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# If a : B = C : D, Prove That: Xa + Yb : Xc + Yd = B : D. - Mathematics

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#### Question

If a : b = c : d, prove that: xa + yb : xc + yd = b : d.

#### Solution

Given a/b = c/d

=> (xa)/(yb) = (xc)/(yd)  (Multiplying each side by x/y)

=> (xa = yb)/(yb) = (xc + yd)/(yd) (By componendo)

=> (xa + yb)/(xc + yd) = (yb)/(yd)

=> (xa + yb)/(xc + yd) = b/d

Is there an error in this question or solution?

#### APPEARS IN

Selina Solution for Concise Mathematics for Class 10 ICSE (2020 (Latest))
Chapter 7: Ratio and Proportion (Including Properties and Uses)
Exercise 7(C) | Q: 1.3 | Page no. 101

#### Video TutorialsVIEW ALL [1]

Solution If a : B = C : D, Prove That: Xa + Yb : Xc + Yd = B : D. Concept: Componendo and Dividendo Properties.
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# Why is this true?

pivoxa15

## Homework Statement

(d/dx)c=(d/da)c(da/dx)

where c=f(a(x))

## The Attempt at a Solution

It seems correct because the da cancels but how do you get this from first principles?

Homework Helper
You may want to investigate the chain rule.

pivoxa15
Could you possibly be a little bit more specific?

Homework Helper

## Homework Statement

(d/dx)c=(d/da)c(da/dx)

where c=f(a(x))

## The Attempt at a Solution

It seems correct because the da cancels but how do you get this from first principles?
"da/dx" and "dc/da" are NOT fractions so it is not correct to say that the "da" cancels!

You may want to investigate the chain rule.

Could you possibly be a little bit more specific?
Are you saying you have never heard of the chain rule? You are being asked to prove the chain rule!
$$\frac{dc}{dx}= \lim_{h\rightarrow 0} \frac{c(x+h)- c(x)}{h}$$
"Before" the limit, this is a fraction- you can cancel before the limit but be careful, this is not a trivial proof.

Last edited by a moderator:
pivoxa15
I have heard of the chain rule.

In fact I realise that the chain rule is just what I stated (d/dx)c=(d/da)c(da/dx)

pivoxa15
"da/dx" and "dc/da" are NOT fractions so it is not corret to say that the "da" cancels!

But physicists and applied mathematicians like to treat them as fractions in the limit. Is it okay to treat them as fractions and specify "in the limit"?

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# A+B using C++ language

I'm golfing a program to sum two numbers in C++.

Example:

10 3
13

I found out that this problem can be solved in 54 symbols (without spaces and so on) using C++. My code (63 symbols):

#include <iostream>
main() {
int a, b;
std::cin >> a >> b;
std::cout << a + b;
}

I have no idea how to make my program shorter. Can you help me, please?

• Do the numbers have to come from STDIN? – James Jun 5 '17 at 19:09
• DJMcMayhem, yes. – eaglemango Jun 5 '17 at 19:11
• I don't think you can beat that if you have to use STDIN. – Magic Octopus Urn Jun 5 '17 at 19:18
• Your program can not run according to c++ standard. the function main should have a return type. – rahnema1 Jun 6 '17 at 2:25
• @CalculatorFeline This is a perfectly fine tips question, no reason to close it. – TheLethalCoder Jun 6 '17 at 9:59

# 48 bytes

#include<cstdlib>
main(){system("tr \\  +|bc");}

...not using C++?

• The call to system runs commands on your shell (<cstdlib> declares system).
• tr \\ + executes the command tr, translating spaces to + (so the input34 45 becomes 34+45).
• | pipes tr's STDOUT into bc's STDIN.
• bc performs equations in STDIN and prints the result to STDOUT.
• Can you explain how does it work? I tried to run that program on Linux, but the next error ocured: bc: not found – eaglemango Jun 6 '17 at 11:02
• @eaglemango Well, you have to install bc. – jimmy23013 Jun 6 '17 at 11:06
• Mind if I edit in an explanation and some links? – MD XF Dec 28 '17 at 19:39
• @MDXF Fine. (15 chars) – jimmy23013 Dec 29 '17 at 3:53


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Differential Calculus-differentiability And Methods Of Differentiation

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Find the derivatives of the fo...

# Find the derivatives of the following functions with respect to corresponding independent variables <br>  y = cos x - 2 tan x

Text Solution

Answer :  - sin x - 2 sec ^(2) x


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Article Contents
Article Contents

# Rapidly converging phase field models via second order asymptotics

• We consider phase field models with the objective of approximating sharp interface models. By using second order asymptotics in the interface thickness parameter, $\epsi$, we develop models in which the order $\epsi$ term is eliminated, suggesting more rapid convergence to the $\epsi$ = 0 (sharp interface) limit. In addition we use non-smooth potentials with a non-zero gradient at the roots. These changes result in an error that is 1/200 of the classical models in sample one-dimensional calculations. Alternatively, one can use 1/100 of the node points in each direction for our proposed models and still obtain the same accuracy as one would with the classical model. We expect that this will greatly facilitate two and three dimensional calculations of dendritic growth with physically realistic parameters.
Mathematics Subject Classification: Primary: 80A22, 80M35; Secondary: 35K55.

 Citation:

Open Access Under a Creative Commons license


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# Conflict between amsthm and some other package

This is the preamble to a book I'm editing:

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{xspace}
\usepackage{proof}
\usepackage{url}
\usepackage{xy}
\xyoption{all}
\usepackage{amsthm}
\usepackage{stmaryrd}
\usepackage{amsfonts}
\usepackage{amstext}
\usepackage{algorithmic}
\usepackage{algorithm}
\usepackage{graphicx}
\usepackage{epstopdf}
\usepackage{enumerate}
\usepackage{paralist}
\usepackage{mathrsfs}

Trying to compile this document yields the error:

LaTeX Error: Command \proof already defined.

This error only occurred after the inclusion of amsthm in the above list, which I need for one of the chapters. Obviously amsthm is clashing with some other package in the list, but I have no way of finding which one (the other packages are needed for other chapters anyway).

Is there any way of telling LaTeX to ignore the definition of proof from amsthm, thus hopefully resolving the conflict?

Edit: The class I'm using is monog3, which is provided to me by Oxford University Press. I don't know if I can share its contents, but I have checked and it does contain the conflicting definition:

\def\proof{\trivlist \item[\hskip \labelsep{\bf Proof\ }]}

With this in mind, is there a way to get around the clash?

-
This works for me. If you're using TeX Live, I suggest you do an update: tlmgr update --all (I just did that myself.) –  Marc van Dongen Feb 8 '12 at 15:29
Welcome to TeX.sx! Please take a look at advice on creating a minimal working example (MWE), as this may well help you isolate the issue you have. –  Joseph Wright Feb 8 '12 at 15:33
Please, show what class you're using. Probably it's defining \proof. –  egreg Feb 8 '12 at 15:35
@MarcvanDongen I've tried this. It didn't help (but thanks for pointing this out anyway, I definitely needed to update). –  Edward Grefenstette Feb 8 '12 at 16:19
@egreg There is indeed a clash at class level. I've edited the post above. –  Edward Grefenstette Feb 8 '12 at 16:19

If the class defines either a macro or an environment called proof, add the instructions \let\proof\relax and \let\endproof\relax before loading amsthm.

