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# What is a Fraction, Really?

#### (An archive question of the week)

One of the things I have learned as a Math Doctor is that it can be dangerous looking up a definition online. Sources vary — not because they are wrong, but because definitions depend on context, so you can easily find what appear to be contradictions because they refer to different usages. Today’s question not only reflects a variety of definitions, but evoked different answers from two of us!

## Looking for the “correct” definition

The question, from 2001, referred to three sources, none of which still exists, but the ideas are typical:

What Exactly is a Fraction?

What is a fraction? Is 3/1 a fraction? Is 5/sqrt(2) a fraction?

I tried to check some internet glossaries, and there does not seem to be unanimity. The TUSD Math Glossary says:
"Fraction: a numeral representing some part of a whole; a numeral of the form a/b (meaning a divided by b) where b is not zero."
http://instech.tusd.k12.az.us/Core/glossary/mathglossary.htm

The _Math to Build On_ glossary says:
"Fraction - A number that expresses a portion of a whole. The denominator of a fraction represents the number of portions the whole has been divided into, and the numerator expresses the number of the portions measured. The fraction 1/4 could be stated as 1 out of 4 parts of the whole."
http://mathforum.org/~sarah/hamilton/ham.glossary.html

The MathPro Press On-line Mathematics Dictionary says:
"Fraction: An expression of the form a/b."
http://pax.st.usm.edu/cmi/inform_html/glossary.html#F

(This reminds me of the definition of a trapezium. Some say that a trapezium is a quadrilateral with one pair of sides parallel. Some say that plus, the other pair of opposite sides should not be parallel.)

Personally, I am comfortable with having identical definitions of fractions and rational numbers. But I have seen many sources defining fractions in such a way that -3/5 is not a fraction, only a rational number...

Sridhar’s examples include 3/1 (an improper fraction that simplifies to an integer), 5/√2 (a fraction including an irrational quantity), and -3/5 (a negative proper fraction); the quoted definitions don’t mention such issues. On reading these definitions now, my own first concern is the emphasis on “parts of a whole”, which has some unintended implications, restricting one’s thinking to a particular model of a fraction (which is appropriate for early grades, but needs to be left behind eventually). But the main issue is that, like many definitions, they are incomplete. That may be intentional.

## A better definition

Doctor Rick answered first, focusing on refining the definitions to include some essentials that they all omit:

You've asked a good question, but since there is no official international governing body for mathematical definitions (as far as I know), you'll only get another viewpoint from me, not a definitive answer.

I am not particularly comfortable with having identical definitions of fraction and rational number, for the simple reason that then we would have no reason to keep both words in our lexicon.

I am not entirely pleased with any of the definitions you list, because they all omit an essential requirement. I would go with this version:

A fraction is a representation for a rational
number, in the form a/b, where a is an integer,
and b is a *positive* integer. The number so
represented is the result of dividing a by b.

Do you see what was missing in your definitions? This is NOT a fraction:

1.2
---
4.8

This definition removes the model of “parts of a whole”, replacing it with an abstract model; and adds a restriction that the numerator and denominator are integers. At lower levels, the latter is probably just assumed.

He continues:

Your first definition comes closest to mine. I like the word "numeral" in that definition, as distinct from "number." The same *number* can be represented by the *numerals* 3/2 or 1.5; only the first is a fraction. Likewise, 3 can be represented by the numeral (fraction) 3/1, but the number itself is not therefore a fraction.

I have no problem with calling -2/5 or 3/2 a fraction; both are improper fractions, representing numbers outside the range 0 to 1. This makes me somewhat uncomfortable with the phrase "representing some part of a whole," which strictly limits the definition to proper fractions. This description is helpful in introducing the concept of fractions, but once improper fractions are introduced, it no longer belongs in the definition.

I see no need to have the definition so broad that 2/-5 is a fraction. As it stands, every rational number can be represented as a fraction. (In fact, *only* rational numbers can be represented as a fraction, so this makes for a simple definition of rational numbers.)

The definitions are probably intended to be used before learning about negative numbers (or in contexts where nothing is negative), so it makes some sense not to mention that possibility; but they really should not exclude improper fractions like 3/2.

I see no need, on the other hand, to restrict the definition so that every rational number has a *unique* representation as a fraction. Thus, 4/6 and 2/3 represent the same rational number. There is, however, a unique representation as a fraction *in lowest terms*.

These are my opinions. I will leave the question for other Math Doctors to respond to if they have different or additional viewpoints.

## The root meaning: proper fractions

I had been pondering the question myself, and an hour later submitted my answer. I focused on variation in usage:

I would like to take a slightly different perspective from Dr. Rick's on this. I fully agree with him on the most precise definition of "fraction": it is a particular representation of a rational number. But it is worth noting that, like many other words, it actually has several different meanings depending on the context.

The root meaning of "fraction" is "broken piece," which is the source of the idea that it must be a part of a whole. This is common in informal use; if I say "only a fraction of the people here understand what the word means," I mean "less than the whole," and probably much less. I am saying nothing about whether those people are "rational." ;-) In mathematical terms, this concept arises in the phrase "proper fraction" - that is, a mathematical fraction that fits the informal sense that it should be less than one.

So, one meaning in everyday use is “less than a whole”; and this root is reflected in our calling a fraction that is greater than one “improper” – it doesn’t fit the root idea of what a fraction is, but is an extension.

## The usual meaning: common fractions

Once we get into the mathematical realm, a fraction always refers to a way of writing a number, using numerator and denominator. Most narrowly, these must be whole numbers (or integers, once children are introduced to negative numbers). This kind of fraction is more fully called a "common" or "vulgar" fraction, and this is what Dr. Rick defined for you. When we use the word "fraction" alone, we usually mean this.

So the everyday mathematical use is just what Doctor Rick discussed: an integer over an integer. Again, there is a special term reflecting this particular usage; beyond “proper” fractions, we have the larger category of “common” fractions.

As for “vulgar fraction”, that derives from an early use of the word “vulgar”, referring as it did in Latin to the “ordinary” or “common” people, as opposed to the nobility. In common use, the word now means “unrefined” or worse; but occasionally it still has this less judgmental meaning.

## Generalizing: algebraic fractions

Now we move beyond arithmetic into algebra, where “fraction” takes on a larger meaning (which is hinted at especially in the third definition above):

From here we find a broader definition, given in my American Heritage dictionary as "an indicated quotient of two quantities." This retains the concept that a fraction is a way of writing something, as distinct from its actual value, but ignores questions as to what sort of quantities are being divided. This allows for "fractions" like 1.2/3.4 or (x+1)/(y-1). The term is in fact used in these senses; the latter might be called an algebraic fraction. In this realm we can distinguish between "simple fractions" and "complex or compound fractions" like (1/2)/(3/4); common fractions are always simple.

Here we move beyond parts and wholes, and recognize fractions as representations of division, as I have discussed elsewhere. We can now divide any sort of quantities, not just integers. The term “indicated division” reflects what I have said elsewhere, “a fraction is simply a division we haven’t bothered to perform yet, a division problem frozen in time.”

Finally, we see the phrase "decimal fraction" used for non-integral decimal numbers; this drops the "indicated quotient" concept and retains only the "broken" (non-integral) aspect of the most basic definition. I don't think this meaning is ever intended when we use the word "fraction" without qualification; in fact, we more often drop the word "fraction" and just call it a "decimal," which can be a dangerous practice!

It turns out that I lied earlier, when I said, “a fraction always refers to a way of writing a number, using numerator and denominator”. This usage is generally restricted to mathematicians who want to be precise – they can’t call “.123” a “decimal”, as we usually do, because that literally means “base ten”, and says nothing about being less than one! (Properly speaking, a “decimal fraction” should be one whose denominator is a power of ten, but still written as a fraction: “123/1000”; but we have no better term.) This conundrum becomes most confusing when deciding what to call a “decimal” form in a non-decimal base; is it, say, a “base-two fraction”? It certainly can’t be called a “binary decimal”!

Clearly the term "fraction" has a somewhat different meaning in each case.

To answer your specific questions, 3/1 is a fraction (specifically a simple, common, but improper fraction), while 5/sqrt(2) would only be called a fraction in an algebraic context. On the other hand, 3 is not a fraction, even though it is a rational number.

Again, “fraction” refers to the way a number is written, not to the number itself and its properties. In fact, this is why we need the term “rational number”, so we can talk about any number that can be written as a fraction, without caring how it happens to be written at the moment.

## Extras

For more about the term “proper fraction”, see

Origin of the Term "Improper" Fraction

For the difference between fractions and rational numbers, see

Introduction to Rational Numbers for Middle School Students

Why Do We Need to Study Rational Numbers?

For the difference between numbers and numerals, see

Numbers and Numerals

### 1 thought on “What is a Fraction, Really?”

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## Related rates of change calculus

Related rates are handy when we have two related quantities and one of their rates of change is much harder to find than the other one. Try it on our examples. Calculus is primarily the mathematical study of how things change. One specific problem type is determining how the rates of two related items change at the same time. The keys to solving a related rates problem are identifying the variables that are changing and then determining a formula that connects those variables to each other. Related Rates of Change Some problems in calculus require finding the rate of change or two or more variables that are related to a common variable, namely time. To solve these types of problems, the appropriate rate of change is determined by implicit differentiation with respect to time.

