form * * * * * True, but that is not a decimal! As a decimal fraction, it is 0.6

### What is 85 over 99 in simplest form?

85/99 is the simplest form as a fraction. In decimal form it is 0.858685/99 is the simplest form.

### Is 2 over 15 in its simplest form?

2/15 is the simplest form. in decimal notation it is 0.133333

### What is 17 over 4 in simplest form?

It is already in its simplest form as 17/4 which is 4.25 as a decimal

### What is 31 over 5 in simplest form?

31/5 is aleady in its simplest form and as a decimal it is 6.2

### What is 66.6 over 100 in simplest form?

66.6/100 is already in its simplest form which is 0.666 as a decimal

### How do you write 21 over 22 in simplest form?

It is already in its simplest form as a fraction.

## Still have questions?

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text:
# Surface intersected by line parallel to x-axis

Problem:

Consider ## I=\int_S \vec{v} \cdot d \vec{S}=\int_S \vec{v} \cdot \hat{n} dS##, where S is a surface that is intersected once by an line parallel to the x-axis. Show that $$dS=\frac{|\nabla f|}{\partial f/ \partial x}dy \ dx$$, and that therefore, $$I=\int_S \vec{v} \cdot \frac{\nabla f}{\partial f/ \partial x} dy \ dz$$

Attempt at a Solution:

Not really sure where to start...


text:
# Math Overflow Recent Questions

## Primary tabs

most recent 30 from mathoverflow.net 2018-06-24T03:46:36Z

### Definitions I don't find

Sun, 06/17/2018 - 19:05

I am reading an article and I didn't find some of the definitions of the terms the author use in the article. If someone know I will appreciate ! What is the prime-to-$p$ part of a torsion subgroup on a module ? (I know what his a torsion subgroup but prime-to-$p$ part ? ) And what is a pro-$p$ quotient ? Thanks !

### Given a representation-infinite algebra, when is every AR component infinite?

Sun, 06/17/2018 - 18:44

Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-modules. The Auslander-Reiten quiver is a locally finite quiver whose vertices are indecomposable modules (up to isomorphism) and whose arrows are irreducible morphisms between indecomposable modules. It is given the structure of a translation quiver via the Auslander-Reiten translate $\tau$.

Two distinct vertices/modules of $\Gamma_A$ are said to belong to the same component if there exists a finite path/composition of irreducible morphisms between them. If $A$ is representation-finite (i.e. the category of finitely generated right $A$-modules contains finitely many non-isomorphic indecomposable objects), then $\Gamma_A$ consists of finitely many components, which each have finitely many vertices. However if $A$ is representation-infinite, then necessarily $\Gamma_A$ is infinite.

It is easy to construct an example of a representation-infinite algebra $A$ for which there exists a finite component of $\Gamma_A$. For example, consider the algebra $KQ/I$, where $KQ$ is the path algebra of the quiver $$Q\colon \; 1 \begin{smallmatrix} \alpha \\ \rightarrow \\ \rightarrow \\ \beta \end{smallmatrix} 2 \begin{smallmatrix} \gamma \\ \rightarrow \\ \color{white} \gamma \end{smallmatrix} 3 \begin{smallmatrix} \delta \\ \rightarrow \\ \color{white} \delta \end{smallmatrix} 4$$ and $I$ is the ideal generated by the set $\{\beta\gamma, \gamma\delta\}$. This is representation-infinite because it contains the Kronecker quiver as a subquiver. But the relation $\gamma\delta$ 'cuts off' the right-hand side of the quiver, which means we have a finite Auslander-Reiten component $$\begin{matrix} S(4) & & & & S(3) \\ & \searrow & & \nearrow & \\ & & P(3)=I(4) & & \end{matrix}$$ (which is essentially $\mathrm{mod}\;K\mathbb{A}_2$) in $\Gamma_A$, where $S(v)$, $P(v)$ and $I(v)$ are the simple, indecomposable projective, and indecomposable injective modules corresponding to the vertex $v$ in $Q$ respectively. (One can also use the fact that this is a string algebra to easily compute $\Gamma_A$.)

My question is this: When is it the case that for a representation-infinite algebra $A$, every component of $\Gamma_A$ contains infinitely many vertices? Is this for example true if the algebra is self-injective? (I imagine the proof of the latter question is straightforward if true, but please point me to a reference if one exists.)

### Upward confluence in the interaction calculus

Sun, 06/17/2018 - 17:29

The lambda calculus is not upward confluent, counterexamples being known for a long time. Now, what about the interaction calculus? Specifically, I am looking for configurations $c_1$ and $c_2$ such that $\exists c: c_1 \rightarrow^* c\ \wedge\ c_2 \rightarrow^* c$, but $\nexists c': c' \rightarrow^* c_1\ \wedge\ c' \rightarrow^* c_2$.

### If a triangle can be displaced without distortion, must the surface have constant curvature?

Sun, 06/17/2018 - 17:17

Suppose $S$ is a surface, a Riemannian manifold in $\mathbb{R}^3$. Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics. If $T$ can be moved around arbitrarily on $S$ while remaining congruent (edge lengths the same, vertex angles the same), does this imply that $S$ has constant curvature?

I realize this is a naive question. If $S$ has constant curvature, then $T$ can be moved around without distortion. I would like to see reasoning for the reverse: If $T$ can be moved around while maintaining congruence, then $S$ must have constant curvature. What is not clear to me is how to formalize "moved around."

### Steiner's inequality reference request

Sun, 06/17/2018 - 15:27

I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have $$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$ where $$\Omega_r=\{x\in \mathbb{R}^2: d(x,\Omega)\leq r\},$$ $A$ denotes the area and $L$ the length. I tried to find a reference for this inequality, but I only found Steiner's formula which states that equality holds when $\Omega$ is convex. Can someone please give me a reference for that inequality?

### Classifying stacks and semidirect product

Sun, 06/17/2018 - 15:11

I am trying to understand the behaviour of quotient stacks under the semidirect product.

Let $G\rtimes H$ be an algebraic group acting on a scheme $X$, over a field $k$. Assume that $G$ acts trivially on $X$. The canonical map $[X/G\rtimes H]\to [X/H]$ is a $G$-gerbe, and it admits a section $s:[X/H]\to[X/G\rtimes H]$ induced by the inclusion $H\subseteq G\rtimes H$. Is $s$ a $G$-torsor? By this I mean the following:

for any scheme $T$ and any map $T\to [X/G\rtimes H]$, the induced map $T\times_{[X/G\rtimes H]}[X/H]\to T$ is a $G$-torsor.

The map $s$ is clearly representable, so at least the question makes sense. Objects of $T\times_{[X/G\rtimes H]}[X/H]$ are $H$-torsors $P$ with an $H$-equivariant map to $X$ such that $P\times^H(G\rtimes H)$ is isomorphic to a fixed $G\rtimes H$-torsor (with compatibility of the maps to $X$), but I cannot define a $G$-action on this set.