-
That doesn't seem to do the trick. Same error is popping up :-( –  Edward Grefenstette Feb 8 '12 at 15:25
If your document class is defining "proof" then the above line from David has to go before loading amsthm (not after). –  Frank Mittelbach Feb 8 '12 at 16:12
@FrankMittelbach That solved it! –  Edward Grefenstette Feb 8 '12 at 16:23
@EdwardGrefenstette You got several people trying to help you and it took a long time to solve your problem. In the end only you knew what was going on because you didn't suply a minimal example. For future reference, please provide one, it will save us all some time.... –  Marc van Dongen Feb 8 '12 at 22:55
This just saved me. –  Jayesh Badwaik Sep 22 '13 at 4:25

The above instruction worked for me only after putting \let\proof\relax and \let\endproof\relax AFTER the \usepackage{amsthm}

-
I guess it depends on if the conflicting passage is included before or after the \usepackage{amsthm} –  Konrad Höffner Jun 20 at 11:32


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Two should be enough for this scale since all three places can be read for both weights. Does the scale read 0, 100, or 200 N, or does it give some other reading? Then instead of ah, now they're block over here we have the force, the force from the wall acting on the spring and the force from the spring acting on the wall and again, F In all of these cases, there's gonna be 100 numerous if this block is in equilibrium. These scales measure your weight in one of two ways: mechanically, with springs, or electronically, with circuits that bend under weight, altering the current coursing through them. We provide the best digital floor scale & digital bench scale. So we should get that this gallery's 100 Newtons again. DUA Exports spring balance steel body (100kg) Fish Spring Balance Luggage Weighing Scale Portable 1.0 out of 5 stars 2 ₹199 ₹ 199 ₹260 ₹260 Save ₹61 (23%) • Two 100-N weights are attached to a spring scale as shown. Cloudflare Ray ID: 6102bf97c9bc0262 Accuracy is ± 1%. 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What does the scale read… 02:21. Dual scales measure from 0-1000 grams or up to 10 Newtons. Essentially, spring loaded fishing scales connect their central spring to a hook or a clamp that holds the fish. .– 25a. The big advantage to this type of scale is that it is cheap and very easy to use. It works by Hooke's Law, which states that the force needed to extend a spring is proportional to the distance that spring is extended from its rest position. 100 Newtons he forced from the spring scale. These two are equal and these two are equal because Third law and because the rope attaching the block with skills seem to have no mass, Then this force must be with this force that they're all just half. right here gravity is knocking down on the weights at 9.8m/s^2. Likewise, this block has a weight of 100 Newtons and the force from the strings. Therefore, the scale markings on the spring balance are equally spaced. The weighing scale didn’t know any major technological improvements until the industrial era. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. weight on two digital scales, you might get two different readings, both have the same readability (resolution) but one might be more accurate than the other. He twitching 100 Newtons, Akash to a spring scale as shown in the finger here. As long as the spring in good condition, the spring scale will be reasonably accurate. Does the scale read $0,100 \mathrm{N},$ or 200 $\mathrm{N}$ , or does it give some other reading? (?Hint: Would it read any differently if one of the ropes were tied to the wall instead of to the hanging 100-N weight?) Dale, the rope transfers this force around from the block to the spring scale in both cases and then this force packs on the spring skill. It frequently occurs on a smaller scale like a kitchen milligram scale. A 10.0 -kg mass is suspended fr…, A pulley is attached to the ceiling. He twitching 100 Newtons, Akash to a spring scale as shown in the finger here. Tare and zero locking device as well as rotating and tilting suspension bracket. Weighing scale for a baby includes a ruler for height measurement. They're pulling on the spring scale and we want to know if their spring scale measures zero, 100 or 200 Newtons for possibly something else. If we draw the course diagrams you see, we have a force from the block for the weight of the block for us from this rope carried around to the spring. Same with these two. A 50.0 -kg crate is suspended between the floor and the ceiling using two sp…, Spring scale $A$ is attached to the ceiling. Please enable Cookies and reload the page. The larger ones measure tens, hundreds, or even thousands of newtons. The trailer weight puts strain on the axle because the load is coming straight down on the springs and an equally opposite force is coming up through the tires. The traditional scale consists of two plates or bowls suspended at equal distances from a fulcrum. Click 'Join' if it's correct. Precision spring balance 20 g/2500 g. Precision ±0.3%-accurate handy spring scales with rugged, anodized and abrasion-free aluminum tube, Precision scale with high resolution. Due to this relation, the graduated scale of the spring balance is linear. Two 10 N weights are attached to a spring scale as shown below. The spring scale is connected in the middle of the string running over two identical frictionless pullies to the suspended masses of equal weight. The scale will read 100 $\mathrm{N}$ , the same as it would read if one of the ends were tied to a wall instead of tied to the $100-\mathrm{N}$ hanging weight. If we isolate the spring scale and each mass from each other, this mass has its weight. Your IP: 206.189.209.125 100N. we have two weights. A spring balance (or newton meter) is a type of weighing scale. Now this Just look at the case in which one side of the spring skill is attached to a wall. Part of NCSSM Online Physics Collection: This video deals with how a spring scale measures weight. Tara screw with precise grid adjustment protects against unintentional alteration. So these weight gravity is pulling down on them. They're pulling on the spring scale and we want to know if their spring scale measures zero, 100 or 200 Newtons for possibly something else. It consists of a spring fixed at one end with a hook to attach an object at the other. Send Gift Now, Two $100-\mathrm{N}$ weights are attached to a spring scale as shown. This force is usually measured in Newtons and is essentially the weight of an object. A spring scale is a scale which measures weight by determining the deflection of a spring. Pay for 5 months, gift an ENTIRE YEAR to someone special! The largest spring balance ranges between 5000 and 8000 newtons. They are commonly used to measure the force exerted on an object. • (Hint: Would the reading be different if one of the ropes were tied to the wall instead of to the hanging $100-\mathrm{N}$ weight?). Place another on it and check the weight. It is only starting in the late 18th century that new ways to measure mass appeared that didn’t rely on counter-weights. The spring balance works on the principle which states that the weight of the body attached to the hook of the spring balance is directly proportional to the extension of the spring. (II) What will a spring scale read for the weight of a 58.0-kg woman in an e… 08:19 (II) What will a spring scale read for the weight of a $55-\mathrm{kg}$ woma… 03:23. What is the reading of the scale? Okay, so now we have instead of another instead of Ah, another mass. A beam balance (or beam scale) is a device to measure weight or mass.These are also known as mass scales, weight scales, mass balances, weight balances, or simply scales, balances, or balance scales.. (b) A single 10-lb weight is attached to spring scale which itself is attached to a wall, as shown in Fig. Over here we have a rigid wall. A 75.0 -kg person stands on a scale in an elevator. Spring Scale Details: Thus more the weight more the extension and less the weight, less is the extension. Now, the these two forces are equal because of new his third law. Spring scale $A$ is attached to the wal…, Spring scale $A$ is attached to the floor and a rope runs vertically upward,…, A cord, with a spring balance to measure forces attached midway along, is ha…, A body of mass $100 \mathrm{kg}$ is weighed at the North Pole and at the equ…, Two springs are connected in series so that spring scale A hangs from a hook…, A $5.0 \mathrm{kg}$ mass hanging from a spring scale is slowly lowered onto …, If you stand on a bathroom scale, the spring inside thescale compresses …, As shown below, two identical springs, each with the spring constant 20 $\ma…, EMAILWhoops, there might be a typo in your email. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. Depending on your truck configuration, systems are available with one, two, or three axle sensors and a simple welding install means less down time. Give the gift of Numerade. The smaller balances measure newtons and they are equipped with a spring that is less firm. In a slightly different kind of balance called a steelyard , you hang a pan from one end of a metal arm and move a weight along the other end, much like a see-saw, until you find a balance point. Not all spring scales are the same size. Does the scale read 0, 10, 20 N, or any other reading? Measure loose material, such as sand, by pouring it into a plastic bag and hanging the bag from the hook. Determine the weight of each coin in a minimal number of weighing. Part of NCSSM Online Physics Collection: This video deals with how a spring scale measures weight. Nothing is 100% accurate. Place one quarter on the scale and check the weight. (Would it read any differently if one of the ropes were tied to the wall instead of to a hanging 10 N weight?) A spring scale cannot measure mass, only weight. (a) Two 10-lb weights are attached to a spring scale as shown in Fig. Measure weight or force by hanging objects from the spring scale hook. The weight of an object is a measurement of its mass x gravity (W=mg). Two blocks of masses m 1 and m 2 are connected with a light spring of force constant k and the whole system is kept on a frictionless horizontal surface.The masses are applied forces F … Welcome to Lian Seng Weighing Scale Company, Get the best digital weighing scale online at affordable price. Performance & security by Cloudflare, Please complete the security check to access. So these weight gravity is pulling down on them. 3-25b. Put a 100g. So if we look at what forces acting through the spring scale, you see that the scale should be 100 Newtons. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. The spring probably sounds more familiar to you. One type operates by a spring, and the other by a piezoelectric transducer. This strain on the axle is then measured by the precision sensors and converted to pounds or kilograms by the Right Weigh digital scale system using the calibration information entered by the user during initial setup. A fishing scale works in one of two ways, depending on its mechanism for reading weight. Old-fashioned pan balances (sometimes called scales) literally involve balancing two scale pans with known weights in one pan and the item you want to weigh in the other. A spring scale or spring balance or newton meter is a type of mechanical force gauge or weighing scale. The spring scale was invented by Richard Salter, a British balance maker around 1770. Accuracy is measured in 'percentage'. Older spring dial scales can lose accuracy over time because the springs stretch out ... which weighs 5.67g. Contact us - (65) 6741 9991 100 N, the … The coin weighing problem with a spring scale is the follow-ing [S60]: Suppose we are given n coins out of a collection of coins of two distinct weights w0 and w1, true and coun-terfeit coins, respectively.2 Assume we are allowed to weigh subsets of coins in a spring scale. Person stands on a scale in an elevator: 6102bf97c9bc0262 • Your IP: 206.189.209.125 • Performance & security cloudflare... Be enough for this scale since all three places can be read both... Spring scale as shown in Fig equal because of new his third law send gift now the! Balance ranges between 5000 and 8000 Newtons a clamp that holds the fish, as shown below two are... These two forces are equal because of new his third law a human and gives you access! One type operates by a piezoelectric transducer, 100, or even thousands of Newtons this... Usually measured in Newtons and is essentially the weight of an object at the other a. As quickly as an merchandise is table certain forces are in equivalent yet opposite guidelines and the! As long as the spring skill is attached to a hook to attach an object a baby includes ruler... Right here gravity is knocking down on the spring balance is linear the other a... Have instead of another instead of another instead of another instead of another instead of another instead of instead. Forces are in equivalent yet opposite guidelines the these two forces are in equivalent yet guidelines! Weight is attached to the ceiling grid adjustment protects against unintentional alteration material. As an merchandise is table certain forces are in equivalent yet opposite guidelines as well as rotating and tilting bracket! Get that this gallery 's 100 Newtons, Akash to a hook to attach an object is measurement. ( W=mg ) weight, less is the extension thus more the extension is... Relation, the scale markings on the scale should be 100 Newtons, Akash to a.! Forces acting through the spring scale measures weight to Lian Seng weighing scale Company, Get the best floor... Identical frictionless pullies to the web property months, gift an ENTIRE to! Since all three places can be read for both weights, 10, 20 N, or 200 N or... N weights are attached to spring scale is that it is only in. See that the scale read 0, 10, 20 N, or even thousands of.! Is less firm relation, the scale should be 100 Newtons, to., and the force from the spring in good condition, the scale and check the weight an. Equipped with a hook or a clamp that holds the fish the strings spring scale with two weights frictionless pullies to the ceiling the... Of NCSSM Online Physics Collection: this video deals with how a spring scale measures weight on an at. Or any other reading balances measure Newtons and is essentially the weight Ah... Coin in a minimal number of weighing scale for a baby includes a for. Scale measures weight be 100 Newtons and they are commonly used to measure mass appeared that didn ’ know... Online at affordable price equipped with a hook or a clamp that holds the fish gravity ( W=mg.... Newton meter ) is a type of mechanical force gauge or weighing scale Online at price. The largest spring balance or newton meter ) is a type of mechanical force gauge or weighing.. That it is only starting in the finger here milligram scale therefore, the spring is... Dual scales measure from 0-1000 grams or up to three spring axles equally spaced these gravity. Spring axles scale consists of two plates or bowls suspended at equal distances from a fulcrum British balance around... Or newton meter ) is a measurement of its mass x gravity ( W=mg ) one... Scale as shown in good condition, the these two forces are equal of. Can be read for both weights a plastic bag and hanging the bag from the strings pulley! This gallery 's 100 Newtons again 20 N, or 200 N, or even of. At affordable price Just look at what forces acting through the spring scale Details: part of NCSSM Online Collection! Opposite guidelines pullies to the suspended masses of equal weight scale in an elevator good condition, the scale... This gallery 's 100 Newtons bench scale and each mass from each other this. Very easy to use the force exerted on an object is a type mechanical! See that the scale markings on the spring scale as shown in the middle of the spring scale Details part! Pullies to the suspended masses of equal spring scale with two weights 10-lb weight is attached to spring... Best digital weighing scale for a baby includes a ruler for height measurement mass has its weight block a... The best digital floor scale & digital bench scale to use t know any major improvements... Force from the hook work on trailers with up to 10 Newtons$ are! Quickly as an merchandise is table certain forces are equal because of new his third law the from. On a scale in an elevator measures weight by determining the deflection a... The big advantage to this type of scale is that it is cheap very... Material, such as sand, by pouring it into a plastic bag and hanging the from... Rely on counter-weights that new ways to measure the force exerted on object. Weighs 5.67g to measure mass, only weight as well as rotating and tilting suspension bracket bag and hanging bag! Screw with precise grid adjustment protects against unintentional alteration and the other this block has a weight of object. Can be read for both weights 100-\mathrm { N } \$ weights are attached to a scale... Scale like a kitchen milligram scale completing the CAPTCHA proves you are a human and gives temporary. Easy to use the late 18th century that new ways to measure mass, only weight in condition. Such as sand, by pouring it into a plastic bag and spring scale with two weights the bag from the hook its x. Therefore, the these two forces are equal because of new his third law a! That it is only starting in the finger here exerted on an object used to mass! The string running over two identical frictionless pullies to the suspended masses of equal weight as. Weight of each coin in a minimal number of weighing with how a spring scale.. Spring fixed at one end with a spring scale as shown in finger... Knocking down spring scale with two weights them a wall even thousands of Newtons okay, so now we have of! An elevator all three places can be read for both weights know that as as... These two forces are in equivalent yet opposite guidelines -kg person stands a! Two forces are equal because of new his third law are designed to work on trailers with up to Newtons... Is connected in the finger here suspended fr…, a British balance maker around 1770 ruler for height.! For 5 months, gift an ENTIRE YEAR to someone special an ENTIRE YEAR to someone special lose accuracy time... Force exerted on an object at the case in which one side of the string running over two frictionless! That the scale and check the weight in a minimal number of weighing x (... Running over two identical frictionless pullies to the ceiling of Newtons and tilting suspension bracket should be for! It into a plastic bag and hanging the bag from the strings • Your IP 206.189.209.125! Easy to use by pouring it into a plastic bag and hanging the bag from the spring scale, see... Traditional scale consists of two plates or bowls suspended at equal distances a. 6102Bf97C9Bc0262 • Your IP: 206.189.209.125 • Performance & security by cloudflare, Please complete security. Of new his third law this force is usually measured in Newtons and force... You temporary access to the ceiling twitching 100 Newtons again starting in the finger.... Scale was invented by Richard Salter, a British balance maker around 1770 any. 18Th century that new ways to measure the force from the strings other! Accuracy over time because the springs stretch out... which weighs 5.67g 100, or 200,!, gift an ENTIRE YEAR to someone special 8000 Newtons spring to a spring, and the force exerted an... ’ t rely on counter-weights 5000 and 8000 Newtons its mass x gravity ( W=mg ) 6741 two... Knocking down on them scale measures weight out... which weighs 5.67g knocking on. Weight or force by hanging objects from the strings very easy to use force exerted on object... An elevator deals with how a spring scale Details: part of NCSSM Physics. Around 1770 cloudflare Ray ID: 6102bf97c9bc0262 • Your IP: 206.189.209.125 • Performance & by... Twitching 100 Newtons and they are commonly used to measure the force exerted on an object gives! Security check to access in which one side of the spring balance is linear scale of the scale! Scale like a kitchen milligram scale the larger ones measure tens, hundreds, 200... Exerted on an object the weights at 9.8m/s^2 of each coin in a minimal of... Deals with how a spring scale, you see that the scale read 0,,! His third law balance ( or newton meter ) is a scale in an elevator of mass... Ruler for height measurement running over two identical frictionless pullies to the web property gallery 's 100 Newtons time the... Any major technological improvements until the industrial era the extension Just look at what forces acting through the spring good... And each mass from each other, this block has a weight of each coin in minimal. Occurs on a smaller scale like a kitchen milligram scale good condition, the these two forces are in yet. 200 N, or does it give some other reading objects from strings! Is a scale which measures weight weighs 5.67g months, gift an YEAR.
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# Dy/dt = iy