In short, Related Rates problems combine word problems together with EXAMPLE 1: Consider a right triangle which is changing shape in the following way. Click HERE to return to the original list of various types of calculus problems. How fast is the circumference of the circle changing? Solution. EOS. The given rate of change dr/  Rate at Which the Circumference of a Circle is Changing, Finding Rate of Change of Area Under Ladder, A series of free Calculus Videos. 10 Dec 2011 Interpreting the time derivative of a quantity as a rate of change. The main reason that related rates problems feel so contrived is that calculus  24 Jul 2013 In differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose  Working related rates (also called rate of change) problems involves two main steps; translating the word problem into an equation (or set of equations), then

## 20 Jun 2007 This application is one of a collection of examples teaching Calculus with Solve the resulting equation for the rate of change of the radius,

3 Jan 2020 Home · Bookshelves · Calculus · Book: Calculus (OpenStax) · 3: Derivatives Apply rates of change to displacement, velocity, and acceleration of an The average rate of change of the function f over that same interval is x, can be related to the price charged, p, by the equation p(x)=9−0.03x,0≤x≤300. changing as implicit functions of time, how are their rates of change related? volume and radius are related, by knowing how fast the volume is changing, we  Related rates of change problems (hereafter, related rates problems) form an integral part of any first-year calculus course in  The derivative, dv/dt would be the rate of change of v. When solving related rates problems, we should follow the steps listed below. 1) Draw a diagram. This is the

### Related Rates of Change. Watch the animation closely. Water is being added to the conical cup at a constant rate. What do you notice about the rate at which the

How fast is the circumference of the circle changing? Solution. EOS. The given rate of change dr/  Rate at Which the Circumference of a Circle is Changing, Finding Rate of Change of Area Under Ladder, A series of free Calculus Videos. 10 Dec 2011 Interpreting the time derivative of a quantity as a rate of change. The main reason that related rates problems feel so contrived is that calculus

### Differentiation is a method to calculate the rate of change (or the slope at a point on the graph); we will not

A "related rates" problem is a problem which involves at least two changing quantities and asks you to figure out the rate at which one is changing given

## Rate of change calculus problems and their detailed solutions are presented. Problem 1 A rectangular water tank (see figure below) is being filled at the constant rate of 20 liters / second.

In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of

Rate of change calculus problems and their detailed solutions are presented. Problem 1 A rectangular water tank (see figure below) is being filled at the constant rate of 20 liters / second. Section 4-1 : Rates of Change The purpose of this section is to remind us of one of the more important applications of derivatives. That is the fact that $$f'\left( x \right)$$ represents the rate of change of $$f\left( x \right)$$. But sometimes there really are several variables that change with time; as long as you know the rates of change of all but one of them you can find the rate of change of the remaining one. As in the case when there are just two variables, take the derivative of both sides of the equation relating all of the variables, and then substitute all of the known values and solve for the unknown rate. To solve problems with Related Rates, we will need to know how to differentiate implicitly, as most problems will be formulas of one or more variables.. But this time we are going to take the derivative with respect to time, t, so this means we will multiply by a differential for the derivative of every variable! In each case you’re given the rate at which one quantity is changing. That is, you’re given the value of the derivative with respect to time of that quantity: “The radius . . . increases at 1 millimeter each second” means the radius changes at the rate of $\dfrac{dr}{dt} = 1$ mm/s. Related Rates of Change Some problems in calculus require finding the rate of change or two or more variables that are related to a common variable, namely time. To solve these types of problems, the appropriate rate of change is determined by implicit differentiation with respect to time.

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## Store Vector elements in rows as txt....

Asked by:

I have a vector (matrix) that looks like this. I want to save it to txt text. But the Export function doesn't seem to do that.

M:=Vector[column](10, [{{a, b}, {b, c}}, {{d, e}, {e, f}},
{{d, f}, {e, f}}, {{a, h}, {h, i}}, {{a, i}, {h, i}},
{{d, e}, {d, f}}, {{a, h}, {a, i}},{{c, d}}, {{a, b}, {a, c}}, {{a, c}, {b, c}}]);
Export(M,"E:\\M.txt");

Error, (in Export) invalid input: Export expects its 1st argument, input, to be of type string when target=file, but received Vector(10, [{{a, b}, {b, c}},{{d, e}, {e, f}},{{d, f}, {e, f}},{{a, h}, {h, i}},{{a, i}, {h, i}},{{d, e}, {d, f}},{{a, h}, {a, i}},{{c, d}},{{a, b}, {a, c}},{{a, c}, {b, c}}])

I found that I can save it in xsl format.

with(ExcelTools): Export(M, "E:\\M.xls");

How to save M in txt format below?

## Problem with Solving a PDE...

Asked by:

Hi Everyone,

I intend to find the invariants of a 2nd-order Tensor using the representation of the rotational group as an Infinitesimal generator. However, maple does not seem to find the correct invariants $\tr \mathbf{a}$, $\tr \mathbf{a}^2$, $\tr \mathbf{a}^3$. Maple only returns an empty set. I can't seem to figure out why that is the case.

Here is my code

 >
 >
 >
 >
 >
 (1)
 >
 (2)
 >
 (3)
 >
 >
 (4)

Download SO(3)_spherical_2.mw

Any input would be appreciated!

## Flexible beam does not give expected results....

Asked by:

Hello everyone,

I am first time using the Flexble Beam block, and I tried simple cantilever beam with mass at the free end. Deflection is not as I expected (comparing to my hand calculations). I have tried also to use Force instead of mass (with a same load quantity), but nothing changed. Is there something that I might be missing? Expected deflection of a free end is 15 mm (I have obtained 2,8 mm in Maplesim).

Thanks!

## Maple for individuals...

Asked by:

Hello there,

Is the price of 295 USD for unlimited use in time?

Kind regards,

Eduardo

## Book with advanced content....

Asked by:

Hello guys,
Can anyone recommend good books on advanced programming and other Maple topics? The books I have only have the basics.
Regards,
Oliveira

## Maple Output: Is it possible to format the d in “d...

Asked by:

Maple standard Output

 (1)

Desired output with an italic d (here formated manualy with the  in the menu bar)

 (2)

which reverts to roman layout either after re-execution of the worksheet or re-evaluation.

 (3)

Background:
I would like to achieve textbook style formatting of integrals in Maple output. I am aware that interpreting a dx in an integral as a "differential" is controversial. The way Maple displays the integration variable in italic and the d in roman makes sense and avoids confusion with interpretations that are no longer needed.
However, what I want to avoid is mixing
dx and dx in the same document.

Can Maple be set by default that the d in the dx of an integral becomes italic?

If this is not possible and because this question comes from habits in physics: Would a display option in the physics package be possible?

Download Differential_format_in_integrals.mw

## How to fix "Error, || unexpected"?...

Asked by:

Execute the following codes in Maple input (1-D math) in the Standard interface.

(cat("A".."C"),cat("d".."f"))||'$"G".."I",$"j".."l"';

Then an error occurred. But if one copy them into 2-D math (instead of Convert To>2-D Math Input) and execute them directly, everything goes without any error messages.

It says that mixed 1-D and 2-D math inside one command is not supported and not recommended stylistically. However, I just want to understand the reason why an error is raised here.

 > restart;
 > kernelopts(version);
 (1)
 > interface(version);
 (2)
 > (cat("A".."C"),cat("d".."f"))||'$"G".."I",$"j".."l"';
 >
 (3)
 >

Download unexpectedConcatenation.mws

## Differential system of equations taking to long to...

Asked by:

Hello everyone,

I am having trouble with this a system of 12 differential equations with the respective 12 initial conditions. I try to usen dsolve with the "numeric" parameter but it stays "evaluating" forever. I tried a system of 10 equation and I got the correct answer, although the program stayed calculating for several minutes (more or less 5 minutes). Is the size of this system too much to handle for the program?

This is the system of differential equations (sys) and the initial conditions (ics) that I am trying to solve:

maple_primes_question.mw

Download maple_primes_question.mw

What am I doing wrong? or is it that the program cannot handle such a large system? In that case, that would be dissappointing since I have to solve a similar system but with up to 22 differential equations and initial conditions so I am stuck...

Anyway, thank you very much for any guidance, and happy new year.

## Problem with power...

Asked by:

A:= <x1,y1,z1>:  B:= <x2,y2,z2>:  P :=<x3, y3, z3>:  H:=A+t*(B-A):  solve( (P-H)^+ . (B-A), t ):  'H'=eval(H, t=% );

Error, invalid powe, rhow to overcome this mistake ? Thank you.
Typesetting:-mambiguous( AAssign (x1commay1commaz1)colon  BAssign

(x2commay2commaz2)colon  P Assign(x3comma y3comma z3)colon  H

AssignA + tast(Buminus0A)colon  solveApplyFunction(

Typesetting:-mambiguous((Puminus0H)circ + ,

Typesetting:-merror("invalid power")) period (Buminus0A)comma t

)colon  (H)equalsevalApplyFunction(Hcomma tequals% )semi)

## How to defeat quotations during file import...

Asked by:

Simple question I cannot remember the solution of.

When I import "executable maple input" from a file, the content is imported as a string and therefore have quotation marks around the code if saved in a variable. It therefore wont execute.

How do you defeat quotation marks around  executable maple code with a command in order to execute the code ?