Added: consider a map $S\to [X/G\rtimes H]$, corresponding to a torsor $P$. We have $S\rtimes_{[X/G\rtimes H]}X=P$, and therefore $S\times_{[X/G\rtimes H]}[X/H]=P/H$. This certainly becomes isomorphic (as a scheme) to $G$ after passing to a cover of $S$, but I don't see how one can define an action on it to make it a $G$-torsor.

### Spectral algebraic geometry vs derived algebraic geometry in positive characteristic?

Sun, 06/17/2018 - 14:58

Let $R$ be a commutative ring. Then there is a forgetful functor from the $\infty$-category of simplicial commutative $R$-algebras to the $\infty$-category of connective $E_{\infty}$-algebras over $R$. It's well-known that this functor admits left and right adjoint. Moreover, it's an equivalence if $R$ contains the field of rational numbers $\mathbb{Q}$. If I understand correctly, this means that SAG and DAG are (more or less) equivalent over such $R$.

On the other hand, if $R$ does not contain $\mathbb{Q}$ (say $R$ is a finite field), then the forgetful functor above is not necessarily an equivalence. This implies that SAG and DAG are very different in this situation. I wonder if there are any examples of problems where using SAG (resp. DAG) is more appropriate? Such examples would help to illustrate the difference between SAG and DAG in positive characteristic.

P.S.: related but not identical questions: $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras, Why do people say DG-algebras behave badly in positive characteristic?

### Which proportion of elliptic curves over Q is proved to satisfy BSD conjecture?

Sun, 06/17/2018 - 14:06

In 2014, Bhargava and other authors proved that more than 66 % of all elliptic curves defined over $\mathbb{Q}$ satisfy the Birch and Swinnerton-Dyer conjecture. Tonight I stumbled upon an article of <NAME> (arXiv:1803.11074) claiming to prove that all elliptic curves with complex multiplication satisfy this conjecture, entailing a solution of the congruent number problem through Tunnell's theorem.

Assuming such a result holds, which lower bound to the proportion of rational elliptic curves satisfying BSD conjecture could be reached ?

### The étale topos of a scheme is the classifying topos of which groupoid?

Sun, 06/17/2018 - 13:28

[Sent here from Math.StackExchange by suggestion of an user.]

By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. <NAME>. and <NAME>. Representing topoi by topological grupoids. Journal of Pure and Applied Algebra 130, 223-235, 1998) that topoi "with enough points" admit actually a representation as classifying topoi of topological groupoids.

Now my question is the following: take a well-known topos, as the étale topos for a scheme. This is the classifying topos of a localic groupoid, acrtually a topological groupoid since it is "coherent" and thus has enough points by Deligne's theorem; but which groupoid is this in concrete? Do you know if someone has ever investigated that? Thank you in advance.

P.S.: I have been suggested to look at the proof and try to reconstruct the particular case, which I am going to try.

### Presentation of special linear group over localizations of the integers

Sun, 06/17/2018 - 13:19

I am looking (for $n,k\in{\mathbb Z}$) for a presentation (in the best of all worlds concretely, as a list of relators) for the group ${\rm SL}_n(R)$ for $R={\mathbb Z}[\frac{1}{k}]=\{\frac{a}{k^l}\mid a\in {\mathbb Z}\}$.

A search in MathSciNet found a paper of Behr and Mennicke(A presentation of the groups PSL(2,p), Can. J. Math. 20, 1432-1438 (1968)) that gives a presentation for the special case of $n=k=2$; (add the diagonal matrix $(2,\frac{1}{2})$ as extra generator and describe its conjugation action on suitable generators of ${\rm SL}_n({\mathbb Z})$); but a reference search did not yield a further result.

Similarly mathOverflow carried the same question question for other rings, but not $R$.

(A follow-up question would be the same question for ${\rm Sp}$)

### Quantum Lie algebra formalism that doesn't violate P symmetry

Sun, 06/17/2018 - 12:45

begin tl;dr: I just read this paper which gives the equations for the structure constants, braiding operators etc. for a generic quantum Lie algebra. I always found it very annoying that in the Kauffman abstract tensor formalism, you need caps and cups (read: creation and destruction operators, if you read it as Feynman diagrams), at least for undirected lines. Since in any graphical formalism (read: birdtracks - what I do since 30 years might be called quantum birdtracks), the "space" and "time" directions are equivalent. (OK, one could gauge in some left and right kink operators equal to cup and cap, but that's very artificial. But then, why should I be able to unify relativity and quantum theory :-)

Now these equations are even worse in that kind - when one writes the tensors graphically, even left-right symmetry is violated. Outside twistor theory, this is a no-no :-) Thus: Is there an equivalent formulation of quantum Lie algebras (as tensors) keeping as many symmetries as possible (as graphs)?

### Cumulative Distribution Function about T

Sun, 06/17/2018 - 12:40

I am writing because I have a doubt about a cumulative distribution function having a pdf.

Function

How can I derive the CDF?

Thanks in advice to everyone who help me.

Have a good day :)

### Priority for lemniscate of Gerono?

Sun, 06/17/2018 - 09:57

The Lemniscate of Gerono is a special case of the Lissajous curves. The dates for the two mathematicians are fairly close: Gerono (1799-1891) and Lissajous (1822-1880). There seems to have been earlier work (by <NAME> in 1647 and <NAME> en 1750) that Gerono and Lissajous don't seem to have been aware of.

Historically which of the two 19th century Frenchmen has priority for the lemniscate? Was Lissajous aware of Gerono's work when he introduced his curves?

The question was posed a few days ago at hsm to little effect.

### Does this sequence contain a nonnegative number?

Sun, 06/17/2018 - 09:21

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to be $\sum\bar{a_g}g^{-1}$ and for $\beta=\sum b_gg\in\mathbb C[G]$, we have the (convolution) product $\alpha\beta:=\sum a_gb_hgh$. Let $\alpha=\alpha^*$ be an element in $\mathbb C[G]$, define $F(\alpha):=a_e-\sum_{g\neq e}|a_g|$. Consider the sequence $(A_n)$ defined by $$A_n:=F\left(\alpha^{2n}\right)\quad(n\in\mathbb N)$$ Does the sequence $(A_n)$ contain a nonnegative number?

### On the Leray spectral sequence and sheaf cohomology

Sun, 06/17/2018 - 07:09

I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two topological spaces (say as nice as we want) spaces, I would like to understand how to construct the $E_2$ term $$E_2^{pq} = H^*(Y,\mathcal{H}^*),$$ where $\mathcal{H}^*$ is the pre-sheaf $U \mapsto H^*(f^{-1}(U))$. I know a bit about sheaf cohomology, but honestly I don't really understand how this pre-sheaf appears, or why/if we need to look at its sheafification.