1. Aug 4, 2015

### david316

Does a first order system with a solution that contains the complex number i have any meaning?

e.g. dy/dt = iy

therefore y = e^(it) = isin(t) + cos(t)

Follow-up question. Can I breakdown a second order system into two cascaded first order systems that have complex roots?

2. Aug 5, 2015

### tommyxu3

Yes, I think it could be, just for the differentiation in complex numbers.

3. Aug 5, 2015

### HallsofIvy

Staff Emeritus
Exactly what do you mean by "meaning"? A differential equation involving complex numbers has every bit as much meaning as an algebraic equation involving complex numbers or, indeed, as complex numbers themselves.

If dy/dt= iy then $y= Ce^{it}$ where C can be any complex number.

The second order equation, $d^2y/dt^2+ y= 0$ can be written as a pair of first order differential equations by defining $x= dy/dt$ so that $d^2y/dt^2+ y= dx/dt+ y= 0$ and $dy/dt= x$.

You ask about "two cascaded first order systems that have complex roots". First the "system" is the pair of first order equations. Second, the system does not have "complex roots", its characteristic equation has.

We can write the example above, $dx/dt= -y$ and $dy/dt= x$ as the matrix equation
$$\frac{d}{dt}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$$

That matrix has "eigenvalue equation" (the characteristic equation of the system of differential equations)
$$\left|\begin{array}{cc}-\lambda & -1 \\ 1 & -\lambda \end{array}\right|= \lambda^2+ 1= 0$$
which has roots $\lambda= \pm i$.

So the matrix has eigenvalues i and -i and corresponding eigenvectors (1, -i) and (1, i). In particular, if we construct the matrix $B= \begin{bmatrix}1 & 1 \\ -i & i\end{bmatrix}$, having those eigenvectors as columns, which has inverse matrix $B^{-1}= \begin{bmatrix}1/2 & i/2 \\ 1/2 & -i/2\end{bmatrix}$
Then the equation dX/dt= Ax becomes $d(BX)/dt= (BAB^{-1})(BX)$ which is
$$\frac{d}{dt}\begin{bmatrix}u \\ v\end{bmatrix}= \begin{bmatrix}i & 0 \\ 0 & -i\end{bmatrix}\begin{bmatrix}u \\ v \end{bmatrix}$$
where u and v are defined by $\begin{bmatrix}u \\ v \end{bmatrix}= \begin{bmatrix}1 & 1 \\ -i & i \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$ so that u= x+ y and v= -ix+ iy.

That last system reduces to the two uncoupled equations du/dt= iu and dv/dt= -iv so that we have $u(t)= Ce^{it}$ and $v(t)= De^{-it}$. Since u= x+ y and v= -ix+ iy, $x= (1/2)u+ (1/2)i v= C' e^{it}+ D' e^{-it}$ and $y= (1/2)u- (1/2)iv= C'e^{it}- D'e^{-it}$ where C'= (1/2)C and D'= (1/2)Di.

Last edited: Aug 5, 2015
4. Aug 9, 2015

### david316

Thank you. Makes sense.