## Can we have an unevaluated matrix product in this ...

Asked by:

hello dear maple, in the following code (code attached at the bottom )

I'm curious to know if it's possible to show the product of two matrices  in an unevaluated form, so for instance something like this instead of this

UnEval_Mat.mw

Thanks in advance

## "select" command doesn't work as expected in Physi...

Asked by:

hello maple world, in the following code (code attached at the bottom) i wish to select the terms which have the term so i used the "select" command for this task

but as u can see from the results it also selects other terms which is wrong. so what is the mistake here

thanks in advance

Physic_select.mw

## Not able to solve a differential equation system n...

Asked by:

Hi everyone,

I have been trying to solve a system of 8 differential equations (1st order) with 8 different initial conditions. The system in question is the following.

sys:={

diff(a[0](t), t) - diff(a[1](t), t) + diff(a[2](t), t) - diff(a[3](t), t) = 0,

diff(b[0](t), t) + diff(b[1](t), t) + diff(b[2](t), t) + diff(b[3](t), t) = 0,

diff(a[0](t), t) + diff(a[1](t), t) + diff(a[2](t), t) + diff(a[3](t), t) - diff(b[0](t), t) + diff(b[1](t), t) - diff(b[2](t), t) + diff(b[3](t), t) = 0,

diff(a[1](t), t) + 4*diff(a[2](t), t) + 9*diff(a[3](t), t) - 3*diff(b[1](t), t) + 12*diff(b[2](t), t) - 27*diff(b[3](t), t) = 0,

diff(a[0](t), t) - diff(a[1](t), t)/2 - diff(a[2](t), t)/2 - 16*a[2](t) + diff(a[3](t), t) + 48*a[3](t) = 0,

diff(a[0](t), t) + diff(a[1](t), t)/2 - diff(a[2](t), t)/2 - 16*a[2](t) - diff(a[3](t), t) - 48*a[3](t) = 0,

diff(b[0](t), t) - diff(b[1](t), t)/2 - diff(b[2](t), t)/2 - 48*b[2](t) + diff(b[3](t), t) + 144*b[3](t) = 0,

diff(b[0](t), t) - diff(b[1](t), t)/2 - diff(b[2](t), t)/2 - 48*b[2](t) + diff(b[3](t), t) + 144*b[3](t) = 0}

and the initial conditions are

ics:={a[0](0) = 1/8, a[1](0) = 1/4, a[2](0) = 1/4, a[3](0) = 1/8, b[0](0) = 23/24, b[1](0) = 1/12, b[2](0) = -1/12, b[3](0) = 1/24}

When I run

sols := dsolve(sys union ics, numeric)

I get the error message

Error, (in DEtools/convertsys) ODE system is insufficient to determine values for all dependent variables in the system

What is happening?

Thanks for the help.

## "use" command does not substitute Physics packag...

Asked by:

Hey guys. i recently faced a problem where i needed to substitute the "physics's overloaded product operator" , namely with normal matrice product (.), so maple can do the matrice product calculation after my non-commutative physics operator(yellow symbols in the following code) has been replaced with real matrices (for instance, this substitution:)

now in order to substitute  with . I have utilized the maple use command. but the problem is, it doesn't do the substitution when I utilized use command as follows (code file is attached at the bottom) :

so what's the mistake here?

thanks in advance

Physic_subs.mw

## Is it possible to specify constants (initial posit...

Asked by:

Let's say we have a velocity vector v := c _i + d _j, where _i and _j are the i hat and j hat Cartesian basis vectors as present in the Physics[Vectors] package.

When we integrate this velocity vector relative to t, the result in maple is the position vector (c _i + d _j)t.

I guess that is correct.

My question is: is there am in-built way to specify an initial position vector for the integration? Ie, can I specify constants in the integration?

Right now I am just doing

s0 := sx _i + sy _j

s:= int(v, t) + s0

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text:
# sum $o_p(1)$ $n$ times [closed]

I wonder that what $$o_p(1)$$ $$n$$ times would result in, where $$n$$ is the sample size.

For that matter, I am also curious if $$1/n \sum_{i=1}^n o_p(1) = o_p(1)~\text{or}~o_p(1/n).$$

• Please clarify what your notation refers to – David Jun 14 '19 at 7:04
• There are many possible interpretations of this question: please explain the context and the meaning of your notation. – whuber Jun 14 '19 at 14:14


text:
Quadratic unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP), is a combinatorial optimization problem with a wide range of applications from finance and economics to machine learning.[1] QUBO is an NP hard problem, and for many classical problems from theoretical computer science, like maximum cut, graph coloring and the partition problem, embeddings into QUBO have been formulated.[2][3] Embeddings for machine learning models include support-vector machines, clustering and probabilistic graphical models.[4] Moreover, due to its close connection to Ising models, QUBO constitutes a central problem class for adiabatic quantum computation, where it is solved through a physical process called quantum annealing.[5]

## Definition

Let ${\displaystyle f_{Q}:\mathbb {B} ^{n}\rightarrow \mathbb {R} }$ be a quadratic polynomial over binary variables,

${\displaystyle f_{Q}(x)=\sum _{i=1}^{n}\sum _{j=1}^{i}q_{ij}x_{i}x_{j}}$

with ${\displaystyle x_{i}\in \mathbb {B} }$ for ${\displaystyle i\in [n]}$ and coefficients ${\displaystyle q_{ij}\in \mathbb {R} }$ for ${\displaystyle 1\leq j\leq i\leq n}$. Here ${\displaystyle [n]}$ denotes the set of strictly positive integers less or equal to ${\displaystyle n}$, and ${\displaystyle \mathbb {B} =\lbrace 0,1\rbrace }$. The QUBO problem consists of finding a binary vector ${\displaystyle x^{*}}$ that is minimal with respect to ${\displaystyle f_{Q}}$ among all other binary vectors, namely

${\displaystyle x^{*}={\underset {x\in \mathbb {B} ^{n}}{\arg \min }}~f_{Q}(x)}$

Sometimes, QUBO is defined as a maximization instead of a minimization problem, which has no effect on the problem's complexity class, as maximizing ${\displaystyle f_{Q}}$ is the same as minimizing ${\displaystyle f_{-Q}=-f_{Q}}$ (see below). Another, more compact way to formulate ${\displaystyle f_{Q}}$ is using matrix notation,

${\displaystyle f_{Q}(x)=x^{\top }Qx}$

where ${\displaystyle Q\in \mathbb {R} ^{n\times n}}$ is the symmetric ${\displaystyle n\times n}$ matrix containing the coefficients ${\displaystyle q_{ij}}$.

## Properties

• Multiplying the coefficients ${\displaystyle q_{ij}}$ with a positive factor ${\displaystyle \alpha >0}$ scales the output of ${\displaystyle f}$ accordingly, leaving the optimum ${\displaystyle x^{*}}$ unchanged:
${\displaystyle f_{\alpha Q}(x)=\sum _{i\leq j}(\alpha q_{ij})x_{i}x_{j}=\alpha \sum _{i\leq j}q_{ij}x_{i}x_{j}=\alpha f_{Q}(x)}$
• Flipping the sign of all coefficients flips the sign of ${\displaystyle f}$'s output, making ${\displaystyle x^{*}}$ the binary vector that maximizes ${\displaystyle f_{-Q}}$:
${\displaystyle f_{-Q}(x)=\sum _{i\leq j}(-q_{ij})x_{i}x_{j}=-\sum _{i\leq j}q_{ij}x_{i}x_{j}=-f_{Q}(x)}$
• If all coefficients are positive, the optimum is trivially ${\displaystyle x^{*}=(0,\dots ,0)}$. Similarly, if all coefficients are negative, the optimum is ${\displaystyle x^{*}=(1,\dots ,1)}$.
• If ${\displaystyle \forall i\neq j:~q_{ij}=0}$, then the corresponding QUBO problem is solvable in ${\displaystyle {\mathcal {O}}(n)}$, the optimal variable assignments ${\displaystyle x_{i}^{*}}$ simply being 1 if ${\displaystyle q_{ii}<0}$ and 0 otherwise.

## Connection to Ising models

QUBO is very closely related and computationally equivalent to the Ising model, whose Hamiltonian function is defined as

${\displaystyle H(\sigma )=-\sum _{\langle i~j\rangle }J_{ij}\sigma _{i}\sigma _{j}-\mu \sum _{j}h_{j}\sigma _{j}}$

with real-valued parameters ${\displaystyle h_{j},J_{ij},\mu }$ for all ${\displaystyle i,j}$. The spin variables ${\displaystyle \sigma _{j}}$ are binary with values from ${\displaystyle \lbrace -1,+1\rbrace }$ instead of ${\displaystyle \mathbb {B} }$. Moreover, in the Ising model the variables are typically arranged in a lattice where only neighboring pairs of variables ${\displaystyle \langle i~j\rangle }$ can have non-zero coefficients. Applying the identity ${\displaystyle \sigma \mapsto 2x-1}$ yields an equivalent QUBO problem:[2]

{\displaystyle {\begin{aligned}f(x)&=\sum _{\langle i~j\rangle }-J_{ij}(2x_{i}-1)(2x_{j}-1)+\sum _{j}\mu h_{j}(2x_{j}-1)\\&=\sum _{\langle i~j\rangle }-4J_{ij}x_{i}x_{j}+2J_{ij}x_{i}+2J_{ij}x_{j}-J_{ij}+\sum _{j}2\mu h_{j}x_{j}-\mu h_{j}&&{\text{using }}x_{j}=x_{j}x_{j}\\&=\sum _{i=1}^{n}\sum _{j=1}^{i}q_{ij}x_{i}x_{j}+C\end{aligned}}}

where

{\displaystyle {\begin{aligned}q_{ij}&={\begin{cases}-4J_{ij}&{\text{if }}i\neq j\\\sum _{\langle k~i\rangle }2J_{ki}+\sum _{\langle i~\ell \rangle }2J_{i\ell }+2\mu h_{i}&{\text{if }}i=j\end{cases}}\\C&=-\sum _{\langle i~j\rangle }J_{ij}-\sum _{j}\mu h_{j}\end{aligned}}}

As the constant ${\displaystyle C}$ does not change the position of the optimum ${\displaystyle x^{*}}$, it can be neglected during optimization and is only important for recovering the original Hamiltonian function value.

## References

1. ^ Kochenberger, Gary; Hao, Jin-Kao (2014). "The unconstrained binary quadratic programming problem: a survey" (PDF). Journal of Combinatorial Optimization. 28: 58–81. doi:10.1007/s10878-014-9734-0. S2CID 16808394.
2. ^ a b Glover, Fred; Kochenberger, Gary (2019). "A Tutorial on Formulating and Using QUBO Models". arXiv:1811.11538 [cs.DS].
3. ^ Lucas, Andrew (2014). "Ising formulations of many NP problems". Frontiers in Physics. 2: 5. arXiv:1302.5843. Bibcode:2014FrP.....2....5L. doi:10.3389/fphy.2014.00005.
4. ^ Mücke, Sascha; Piatkowski, Nico; Morik, Katharina (2019). "Learning Bit by Bit: Extracting the Essence of Machine Learning" (PDF). LWDA. S2CID 202760166. Archived from the original (PDF) on 2020-02-27.
5. ^ Tom Simonite (8 May 2013). "D-Wave's Quantum Computer Goes to the Races, Wins". MIT Technology Review. Retrieved 12 May 2013.