I have several general questions regarding all this, some being even more general:

1. Why are we always talking about the $E_2$-term and not the $E_1$-term of a spectral sequence?
2. From where to I need to start in order to construct this $E_2$-term ?
3. Suppose that there exists an open cover $\{ U_i \}$ of $Y$ such that $\mathcal{H}^*(U_i) = \mathcal{G}^*(U_i)$, where $\mathcal{G}^*$ is another pre-sheaf (for instance the constant one). Under what condition could we have: $$H^*(Y,\mathcal{H}^*) = H^*(Y, \mathcal{G}^*) ?$$

Does anyone have a nice reference to these concepts (important to note that I'm no specialist in algebraic geometry :))

Thanks a lot for your help !

### Bounded over time versus bounded over stopping times

Sun, 06/17/2018 - 05:41

Consider the expectation $E(G(v,X_v)|\mathcal{F}_t)$ for $t\leq v \leq T$ for a stochastic process $X_t$. We can impose one of the two following conditions :

• $E(G(v,X_v)|\mathcal{F}_t)$ has a uniform bound which is independent of $v$ for $t\leq v \leq T$
• $E(G(\tau,X_\tau)|\mathcal{F}_t)$ is bounded for all stopping times $\tau$

Are the two conditions above equivalent? If not then does one imply the other?

### Lowest Dimension for Counterexample in Topological Manifold Factorization

Sun, 06/17/2018 - 00:44

Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is best possible? In other words, if $X \times Y =Z$ where $X$ is a manifold and $Z$ is a 3-manifold, then is $Y$ a manifold?

In Bing's example, $Z$ is not compact. Is there a compact example in dimension $4$?

### An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices

Sat, 06/16/2018 - 15:06

Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N$, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}|$$ Denote $z^*_n := \min_{S^{n-1}} f_n$.

Is it true that for sufficiently large $n$s (which depend on $S$), we have $z_n^* = 2(n-1)$?

Remarks:

• $f_n(x)$ is sum of the absolute values of eigenvalues of a circulant matrix generated by $(0,x_1,x_2,\ldots,x_{n-1})$.

• $f_n(x) = 2(n-1)$, when $x$ is all ones vector.

• Above question stems from this question and may be related to the Littlewood problem, as a comment of Tao to the later linked question.

### Find a formula for the recurrent sequence $q_{n+1}=q_n(q_n+1)+1$

Sat, 06/16/2018 - 04:23

Find an analytic formula for the recurrent sequence $$q_{n+1}=q_n(q_n+1)+1,\;\;q_0\in\mathbb N.$$

(The question was asked on 03.05.2018 by <NAME>, see page 109 of Lviv Scottish Book).

### Jordan form on an invariant vector subspace

Sat, 06/16/2018 - 04:18

Let $\mathbb{F}$ be a field and $V$ an $\mathbb{F}$-vector space. Let $\operatorname{T}\in\mathrm{End}(V)$ be an $\mathbb{F}$-linear operator. It is well known that if $\dim V<\infty$ then $\operatorname{T}$ has a Jordan canonical form, i.e., it is similar to the direct sum of a number of Jordan blocks. If a certain eigenvalue (i.e., root of a polynomial) is not inside $\mathbb{F}$ then in the corresponding Jordan block it is represented by its $\mathbb{F}$-matrix representation (that is, the Jordan block is a block matrix).

In infinite dimensions the existence of Jordan canonical form is not guaranteed. I consider only those operators for which it exists.

Question:

Let $V$ have arbitrary dimensions, and assume that $\operatorname{T}$ does have a Jordan canonical form, i.e., it is similar to the direct sum of (an arbitrary set of) Jordan blocks. Let now $W\subset V$ be an $\mathbb{F}$-vector subspace which is $\operatorname{T}$-invariant, $\operatorname{T}W\subset W$. Is it true that the restriction $\operatorname{T}|_W\in\mathrm{End}(W)$ also has a Jordan canonical form?

Thank you.


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# Thread: Formula for sequence and Prove de Morgan's law for unions

1. ## Formula for sequence and Prove de Morgan's law for unions

I'm trying figure out a formula that would work with this sequence defined by a0=1, a1=2, and ak = 2ak-1 + 3ak-2, for all integers k is equal or greater than 2.

I found the values for the rest of the a subscripts
a2 = 7
a3 = 20
a4 = 61
a5 = 182
Seems like it's the values are 3 times more than previous value plus 1 or minus 1

As for de Morgan's law for unions.
If A and B are subsets of the universal set U, then (AUB)^c = (A^c)⋂(B^c).

Would i prove this law by showing a universal set then select random values from the universal set to set A and set B? After I would show AUB, A^c, B^c, etc all the way to (AUB)^c = (A^c)⋂(B^c)?

or would i need to prove this by some form of induction?....

2. I'm trying figure out a formula that would work with this sequence defined by a0=1, a1=2, and ak = 2ak-1 + 3ak-2, for all integers k is equal or greater than 2.
This is a so-called linear homogeneous recurrence relations with constant coefficients. See further down the same page how to solve it.

If A and B are subsets of the universal set U, then (AUB)^c = (A^c)⋂(B^c).
You can see this from a Venn diagram. Or you follow the usual method: proving $(A\cup B)^c\subseteq A^c\cap B^c$ and $A^c\cap B^c\subseteq (A\cup B)^c$. For each inclusion, assume that some x is in the smaller set; then you need to show that x is in the bigger set.

Edit.
or would i need to prove this by some form of induction?....
No, one proves by induction facts about objects that are inductively generated. For example, natural numbers are generated from 0 by repeatedly applying "plus one" function. Sets in general are not generated gradually in this way, so one can't use induction here.

3. Ok the linear homogeneous recurrence relations with constant coefficients is still a bit confusing from wiki. My book has a smiliar example but doesn't really show how to obtain a formula by it.

Also, is the Fibonacci squence similar for this problem?

Suppose Fn, n is greater or equal to 0 is the Fibonacci sequence (i.e. F0=F1 = 1, and Fn = Fn-1 + Fn-2 for n is greater or equal to 2). Use mathematical induction to prove that
(Fn+2)(Fn)-F^2 * n+1 = (-1)^n for all n greater or equal to 0.

4. Ok the linear homogeneous recurrence relations with constant coefficients is still a bit confusing from wiki.
Take the equation $a_k = 2a_{k-1} + 3a_{k-2}$. Replace $a_k$ by $x^k$ for some variable x; you get $x^k=2x^{k-1}+3x^{k-2}$. Divide both sides by $x^{k-2}$ to get $x^2=2x+3$. This is called the characteristic equation of the original equation. Solving it gives you two roots $x_1$ and $x_2$. The solution to the original equation is $a_k=Cx_1^k+Dx_2^k$ for some constants C and D. To find C and D, substitute k = 0 and k = 1 in $a_k=Cx_1^k+Dx_2^k$ and use the given $a_0$ and $a_1$.

Also, is the Fibonacci sequence similar for this problem?
Yes.