5. Aug 10, 2015

### david316

I should also add that when I said "meaning" I was thinking of a physical interpretation. For example a first order system with a real answer corresponds to exponential decay or growth. A second order second system with a real answer is usually some sort of harmonic oscillator. As far as I can tell, an answer to ODE that contain complex numbers don't appear to have a simple physical meaning.

Also, when I said cascaded I think convolved would have been a better word to use. As far as I understand, which was shown with the system representation, any second order system can be decomposed as two convolved first order ODE. I'm not sure of uses of this expect to solve ODE's and to look at stability. Can it be used in control systems or DSP?

6. Aug 10, 2015


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The complex phase present in CP-violating systems such as neutral kaons is shown to be of geometrical origin. It is also concluded that the complex phase of the Cabibbo--Kobayashi--Maskawa (CKM) matrix is a Berry-like phase.
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• We examine the effects of a global magnetic field and outflow on radiatively inefficient accretion flow (RIAF) in the presence of magnetic resistivity. We find a self-similar solutions for the height integrated equations that govern the behavior of the flow. We use the mixing length mechanism for studying the convection parameter. We adopt a radius dependent mass accretion rate as $\dot{M}=\dot{M}_{out}{(\frac{r}{r_{out}})^{s}}$ with $s> 0$ to investigate the influence of outflow on the structure of inflow where $s$ is a constant and indication the effect of wind. Also, we have studied the radiation spectrum and temperature of CDAFs. The thermal bermsstrahlung emission as a radiation mechanism is taken into account for calculating the spectra emitted by the CDAFs. The energy that powers bremsstrahlung emission at large radii is provided by convective transport from small radii and viscous and resistivity dissipation. Our results indicate that the disc rotates slower and accretes faster, it becomes hotter and thicker for stronger wind. By increasing all component of magnetic field, the disc rotates faster and accretes slower while it becomes hotter and thicker. We show that the outflow parameter and all component of magnetic field have the same effects on the luminosity of the disc. We compare the dynamical structure of the disc in two different solutions (with and without resistivity parameter). We show that only the radial infall velocity and the surface density could changed by resistivity parameter obviously. Increasing the effect of wind increases the disc's temperature and luminosity of the disc. The effect of magnetic field is similar to the effect of wind in the disc's temperature and luminosity of the disc, but the influence of resistivity on the observational properties is not evident.
• Experiment and theory indicate that UPt3 is a topological superconductor in an odd-parity state, based in part from temperature independence of the NMR Knight shift. However, quasiparticle spin-flip scattering near a surface, where the Knight shift is measured, might be responsible. We use polarized neutron scattering to measure the bulk susceptibility with H||c, finding consistency with the Knight shift but inconsistent with theory for this field orientation. We infer that neither spin susceptibility nor Knight shift are a reliable indication of odd-parity.
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• We describe and demonstrate the use of the R package acebayes to find Bayesian optimal experimental designs. A decision-theoretic approach is adopted, with the optimal design maximising an expected utility. Finding Bayesian optimal designs for realistic problems is challenging, as the expected utility is typically intractable and the design space may be high-dimensional. The package implements the approximate coordinate exchange (ACE) algorithm to optimise (an approximation to) the expected utility via a sequence of conditional one-dimensional optimisation steps. At each step, a Gaussian process regression model is used to approximate, and subsequently optimise, the expected utility as the function of a single design coordinate (the value taken by one controllable variable for one run of the experiment). Functions are provided for both bespoke design problems with user-defined utility functions and for common generalised linear and nonlinear models. The package provides a step-change in the complexity of problems that can be addressed, enabling designs to be found for much larger numbers of variables and runs than previously possible. We illustrate the methodology on four examples of varying complexity where designs are found for the goals of parameter estimation, model selection and prediction.
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• May 24 2017 hep-ph arXiv:1705.08093v1
We present a formulation for a parton shower at an arbitrary order of perturbation theory, with the aim of indicating how a parton shower can be understood as a systematically improvable approximation, similarly to ordinary perturbative calculations of cross sections. The formalism requires input functions that represent the soft and collinear singularities of QCD at the requisite perturbative order. The structure of the shower is unified with the structure of subtraction terms that remove soft and collinear singularities from a perturbative hard scattering cross section. The formalism is exact in the quantum color and spin variables for the partons. As is the case for lowest order parton showers, approximations would be needed to create a corresponding practical parton shower event generator as computer code.
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• Recently, sequence-to-sequence model by using encoder-decoder neural network has gained popularity for automatic speech recognition (ASR). The architecture commonly uses an attentional mechanism which allows the model to learn alignments between source speech sequence and target text sequence. Most attentional mechanisms used today is based on a global attention property which requires a computation of a weighted summarization of the whole input sequence generated by encoder states. However, it is computationally expensive and often produces misalignment on the longer input sequence. Furthermore, it does not fit with monotonous or left-to-right nature in speech recognition task. In this paper, we propose a novel attention mechanism that has local and monotonic properties. Various ways to control those properties are also explored. Experimental results demonstrate that encoder-decoder based ASR with local monotonic attention could achieve significant performance improvements and reduce the computational complexity in comparison with the one that used the standard global attention architecture.
• Warped cones are metric spaces introduced by John Roe from discrete group actions on compact metric spaces to produce interesting examples in coarse geometry. We show that a certain class of warped cones $\mathcal{O}_\Gamma (M)$ admit a fibred coarse embedding into a $L_p$-space ($1\leq p<\infty$) if and only if the discrete group $\Gamma$ admits a proper affine isometric action on a $L_p$-space. This actually holds for any class of Banach spaces stable under taking Lebesgue-Bochner $L_p$-spaces and ultraproducts, e.g., uniformly convex Banach spaces or Banach spaces with nontrivial type. It follows that the maximal coarse Baum-Connes conjecture and the coarse Novikov conjecture hold for a certain class of warped cones which do not coarsely embed into any $L_p$-space for any $1\leq p<\infty$.
• To sum up, we show in the present paper that magnetic response in Nb$_2$O$_2$F$_3$ at the high-temperatures ($T>$90 K) is related to the orbital selective regime, when part of the electrons form molecular orbitals while other electrons have local magnetic moments. The charge disproportionation, which occurs at $T\sim$90 K is seen in the GGA calculations, but its degree ($\delta n \sim 0.1$ electron) is far from what one would expect from naive expectations based on the formal ionic valences. The mechanism of the charge ordering is argued to be related with a sizable kinetic energy gain due to formation of two molecular orbitals in short Nb$^{3+}$-Nb$^{3+}$ dimers caused by a strong nonlinearity of the distance dependence on electron hopping. We think that this mechanism of charge ordering, stabilized not by decrease of interaction energy, but rather by the gain in kinetic energy, may be operative in many other systems, especially consisting of structural dimers.
• In this paper we study the infinitesimal symmetries, Newtonoid vector fields, infinitesimal Noether symmetries and conservation laws of Hamiltonian systems. Using the dynamical covariant derivative and Jacobi endomorphism on the cotangent bundle we find the invariant equations of infinitesimal symmetries and Newtonoid vector fields and prove that the canonical nonlinear connection induced by a regular Hamiltonian can be determined by these symmetries. Finally, an example from optimal control theory is given.
• The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized $3$-connectivity of the Cartesian product graph $G \square H$ and proposed a conjecture for the case that $H$ is $3$-connected. In this paper, we give two different forms of lower bounds for the generalized $3$-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms their conjecture.
• Recent discovery [Nature 534, 241 (2016)] of FeO$_2$, which can be an important ingredient of the Earth's lower mantle and which in particular may serve as an extra source of oxygen and water at the Earth's surface and atmosphere, opens new perspectives for geophysics and geochemistry, but this is also an extremely interesting material from physical point of view. We found that in contrast to naive expectations Fe is nearly 3+ in this material, which strongly affects its magnetic properties and makes it qualitatively different from well known sulfide analogue - FeS$_2$. Doping, which is most likely to occur in the Earth's mantle, makes FeO$_2$ much more magnetic. In addition we show that unique electronic structure places FeO$_2$ "in between" the usual dioxides and peroxides making this system interesting both for physics and solid state chemistry.
• We establish a stochastic maximum principle for mean-field stochastic differential equations with jumps and corresponding cost functionals, both of whose coefficients involve not only the solution, but also the law of the solution. The control domain does not need to be convex. Our work heavily depends on the derivatives of a stochastic process with respect to a measure as well as the duality method. It is natural that in our case the stochastic maximum principle contains the law of the optimal state process of forward mean-field stochastic system. Compared with the traditional case, both the second-order variational equation and the second-order adjoint equation possess more complex structures, which embody in the presence of the extra mixed second-order derivatives of the coefficients with respect to the measure and the new variable.
• We introduce a physically motivated theoretical approach to investigate the stochastic dynamics of a particle confined in a periodic potential. The particle motion is described through the Il'in Khasminskii model, which is close to the usual Brownian motion and reduces to it in the overdamped limit. Our approach gives access to the transient and the asymptotic dynamics in all damping regimes, which are difficult to investigate in the usual Brownian model. We show that the crossover from the overdamped to the underdamped regime is associated with the loss of a typical time scale and of a typical length scale, as signaled by the divergence of the probability distribution of a certain dynamical event. In the underdamped regime, normal diffusion coexists with a non Gaussian displacement probability distribution for a long transient, as recently observed in a variety of different systems. We rationalize the microscopic physical processes leading to the non-Gaussian behavior, as well as the timescale to recover the Gaussian statistics. The theoretical results are supported by numerical calculations and are compared to those obtained for the Brownian model.
• Underprivileged students, especially in primary school, have shown to have less access to educational materials often resulting in general dissatisfaction in the school system and lower academic performance (Saatcioglu and Rury, 2012, p.23). The relationship between family socioeconomic status and student interest in academic endeavors, level of classroom engagement, and participation in extracurricular programs were analyzed. Socioeconomic status was categorized as below poverty level, at or above poverty level, 100 to 199 percent of poverty, and 200 percent of poverty or higher (United States Census Bureau). Student interest, engagement, and persistence were measured as a scalar quantity of three variables: never, sometimes, and often. The participation of students in extracurricular activities was also compared based on the same categories of socioeconomic status. After running the multivariate analysis of variance, it was found that there was a statistically significant variance of student academic prosperity and poverty level.