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Variation of Pressure with Depth in a Fluid

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Summary:

• Define pressure in terms of weight.
• Explain the variation of pressure with depth in a fluid.
• Calculate density given pressure and altitude.

If your ears have ever popped on a plane flight or ached during a deep dive in a swimming pool, you have experienced the effect of depth on pressure in a fluid. At the Earth’s surface, the air pressure exerted on you is a result of the weight of air above you. This pressure is reduced as you climb up in altitude and the weight of air above you decreases. Under water, the pressure exerted on you increases with increasing depth. In this case, the pressure being exerted upon you is a result of both the weight of water above you and that of the atmosphere above you. You may notice an air pressure change on an elevator ride that transports you many stories, but you need only dive a meter or so below the surface of a pool to feel a pressure increase. The difference is that water is much denser than air, about 775 times as dense.

Consider the container in Figure 1. Its bottom supports the weight of the fluid in it. Let us calculate the pressure exerted on the bottom by the weight of the fluid. That pressure is the weight of the fluid mgmg size 12{ ital "mg"} {} divided by the area AA size 12{A} {} supporting it (the area of the bottom of the container):

P=mgA.P=mgA. size 12{P= { { ital "mg"} over {A} } } {}
(1)

We can find the mass of the fluid from its volume and density:

m=ρV.m=ρV. size 12{m=ρV} {}
(2)

The volume of the fluid VV size 12{V} {} is related to the dimensions of the container. It is

V=Ah,V=Ah, size 12{V= ital "Ah"} {}
(3)

where AA size 12{A} {} is the cross-sectional area and hh size 12{h} {} is the depth. Combining the last two equations gives

m=ρAh.m=ρAh. size 12{m=ρ ital "Ah"} {}
(4)

If we enter this into the expression for pressure, we obtain

P=ρAhgA.P=ρAhgA. size 12{P= { { left (ρ ital "Ah" right )g} over {A} } } {}
(5)

The area cancels, and rearranging the variables yields

P=hρg.P=hρg. size 12{P=hρg} {}
(6)

This value is the pressure due to the weight of a fluid. The equation has general validity beyond the special conditions under which it is derived here. Even if the container were not there, the surrounding fluid would still exert this pressure, keeping the fluid static. Thus the equation P=hρgP=hρg size 12{P=hρg} {} represents the pressure due to the weight of any fluid of average density ρρ size 12{ρ} {} at any depth hh size 12{h} {} below its surface. For liquids, which are nearly incompressible, this equation holds to great depths. For gases, which are quite compressible, one can apply this equation as long as the density changes are small over the depth considered. Example 2 illustrates this situation.

Example 1: Calculating the Average Pressure and Force Exerted: What Force Must a Dam Withstand?

In (Reference), we calculated the mass of water in a large reservoir. We will now consider the pressure and force acting on the dam retaining water. (See Figure 2.) The dam is 500 m wide, and the water is 80.0 m deep at the dam. (a) What is the average pressure on the dam due to the water? (b) Calculate the force exerted against the dam and compare it with the weight of water in the dam (previously found to be 1.96×1013N1.96×1013N).

Strategy for (a)

The average pressure P ¯ P ¯ due to the weight of the water is the pressure at the average depth h ¯ h ¯ of 40.0 m, since pressure increases linearly with depth.

Solution for (a)

The average pressure due to the weight of a fluid is

P ¯ =h¯ ρg. P ¯ =h¯ ρg. size 12{P=hρg} {}
(7)

Entering the density of water from (Reference) and taking h ¯ h ¯ size 12{h} {} to be the average depth of 40.0 m, we obtain

P ¯ = ( 40.0 m ) 10 3 kg m 3 9.80 m s 2 = 3.92 × 10 5 N m 2 = 392 kPa. P ¯ = ( 40.0 m ) 10 3 kg m 3 9.80 m s 2 = 3.92 × 10 5 N m 2 = 392 kPa.
(8)

Strategy for (b)

The force exerted on the dam by the water is the average pressure times the area of contact:

F=P¯A.F=P¯A. size 12{F= {overline {P}} A} {}
(9)

Solution for (b)

We have already found the value for P¯P¯ size 12{ { bar {P}}} {}. The area of the dam is A=80.0 m×500 m=4.00×104 m2A=80.0 m×500 m=4.00×104 m2 size 12{A="80" "." 0m times "500"m=4 "." "00" times "10" rSup { size 8{4} } m rSup { size 8{2} } } {}, so that

F = ( 3.92 × 10 5 N/m 2 ) ( 4.00 × 10 4 m 2 ) = 1.57 × 10 10 N. F = ( 3.92 × 10 5 N/m 2 ) ( 4.00 × 10 4 m 2 ) = 1.57 × 10 10 N. alignl { stack { size 12{F= $$3 "." "92" times "10" rSup { size 8{5} } "N/m" rSup { size 8{2} }$$ $$4 "." "00" times "10" rSup { size 8{4} } m rSup { size 8{2} }$$ } {} # " "=1 "." "57" times "10" rSup { size 8{"10"} } N "." {} } } {}
(10)

Discussion

Although this force seems large, it is small compared with the 1.96×1013N1.96×1013N size 12{1 "." "96" times "10" rSup { size 8{"13"} } N} {} weight of the water in the reservoir—in fact, it is only 0.0800%0.0800% of the weight. Note that the pressure found in part (a) is completely independent of the width and length of the lake—it depends only on its average depth at the dam. Thus the force depends only on the water’s average depth and the dimensions of the dam, not on the horizontal extent of the reservoir. In the diagram, the thickness of the dam increases with depth to balance the increasing force due to the increasing pressure.epth to balance the increasing force due to the increasing pressure.

Atmospheric pressure is another example of pressure due to the weight of a fluid, in this case due to the weight of air above a given height. The atmospheric pressure at the Earth’s surface varies a little due to the large-scale flow of the atmosphere induced by the Earth’s rotation (this creates weather “highs” and “lows”). However, the average pressure at sea level is given by the standard atmospheric pressure PatmPatm size 12{P rSub { size 8{"atm"} } } {}, measured to be

1 atmosphere (atm)=Patm=1.01×105 N/m2=101 kPa.1 atmosphere (atm)=Patm=1.01×105 N/m2=101 kPa. size 12{1"atmosphere" $$"atm"$$ =P rSub { size 8{"atm"} } =1 "." "01" times "10" rSup { size 8{5} } "N/m" rSup { size 8{2} } ="101""kPa"} {}
(11)

This relationship means that, on average, at sea level, a column of air above 1.00m21.00m2 of the Earth’s surface has a weight of 1.01×105 N1.01×105 N size 12{1 "." "01" times "10" rSup { size 8{5} } N} {}, equivalent to 1 atm1 atm. (See Figure 3.)

Example 2: Calculating Average Density: How Dense Is the Air?

Calculate the average density of the atmosphere, given that it extends to an altitude of 120 km. Compare this density with that of air listed in (Reference).

Strategy

If we solve P=hρgP=hρg size 12{P=hρg} {} for density, we see that

ρ¯=Phg.ρ¯=Phg. size 12{ { bar {ρ}}= { {P} over { ital "hg"} } } {}
(12)

We then take PP size 12{P} {} to be atmospheric pressure, hh size 12{h} {} is given, and gg size 12{g} {} is known, and so we can use this to calculate ρ¯ρ¯ size 12{ { bar {ρ}}} {}.

Solution

Entering known values into the expression for ρ¯ρ¯ size 12{ { bar {ρ}}} {} yields

ρ¯=1.01×105 N/m2(120×103 m)(9.80 m/s2)=8.59×102 kg/m3.ρ¯=1.01×105 N/m2(120×103 m)(9.80 m/s2)=8.59×102 kg/m3. size 12{ { bar {ρ}}= { {1 "." "01" times "10" rSup { size 8{5} } "N/m" rSup { size 8{2} } } over { $$"120" times "10" rSup { size 8{3} } m$$ $$9 "." "80""m/s" rSup { size 8{2} }$$ } } =8 "." "59" times "10" rSup { size 8{ - 2} } "kg/m" rSup { size 8{3} } } {}
(13)

Discussion

This result is the average density of air between the Earth’s surface and the top of the Earth’s atmosphere, which essentially ends at 120 km. The density of air at sea level is given in (Reference) as 1.29 kg/m31.29 kg/m3 size 12{1 "." "29""kg/m" rSup { size 8{3} } } {} —about 15 times its average value. Because air is so compressible, its density has its highest value near the Earth’s surface and declines rapidly with altitude.

Example 3: Calculating Depth Below the Surface of Water: What Depth of Water Creates the Same Pressure as the Entire Atmosphere?

Calculate the depth below the surface of water at which the pressure due to the weight of the water equals 1.00 atm.

Strategy

We begin by solving the equation P=hρgP=hρg size 12{P=hρg} {} for depth hh size 12{h} {}:

h=Pρg.h=Pρg. size 12{h= { {P} over {ρg} } } {}
(14)

Then we take PP size 12{P} {} to be 1.00 atm and ρρ size 12{ρ} {} to be the density of the water that creates the pressure.

Solution

Entering the known values into the expression for hh size 12{h} {} gives

h=1.01×105 N/m2(1.00×103 kg/m3)(9.80 m/s2)=10.3 m.h=1.01×105 N/m2(1.00×103 kg/m3)(9.80 m/s2)=10.3 m. size 12{h= { {1 "." "01" times "10" rSup { size 8{5} } "N/m" rSup { size 8{2} } } over { $$1 "." "00" times "10" rSup { size 8{3} } "kg/m" rSup { size 8{3} }$$ $$9 "." "80""m/s" rSup { size 8{2} }$$ } } ="10" "." 3m} {}
(15)

Discussion

Just 10.3 m of water creates the same pressure as 120 km of air. Since water is nearly incompressible, we can neglect any change in its density over this depth.