Suppose Fn, n is greater or equal to 0 is the Fibonacci sequence (i.e. F0=F1 = 1, and Fn = Fn-1 + Fn-2 for n is greater or equal to 2). Use mathematical induction to prove that
(Fn+2)(Fn)-F^2 * n+1 = (-1)^n for all n greater or equal to 0.
It is recommended to start a new thread for new questions. See rules #8 and 9 here.

5. Originally Posted by emakarov
Take the equation $a_k = 2a_{k-1} + 3a_{k-2}$. Replace $a_k$ by $x^k$ for some variable x; you get $x^k=2x^{k-1}+3x^{k-2}$. Divide both sides by $x^{k-2}$ to get $x^2=2x+3$. This is called the characteristic equation of the original equation. Solving it gives you two roots $x_1$ and $x_2$. The solution to the original equation is $a_k=Cx_1^k+Dx_2^k$ for some constants C and D. To find C and D, substitute k = 0 and k = 1 in $a_k=Cx_1^k+Dx_2^k$ and use the given $a_0$ and $a_1$.

Yes.

It is recommended to start a new thread for new questions. See rules #8 and 9 here.
Thanks, yeah I'll start a new thread if I have problem with that question sorry. I need to work out the first one first. Will be back, thanks.

6. is the answer:

(3/4)*3^k + (1/4)*(-1)^k ?

is there a simplified version of it?

7. Hello, hellfire127!

$\text{Find a formula for the sequence de{f}ined by:}$

$a_0=1,\; a_1=2,\text{ and }a_k \,=\, 2a_{k-1} + 3a_{k-2},\text{ for all integers }k \ge 2.$

$\text{I found the values for the rest of the }a_n:$

. . $\begin{array}{ccc}a_0 &=& 1 \\ a_1 &=& 2 \\a_2 &=& 7 \\ a_3 &=& 20 \\ a_4 &=& 61 \\ a_5 &=& 182 \\ \vdots && \vdots\end{array}$

$\text{Seems like the values are 3 times the previous value plus 1 or minus 1}$
. . Good!

You are correct: . $a_n \;=\;3a_{n-1} + (-1)^n$
. . Yet this is just another recurrence relation, not a "closed form".
[But you can be proud of seeing that relationship.]

Here is one procedure for finding the closed form . . .

We have: . $a_k - 2a_{k-1} - 3a_{k-2} \:=\:0$

Let $X^k \,=\,a_k$
. . We conjecture that the function is exponential.

Then we have: . $X^k - 2X^{k-1} - 3X^k \:=\:0$

Divide by $X^{k-2}\!:\;\;X^2 - 2X - 3 \:=\:0$

. . . . . . . . . . . $(X - 3)(X + 1) \:=\:0 \quad\Rightarrow\quad X \:=\:3,\,\text{-}1$

We have two functions: . $3^n\,\text{ and }\,(\text{-}1)^n$

Form a linear combination: . $f(n) \;=\;A\!\cdot\!3^n + B\!\cdot\!(\text{-}1)^n$

Use the first two terms of the sequence and form a system of equations:

. $\begin{array}{ccccccccc}
f(0) = 1\!: & A + B &=& 1 \\ f(1) = 2\!: & 3A - B ^&=& 2 \end{array}$

Solve the system: . $A = \frac{3}{4},\:B = \frac{1}{4}$

Therefore: . $\boxed{f(n) \;=\;\tfrac{3}{4}\left(3^n\right) + \tfrac{1}{4}(\text{-}1)^n}$

Edit: Ah . . . emakarov beat me to it.
. . . .Will I delete all this? . . . No way!


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# Finding time-varying coefficients for a VAR model by using the Kalman Filter

I'm posting this again, since after my last post i've been able to advance the code quite alot. I'm still trying to write some code in R to reproduce the model i found in this article.

The idea is to model the signal as a VAR model, but fit the coefficients by a Kalman-filter model. As i understand this would enable me to deal with non-stationary data to a degree.

The model to track the coefficients is:

X(t) = F(t) X(t− 1) +W(t)

Y(t) = H(t) X(t) + E(t),

where H(t) is the Kronecker product between lagged measurements in my time-series Y and a unit vector, and X(t) fills the role of regression-coefficients. F(t) is taken to be an identity matrix, as that should mean we assume coefficients to evolve as a random walk.

In the article, from W(T), the state noise covariance matrix Q(t) is chosen at 10^-3 at first and then fitted based on some iteration scheme. From E(t) the state noise covariance matrix is R(t) substituted by the covariance of the noise term unexplained by the model: Y(t) - H(t)Xhat(t)

I have the a priori covariance matrix of estimation error (denoted Σ in the article) written as P (based on other sources) and the a posteriori as Pmin, since it will be used in the next recursion as a priori, if that makes sense.

So far i've written the following, based on the articles Appendix A 1.2

Y <-                                  *my timeseries, for test purposes two channels of 3000 points*
F <- diag(8)                          # F is (m^2*p by m^2 *p) where m=2 dimensions and p =2 lags
H <- diag(2) %x% t(vec(Y[,1:2]))      #the Kronecker product of vectorized lags Y-1 and Y-2
Xhatminus <- matrix(1,8,1)            # an arbitrary *a priori* coefficient matrix
Q <- diag(8)%x%(10**-7)               #just a number a really low number diagonal matrix, found it used in some examples
R<- 1                                 #Didnt know what else to put here just yet
Pmin = diag(8)                        #*a priori* error estimate, just some 1-s...

Now should start the reccursion. To test i just took the first 3000 points of one trial of my data.

   Xhatstorage <- matrix(0,8,3000)

for(j in 3:3000){
H <- diag(2) %x% t(vec(Y[,(j-2):(j-1)]))
K <- (Pmin %*% t(H)) %*% solve((H%*%Pmin%*%t(H) + R)) ##Solve gives inverse matrix ()^-1
P <- Pmin - K%*% H %*% Pmin
Xhatplus <- F%*%( Xhatminus + K%*%(Y[,j]-H%*%Xhatminus) )
Pplus <- (F%*% P %*% F)  + Q
Xhatminus <- Xhatplus
Xhatstorage[,j] <- Xhatplus
Pmin <- Pplus
}

I extracted Xhatplus values into a storage matrix and used them to write this primitive VAR model with them:

Yhat<-array(0,3000)
for(t in 3:3000){
Yhat[t]<- t(vec(Y[,(t-2)])) %*% Xhatstorage[c(1,3),t] + t(vec(Y[,(t-1)])) %*% Xhatstorage[c(2,4),t]
}

The plot looks like

The blue line is VAR with Kalman filter found coefficients, Black is original data.. Only showing the first variable plot here, the other one fits a bit better actually.

I'm having issue understanding how i can better evaluate my coefficients? Why is it so off?

Besides this, how should i better choose the first a priori and a posteriori estimates to start the recursion? Most places i pieced this together from start from arbitrary 0 assumptions in toy models.

Lastly, is this recursion even a correct implementation of Oya et al describe in the article?