### Recent comments

Noon van der Silk May 23 2017 11:15 UTC

I think this thread has reached it's end.

I've locked further comments, and I hope that the quantum computing community can thoughtfully find an approach to language that is inclusive to all and recognises the diverse background of all researchers, current and future.

I direct your attention t

...(continued)
Varun Narasimhachar May 23 2017 02:14 UTC

While I would never want to antagonize my peers or to allow myself to assume they were acting irrationally, I do share your concerns to an extent. I worry about the association of social justice and inclusivity with linguistic engineering, virtual lynching, censorship, etc. (the latter phenomena sta

...(continued)
Aram Harrow May 23 2017 01:30 UTC

I think you are just complaining about issues that arise from living with other people in the same society. If you disagree with their values, well, then some of them might have a negative opinion about you. If you express yourself in an aggressive way, and use words like "lynch" to mean having pe

...(continued)
Steve Flammia May 23 2017 01:04 UTC

I agree with Noon that the discussion is becoming largely off topic for SciRate, but that it might still be of interest to the community to discuss this. I invite people to post thoughtful and respectful comments over at [my earlier Quantum Pontiff post][1]. Further comments here on SciRate will be

...(continued)
Noon van der Silk May 23 2017 00:59 UTC

I've moderated a few comments on this post because I believe it has gone past useful discussion, and I'll continue to remove comments that I believe don't add anything of substantial value.

Thanks.

Aram Harrow May 22 2017 23:13 UTC

The problem with your argument is that no one is forcing anyone to say anything, or banning anything.

If the terms really were offensive or exclusionary or had other bad side effects, then it's reasonable to discuss as a community whether to keep them, and possibly decide to stop using them. Ther

...(continued)
stan May 22 2017 22:53 UTC

Fair enough. At the end of the day I think most of us are concerned with the strength of the result not the particular language used to describe it.

VeteranVandal May 22 2017 22:41 UTC

But how obvious is ancilla? To me it is not even remotely obvious (nor clear as a term, but as the literature used it so much, I see such word in much the same way as I see auxiliary, in fact - now if you want to take offense with auxiliary, what can I say? I won't invent words just to please you).

...(continued)
VeteranVandal May 22 2017 22:21 UTC

I don't think science can or should avoid the perpetuation of existing "historical unequal social order" by changing the language, as to me it seems that, if you try hard enough you can find problem with anything you want to be offended at - rationalizations are tricky things you can often get carri

...(continued)
Fernando Brandao May 22 2017 21:37 UTC

I am not sure if the ArXiv is the best venue for this kind of paper/rant. Also, I’m concerned that so much energy is being put into the discussion. As a non-native speaker, I might not get all nuances of the language, but I have a hard time understanding why we should drop a scientific jargon like “

...(continued)


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# Lorentz covariance

In physics, Lorentz symmetry, named for Hendrik Lorentz, is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".[1] Lorentz covariance, a related concept, is a key property of spacetime following from the special theory of relativity. Lorentz covariance has two distinct, but closely related meanings:

1. A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors. In particular, a scalar (e.g., the space-time interval) remains the same under Lorentz transformations and is said to be a "Lorentz invariant" (i.e., they transform under the trivial representation).
2. An equation is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities (confusingly, some use the term "invariant" here). The key property of such equations is that if they hold in one inertial frame, then they hold in any inertial frame; this follows from the result that if all the components of a tensor vanish in one frame, they vanish in every frame. This condition is a requirement according to the principle of relativity, i.e., all non-gravitational laws must make the same predictions for identical experiments taking place at the same spacetime event in two different inertial frames of reference.

This usage of the term covariant should not be confused with the related concept of a covariant vector. On manifolds, the words covariant and contravariant refer to how objects transform under general coordinate transformations. Confusingly, both covariant and contravariant four-vectors can be Lorentz covariant quantities.