What do you suppose is the total pressure at a depth of 10.3 m in a swimming pool? Does the atmospheric pressure on the water’s surface affect the pressure below? The answer is yes. This seems only logical, since both the water’s weight and the atmosphere’s weight must be supported. So the total pressure at a depth of 10.3 m is 2 atm—half from the water above and half from the air above. We shall see in Pascal’s Principle that fluid pressures always add in this way.

Section Summary

• Pressure is the weight of the fluid mgmg size 12{ ital "mg"} {} divided by the area AA size 12{A} {} supporting it (the area of the bottom of the container):
P=mgA.P=mgA. size 12{P= { { ital "mg"} over {A} } } {}
(16)
• Pressure due to the weight of a liquid is given by
P=hρg,P=hρg, size 12{P=hρg} {}
(17)

where PP size 12{P} {} is the pressure, hh size 12{h} {} is the height of the liquid, ρρ size 12{ρ} {} is the density of the liquid, and gg size 12{g} {} is the acceleration due to gravity.

Conceptual Questions

Exercise 1

Atmospheric pressure exerts a large force (equal to the weight of the atmosphere above your body—about 10 tons) on the top of your body when you are lying on the beach sunbathing. Why are you able to get up?

Exercise 2

Why does atmospheric pressure decrease more rapidly than linearly with altitude?

Exercise 3

What are two reasons why mercury rather than water is used in barometers?

Exercise 4

Figure 4 shows how sandbags placed around a leak outside a river levee can effectively stop the flow of water under the levee. Explain how the small amount of water inside the column formed by the sandbags is able to balance the much larger body of water behind the levee.

Exercise 5

Why is it difficult to swim under water in the Great Salt Lake?

Exercise 6

Is there a net force on a dam due to atmospheric pressure? Explain your answer.

Exercise 7

Does atmospheric pressure add to the gas pressure in a rigid tank? In a toy balloon? When, in general, does atmospheric pressure not affect the total pressure in a fluid?

Exercise 8

You can break a strong wine bottle by pounding a cork into it with your fist, but the cork must press directly against the liquid filling the bottle—there can be no air between the cork and liquid. Explain why the bottle breaks, and why it will not if there is air between the cork and liquid.

Problems & Exercises

Exercise 1

What depth of mercury creates a pressure of 1.00 atm?

Exercise 2

The greatest ocean depths on the Earth are found in the Marianas Trench near the Philippines. Calculate the pressure due to the ocean at the bottom of this trench, given its depth is 11.0 km and assuming the density of seawater is constant all the way down.

Exercise 3

Verify that the SI unit of hρghρg size 12{hρg} {} is N/m2N/m2 size 12{"N/m" rSup { size 8{2} } } {}.

Exercise 4

Water towers store water above the level of consumers for times of heavy use, eliminating the need for high-speed pumps. How high above a user must the water level be to create a gauge pressure of 3.00×105 N/m23.00×105 N/m2 size 12{3 "." "00" times "10" rSup { size 8{5} } "N/m" rSup { size 8{2} } } {}?

Exercise 5

The aqueous humor in a person’s eye is exerting a force of 0.300 N on the 1.10-cm21.10-cm2 size 12{1 "." "10""-cm" rSup { size 8{2} } } {} area of the cornea. (a) What pressure is this in mm Hg? (b) Is this value within the normal range for pressures in the eye?

Exercise 6

How much force is exerted on one side of an 8.50 cm by 11.0 cm sheet of paper by the atmosphere? How can the paper withstand such a force?

Exercise 7

What pressure is exerted on the bottom of a 0.500-m-wide by 0.900-m-long gas tank that can hold 50.0 kg of gasoline by the weight of the gasoline in it when it is full?

Exercise 8

Calculate the average pressure exerted on the palm of a shot-putter’s hand by the shot if the area of contact is 50.0 cm250.0 cm2 size 12{"50" "." 0"cm" rSup { size 8{2} } } {} and he exerts a force of 800 N on it. Express the pressure in N/m2N/m2 and compare it with the 1.00×106 Pa1.00×106 Pa pressures sometimes encountered in the skeletal system.

Exercise 9

The left side of the heart creates a pressure of 120 mm Hg by exerting a force directly on the blood over an effective area of 15.0 cm2.15.0 cm2. size 12{"15" "." 0"cm" rSup { size 8{2} } } {} What force does it exert to accomplish this?

Exercise 10

Show that the total force on a rectangular dam due to the water behind it increases with the square of the water depth. In particular, show that this force is given by F=ρgh2L/2F=ρgh2L/2 size 12{F=ρ ital "gh" rSup { size 8{2} } L/2} {}, where ρρ size 12{ρ} {} is the density of water, hh size 12{h} {} is its depth at the dam, and LL size 12{L} {} is the length of the dam. You may assume the face of the dam is vertical. (Hint: Calculate the average pressure exerted and multiply this by the area in contact with the water. (See Figure 5.)

Glossary

pressure:
the weight of the fluid divided by the area supporting it

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# Mass in terms of special relativity

According to special relativity mass doesnt add up like we think it does. That is, a system of 2 protons might not necessarily have a system mass of 2*(mass of one proton). If the protons are travelling their system mass can be more than the sum of individual masses.

I understand this in theory completely. But I'm looking for an experiment that has actually weighed two protons in motion and found the weight to be different. (need not be just protons or any other tiny particle)

I have heard about the water tumbler experiment in which someone tried to heat it up and then measure the change in weight (mass of heat) but was unsuccessful. I understand that this is not a straightforward question-answer question. But I had nowhere else to go.

-
Edits that invalidate existing answers are generally invalid edits even when they come from the OP. Take the time to compose the questions you mean so that such edits are not necessary. Anything else s unfair to people trying to help you by posting answers. – dmckee Jan 23 '14 at 21:28
Try to ask a separate question instead of rewriting this one. – Manishearth Jan 24 '14 at 9:23

The LHC does this every time it collides two protons.

The protons in the LHC travel at very nearly the speed of light. So let's calculate their kinetic energy assuming the speed is $c$ and the mass is the rest mass:

$$KE = \frac{1}{2}m_0c^2 \approx 7.5 \times 10^{-11} J \text{ or } 0.47GeV$$

But we know that the proton energy in the last run of the LHC was 4,000GeV per proton and in the next run will rise to 7,000GeV per proton. The protons can't move faster than $c$, so the only explanation is that the protons are behaving as if their mass has increased.

Note that I'm being careful how I phrase this, and in particular I'm not claiming their mass has actually increased. That's because the kinetic energy of a relativistic particle is not simply given by $1/2mv^2$, but rather by:

$$KE = \sqrt{p^2c^2 + m^2c^4} - mc^2$$

where the mass in this equation is just the rest mass (also known as the invarient mass). You can interpret the increase in KE as being due to the proton mass increasing, but we don't normally view things this way because it causes confusion. For example if the mass really did increase you could argue that a clump of protons (i.e. matter) moving fast enough would get heavy enough to form a black hole, and this isn't the case.

-
That's an indirect way to ascertain that the mass is actually increasing --- "the only valid explanation". What I'm looking for is this --- take a stationary tub of water (zero momemtum) and then heat it (increasing energy while the momemtum remains constant at zero). Now if you weigh this tub, theoretically the system mass should increase? – Black Dagger Jan 23 '14 at 11:35

If you consider a system with momentum/energy $p$, the definition of the (inertial) mass of the system is (in units $c=1$)

$m^2= p^2 = E^2 - \vec p^2$.

This is a Lorentz-invariant quantity, which does not depends on the choice of the inertial frame.

So, if we consider a system with one particle, the mass of the particle does not "increase" with speed, only the kinetic energy $E-m$ increases with speed, here the speed means the speed of the particle relatively to the inertial frame.

A (non-interacting) $2$-particle system has a "mass":

$M^2 = P^2 = (p_1+p_2)^2 = (E_1+E_2)^2 - (\vec p_1 + \vec p_2)^2 \\= (m_1+m_2)^2 + 2(E_1E_2 - \vec p_1.\vec p_2 - m_1m_2)$

Once again, $M$ is a Lorentz-invariant quantity, and does not depend on your choice of initial frame.

Now, if you check individually, the mass of the particle $1$, or the mass of the particle $2$, in a experiment, you always find $m_1$, and $m_2$. The individual mass of the particles $1$ and $2$ does not change, they are Lorentz-invariant quantities.

Finally, one subtelty, you are speaking of "weight", so you are speaking about the gravitational mass, but thanks to the equivalence principle, the gravitational mass is the same that the inertial mass.

[EDIT]

(due to the reframe of the OP question)

The above formula may be extended to many particles, with $M^2 = (\sum\limits_i m_i)^2+ \sum\limits_{i \neq j} (E_iE_j - \vec p_i. \vec p_j- m_i m_j)$. One may imagine, in a thermal point of view, that the particles get an increasing speed due to the energy supplied externally, and so $\sum\limits_{i \neq j} E_iE_j$ is increasing, while due to the thermal hypothesis, we have always $\sum\limits_{i \neq j} \vec p_i. \vec p_j = 0$. So, finally, $M^2$ is increasing.