• Hi: I can't comment on the specific code for the VAR application but your question seems more general in the sense that you are asking:: Are there are there methodologies for estimatiing the system and observation variances in both the A) "classical" version of the KF and the B) "bayesian" version. For A) check out the prediction error decomposition approach of Harvey in his blue book titled: "structural models and the KF". ( his 1990 book ). For B), check out the yellow book by West and Harrison. I think the title of the latter it dynamic linear models. . – mark leeds Apr 7 at 15:55


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# What are the odds of winning the pokemon lottery?

Level pending

In the pokemon game owned by Nintendo, players are assigned a$$5$$-digit trainer ID ranging from $$00000$$ to $$65535$$ which is randomly generated the moment they start the game. As they progress through the game, players can participate in the pokemon lottery, in which a lottery ticket with a random $$5$$-digit number ranging from $$00000$$ to $$65535$$ is drawn. Players are then awarded prizes based on the number of digits on the lottery ticket which matches their trainer ID. The prizes awarded are as follows:

$$2$$ digits matched- PP Up

$$3$$ digits matched- PP Max

$$4$$ digits matched- Rare Candy

$$5$$ digits matched- Master Ball

What is the probability of winning a Master Ball or Rare Candy in a single draw?

Details and assumptions:

• You might want to use a calculator or Mathematical software to ease the computational process once you figure out the way to solve the problem.
×


text:
# Revision history [back]

### MQTT over Websocket?

Hello,

I'm trying to use wireshark to decode MQTT over Websocket traffic...considering Wireshark already decodes both websocket and MQTT traffic over TCP, this doesn't sound too hard, but I haven't been able to do it.

I've tried many iterations and permutations, but in it's simplest form, I want to get a handle to the existing MQTT dissector from TCP port 1883 (standard port for MQTT over TCP) and install it after the Websocket dissector running on port 8083 (stard port for MQTT over Websocket)...although this LUA does not compile, the idea is something like this...

local tcp_dissector_table = DissectorTable.get("tcp.port")

local ws_dissector_table = DissectorTable.get("ws.port")

local mqtt_dissector = tcp_dissector_table:get_dissector(1883)

Am I even approaching this problem the right way?

### MQTT over Websocket?

Hello,

I'm trying to use wireshark to decode MQTT over Websocket traffic...considering Wireshark already decodes both websocket and MQTT traffic over TCP, this doesn't sound too hard, but I haven't been able to do it.

I've tried many iterations and permutations, but in it's simplest form, I want to get a handle to the existing MQTT dissector from TCP port 1883 (standard port for MQTT over TCP) and install it after the Websocket dissector running on port 8083 (stard (standard port for MQTT over Websocket)...although this LUA does not compile, the idea is something like this...

local tcp_dissector_table = DissectorTable.get("tcp.port")

local ws_dissector_table = DissectorTable.get("ws.port")

local mqtt_dissector = tcp_dissector_table:get_dissector(1883)

Am I even approaching this problem the right way?

 3 None grahamb 23476 ●4 ●805 ●226 https://www.wireshark.org

### MQTT over Websocket?

Hello,

I'm trying to use wireshark to decode MQTT over Websocket traffic...considering Wireshark already decodes both websocket and MQTT traffic over TCP, this doesn't sound too hard, but I haven't been able to do it.

I've tried many iterations and permutations, but in it's simplest form, I want to get a handle to the existing MQTT dissector from TCP port 1883 (standard port for MQTT over TCP) and install it after the Websocket dissector running on port 8083 (standard port for MQTT over Websocket)...although this LUA does not compile, the idea is something like this...

local tcp_dissector_table = DissectorTable.get("tcp.port") DissectorTable.get("tcp.port")
local ws_dissector_table = DissectorTable.get("ws.port") DissectorTable.get("ws.port")
local mqtt_dissector = tcp_dissector_table:get_dissector(1883) tcp_dissector_table:get_dissector(1883)


text:
# How to ask a good question?

## How to ask a good question?

This thread has advice on the following aspects of writing a good question on this site. Each item in this list links to an answer below about that specific aspect of question writing.

-
Further, there is this old thread filled with good snippets. –  Lord_Farin Jun 18 '13 at 9:01
@Lord_Farin: wow, that thread predated my participation on MSE. Thanks for digging that up! –  <NAME> Jun 18 '13 at 9:05

## Provide Context

Context matters. A question can sometimes be answered in one sentence when the discussion is between two experts familiar with each other's background, while the same question may take many paragraphs of detailed computation when being shown to an undergraduate student. By providing a context you help the potential responders to your question give you the best help you need.

Some different ways you can add context to your question

• Include your work

You have a question, and if you post it here, you've probably attempted, and failed, to solve it yourself. It is much easier for others to judge the most appropriate "level" for an answer to your question if you provide these attempts. So you'll receive answers better suited to your specific needs.

Including your work also shows to the community that you're not using this website as an answer machine -- as such, your question will be received more positively.

A further benefit of writing down precisely what you've tried is that, in the process of doing so, you're not unlikely to spot that crucial error and solving your problem yourself. Bonus!

• You can provide some motivation to your question.

Instead of just asking us to find the roots of an equation, tell us where the equation comes from. This is especially the case when your equation comes from models of the physical worlds: those kinds of intuition are great guiding principles for formulating an answer.

• You can tell us where the question comes from.

If your question comes from studying a textbook, let us know which book. This way the answers can be phrased in a manner and in a notation more familiar to you. Exposition varies from one book to another, affecting which theorems are appropriate to cite in answers, and which definitions you are starting from (see below).

• Indicate your own background

In order to address your question in a useful manner, we need to be able to estimate your background to some degree. (Briefly) Indicate your familiarity with the subject matter so that the answerers have an easier job assessing the audience, and can adjust the level of their answer accordingly.

• Give full references.

If you run across a question when reading a scientific paper, be sure to link to that paper using its doi link, or provide a proper bibliographic information. A question that reads "A theorem of Smith says that Widget X is a type of Gadget Y, but I don't see why Property Z must hold" is likely not going to be very comprehensible to other users without telling us which Smith said what when and where.

• Give definitions.

Something that you are familiar with may not be so to another user. One should of course use one's best judgment in deciding what objects are sufficiently well-known to not need defining. But when in doubt, either provide the definition or provide a link to a resource that gives the definitions.

Another case where this can be useful is when the same mathematical object can be defined in many ways, and the answer to your question may depend on the precise definitions used. For example, Widget X may be defined by Author A to satisfy property T. Practically everyone else may prefer to define it as satisfying property S. Showing the equivalence between property T and property S may happen to be one of the harder but lesser known theorems in the past fifty years. If you ask the question, after reading a treatise by Author A, that "Why is property S true for Widget X?"; the common answer "duh, that's by definition" will probably not be very useful to you.