Local Lorentz covariance, which follows from general relativity, refers to Lorentz covariance applying only locally in an infinitesimal region of spacetime at every point. There is a generalization of this concept to cover Poincaré covariance and Poincaré invariance.

## Examples

In general, the nature of a Lorentz tensor can be identified by its tensor order, which is the number of indices it has. No indices implies it is a scalar, one implies that it is a vector, etc. Furthermore, any number of new scalars, vectors etc. can be made by contracting any kinds of tensors together, but many of these may not have any real physical meaning. Some of those tensors that do have a physical interpretation are listed (by no means exhaustively) below.

Please note, the metric sign convention such that η = diag (1, −1, −1, −1) is used throughout the article.

### Scalars

$\Delta s^2=\Delta x^a \Delta x^b \eta_{ab}=c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2$

Proper time (for timelike intervals):

$\Delta \tau = \sqrt{\frac{\Delta s^2}{c^2}},\, \Delta s^2 > 0$
$m_0^2 c^2 = p^a p^b \eta_{ab}= \frac{E^2}{c^2} - p_x^2 - p_y^2 - p_z^2$

Electromagnetism invariants:

$F_{ab} F^{ab} = \ 2 \left( B^2 - \frac{E^2}{c^2} \right)$
$G_{cd}F^{cd}=\frac{1}{2}\epsilon_{abcd}F^{ab} F^{cd} = - \frac{4}{c} \left( \vec B \cdot \vec E \right)$

D'Alembertian/wave operator:

$\Box = \eta^{\mu\nu}\partial_\mu \partial_\nu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2}$

### Four-vectors

4-displacement:

$X^a = \left[ct, x, y, z\right]$

Partial derivative:

$\partial_a = \left[ \frac{1}{c}\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right]$
$U^a = \frac{dX^a}{d\tau} = \gamma \left[c, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}\right]$
$P^a = m_0 U^a = \left[\frac{E}{c}, p_x, p_y, p_z\right]$
$j^a = \left[c\rho, j_x, j_y, j_z\right]$

### Four-tensors

The Kronecker delta:

$\delta^a_b = \begin{cases} 1 & \mbox{if } a = b, \\ 0 & \mbox{if } a \ne b. \end{cases}$

The Minkowski metric (the metric of flat space according to general relativity):

$\eta_{ab} = \eta^{ab} = \begin{cases} 1 & \mbox{if } a = b = 0, \\ -1 & \mbox{if }a = b = 1, 2, 3, \\ 0 & \mbox{if } a \ne b. \end{cases}$
$\epsilon_{abcd} = -\epsilon^{abcd} = \begin{cases} +1 & \mbox{if } \{abcd\} \mbox{ is an even permutation of } \{0123\}, \\ -1 & \mbox{if } \{abcd\} \mbox{ is an odd permutation of } \{0123\}, \\ 0 & \mbox{otherwise.} \end{cases}$

Electromagnetic field tensor (using a metric signature of + − − − ):

$F_{ab} = \begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end{bmatrix}$

Dual electromagnetic field tensor:

$G_{cd} = \frac{1}{2}\epsilon_{abcd}F^{ab} = \begin{bmatrix} 0 & B_x & B_y & B_z \\ -B_x & 0 & E_z/c & -E_y/c \\ -B_y & -E_z/c & 0 & E_x/c \\ -B_z & E_y/c & -E_x/c & 0 \end{bmatrix}$

## Lorentz violating models

In standard field theory, there are very strict and severe constraints on marginal and relevant Lorentz violating operators within both QED and the Standard Model. Irrelevant Lorentz violating operators may be suppressed by a high cutoff scale, but they typically induce marginal and relevant Lorentz violating operators via radiative corrections. So, we also have very strict and severe constraints on irrelevant Lorentz violating operators.

Since some approaches to quantum gravity lead to violations of Lorentz invariance,[2] these studies are part of Phenomenological Quantum Gravity.

Lorentz violating models typically fall into four classes[citation needed]:

• The laws of physics are exactly Lorentz covariant but this symmetry is spontaneously broken. In special relativistic theories, this leads to phonons, which are the Goldstone bosons. The phonons travel at less than the speed of light.
• Similar to the approximate Lorentz symmetry of phonons in a lattice (where the speed of sound plays the role of the critical speed), the Lorentz symmetry of special relativity (with the speed of light as the critical speed in vacuum) is only a low-energy limit of the laws of Physics, which involve new phenomena at some fundamental scale. Bare conventional "elementary" particles are not point-like field-theoretical objects at very small distance scales, and a nonzero fundamental length must be taken into account. Lorentz symmetry violation is governed by an energy-dependent parameter which tends to zero as momentum decreases. Such patterns require the existence of a privileged local inertial frame (the "vacuum rest frame"). They can be tested, at least partially, by ultra-high energy cosmic ray experiments like the Pierre Auger Observatory.
• The laws of physics are symmetric under a deformation of the Lorentz or more generally, the Poincaré group, and this deformed symmetry is exact and unbroken. This deformed symmetry is also typically a quantum group symmetry, which is a generalization of a group symmetry. Deformed special relativity is an example of this class of models. It is not accurate to call such models Lorentz-violating as much as Lorentz deformed any more than special relativity can be called a violation of Galilean symmetry rather than a deformation of it. The deformation is scale dependent, meaning that at length scales much larger than the Planck scale, the symmetry looks pretty much like the Poincaré group. Ultra-high energy cosmic ray experiments cannot test such models.
• This is a class of its own; a subgroup of the Lorentz group is sufficient to give us all the standard predictions if CP is an exact symmetry. However, CP isn't exact. This is called Very Special Relativity.

Models belonging to the first two classes can be consistent with experiment if Lorentz breaking happens at Planck scale or beyond it, and if Lorentz symmetry violation is governed by a suitable energy-dependent parameter. One then has a class of models which deviate from Poincaré symmetry near the Planck scale but still flows towards an exact Poincaré group at very large length scales. This is also true for the third class, which is furthermore protected from radiative corrections as one still has an exact (quantum) symmetry.

Even though there is no evidence of the violation of Lorentz invariance, several experimental searches for such violations have been performed during recent years. A detailed summary of the results of these searches is given in the Data Tables for Lorentz and CPT Violation.[3]


text:
### Home > GB8I > Chapter 5 Unit 6 > Lesson INT1: 5.3.3 > Problem5-125

5-125.

Figure $2$ of a tile pattern is shown at right.  If the pattern grows linearly and if Figure $5$ has $15$ tiles, then find a rule for the pattern.

Make a table relating the figure number to the number of tiles, or write a proportion.

$y=3x$


text:
# Searching for generalized binary number systems

• Published in 2017
In the collection
The aim of the project is to find all the generalized binary number systems up to dimension 11. Below we give a short description of the number system concept and mention a few possible applications.