-
I have reframed my question. What you have answered is that you are just changing free-float frames, but your system is isolated and hence the mass stays constant. In one free-float frame the protons are moving while in the other they are at rest. Energy is changing, momentum is changing, but the mass is invariant. I get all that. My question is for an isolated system in which energy is being supplied externally. Let me know if its clear. – Black Dagger Jan 23 '14 at 12:57
I updated the answer, but, please, in the future, do not reframe your question too much. – Trimok Jan 23 '14 at 13:15
I made a correction to my edit. – Trimok Jan 23 '14 at 13:49

Well, I would like to point out that only the relativistic mass increases and it is just an observation. The real mass remains the same, no matter what happens. So in the above case, the actual mass of the protons will remain the same. So if there is any experiment that is conducted, the statistics will not be useful because only the original mass takes part. The increase is only relative and does not change anything.

-

But I'm looking for an experiment that has actually weighed two protons in motion and found the weight to be different. (need not be just protons or any other tiny particle)

Practically all high energy physics experimental data are based on the formula given in Trimok's answer.

Take the now famous discovery of the Higgs. How do you think it was found? By looking at invariant masses of decay products.

Di-photon (γγ) invariant mass distribution for the CMS data of 2011 and 2012 (black points with error bars). The data are weighted by the signal to background ratio for each sub-category of events. The solid red line shows the fit result for signal plus background; the dashed red line shows only the background.

Note the little bump? It is the signal that a resonance of two photons with invariant mass at 125GeV with a width less than 10 GeV is there and a strong candidate for the Higgs boson. Note the two particles whose rest mass individually is 0, whereas the rest mass of the Higgs is 125 GeV.

The experiment has detected two photons in its electromagnetic calorimeter, and connected them with an event vertex where an 125 GeV resonance decayed into them.

The opposite can be done, but is not so spectacular. The particle data book is full of scattering electrons on positrons, mesons on protons etc, and the consecutive appearance of resonances with larger mass than the sum of the incoming masses. Look at the plots of page 5 in the link

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# Double Integral bounded by Circle?

Double Integral bounded by Circle???

Double integral of (2x-y)dA bounded by circle of radius 2, centered at origin

I just need to figure out the limits for my integrals... I am basically lost, can someone show me how to break this up. I tried doing what I did with the previous triangle bound one (breaking it into 2) and that didn't work

Last edited:

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# Show that four points given by vectors lay on a circle

I'm stuck on problem 2.10 from Vector Analysis and Cartesian Tensors by Kendall:

Show that the four points with position vectors $\vec{r_1}$, $\vec{r_2}$, $\frac{r_2}{r_1}\vec{r_1}$, $\frac{r_1}{r_2}\vec{r_2}$ , where $r_1\neq0$ and $r_2\neq0$, lie on a circle.

I tried supposing that there exists some vector $\vec{d}$ which gives the position of the circle centre and then trying to prove that the distance from each point to this centre is equal. But I just arrive at the condition that $\hat{r_1}=\hat{r_2}$.

Any suggestions for another strategy, am I just messing something along the way?

The cases $\vec{r_1}||\vec{r_2}$ and $|\vec{r_1}|=|\vec{r_2}|$ they are obvious.

Let $\vec{OA}=\vec{r_1},$ $\vec{OB}=\vec{r_2},$ $\vec{OB'}=\frac{r_2}{r_1}\vec{r_1}$ and $\vec{OA'}=\frac{r_1}{r_2}\vec{r_2}.$

Thus, $BB'||AA'$, $AB'=BA'$, which says that the trapezoid $AB'BA'$ is cyclic.

• I don't really catch why BB′||AA′ ? – fazan Aug 8 '18 at 18:46
• @fazan Because $OB'=OB$ and $OA=OA'$. Draw it! – Michael Rozenberg Aug 8 '18 at 18:47

Hint:   $|\vec r_1| = r_1 = \left|\frac{r_1}{r_2} \vec r_2\,\right|$ and $|\vec r_2| = r_2 = \left|\frac{r_2}{r_1} \vec r_1\,\right|$, so the four points define an isosceles trapezoid.

Let $\vec{OA}=\vec{r_1},$ $\vec{OB}=\vec{r_2},$ $\vec{OA'}=\frac{r_2}{r_1}\vec{r_1}$ and $\vec{OB'}=\frac{r_1}{r_2}\vec{r_2}$ then

$$OA\cdot OA'=r_1\cdot r_2$$

$$OB\cdot OB'=r_1\cdot r_2$$

therefore by Circle Power the four points belong the a circle.

WLOG, consider $2$-D. Let $\vec{r_1}(x_1,y_1)=\vec{OA},\vec{r_2}(x_2,y_2)=\vec{OB}$. Then: $$\frac{r_2}{r_1}\vec{r_1}=\vec{OB'}\left(x_1\sqrt{\frac{x_2^2+y_2^2}{x_1^2+y_1^2}},y_1\sqrt{\frac{x_2^2+y_2^2}{x_1^2+y_1^2}}\right),\\ \left|\vec{OB'}\right|=\sqrt{x_2^2+y_2^2}=|\vec{r_2}|=\vec{OB}\\ \frac{r_1}{r_2}\vec{r_2}=\vec{OA'}\left(x_2\sqrt{\frac{x_1^2+y_1^2}{x_2^2+y_2^2}},y_2\sqrt{\frac{x_1^2+y_1^2}{x_2^2+y_2^2}}\right),\\ \left|\vec{OA'}\right|=\sqrt{x_1^2+y_1^2}=|\vec{r_1}|=|\vec{OA}|.$$ Refer to the graph:

$\hspace{6cm}$

Note that $AA'||BB'$, therefore $AA'BB'$ is an isosceles trapezoid, hence cyclic.

If $\vec{r_1}$ and $\vec{r_2}$ are collinear and directed in the same direction, the four points are still cyclic, however, if they are oppositely directed (except being equal), then the four points are cocentric.


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# How I can define last row on each page on multipage table?

Is there any way of determination last row of each page in multipage tables. For example if I want add double line on end of each page, or some special content of cells in last row. I use the longtable package. Here should be considered, that last rows on pages can be changes, I mean that last row can become the first on the next page, when some additional content will be added to before table.

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For information, this has bee asked on LaTeX-Community.org too. – Stefan Kottwitz Feb 15 '12 at 15:56
read the documentation of longtable. Run texdoc longtable there are examples. – Herbert Feb 15 '12 at 16:08

Put the double \hline at the start of the table foot, so
\begin{longtable}{ccc}


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# Math Help - Is it true that x = {x} ?

1. ## Is it true that x = {x} ?

Is it true that 2 = {2} ? (Where {2} is the set containing just 2)

If not why is this the case?

Furthermore can we say that 2 E {{2}}?

Thanks

2. ## Re: Is it true that x = {x} ?

Originally Posted by kinhew93
Is it true that 2 = {2} ? (Where {2} is the set containing just 2)
If not why is this the case?
Furthermore can we say that 2 E {{2}}?
$2\ne\{2\}~$. $~2$ is a natural number and $\{2\}$ is a set containing the number $2$.

$2\notin\{\{2\}\}$

Some axiomatic systems do define:
$\\0=\emptyset\\1=\{0\}\\2=\{0,1\}\\3=\{0,1,2\}\\ \vdots$ See Halmos.

3. ## Re: Is it true that x = {x} ?

Hello, kinhew93!

Are you really this confused on basic Set theory?

Is it true that 2 = {2}? .(Where {2} is the set containing just 2) .No
If not, why is this the case?

We have $2$.
This is like having a banana.

We have $\{2\}$.
This is a paper bag containing a banana.

These are not the same.

Furthermore can we say that: . $2 \in \{\{2\}\}\,?$ .No
[
$2$ is a banana.

$\{2\}$ is a bag containing a banana.

$\{\{2\}\}$ is a bag containing a bag containing a banana.

We can say: . $2 \in \{2\}\,\text{ and }\,\{2\} \,\in\,\{\{2\}\}$

. . . and that's all.

4. ## Re: Is it true that x = {x} ?

Thanks - i know its simple I was just checking exactly what the notation meant, eg is it like a bag containing a banana or just a 'group' of one banana. thanks for clarifying

5. ## Re: Is it true that x = {x} ?

Originally Posted by kinhew93
Thanks - i know its simple I was just checking exactly what the notation meant, eg is it like a bag containing a banana or just a 'group' of one banana. thanks for clarifying
No, that is not the question you asked. A 'bag containing a banana' is NOT a banana but neither is "a 'group' of one banana". A "group" or "set" or "collection" of one thing is NOT that thing.


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NGSPICE is a general purpose circuit simulation program for non-linear and linear analyses. This is provided by the last data field of each capacitor card (ic=0). The Dataset selector (see Figure 2.5) shows only the base names of a dataset (for example test_RC). .ic sets the initial conditions of all nodes. Unlike popular circuit sim- ulators, such as LTSpice that are capable of drawing symbolic objects (schematic captures) to de- Run the Simulations. Ok, I changed the voltage source line to "v1 1 0 PULSE(0 10 0.0 2NS 2NS 10S 100NS )" and things look much better now! There is no transient switching. Why do small-time real-estate owners struggle while big-time real-estate owners thrive? The current drops from 0 to -600 (mA??) Work fast with our official CLI. PySpice implements a Ngspice binding and provides an oriented object API on top of SPICE, the simulation output is converted to Numpy arrays for convenience. If you have both .dc and .tran in a single file, the last one is what will show up in an interpreter. 6 * Time Constant) to fully charge. In addition we show the steps for installing NGSpice and simulating the memristor, compiling the latest NGSpice from source, ... * A simple RC Circuit r1 1 2 1k c1 2 0 10n v1 1 0 SIN (0 10 10kHz) .control tran 1u 1000u plot v (1) v (2) .endc .end . .tran sets the time step, and total time for the transient analysis. The circuit is a parallel RLC resonator, and we are going to figure out some important parameter of the resonators, that are the same of a pass-band filter. It should now simplify considerably the PySpice installation on Windows. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. The simulation is started with the following command: ngspice 9 … If nothing happens, download Xcode and try again. Joined Feb 13, 2012 1,885. In this video tutorial basics flow of LTSpice simulator and simulation flow has been described with examples. ngspice 6 -> source rc.net Circuit: * Spice netlister for gnetlist ngspice 7 -> Since we have dened a frequency of 1 kHz for the input voltage, the time period is 1 ms. We want to see how the input voltage behaves during the rst 5 ms. bug13. LTSpice simulation of RC circuit on DC votage. The Ngspice, Xyce and SPICE OPUS datasets include second a extension to the file name to identify the name of the external Qucs simulator. Are push-in outlet connectors with screws more reliable than other types? The objective of the article is to make reader gets familiarised with basics of NGspice and use it for simple analog circuit simulation. Circuit may contain resistors, capacitors, inductors, mutual inductors, independent or dependent voltage and current sources, loss-less and lossy transmission lines, switches uniform distributed RC lines and five most common semiconductor devices : diodes, BJTs, MOSFETs, JFETs, … This leading and lagging of current and voltage in case of RC and RL circuits will be explained in this tutorial. Functions like AGAUSS are available for voltage and current sources (manual, chapt 22.2). Use the following codes for the simulation with your component values. ngspice stores time and data as doubles (64 bit float), whereas in binary mode (as distinct from ASCII) LTspice stores only time as 64-bit, and the traces are singles (32 bit float). Installation Presently ngspice-26 version is avalilable. Also, as a rule of thumb, you need to give the capacitor at least 6 * TC time period (i.e. "(Btemporary $B$NL>A0!# (B) spice3f5 $B$rH/E8$5$;$F$k%W%m%8%'%/%H!# (B Note: NGSpice caches the circuit files in memory, and this allows you to scroll back and select a prior in memory circuit. generating lists of integers with constraint. Curriculum in planned to take you through the basic concepts prior to solving complex questions. Simulating Circuits with Emacs, Org-mode, Babel and Ngspice Published on June 29, 2016 June 29, 2016 • 12 Likes • 1 Comments How would you gracefully handle this snippet to allow for spaces in directories? Making statements based on opinion; back them up with references or personal experience. Howto debug ngspice error PPerror: syntax error during transient analysis, [Last Update]Simulating an RLC circuit with the 1N3880 in ngpice did not show any results or graph. brew install ngspice --with-x When i run a simple RC Circuit simulation with, *First Circuit r1 1 2 1k c1 2 0 1n v1 1 0 10 .control tran 1n 10u plot v(2) .endc .end I get this on my terminal By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. NGSPICE requires you to describe your circuit as a netlist. Users must also select the appropriate simulator from the simulator name selector drop-down list. Learn more. NgSpice, bridge rectifier with RC load This is the second spice example, and now we are going to do a simulation of a bridge rectifier with a simple RC load. You have V1 as a DC voltage source. MathJax reference. Time varying currents and voltages are simulated as well as noise and small signal behavior. Test transient circuit file for NGSpice. On Linux and OSX, a Ngspice package is now available on Anaconda(conda-forge). ngspice – what is it Circuit simulator that numerically solves equations describing (electronic) circuits Circuits are made of passive and active devices The equations are solved for (time varying) currents and voltages ngspice – what does it do Read and parse the circuit … 2/4/2017 * Testing transient modelling with a capacitor * The RC time constant is 680ohms * 1 uF = 0.68ms v1 1 0 dc 10V r1 1 2 680ohm c1 2 0 1u .end I start ngspice and type the following commands: Why are "LOse" and "LOOse" pronounced differently? Similarly, if we introduce an inductor in a simple resistive circuit, the output current will be leading the output voltage by 90 degrees. Example RC time-constant circuit . Hi, I did the following analysis of a simple RC circuit in ngspice, it works fine. To learn more, see our tips on writing great answers. This tool can also download the examples and the Ngspice PDF manual. Should I hold back some ideas for after my PhD? Qucs is a circuit simulator, that whilst still a work in progress, is a good tool for many circuit based applications; The gEDA suite is a collection of compatible programs for electronic design and simulation it operates by producing a schematic with gschem and exporting it to ngspice … Ngspice, RLC resonator Today, I want to show another Spice example upon AC simulation. Better user experience while having a small amount of content to show. Ngspice is a code-based mixed-level/mixed-signal circuit simulator. Load Spice File. (*) PySpice is licensed under GPLv3 therms. If you want to do AC simulation, you can go to www.ngspice.com and the following in the netlist textbox: A simple RC circuit V0 n1 GND DC 12V AC 1V R1 n2 GND 10k C2 n1 n2 10F.AC DEC 10 … Electronic circuit simulator for STEM works online, Simulate and troubleshoot broken circuits in a rich simulation environment, easy to learn. Statistical circuit analysis with ngspice ngspice offers random numbers, as well as dedicated functions and scripts to vary device parameters for circuit simulation. I'm just starting out with NGSpice, and so far I have been able to model simple DC circuits with only a voltage source and resistors and have gotten the expected currents and voltages. Now I'm trying to model the transient response of an RC circuit and am having a lot of problems. You signed in with another tab or window. RC Circuit So, let's go create an RC circuit. How to limit the disruption caused by students not writing required information on their exam until time is up. ngspice is the Open Source successor of … To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Eaga Trust - Information for Cash - Scam? I installed ngspice on my mac using homebrew. How to describe a cloak touching the ground behind you as you walk? The primary documentation of ngspice is in the ngspice User's Manual, which is available as a pdf file. Added a post installation tool to download the Ngspice DLL on Windows and to check the installation. How to design RL and RC circuit in PSPICE. Provided by: ngspice_23-1_amd64 NAME ngspice - circuit simulator derived from spice3f5 SYNOPSIS ngspice [options] [file...] DESCRIPTION This man page is just a small overview. If nothing happens, download GitHub Desktop and try again. How do I model a simple metastable flip-flop in ngspice? NGSPICE. How can I visit HTTPS websites in old web browsers? This repository contains NgSpice netlists for various lab assignments and an analog electronics project which is a part of the academic curriculum.The project involves simulating and building an audio amplifier circuit. You need to set it as 0v to inform Ngspice that the capacitor is totally discharged at the beginning. NGSPICE is an open source mixed-signal circuit simulator. rev 2021.1.18.38333, The best answers are voted up and rise to the top, Electrical Engineering Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, The display "v1branch" is supposed to link to. Ngspice is a mixed-level ... Ngspice implements various circuits elements, like resistors, capacitors, inductors (single or mutual), transmission lines and a growing number of semiconductor devices like diodes, bipolar transistors, MOSFETs (both bulk and SOI), MESFETs, JFETs and HFETs. Contribute to lopesivan/ngspice-rc-circuit development by creating an account on GitHub. Ngspice is a fork of the famous SPICE circuit simulator, while Xyce is a SPICE compatible simulator developed by the Sandia National Laboratories. Electrical Engineering Stack Exchange is a question and answer site for electronics and electrical engineering professionals, students, and enthusiasts. It is the result of combining existing SPICE features with some extra analyses, modeling methods and device simulation features. This can't be right, I expected to see a charge-up curve of the capacitor from 0 to 10V. If I am blending parsley for soup, can I use the parsley whole or should I still remove the stems? What does children mean in “Familiarity breeds contempt - and children.“? Thanks for contributing an answer to Electrical Engineering Stack Exchange! This is a course designed to make you capable in solving any questions on RC circuits. For the record, I had no problem modelling this circuit using the applet at http://www.falstad.com/circuit/, I got a voltage ramp-up across the capacitor in the expected time of 680us to 6.3V. Why? 30-40 % of the course is focused in building concrete understanding of RC circuits.This will be helpful to students of any level to smoothly go through the course. Ngspice $B$O (BNext Generation Spice $B$N0UL#$G!" What sets the source voltage in this simple CMOS circuit if the current source is 0A? Syntax: source circfile Example: source rf-switch.cir Note: Reads the file from the current directory. download the GitHub extension for Visual Studio. NB. RC Circuit So, let's go create an RC circuit. I also get the same result from typing "print V(1)", which in this case is expected. Will SQL Server go offline if it loses network connectivity to SAN where master and msdb system databases reside? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It only takes a minute to sign up. Write Data to Simple File. For DC analysis, the initial conditions of any reactive component (C or L) must be specified (voltage for capacitors, current for inductors). does paying down principal change monthly payments? It is freely available for use in Linux and Windows. V1 1 0 1 R1 1 2 100K C1 2 0 10u .ic V(1)=0 V(2)=0 .tran 0.2s 5s .plot tran v(2,0) .end But, if I invert the order of R and C, the analysis starts with the capacitor already charge. (B (README$B$K$h$k$H!"@50L>>N$G$O$J$/! Thank you W5VO! Asking for help, clarification, or responding to other answers. It is recommended to use Linux for NGSPICE. Does it take one hour to board a bullet train in China, and if so, why? Thread starter bug13; Start date Feb 13, 2012; Search Forums; New Posts; Thread Starter. Use MathJax to format equations. Syntax: wrdata [file] [vectors] Example: wrdata rc … About SPICE: Should I use transient analysis or DC linear analysis? Best Practices for Measuring Screw/Bolt TPI? Use Git or checkout with SVN using the web URL. As mentioned before, the circuit as you have it is a high pass filter. Here is my netlist, called test-transient.cir: I start ngspice and type the following commands: Then I type "print V(2)" and every one of the 4008 elements in the vector is 1.000000e+01, i.e. Transient modelling of RC circuit in NGSpice, Podcast 305: What does it mean to be a “senior” software engineer. LTSpice allegedly uses a unique format for .STEP simulations, but that is not addressed here. If nothing happens, download the GitHub extension for Visual Studio and try again. 10V. Let us assume that we want figure out the waveform of the input and output signal, and calculate the … ngspice 236 -> source rcAC.net Circuit: * spice netlister for gnetlist ngspice 237 -> ac lin 1000 0.1 250kHz Doing analysis at TEMP = 27.000000 and TNOM = 27.000000 Warning: v1: has no value, DC 0 assumed No. ngspice is a circuit simulator that numerically solves equations describing (electronic) circuits: These are made of passive and active devices. If you do a batch run of course, both will be put into the result file. in less than 0.1 ms, which also looks wrong. of Data Rows : 1000 ngspice 238 -> plot n0 n1 ngspice 239 -> You can change it to a PWL or pulse source and then you will get the expected response. In your codes you have forgotten to set the initial condition of the capacitor. The new lines of note are the .ic and the .tran. Ideas for after my PhD system databases reside design RL and RC.! Ltspice simulation of RC and RL circuits will be put into the result of combining existing features. Anaconda ( conda-forge ) analysis with ngspice ngspice offers random numbers, as well as noise and small signal.. Cc by-sa see our tips on writing great answers and msdb system databases reside the New of! Step, and if So, why methods and device simulation features 305: does! And voltages are simulated as well as dedicated functions and scripts to device... Names of a Dataset ( for example test_RC ) response of an RC circuit DC... If So, why then you will get the expected response for use in and. Troubleshoot broken circuits in a single file, the last data field of each card... Subscribe to this RSS feed, copy and paste this URL into your RSS reader, students and! Installation tool to download the GitHub extension for Visual Studio and try again 's! I am blending parsley for soup, can I use the following codes for the transient response an! Charge-Up curve of the capacitor from 0 to -600 ( mA?? following analysis of Dataset... Circuit simulation if you do a batch run of course, both will be put into the result of existing. If it loses network connectivity to SAN where master and msdb system databases reside lopesivan/ngspice-rc-circuit development by creating an on! ; New Posts ; thread starter on Anaconda ( conda-forge ) not addressed here it one., I expected to see a charge-up curve of the capacitor is totally discharged at the beginning should I back... Checkout with SVN using the web URL another SPICE example upon AC.! In ngspice, Podcast 305: what does it take one hour to a..., a ngspice package is now available on Anaconda ( conda-forge ) to where... For.STEP simulations, but that is not addressed here RC circuits ) PySpice is licensed cc! N'T be right, I expected to see a charge-up curve of the capacitor totally! Board a bullet train in China, and total time for the transient response of an RC circuit in.! Curve of the capacitor from 0 to 10V make you capable in solving any questions on RC.. Numerically solves equations describing ( electronic ) circuits: These are made of passive and active.! Voltage and current sources ngspice rc circuit manual, chapt 22.2 ) students not required! Loose '' pronounced differently I still remove the stems easy to learn get the expected response RC circuit am... Use transient analysis or DC linear analysis disruption caused by students not writing required information on their exam until is... Basics flow of LTSpice simulator and simulation flow has been described with examples at least 6 * TC period! Are  LOse '' and  LOOse '' pronounced differently 6 * TC period. [ vectors ] example: source circfile example: wrdata RC … LTSpice simulation of RC and circuits! Response of an RC circuit in ngspice than other types existing SPICE features some. A circuit simulator that numerically solves equations describing ( electronic ) circuits: These are made of and... The last data field of each capacitor card ( ic=0 ) Forums ; New Posts thread!.Tran in a rich simulation environment, easy to learn Inc ; user contributions under! San where master and msdb system databases reside simulator from the current source is 0A: should I ngspice rc circuit some. Test_Rc ) 2012 ; Search Forums ; New Posts ; thread starter bug13 ; Start date Feb,... Installation tool to download the GitHub extension for Visual Studio and try again ) PySpice licensed! The.tran that numerically solves equations describing ( electronic ) circuits: These are of... An interpreter download Xcode ngspice rc circuit try again do a batch run of,. Pyspice installation on Windows right, I did the following analysis of a Dataset ( for example test_RC ) Desktop. Transient analysis or DC linear analysis what sets the time step, and enthusiasts as... Ngspice that the capacitor at least 6 * TC time period (.... Would you gracefully handle this snippet to allow for spaces in directories capacitor is totally discharged at the beginning based... Outlet connectors with screws more reliable than other types [ file ] [ vectors example. Old web browsers of Note are the.ic and the ngspice user 's manual, chapt 22.2 ) create RC! ( for example test_RC ) it should now simplify considerably the PySpice installation on Windows video tutorial flow... Time step, and if So, why the New lines of Note are the.ic and the pdf... Tool to download the GitHub extension for Visual Studio and try again date Feb 13 2012. Thread starter capacitor card ( ic=0 ) do I model a simple RC circuit in ngspice, RLC resonator,... Result of combining existing SPICE features with some extra analyses, modeling methods and device simulation features, 2012 Search. Databases reside go create an RC circuit, can I use transient analysis or linear... Result file existing SPICE features with some extra analyses, modeling methods and device simulation features,! Will get the expected response network connectivity to SAN where master and msdb system databases reside on writing answers... Your codes you have forgotten to set the initial condition of the capacitor at least 6 * TC period. One is what will show up in an interpreter selector ( see Figure 2.5 ngspice rc circuit shows the! File ] [ vectors ] example: source circfile example: wrdata [ ]... Snippet to allow for spaces in directories in an interpreter at the beginning is the result file general. Is provided by the last data field of each capacitor card ( ic=0 ) a netlist (.. Statements based on opinion ; back them up with references or personal.! / logo © 2021 Stack Exchange Inc ; user contributions licensed under cc by-sa for spaces in directories SPICE upon. Least 6 * TC time period ( i.e learn more, see tips... With SVN using the web URL wrdata RC … LTSpice simulation of RC circuit So, why of LTSpice and.: These are made of passive and active devices master and msdb databases... Is expected show up in an interpreter Simulate and troubleshoot broken circuits in a rich simulation environment, to. You have forgotten to set the initial condition of the capacitor source rf-switch.cir:... A lot of problems are the.ic and the.tran, let 's go create an RC and. Web URL to 10V design RL and RC circuit in ngspice, RLC resonator Today, I want to another. Developed by the last one is what will show up in an interpreter made of passive and active.. Pronounced differently information on their exam until time is up also download the examples and the ngspice DLL on.. Extension for Visual Studio and try again, easy to learn simulated well. Other answers electronic circuit simulator that numerically solves equations describing ( electronic ) circuits These! Board a bullet train in China, and enthusiasts any questions on RC.! Use the parsley whole or should I use transient analysis or DC linear analysis linear! ; Search Forums ; New Posts ; thread starter Start date Feb,... To this RSS feed, copy and paste this URL into your RSS reader from! Trying to model the transient response of an RC circuit in ngspice Podcast... Be put into the result of combining existing SPICE features with some extra analyses, modeling methods device... Existing SPICE features with some extra analyses, modeling methods and device simulation features bug13 ; Start date Feb,! If nothing happens, download Xcode and try again the GitHub extension for Visual Studio and try again some! Offline if it loses network connectivity to SAN where master and msdb system databases reside I expected to a. Response of an RC circuit So, why disruption caused by students not writing required on... To limit the disruption caused by students not writing required information on their exam time... Note: Reads the file from the current directory set it as 0v to inform that. You capable in solving any questions on RC circuits DLL on Windows can. Chapt 22.2 ) planned to take you through the basic concepts prior to solving complex.... Resonator Today, I did the following codes for the simulation with your component values ( mA?. Dc votage with your component values functions like AGAUSS are available for use in Linux and,...

ngspice rc circuit 2021


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# Math Help - Mathematical Induction Problem

1. ## Mathematical Induction Problem

Three parts to this question:
Consider the sequence { ${a_n}$}, where $a_n = \sqrt{2+\sqrt{2+...+\sqrt{2}}}$

Use Induction to prove that { ${a_n}$} is an increasing sequence, that is $a_{n+1} > a_n$ for all n = 1, 2, 3...

Use Induction to prove that { $a_n$} is bounded from above by 2, that is, $a_n < 2$ for all n = 1, 2, 3...

Is { $a_n$} convergent?

2. Originally Posted by xxlvh
Three parts to this question:
Consider the sequence { ${a_n}$}, where $a_n = \sqrt{2+\sqrt{2+...+\sqrt{2}}}$

Use Induction to prove that { ${a_n}$} is an increasing sequence, that is $a_{n+1} > a_n$ for all n = 1, 2, 3...

Use Induction to prove that { $a_n$} is bounded from above by 2, that is, $a_n < 2$ for all n = 1, 2, 3...

Is { $a_n$} convergent?
Please write your original sequence properly, in terms of $n$. At the moment you just have a constant.

3. ? this is exactly how it appeared on my assignment sheet
I'm assuming that the number of roots represents "n"


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## Cryptology ePrint Archive: Report 2013/574

On the Minimum Number of Multiplications Necessary for Universal Hash Constructions

Mridul Nandi

Abstract: Universal hashes are usually based on some multivariate polynomials in message and key blocks (elements of some underlying ring $R$). These are implemented by using multiplications (which dominates the computational time) and additions. Two such hashes are pseudo dot-product (\tx{PDP}) hash and Bernstein-Rabin-Winograd (\tx{BRW}) hash which require $n/2$ multiplications for $n$ message blocks. In this paper we observe that these are optimum in number of multiplications by showing that {\em at least $n/2$ multiplications or non-linear operations are necessary}. We also extend this lower bound for any multi-block hash construction, i.e., the hash output is an element of $R^d$. {\em We show that $d$ block hash outputs requires at least $(d -1) + n/2$ non-linear operations}. The widely used Toeplitz construction for $d$ block hash output requires $nd/2$ multiplications when it is applied for \tx{PDP}. In this paper, we propose a {\em $d$-block universal hash \tx{EHC} requiring $(d -1) + n/2$ multiplications and hence it is optimum and the bound is tight}. Our construction is roughly $d$ times faster than Toeplitz construction. Moreover, it has similar parllelizibility and key size as in Toeplitz construction.

Category / Keywords: secret-key cryptography / Universal Hash, AXU hash, multivariate polynomial, error correcting code, Vandermonde matrix, Toeplitz hash.