-
I think an answer should demand as little knowledge as possible to understand it regardless the OP's knowledge. This is because each question is not the OP's property but the site's which should serve for the benefits of as many users as possible. Therefore I think giving motivation or context for a question is not necessary as long as it is mathematically clear. –  user157678 Aug 13 at 0:26
@Deep: People capable of writing "mathematically clear" questions are of course free to disregard the suggestions on this page. The fact remains that many questions asked on this site will benefit from (some of) the advice given in this Q+A item. –  <NAME> Aug 19 at 7:55

## A good title

The title of a question is the first thing people see. Like headings in newspapers, book, song and album titles, their importance is not to be underestimated -- the presence of a good, descriptive title for your question often greatly improves the exposure (and hence the amount and quality of answers) it gets. To ensure maximal descriptiveness of your question's title, review it before posting and ensure that it (still) adequately describes your question's content.

How to choose a good title

Use your title to convey as much information about your question as possible. Since the tags already convey the general subject area of your question, the title should communicate the question itself as faithfully as possible. If necessary, leave out hypotheses in the title, and in the body of the question, explain why the question requires those hypotheses.

• MathJax works in titles

Titles have MathJax support. This means you can e.g. include the integral your question is about in the title, and do not have to resort to vague descriptions like "difficult integral". In your use of MathJax, please adhere to the community guidelines for MathJax in titles. Most importantly, keep the vertical space your title uses to a minimum, and be sure to include at least some plain words.

• Don't be afraid to make the title long

Titles are allowed to be anywhere from 15 to 150 characters long. 140 characters (the length of a tweet) of plain text take up about two full lines on the home page, so try to keep it less than that. But 140 characters is a lot longer than you might think. Too many people restrict themselves to 20 character titles. They're trying not to waste your time by making you read a long title, but they end up wasting more of your time because you have to actually open the question to see if it's interesting to you.

• Make your title interesting for others

Mathematics.SE is designed to be a repository of good mathematical questions and answers. Thus, there is no need to refer to your personal situation in the title. Make your title a question of universal value. For example, the title Help me solve $a^2+b^2=c^2$ for my exam preparation is very specific to your personal situation. Deriving the formula for Pythagorean triples would be a more universal, better title for the same question.

• Your question should be clear without the title

After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title.

### Other Tips

When you are posting a question, write your title first. The system will then suggest possible duplicates: take a look at them, opening links in another tab. If none of those are actual duplicates, write out the body of your question. Then go back and put in a better title for the body that you wrote. (From <NAME>)

-

## Tagging

Tags are a way to help us organize posts on this website. People also use them to locate the questions they will find the most interesting. Thus, good tagging helps to attract the best potential answerers to your question.

How do you select the best tags for your question

• Tags are about content. Tags (except "meta" tags, see below) are supposed to refer to the content of your question, and not so much to the context in which you encountered them (but please, do add context). For example, if you have trouble solving a polynomial equation, use , even if you encountered it while solving an problem.

• Use many tags. You can add up to five tags to your question. A combination of tags gives more information about the question, and so increases the usefulness of the tagging process.

• Tag Wikis. Not sure if a tag fits? Move your mouse over the tag name to see its tag wiki excerpt. Read through it to see if the tag actually fits your question. Quite often the excerpts will also contain suggestions to other, more suitable tags.

• Be wary of the meta tags. Meta tags like and do not give information about the mathematics in the question. They should be used as assistance to the tags describing content, not be the main, or only, tag to your question.

-

## Mathematical typesetting using MathJax

Mathematics.SE uses the emulation engine MathJax for providing $\TeX$-like mathematical typesetting. This means that you can use mathematical notation in your questions in a visually appealing way.

How can I use MathJax?

• Basic information: This gives a quick start for people familiar with the $\TeX$-family of markup languages.
• Specific information: Not every $\LaTeX$ command is supported in MathJax. For extensive documentation of commonly used constructs, see here. That thread is also good for quickly getting to grips with MathJax if you're new to $\TeX$-like typesetting.
• Mathematical expressions in titles: The title of your question supports MathJax as well, so use it as deemed fit! However, please adhere to the specific community guidelines for their use in titles.
-

## Formatting and writing

This is not some random internet forum. We strive for well-composed questions and answers of lasting value -- so keep in mind that your question can be of interest to others as well.

Posing a well-formatted question

• Use proper English to the best of your ability

The use of proper spelling, grammar and punctuation makes your question easier to understand, more appealing, and more likely to attract knowledgeable experts to answer your question. If English is not your first language and you are concerned that you will be unable to express your question clearly in English, it might be better to post your question in your native language; it is likely that another user will be able to translate it for you.

• Make your actual question stand out

If your question contains a lot of context, your work, or background information, others may find it difficult to figure out what exactly your question is. You can use bold text: **bold text**, a "quote":

My question

by > My question (on a new line), or a horizontal line: --- (preceded by an empty line) to make your question easily identifiable. See also "MarkDown formatting" below.

• Use paragraphs

Nobody likes to read a densely packed monolith of text. You can enter a blank line in the editor to start a new paragraph at natural places. Add them in a natural frequency that makes for a pleasant read.

• MarkDown formatting

Markdown is the markup language used to format posts on this site.
Most frequently used things are italics and bold, which you can achieve using *italics* and **bold**. To include links you can use this syntax: [Markdown](http://en.wikipedia.org/wiki/Markdown), which gives you Markdown.
For more details and other stuff you can do using Markdown see help.

-


text:
Friday, August 31, 2012

Publishing Your Dissertation With LaTeX

I published my doctoral dissertation. As I used LaTeX to write it in the first place the publisher agreed to let me do the typesetting myself. This way I believed I would avoid a load of extra work converting it all to MS Word and all the annoyances that come along with it. While I am happy with the result in terms of typesetting quality, doing it all in LaTeX turned out to be quite a messy process that took me almost a year.

1. Fundamentals: jurabib hacking

Most of the complexity was hidden in subtle problems I had not foreseen. The main obstacle that consumed most of my time was caused by a decision I made years ago, when I decided to use the jurabib package to manage my references. By the time I was ready to publish the dissertation the package was not actively supported anymore. Instead there was biblatex, which looked very promising but did lack one feature I could not live without (support for archival records is essential for historians). I ended up adapting the official jurabib package to my needs (which were not fully covered by the original package), which took me at least 3 months —working in my spare time after my full-time job. Before that I had only been a user of LaTeX. While I felt comfortable using the higher level API, I had to learn La/TeX from scratch in order to do some more serious stuff. I got there mostly by reading parts of Knuth's TexBook and getting hints from the Internets. What I found out about jurabib was not very encouraging. Some implementation details seemed like a hack to me once I started understanding what was going on. My modifications did not help in any way making it look nicer or more robust. I just tried to coerce it somehow to follow my will. The result is probably not worth sharing but it did what I needed it to do.

2. XeLaTeX or LaTex?

I first jumped on to the XeLaTeX train, which seemed to make my life easier because I was able to use the OpenType fonts installed on my system. At same time I had to convert everything into UTF-8 because for some unknown reason I could not get the

\XeTeXinputencoding
switch to work. But the showstopper was that the quality of the typesetting simply did not match the printed books I used as references. The main reason for that is that XeLaTex does not yet support the microtypography features that are available in LaTaX through the microtype package. I reverted back to LaTeX and used fontools' autoinst, which is basically a wrapper around LCDF TypeTools. It turned out to be not hard at all to use my system's OpenType fonts that way.

3. Adaptation to my publisher's requirements

My only guidance as to what the result should look like was an example of another book published in the same series and some general information from my publisher. I used that and Mac OS X's Preview program to get a feel for sizes and proportions of the intended result and used What The Font to identify the fonts. It felt weird figuring out the layout that way. I think this awkwardness was due to the publisher's workflow clearly being geared towards a MS Word centric approach, where all the typesetting happens at the publishing house. I had used KOMAscript to typeset the initial version of the text, which I submitted to my university. This turned out to be the next roadblock. There was a stark dissonance between what my publisher demanded and what the author of the KOMA templates deemed acceptable in an effort to educate his users towards his understanding of typesetting. Again I was reluctant to abandon the KOMA packages completely as they also offered much valued functionality in other areas.

My main difficulty was font sizes where I had to do nasty things like this:

\KOMAoption{fontsize}{10.5pt}\makeatletter%hacking to 11pt\def\normalsize{% \@setfontsize\normalsize\@xipt{12.75}% \abovedisplayskip 11\p@ \@plus3\p@ \@minus6\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ \belowdisplayskip \abovedisplayskip \let\@listi\@listI}\makeatother

I just did not find another solution. If I had had the time, the clean solution might have been to write my own template based on one of the default LaTeX templates instead.

4. Indices

Just one thing: Make sure they are ordered correctly even if you have got words/titles with umlauts in them ...

Conclusion

What does this all mean? Don't do LaTeX? Not at all, quite the opposite! Do it! Apart from the jurabib issue which was really painful, the other points mentioned here were lessons to be learned rather than unsurmountable obstacles. So by all means do it, maybe choose your packages wisely. Using TeX saved me a lot of tedious manual work while preparing indices and managing references and it also gave me professional grade typesetting on top of that.

Thursday, August 30, 2012

The Cost of Apache Commons HashCode

I had a discussion at work today about the cost of object creation in Java. It was about the way you are supposed to use Apache Commons HashCodeBuilder and EqualsBuilder creating a new object on every single call to hashcode() or equals():

@Override
public int hashCode(){
return new HashCodeBuilder()
.append(name)
.append(length)
.append(children)
.toHashCode();
}

While I was pretty sure that the cost of creating these short lived objects on a modern JVM was pretty negligible in most use cases, I had no numbers to prove it.

Time for a biased benchmark of the simplest kind like the following:

package org.blogspot.pbrc.benchmarks;

import java.util.HashSet;
import java.util.Random;
import java.util.Set;

public class Runner {

private static final int NUM_ELEMS = 100000;
private static final int NUM_RUNS = 100;

public static void main(String[] args) {
String[] keys = createKeys();
Integer[] numerics = createNumericValues();

// priming the jvm
for (int i = 0; i < 5; i++) {
manualEqualsTest(i, keys, numerics);
autoEqualsTest(i, keys, numerics);
}

long time = 0l;
for (int i = 0; i < NUM_RUNS; i++) {
time += manualEqualsTest(i, keys, numerics);
}
System.out.format("%s %.4fsecs\n", "Manual avg:",
(time / Double.valueOf(NUM_RUNS)) / 1000000000.0);

long autotime = 0l;
for (int i = 0; i < NUM_RUNS; i++) {
autotime += autoEqualsTest(i, keys, numerics);
}
System.out.format("%s %.4fsecs\n", "Apache commons avg:",
(autotime / Double.valueOf(NUM_RUNS)) / 1000000000.0);

}

private static long autoEqualsTest(int runNumber, String[] keys,
Integer[] numerics) {
Set<CommonsEqualsAndHashcode> autoObjs = new HashSet<CommonsEqualsAndHashcode>(
NUM_ELEMS);

long start = System.nanoTime();
for (int i = 0; i < NUM_ELEMS; i++) {
}

return System.nanoTime() - start;
}

private static long manualEqualsTest(int runNumber, String[] keys,
Integer[] numerics) {
Set<ManualEqualsAndHashCode> valueObjs = new HashSet<ManualEqualsAndHashCode>(
NUM_ELEMS);
long start = System.nanoTime();

for (int i = 0; i < NUM_ELEMS; i++) {
}

return System.nanoTime() - start;
}

private static Integer[] createNumericValues() {
Integer[] result = new Integer[NUM_ELEMS];
Random rand = new Random();
for (int i = 0; i < NUM_ELEMS; i++) {
result[i] = rand.nextInt();
}
return result;
}

private static String[] createKeys() {
String[] result = new String[NUM_ELEMS];
RandomString rand = new RandomString(32);
for (int i = 0; i < NUM_ELEMS; i++) {
result[i] = rand.nextString();
}
return result;
}

}

I used two—otherwise identical—immutable value types: one with hand-crafted equals() and hashcode(), the other using Apache Commons HashCodeBuilder and EqualsBuilder.

package org.blogspot.pbrc.benchmarks;
public class ManualEqualsAndHashCode {

public final String val;
public final Integer numeric;

public ManualEqualsAndHashCode(String val, Integer numeric) {
this.val = val;
this.numeric = numeric;

}

@Override
public boolean equals(Object obj) {
if (obj == this) {
return true;
}
if (!(obj instanceof ManualEqualsAndHashCode)) {
return false;
}
ManualEqualsAndHashCode other = (ManualEqualsAndHashCode) obj;
return other.val.equals(this.val) && other.numeric.equals(this.numeric);
}

@Override
public int hashCode() {
int result = 17;
result = 31 * result + val.hashCode();
result = 31 * result + numeric;
return result;
}

}

I got the following results on a 2 GHz Intel Core 2 Duo with 4 GB 1067 MHz DDR3 running OS X 10.8.1 (12B19) with Java 1.7.06

Manual avg: 0.4052secs
Apache commons avg: 0.4508secs

In this particular scenario the overhead—when using the Apache Commons classes—amounts to about ten percent. Depending on where you're coming from, this might be a price worth paying. But, of course, this test scenario of just putting items into a collection is highly synthetic so your mileage may vary.

Now go and find more flaws in the benchmark!


text:
Top

Rationalize the Denominator Calculator
Top
Rationalize the Denominator means eliminating the roots in the fractions. Online Rationalize the Denominator Calculator is a tool which makes calculations easy and fun. It is used to rationalize the denominator. Try our Rationalize the Denominator Calculator and get your problems solved instantly.

Below is given a default rational expression, you could see how to rationalize the denominator using  this calculator.

## Steps for Rationalize the Denominator Calculator

Step 1 :

To rationalize the denominator, first multiply both the numerator and the denominator by the conjugate of the denominator.

Step 2 :

Use distribute property (or FOIL) on both the numerator and the denominator.

Step 3 :

Combine like terms and then simply it further.

## Examples on Rationalize the Denominator Calculator

1. ### Rationalize the Denominator:4/(sqrt(6)+ sqrt(5)).

Step 1 :

(4/(sqrt(6)+ sqrt(5)))*((sqrt(6)-sqrt(5)))/((sqrt(6)-sqrt(5)))

Step 2 :

4*(sqrt(6)-sqrt(5))/(sqrt(6)+ sqrt(5))*(sqrt(6)-sqrt(5)) =>(4*sqrt(6)- 4*sqrt(5))/(sqrt(6))^2-(sqrt(5))^2 [(a+b)(a-b)=a^2-b^2] =>(4*sqrt(6)- 4*sqrt(5))/(6-5)

Step 3 :

=>(4*sqrt(6)- 4*sqrt(5))/1 => 4*(2.449)- 4*(2.236) => 9.798 - 8.944 => 0.854

0.854

2. ### Rationalize the Denominator:6/(sqrt(11)+ sqrt(13)).

Step 1 :

(6/(sqrt(11)+ sqrt(13)))*((sqrt(11)-sqrt(13)))/((sqrt(11)-sqrt(13)))

Step 2 :

6*(sqrt(11)-sqrt(13))/(sqrt(11)+ sqrt(13))*(sqrt(11)-sqrt(13)) =>(6*sqrt(11)- 6*sqrt(13))/(sqrt(11))^2-(sqrt(13))^2 [(a+b)(a-b)=a^2-b^2] =>(6*sqrt(11)- 6*sqrt(13))/(11-13)

Step 3 :

=>(6*sqrt(11)- 6*sqrt(13))/-2 => 6*(3.317)- 6*(3.606)/(-2) => -(1.734)/(-2) => 0.867


text:
• # question_answer 30) The mean of the following distribution is 18. Find the frequency f of the class 19 ? 21. Class 11 ? 13 13 ? 15 15 ? 17 17 ? 19 19 ? 21 21 ? 23 23 ? 25 Frequency 3 6 9 13 f 5 4 OR The following distribution gives the daily income of 50 workers of a factory: Daily Income (in Rs.) 100 ? 120 120 ? 140 140 ? 160 160 ? 180 180 ? 200 Number of workers 12 14 8 6 10 Convert the distribution above to a less than type cumulative frequency distribution and draw it?s give.

 C.I. Mid value ${{x}_{i}}$ ${{f}_{i}}$ ${{f}_{i}}{{x}_{i}}$ 11 ? 13 12 3 36 13 ? 15 14 6 84 15 ? 17 16 9 144 17 ? 19 18 13 234 19 ? 21 20 f 20f 21 ? 23 22 5 110 23 ? 25 24 4 96 Total $\sum{{{f}_{i}}=40+f}$ $\sum{{{f}_{i}}{{x}_{i}}=704+20f}$ Now,                 Mean = 18                    (Given) $\Rightarrow$               $\frac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}=18$ $\therefore$                  $\frac{704+20f}{40+f}=18$ $\Rightarrow$               $704+20f=18(40+f)$ $\Rightarrow$               $704+20f=720+18f$ $\Rightarrow$               $20f-18f=720-704$ $\Rightarrow$                          $2f=16$ $\Rightarrow$                            $f=8$ OR Less than type cumulative frequency distribution: Daily Income No. of workers Less than 120 12 Less than 140 26 Less than 160 34 Less than 180 40 Less than 200 50


text:
## RAG

This is the RAG-Token Model of the the paper Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks by <NAME>, <NAME>, <NAME> et al.

The model is a uncased model, which means that capital letters are simply converted to lower-case letters.

The model consits of a question_encoder, retriever and a generator. The retriever extracts relevant passages from the wiki_dpr train datasets, which is linked above. The question_encoder and retriever are based on facebook/dpr-question_encoder-single-nq-base and facebook/bart-large, which were jointly finetuned on on the wiki_dpr QA dataset in an end-to-end fashion.

## Usage:

Note: In the usage example below only the dummy retriever of wiki_dpr is used because the complete lecagy index requires over 75 GB of RAM. The model can generate answers to any factoid question as follows:

from transformers import RagTokenizer, RagRetriever, RagTokenForGeneration


text:
Simplified Processes Matrix

There is a lot emphasis on matrices in PMI project management and the word “matrix” is an important one. You probably know the Requirement Traceability Matrix, the Responsibility Assignment Matrix, RACI chart, sometimes called matrix, yet arguably the most prominent of them all – the one containing processes – stands somewhat in the background. It is however priceless if you can picture it in your mind each time processes, inputs, outputs and tools and techniques are mentioned. And they are mentioned very frequently. In fact the better you understand them, the better your chance of succeeding in the exam. And once in the exam, you will have enough other worries than recalling where this or that process stands within the structure. Let’s unofficially call it “Simplified Processes Matrix” and the objective is to write it down as part of your brain dump in the few minutes available before the exam. It will take a lot of strain out of your studies.

The idea is: you don’t have to rigidly memorize all the names. You rather remember WHAT they stand for and WHERE they are in the overall picture. This is best done when you have some sort of equivalent SHORT “placeholder” for them.
In saying that, let’s assume that:
a = Initiating
b = Planning
c = Executing
d = Monitoring and Controling
e = Closing
and
1 = Integration Management
2 = Scope Management
3 = Time Management
4 = Cost Management
5 = Quality Management
6 = HR Management
7 = Communication Management
8 = Risk Management
9 = Procurement Management
10 = Stakeholder Management
(Look at it and try to memorize it. Believe me, it will become very handy later.)
Now you can sketch a chart.
Once you have that, stay assured that ‘nods’ that will give you most hard time are the ones with multiple processes. And this is where graphics come extremely handy.
Now focus on the following:
2b —-> 4 processes
3b —-> 6 processes
4b —-> 3 processes
5c —-> 3 processes
1d —-> 2 processes
2d —-> 2 processes
Tip: Get used to the single sheet of paper and simple pencil right from the beginning and try not to write more than 2 pages. Just like a pro tennis player, look after the routine, eat your banana :), use the right gear in training, you know, don’t let anything take you by surprise when you’re in the real match.

Rejoice now! All the rest of them will have only one process per node which will be easy to remember.
If you want to exercise your brain a bit more, you can even train yourself in the following coding:
e.g.:
Which process is 2b3/4? Yes, you guessed it! It’s Define Scope.
What is 3b3/6? Yes, Sequence Activities.
4b2/3? Yes, it’s Estimate Costs.
5c2/3? Develop Team, that’s right.

But don’t push it. It doesn’t necessarily have to work for you. The simple process chart will be enough. You can all find it there and look it up in an instant.
Now, why am I going through all that trouble? Why make it even more complicated? – as if it wasn’t complicated enough already!
Well, I suggest that you simply use it each time you have to refer to a process, its input/output or T&T. Try that and you will see the benefits soon enough.