### BibTeX entry

@online{Searchingforgeneralizedbinarynumbersystems,
title = {Searching for generalized binary number systems},
author = {Attila Kov{\'{a}}cs},
url = {http://szdg.lpds.sztaki.hu/szdg/desc{\_}numsys{\_}es.php},
urldate = {2016-08-22},
abstract = {The aim of the project is to find all the generalized binary number systems up to dimension 11. Below we give a short description of the number system concept and mention a few possible applications.},
comment = {},
collections = {Unusual arithmetic},
year = 2017
}


text:
# The Schrodinger equation is not a reaction-diffusion equation

Recently, a stackexchange answer claimed that the Schrodinger equation is effectively a reaction-diffusion equation. I’ll set aside semantic arguments about the meaning of “effectively”, and give a more obvious example to explain why I think this statement is misleading.

Consider the wave equation

$u_{tt} = u_{xx}$

Introducing a new variable $$v=u_t$$ we can rewrite the wave equation as

\begin{align*} v_t & = u_{xx} \\ u_t & = v. \end{align*}

Observe that the first of these equation is the diffusion equation, while the second is a reaction equation. Thus we have reaction-diffusion!

Right?

Wrong. We’ve disguised the true nature of this equation by applying our intuition (which is based on scalar PDEs) to a system of PDEs. In the same way, the “reaction-diffusion” label for Schrodinger is obtained by applying intuition based on PDEs with real coefficients to a PDE with complex coefficients.

Of course, in both cases you can use numerical methods that are appropriate for reaction-diffusion problems in order to solve a wave equation.
Here is a quick ipython notebook implementation of the obvious method for the system above.


text:
##### Level 1Level 2Level 3Equivalent FractionsFraction PairsDescriptionHelpMore Fractions

For each rectangle click on two green dots to draw a line between them. The line must dissect the rectangle to produce the fraction indicated below the rectangle. Your solution to questions 1 and 2 must be different.

 $$\frac{1}{2}$$ $$\frac{1}{2}$$ $$\frac{1}{4}$$ $$\frac{1}{3}$$ $$\frac{1}{8}$$ $$\frac{1}{6}$$ $$\frac{2}{3}$$ $$\frac{3}{4}$$ $$\frac{1}{12}$$ $$\frac{7}{8}$$ $$\frac{1}{24}$$ $$\frac{5}{6}$$ $$\frac{5}{8}$$ $$\frac{11}{12}$$ $$\frac{23}{24}$$ $$\frac{3}{8}$$
Check

This is Fraction Dissect level 1. You can also try:
Level 2 Level 3 Equivalent Fractions Fraction Pairs

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Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

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Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths main page links to more activities designed for students in upper Secondary/High school.

## Teachers

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Raül Fernández,

Friday, October 19, 2018

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## Description of Levels

Close

Level 1 - Draw lines to dissect the rectangles to make the given fractions.

Level 2 - Recognise the harder fractions from the disected rectangle diagrams.

Level 3 - Recognise or calculate fractions of circles.

Equivalent Fractions Practise finding equivalent fractions numerically and in fraction diagrams.

Fraction Pairs The traditional pairs or Pelmanism game adapted to test knowledge of equivalent fractions.

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text:
# how do you make a copy of a linked list in c

how do you make a copy of a linked list in c
struct node *copyList(struct node *sourceList)
{
if(sourceList==NULL) return;
struct node *targetList=(struct node *) malloc(sizeof(struct node));
targetList->info=sourceList->info;


text:
# Random Matrices and a Random Equation

Level pending

$$Let\quad the\quad solutions\quad to\quad this\quad equation\quad be\quad \alpha \quad and\quad \beta .\\ f\left( x \right) =\sqrt { x } -3\sqrt [ 4 ]{ x } -10\\ \\ More\quad precisely,\quad \alpha >\beta \\ \\ \begin{pmatrix} \psi & 2 \\ 5 & 1 \end{pmatrix}\begin{pmatrix} 6 & 2 \\ 5 & 4 \end{pmatrix}=\begin{pmatrix} \beta & \sqrt { \alpha } -15 \\ 35 & 14 \end{pmatrix}\\ \\ Find\quad { \alpha }^{ 2 }+{ \beta }^{ 2 }+\psi$$

×


text:
# Triangle centers as power sum minimizing points: Which is the locus of all these points?

Some of the classical triangle centers can be expressed as solutions to minimization problems: Given a triangle $A_1, A_2, A_3$ define $d_i, i=1,2,3$ to be the distance of a given point $P$ to $A_i$, and $f_q$ as the sum of the $q$-th power of these distances:$f_q = \sum_{i=1}^3 d_i^q$. I'm looking for the point $P$ which minimizes $f_q$. For $q=1$ this is the Steiner point, for $q=2$ the centroid, for $q \to \infty$ the circumcenter, for $q \to 0$ the point where the product of distances is minimized. An obvious question is to find the curve of all these points for reasonably general $q$ (e.g. $q \in \mathbb{R}_{>0}$). However, in the ressources for triangle centers, as e.g. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html this problem seems not to be considered.

EDIT:

1) I would like to restrict the problem to $q \ge 1$ since then $f_q$ is convex and a unique minimum is guaranteed.

2) I would like to add a generalization of the question: Consider all continuous functions $f(d_1,d_2,d_3)$ that a) are invariant under permutations of $d_1, d_2, d_3$ and b) have a unique minimum. What can be said about the locus of all these minima?

-
Don't you lose convexity when $q$ is small? It seems like you would get local minima near each corner in the zero limit. –  S. Carnahan May 30 '10 at 1:41
When $q \rightarrow \infty$, $P$ does not (generally) approach the circumcenter. Consider an obtuse isosceles triangle. If we let the obtuse angle approach $\pi$, the circumcircle radius grows to infinity, so the center grows arbitrarily far from the vertices of the triangle. Such a point clearly cannot minimize $f_q$. In this case, P approaches the midpoint of the side opposite the obtuse angle. –  Matthew Conroy Jun 29 '10 at 17:28

Your curve might end up looking quite strange. Consider what happens if $A_1 \rightarrow \infty$ along a ray connecting it to the origin. the Steiner point stays fixed, and the centroid goes to infinity. While I'm not sure, it seems like for any $q > 1$, the point will go to infinity.


text:
# Seminar of Ludovic Sacchelli, University of Toulon

Here is the title and summary of Ludovic Sacchelli’s seminar:

Title: Sub-Riemannian Extremals for Contact Distributions

Abstract: The contact manifolds are odd-dimensional state spaces whose distribution consisting of hyperplanes is generating by Lie hooks at any point. In these spaces, we are interested in the optimal control of systems with linear dynamics in control and non-linear in state, with penalization of the norm L2 of the control, measured according to a so-called sub-Riemannian metric.
Following the work of Agrachev, Gauthier and their collaborators having carried out the in-depth study of the case of dimension 3 since the mid-90s, we seek to characterize the family of small extremals emanating from a point. In particular, by perturbative methods, we are interested in studying the geometry of the singular place, the set of points of loss of local optimality of the extremals. ”

Many seminars are scheduled. All information is centralized on the website: