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# Talk:Arc length

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I think it should be redirected to curve

Tosha 01:25, 15 Mar 2004 (UTC)

I completely disagree with you, Tosha. I think this deserves a page on its own.
This means that in discussion of the geometry of the curves, any links to arc length now jump directly to the curves article. This is a disservice to readers navigating these articles who expect to go directly to a discussion of the concept of arc length, rather than to to the article on curves. -- Decumanus 19:39, 15 Mar 2004 (UTC)
I agree; this page can stand on its own. Michael Hardy 21:34, 15 Mar 2004 (UTC)

Ok, I think this way it is harder to keep it uptodate, but do as you wish.Tosha 23:56, 15 Mar 2004 (UTC)

This article is incomplete. It needs cleanup. I wikified the maths, but some of the formulas didn't seem to make sense the way they were written, so they should be checked.

The article is also in sore need of some images.

I reordered the article and fixed some of the math. It still needs more work though -Jacob 01:38, 14 June 2005 (UTC)

I've been fixated on the text bit - now that I look closely, that explanation of the Fermat math doesn't look right. It keeps sprouting unexplained new variables. Tearlach 19:37, 16 July 2005 (UTC)
I only fixed the LaTex formating of the math -not the content- now that I look criticaly, I see what you mean. It looks like the original author of the page just rewrote it from the pdf he cites. Reading the file explains what he was getting at. We I get some more time, I'll make a plot so one can follow along. We should add the final part too (from the pdf) where the formula shows that Fermat really did find the integral and his formula, he just didn't understand the implications. I'd like another source for verification that this is how it historicaly happened.
As far as I understand it, both van Heuraet and Fermat derived that if you have a function f(x), then the arc length is given by the area under a new function g(x) = sqrt(1 + f'(x)^2). So they had the function to be integrated, but not the integral calculus to apply to it. Tearlach 11:49, 17 July 2005 (UTC)

A bit added about the problems. As it stood, it was overoptimistic about the benefits of applying calculus. PS Semicubical parabola is y^3 = a*x^2. Tearlach 23:24, 15 July 2005 (UTC)

More bibliographic detail: Heuraet at MacTutor, Rida T Farouki reference here (PostScript) and here (text). Tearlach 07:26, 16 July 2005 (UTC)

And this Wallis biography from WW Rouse Ball describes van Heuraet's method for curve rectification. Tearlach 13:07, 16 July 2005 (UTC)

I added a graph for Fermat's method. I had different colors for the line originaly, but somehow, gnuplot stoped changing them as the code evolved. See Image:Arc length, Fermat.png for the code and please help me fix it. --Jacob 18:09, 22 July 2005 (UTC)

The location "Arc length" is arrived at from "Curve length". No idea how history brought us to this situation. Clearly, this article should be "Curve length", and "Arc length" should redirect to "Arc_(geometry)". 68.84.233.37 (talk) 10:50, 30 June 2013 (UTC)

## Article title

According to the article Arc, an arc is a continuous portion of a circle; part of a circle's circumference (also called a circle segment). The present article is about curves. It uses the undefined term "irregular arc", but in fact even in the present text the use of the term "curve" dominates the use of the term "arc". My conclusion is that the title ought to be: Length of a curve. LambiamTalk 03:09, 9 April 2006 (UTC)

Actually, I prefer arc length, as that is what the concept is usually called in calculus books and more advanced books on differential geometry. The other names should just redirect to arc length. I also suspect that when writing an article, most mathematicians would use arc length, so this avoids unnecessary redirects.
There are terminology issues, as you've noticed. "Arc" actually is used to refer to more than what is currently in arc. In geometry/topology areas, it actually means a continuous map of the unit interval, and often in differential geometry, "arc" refers to "rectifiable arc". This usage of "arc" is actually fairly old, and that is the "arc" that is referred to by "arc length"; it may be tied to the historical "irregular arc" mentioned in the article. I think, though, that more people prefer "path" or "curve" over "arc" nowadays in topology.
In answer to your objection, while it may be strange that "curve" dominates "arc" in the article, it's just an artifact of history and usage that "arc length" has become the standard term rather than "curve length", although of course, people would probably understand what you mean by the latter in the appropriate context. --C S (Talk) 03:58, 9 April 2006 (UTC)
I did a bit of research on other treatments. Mathworld, under "arc length", defines "Arc length is defined as the length along a curve"[1] and proceeds to give a treatment that will blow away all non-mathematicians; it appears to presuppose differentiability. PlanetMath, under "arc length", defines: "Arclength is the length of a section of a differentiable curve"[2] (italics added, L.), and the treatment given corresponds more or less to that of our Length of an arc. We also have Curve#Lengths_of_curves, which is somewhat unreadable and probably quite incomprehensible to the mathematically untrained, but in essence what I believe the Length article should cover: length as a kind of limit, with differentiable curves in Euclidean 2-space as a special case. Alternatively, we should make clear that this article covers only an important but nevertheless special case and insert a link to the more general, unfortunately unreadable, treatment. LambiamTalk 06:19, 9 April 2006 (UTC)

I prefer arc length, I'd like to see this article moved to there. -lethe talk + 17:43, 9 April 2006 (UTC)

I also prefer arc length. The more general stuff at curve I think is also called arc length, but in the case of metric spaces, so it should be treated here. --MarSch 10:24, 11 April 2006 (UTC)
I have moved the article. This required deleting and merging old history, since there was once an article arc length, which has a history, and was improperly made into a redirect by Tosha in 2004, in violation of the GFDL. -lethe talk + 19:42, 11 April 2006 (UTC)

## Length of Circular Arc Segments

As this article is named Arc Length, I believe it should at its outset completely and clearly address the generally understood notion of Arc pertaining to a segment of a circle before it addresses the arc lengths of curves. The current definition of a circular arc, while technically correct is woefully inadequate to general users and difficult to find on the page. In order to make Wikipedia useful to the general populace, think like and write to non-mathematicians first, then delve into the higher notions of the concept.

I completely agree with this. This is an endemic issue in most of the math sections on Wikipedia. Usually little to no regard is given to those who are seeking to understand the basics of the concepts and the articles dive headlong into the higher math without giving adequate explanations of what the formulas mean. Most people who are looking for information on Wikipedia don't "speak math" like we do and often skip right over the formulas. I know that the formulas are the most concise definition, but only for those that already understand the concepts or know how to speak mathematics. In the effort to be concise and complete the usefulness of the encyclopedic information often becomes nullified. It is easy to forget who our customers are, who uses the information and what is its purpose? Start with the most basic commonly held definitions give them a solid foundation and build on them. Dissymmetry (talk) 14:46, 7 June 2012 (UTC)

## standard definition

The standard definition of arc length---the sup of polygonal path lengths---is not even mentioned here. I'll be back..... Michael Hardy 22:51, 7 May 2006 (UTC)

That definition applies to all curves in a metric space, not just rectifiable arc segments (differentiable curves?) in Euclidean space, in which case one might wonder why the article is not called "Curve length" or something like that. See also the above discussion on the article title. If someone feels like improving the present article, then note that it never defines what an "irregular arc segment" is, does not give any definition of arc length but only ways to determine it, and further confuses functions with their graphs plotted as curves in space. I also find the account under "Ancient" rather dubious and doubt that it is true as stated.
If general curves are treated, it would be reasonable to give fractal curves as an example of curves of infinite length; see also How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. --LambiamTalk 23:25, 7 May 2006 (UTC)

"Rectifiable" does not imply "differentiable". I thought "rectifiable" meant the total length is finite, regardless of differentiability, or at least the length between any two points on the curve is finite (as with a line). Michael Hardy 03:13, 9 May 2006 (UTC)

The parenthetical question was a guess at a possibly intended meaning of "arc segment", given PlanetMath's definition of "arc length" as "the length of a section of a differentiable curve". --LambiamTalk 08:44, 9 May 2006 (UTC)

In regard to curves that have infinite length between two points on the curve, one could see also space-filling curve. Michael Hardy 21:35, 9 May 2006 (UTC)

Indeed, a special but well-known case of fractals having Hausdorff dimension equal to 2.

Isn't there also a definition based on the limits of a cover of balls? --njh 11:51, 16 May 2006 (UTC)

Both Hausdorff dimension and Minkowski-Bouligand dimension say something like: I am what you get as a limit of a cover of balls. Both are written in a way that makes it hard to see how this relates to the more formal definitions given. How they can be "informally" the same but actually different is not immediately clear to me. --LambiamTalk 13:12, 16 May 2006 (UTC)
They talk about computing the dimension of a structure. I recall there being a way to define arc length itself as a covering. --njh 02:19, 17 May 2006 (UTC)

## Derivation

A representative linear element of the function ${\displaystyle {\begin{cases}y=t^{5}\\x=t^{3}\end{cases}}}$

I would like to show the derivation of at least the fundamental parametric arc length formula, however since I'm new to this math format and it is a moderately long explination, I'm going to be working on it here and moving it into the article once it is complete and has been peer reviewed. Please let me know of any errors that you may see, I will move it into the main page in the near future. If anyone might be able to explain the conversion of this integral into the one used for ordinary functions (f(x)) it would be very useful.

In order to approximate the arc length of the curve, it is split into many linear segments. To make the value exact, and not an approximation, we will need infinitely many linear elements. This means that each element is infinately small. This fact manifests itself later on when we use an integral.

We start by looking at a representative linear segment (see image) and observe that its length (element of the arc length) will be the differential ds. We will call the horizontal element of this distance dx, and the vertical element dy.

Now, recall the distance formula which tells us that

${\displaystyle ds={\sqrt {dx^{2}+dy^{2}}}}$.

Since the function is defined in time, we add up the segments (ds) across infintessimally small intervals of time (dt) yielding the integral:

${\displaystyle \int _{a}^{b}{\sqrt {{\bigg (}{\frac {dx}{dt}}{\bigg )}^{2}+{\bigg (}{\frac {dy}{dt}}{\bigg )}^{2}}}\,dt}$

Which is the arc length from ${\displaystyle t=a}$ to ${\displaystyle t=b}$ of the parametric function f(t).

For example, the curve in this figure if defined by:

${\displaystyle {\begin{cases}y=t^{5}\\x=t^{3}\end{cases}}}$

Subsequently, the arc length integral fo values of t from −1 to 1 is:

${\displaystyle \int _{-1}^{1}{\sqrt {(3t^{2})^{2}+(5t^{4})^{2}}}\,dt=\int _{-1}^{1}{\sqrt {9t^{4}+25t^{8}}}\,dt}$

Using computational approximations, we can obtain a very accurate (but still approximate) arc length of 2.905.

48v 03:18, 12 July 2006 (UTC)

This has now been added to the article. 48v 21:30, 14 July 2006 (UTC)

A revision of the first image, swapping labels dx and dy is at http://en.wikipedia.org/wiki/File:Arc_length_approximation.svg. It also has the ds label moved. This article's preceding section says, "Consider an infinitesimal part of the curve ds (or consider this as a limit in which the change in s approaches ds)." which seems to allow either the portion of curve within the triangle or the straight line which approximates it to be labeled ds. I'm not sure which is more proper. Should the revised images have the ds label at the original location?

96.48.157.217 (talk) 13:59, 16 April 2013 (UTC)

Whbny (talk) 21:06, 4 February 2018 (UTC)whbny %%% The last derivation regarding arc length being independent of parametrization has a few serious problems. First, a continuously differentiable bijection does not need to be non-decreasing: for example, \phi(x)=-x. Second, the inverse of a continuously differentiable bijection need not be differentiable: consider the bijection \phi(x)=x^3, whose inverse bijection fails to be differentiable at x=0. In fact, the derivative of \phi^{-1}(x) becomes unbounded around x=0. These matters can be fixed (I can fix them but I need to get more accustomed to using this Talk page and format).

## Notations used in formulae

The lowercase 'e' in italics is used for Euler's Number. The symbol used in this context should be 'ε' or 'є,' not e.

The Roc 1217 22:28, 4 April 2007 (UTC)

## Derivation without parametric

I think the article is missing a good demonstration not involving the parametric form. This is my work on it, and I think it could be included in the article, though a more rigorous treatment is welcome.

We have a function f(x). It is continuous in the interval [a,b]. Now, let us divide this interval in equal parts, Δx. Starting from a, each successive increment in x, will also increment the f(x), and the arc lenght of that part will be given as:

${\displaystyle L_{i}={\sqrt {\Delta x^{2}+[f(a+i\Delta x))-f(a+(i-1)\Delta x]^{2}}}}$

Foe example, the first increment will result in an arc length of approximately:

${\displaystyle L_{1}={\sqrt {\Delta x^{2}+[f(a+\Delta x))-f(a)]^{2}}}}$

The total arc length will be:

${\displaystyle L_{T}=\sum _{a}^{b}{\sqrt {\Delta x^{2}+[f(a+i\Delta x))-f(a+(i-1)\Delta x]^{2}}}}$
${\displaystyle L_{T}=\sum _{a}^{b}{\sqrt {\Delta x^{2}\left(1+\left[{\frac {f(a+i\Delta x)-f(a+(i-1)\delta x]}{\Delta x}}\right]^{2}\right)}}}$
${\displaystyle L_{T}=\sum _{a}^{b}\Delta x{\sqrt {1+\left[{\frac {f(a+i\Delta x)-f(a+(i-1)\delta x]}{\Delta x}}\right]^{2}}}}$

Now, taking the limit as Δx becomes infinitesimal, and summing up all the parts, we get the expression:

${\displaystyle L_{T}=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\,dx}$

Goldencako 00:41, 7 November 2007 (UTC)

## First section, "General approach", needs work

"A curve in, say, the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment using the theorem of Pythagoras, the total length of the approximation can be found by summing the lengths of each linear segment.

Using a larger number of segments (with each segment often of a smaller length) usually provides a better approximation to the curve, and the length of such an approximation will be greater than one using fewer segments.

For some curves it is the case that none of the lengths of the approximations exceeds a certain smallest number L. If such a number exists, then the curve is said to be rectifiable and the curve is defined to have arc length L."

The first paragraph should be removed entirely, since above all, how to calculate arclength for practical purposes is a separate issue from how arclength is defined -- the definition is more basic and should come first and be clearly labeled as such. (Even if it were appropriate to begin with a discussion of how to *calculate* arclength, this paragraph makes no statement that the cartesian coordinates of the segments' endpoints are known -- which makes the mention of "using the theorem of Pythagoras" irrelevant.)

The word "usually" in the second paragraph seems strange. To make sense, it needs to be stated what a "better approximation to the curve" means, if this phrase is to be used.

The third paragraph is false as stated. (E.g., the length of each polygonal approximation to a unit circle does not exceed 99, but 99 exceeds 2π, the arclength of this circle.)Daqu (talk) 09:08, 17 August 2008 (UTC)

In order to create this new section I separated out statements about the intuition behind calculating arc length, and provided a better definition (see the history for more details), so some of it is a re-write of stuff previously included. The previous version had intuitive statements about arc length being defined for R2, and I thought that keeping these ideas in the context of the plane was a good way of introducing the idea, since, perhaps, most people might find the idea of a curve in n-dimensional space a little intimidating.
I disagree that the definition is more basic than the intuitive version, for two reasons: (1) It's not more basic for non-mathematicians; (2) The definition is precisely that intuitive concept given symbolic form.
The language for this section is written informally: if you think it needs work, go ahead and make the changes. And if it's imprecise, go ahead and make those changes too. It's anybody's article.
Finally, the cartesian coordiates of the segments' endpoints is relevant only if one is working in Euclidean space; if X were a general metric space, I don't know what "cartesian coordinates" would mean. Xantharius (talk) 21:37, 17 August 2008 (UTC)

## Slightly different way should be removed

This section is flagged with a cleanup, but it's not worth cleaning up. It's not substantially different from the main derivation, and full of grammatical and notational errors. I suggest it be removed. Bryanclair (talk) 20:29, 31 March 2009 (UTC)

I second this removal.—Tetracube (talk) 20:35, 31 March 2009 (UTC)

## Time?

Would it not be better for the derivation to call t a parameter rather than saying "time"? 149.157.1.184 (talk) 16:53, 16 May 2009 (UTC)

## Incorrect use of term "Straight Line"

Within section "Historical methods", subsection "Ancient", a 'straight line' is said to have a definite length. According to MathWorld a line (sometimes called a 'straight line') is "a straight one-dimensional figure having no thickness and extending infinitely in both directions".

In the context of this article, I believe the correct term should be 'line segment' which does have a definite length. Fhv1374 (talk) 08:02, 3 December 2009 (UTC)

## petal

Hi. What means petal here ? petal of Lea-Fatou flower  ? --Adam majewski (talk) 18:08, 18 February 2013 (UTC)

## incorrect edit by prodigy

The recent change to the formula for arclength is in error. Tkuvho (talk) 09:12, 29 December 2014 (UTC)

## General approach section

I removed the following paragraph from this section:

Polygonal approximations are linearly dependent on the curve in a few select cases. One of these cases is when the curve is simply a point function as is its polygonal approximation. Another case where the polygonal approximation is linearly dependent on the curve is when the curve is linear. This would mean the approximation is also linear and the curve and its approximation overlap. Both of these two circumstances result in an eigenvalue equal to one. There are also a set of circumstances where the polygonal approximation is still linearly dependent but the eigenvalue is equal to zero. This case is a function with petals where all points for the polygonal approximation are at the origin.

"eigenvalue" and "linearly dependent" make no sense here. The entire paragraph is confusing, and the point it seems to be trying to make is inappropriate in a section that's trying to give a gentle introduction by explaining the basic idea. — Preceding unsigned comment added by 24.56.116.202 (talk) 17:00, 10 August 2015 (UTC)

## Definition

Tosha, do you mean the curve should be defined as a piecewise differentiable mapping from an interval to ${\displaystyle \mathbb {R} ^{n}?}$ This would certainly be a more useful definition for the vast majority of applications. Jrheller1 (talk) 23:38, 15 February 2016 (UTC)

## Example of numerical integration

This information would be useful to anybody who writes a program that needs to calculate the arc length of a curve. It shows that the fact that there is usually no closed form solution for the arc length integral doesn't really matter because numerical integration (even very accurate numerical integration) is usually very efficient. Gauss-Kronrod rules and Gaussian quadrature rules are just the standard ways of doing numerical integration of a non-singular integrand. Jrheller1 (talk) 05:13, 5 March 2016 (UTC)

For theoretical purposes, especially on a general curve of a particular family (such as an arbitrary parabola for which you don't know its parameters), a closed-form solution is much preferable, even if given in terms of special functions like elliptic integrals. Anyone writing a program to compute arc length would probably use numerical integration even if the curve possessed a closed-form arc length formula, since even if it exists, it may be unsuitable for numerical purposes. More generally, the efficiency of numerical integration does no good for theoretical purposes for any definite integral in general. As someone focusing on pure math (at least for now), it only distracts me when I see a remark on numerical integration.--Jasper Deng (talk) 07:59, 5 March 2016 (UTC)
I find the practical information useful; the article needs not be just theoretical. fgnievinski (talk) 00:51, 12 March 2016 (UTC)
If someone wants to do numerical integration they can head over to the article on that. Why is adding numerical integration here any more useful than other situations not determinable in closed form, such as elliptic integrals and the error function?--Jasper Deng (talk) 21:19, 15 March 2016 (UTC)
Numerical integration is applied to many many different mathematical problems; it is impractical for all of them to be covered in a single article. As the OP explained, there are some nuances to this application, that are not obvious to the average reader. The error function can be consider a closed-form_expression#Alternative definitions. fgnievinski (talk) 00:59, 16 March 2016 (UTC)
But how are any of these nuances specific to the problem of arc length? The OP only mentions that these are generally applicable numerical integration methods, without discussion of how arc length integrands are any more suited to those methods than general integrands.--Jasper Deng (talk) 08:25, 16 March 2016 (UTC)
I'd support summarizing that sub-section as a single sentence, just mentioning that quadrature can be readily applied after finding a suitable parameterization; you're right in that currently not much is said about its details (which can be quite complicated); the lengthy numerical example is not particularly illuminating. fgnievinski (talk) 09:42, 16 March 2016 (UTC)
I think that would be satisfiable for me too. How about ?--Jasper Deng (talk) 07:19, 17 March 2016 (UTC)

The history of the page shows that there was an example of numerical integration for more than 9 years and nobody had a problem with it all that time. While removing several redundant and not very good derivations of the arc length integral, I removed this example temporarily back in February 2016. Then I replaced the previous example with a better example. In this new example, the integral has an easily computed transcendental value, so it is easier to analyze the error of Gauss or Gauss-Kronrod quadrature (or any other numerical integration method). How could anybody possibly have a problem with this improved example? Jrheller1 (talk) 04:16, 28 April 2016 (UTC)

That's not relevant. My issue is with including an example at all.--Jasper Deng (talk) 15:23, 28 April 2016 (UTC)

## Smooth or C1

In the section Arc length#Definition for a smooth curve it starts by saying

Let ${\displaystyle f:[a,b]\to \mathbb {R} ^{n}}$ be a continuously differentiable function.

which means that the function is ${\displaystyle C^{1}}$, the remainder of the section seems to work for ${\displaystyle C^{1}}$ curves. The definition of a "smooth" function is that it is ${\displaystyle C^{\infty }}$. Should the section be renamed "Definition for a differentiable curve", or am I missing something?--Salix alba (talk): 05:10, 10 May 2016 (UTC)

"Once continuously differentiable" or ${\displaystyle C^{1}}$ is more precise than "smooth". Sometimes ${\displaystyle C^{1}}$ or ${\displaystyle C^{2}}$ curves or surfaces are referred to as "smooth", though. A ${\displaystyle C^{1}}$ or ${\displaystyle C^{2}}$ curve appears to be "smooth" (using the word smooth in a physical sense rather than strict mathematical sense). Jrheller1 (talk) 02:00, 13 May 2016 (UTC)
I'm agnostic whether narrowing the discussion to ${\displaystyle C^{1}}$ is important (Wikipedia generally eschews highly technical arguments, and "smooth" is often understood by context), but I object to the use of ${\displaystyle |f'(t)|}$ rather than ${\displaystyle ||f'(t)||}$. The function ${\displaystyle f(t)}$ and its derivative are vector valued, so the (Euclidean) norm rather than absolute value is needed in the arc length formula.Hardmath (talk) 06:12, 27 December 2017 (UTC)
Single vertical bars are often used for vector norm, for example in Rudin (one of the sources cited). Jrheller1 (talk) 16:40, 28 December 2017 (UTC)

## Unnecessary verbiage

The words "where the supremum is taken over all possible partitions ${\displaystyle a=t_{0} of ${\displaystyle [a,b]}$" are perfectly clear. But someone keeps inserting the unnecessary and distracting verbiage "please note ${\displaystyle N}$ is not a fixed number, i.e., the supemum is taken over all possible finite partitions of the interval ${\displaystyle [a,b]}$, where the curve ${\displaystyle f}$ is defined." Jrheller1 (talk) 04:52, 27 September 2016 (UTC)

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# Difference between revisions of "Main Page"

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## Latest comic

 Modified Bayes' Theorem Title text: Don't forget to add another term for "probability that the Modified Bayes' Theorem is correct."

## Explanation

 This explanation may be incomplete or incorrect: When using the Math-syntax please also care for a proper layout. Please edit the explanation below and only mention here why it isn't complete. Do NOT delete this tag too soon.

Bayes' Theorem is an equation in statistics that gives the probability of a given hypothesis accounting not only for a single experiment or observation but also for your existing knowledge about the hypothesis, i.e. its prior probability. Randall's modified form of the equation also purports to account for the probability that you are indeed applying Bayes' Theorem itself correctly by including that as a term in the equation.

Bayes' theorem is:

$P(H \mid X) = \frac{P(X \mid H) \, P(H)}{P(X)}$, where

• $P(H \mid X)$ is the probability that $H$, the hypothesis, is true given observation $X$. This is called the posterior probability.
• $P(X \mid H)$ is the probability that observation $X$ will appear given the truth of hypothesis $H$. This term is often called the likelihood.
• $P(H)$ is the probability that hypothesis $H$ is true before any observations. This is called the prior, or belief.
• $P(X)$ is the probability of the observation $X$ regardless of any hypothesis might have produced it. This term is called the marginal likelihood.

The purpose of Bayesian inference is to discover something we want to know (how likely is it that our explanation is correct given the evidence we've seen) by mathematically expressing it in terms of things we can find out: how likely are our observations, how likely is our hypothesis a priori, and how likely are we to see the observations we've seen assuming our hypothesis is true. A Bayesian learning system will iterate over available observations, each time using the likelihood of new observations to update its priors (beliefs) with the hope that, after seeing enough data points, the prior and posterior will converge to a single model.

If $P(C)=1$ the modified theorem reverts to the original Bayes' theorem (which makes sense, as a probability one would mean certainty that you are using Bayes' theorem correctly).

If $P(C)=0$ the modified theorem becomes $P(H \mid X) = P(H)$, which says that the belief in your hypothesis is not affected by the result of the observation (which makes sense because you're certain you're misapplying the theorem so the outcome of the calculation shouldn't affect your belief.)

This happens because, if you apply the original theorem, the modified theorem can be rewritten as: $P(H \mid X) = P(H)(1-P(C)) + P(H \mid X)P(C)$. This is the linear-interpolated weighted average of the belief you had before the calculation and the belief you would have if you applied the theorem correctly. This goes smoothly from the not believing your calculation at all, keeping the same belief as before if $P(C)=0$ to changing your belief exactly as Bayes' theorem suggests when $P(C)=1$.

$1-P(C)$ is the probability that you are using the theorem incorrectly.

As an equation, the rewritten form makes no sense. $P(H \mid X) = P(H)(1-P(C)) + P(H \mid X)P(C)$ is strangely self-referential and reduces to the piecewise equation $\begin{cases}P(H \mid X) = P(H) & P(C) \neq 1 \\ 0 = 0 & P(C) = 1 \end{cases}$. However, the Modified Bayes Theorem includes an extra variable not listed in the conditioning, so a person with an AI background might understand that Randal was trying to write an expression for updating $P(H \mid X)$ with knowledge of $C$ i.e. $P(H \mid X,C)$, the belief in the hypothesis given the observation $X$ and the confidence that you were applying Bayes' theorem correctly $C$, for which the expression $P(H \mid X,C) = P(H)(1-P(C)) + P(H \mid X)P(C)$ makes some intuitive sense.

The title text suggests that an additional term should be added for the probability that the Modified Bayes Theorem is correct. But that's *this* equation, so it would make the formula self-referential. It could also result in an infinite regress -- we'd need another term for the probability that the version with the probability added is correct, and another term for that version, and so on. It's also unclear what the point of using an equation we're not sure of is (although sometimes we can: Newton's Laws are not as correct as Einstein's Theory of Relativity but they're a reasonable approximation in most circumstances. Alternatively, ask any student taking a difficult exam with a formula sheet.).

## Transcript

Modified Bayes' theorem:
P(H|X) = P(H) × (1 + P(C) × ( P(X|H)/P(X) - 1 ))
H: Hypothesis
X: Observation
P(H): Prior probability that H is true
P(X): Prior probability of observing X
P(C): Probability that you're using Bayesian statistics correctly

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# Homework Help: I want check my answer ()()()

1. Nov 12, 2009

### r-soy

Hi

I want check my answer two Question

Question 1

Question 2

2. Nov 12, 2009

### Staff: Mentor

You're starting with an equation, so each step should be a new equation.
xlog x = 100x
log(xlog x) = log(100x)
(log x)(log x) = log 100 + log x
Continue from there. You should end up with a value of x that is correct to three significant digits.

3. Nov 12, 2009

### r-soy

Hi mark

why we said log 100 + log x ?

why + log x ?????

4. Nov 12, 2009

Because

$$\log(ab) = \log(a) + \log(b)$$

5. Nov 14, 2009

### r-soy

thanks >>

6. Nov 14, 2009

### Staff: Mentor

For Q2, your answer is close, but should be rounded to 34.7 years, to the nearest tenth of a year.

Also, there is no In operation. This is Ln, and stands for natural Logarithm, which in Latin is logarithmus naturalis.

BTW, you might get quicker help if you posted the problem and your work rather than posting a scanned copy of your work. Speaking for myself, it's much easier when I can write my response while looking at the work, rather than having to have two windows open and jump back and forth between them.


text:
## anonymous 5 years ago Find the zeros of the function x^2 - x-4

Do you know how to factor this equation?

2. anonymous

Yes but my problem is, is that I can't think of anything that added toger is 1 and multi to get 4

Hm. Fair enough. So there are a couple of ways to approach this. Do you know the quadratic formula?

4. anonymous

yes -2 +- $-b +_\sqrt{b^2 - 4ac} \over 2a$

Right. Do you know what a, b, and c are in your equation?

6. anonymous

yes a is 1x^2 b is 1x and c is 4

Close, a, b, and c are the coefficients -- so a is 1, b is -1, and c is -4. Remember that a minus sign is part of the coefficient :) Can you plug those in?

8. anonymous

yeah I get $1 \pm \sqrt{15} \over 2$

Right-o. And there are your two zeros.

10. anonymous

what are the two zeros?

The quadratic equation describes two results: $x = \frac{1 + \sqrt{15}}{2}$ and $x = \frac{1 - \sqrt{15}}{2}$ Those are the two zeros of the equation.

12. anonymous

oh, ok thanks alot

No problem, glad to help :)

14. anonymous

wouldn't this be 17 instead of 15?


text:
MathSciNet bibliographic data MR1165063 (93k:54061) 54E30 Mizokami, T. On $\phi$$\phi$-extensions of developable spaces. Proc. Amer. Math. Soc. 119 (1993), no. 1, 331–336. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# THE AUDITORY MODELING TOOLBOX

Applies to version: 0.9.8

Go to function

# takanen2013_cueconsistency - Check the consistency before the cues are combined

## Usage

[directionCues cueEnergies] = takanen2013_cueconsistency(directionCues, cueEnergies,fc)

## Input parameters

 directionCues sturcture containing the directional cues obtained from the model of the MSO, LSO, and wideband MSO of the two hemispheres% cueEnergies sturcture containing the energies of the directional cues fc characteristic frequencies

## Output parameters

 directionCues sturcture containing the directional cues obtained from the model of the MSO, LSO, and wideband MSO of the two hemispheres% cueEnergies sturcture containing the energies of the directional cues

The role of the interaural time shifts, envelope time shifts, and level differences in localization are different in different frequency ranges. In order to emulate this when the directional cues are combined, the consistency of the cues is checked and the energies of the different cues are affected. More detailed description about the process can be found in Takanen, Santala, Pulkk 2013 (Sec. 3.2.4)

## References:

M. Takanen, O. Santala, and V. Pulkki. Visualization of functional count-comparison-based binaural auditory model output. Hearing research, 309:147-163, 2014. PMID: 24513586. [ DOI ]


text:
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Категории сайта
1/ Взять подсказку
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# Вопросы и ответы к Seagull CES тесту для моряков по международному кодексу морской перевозки опасных грузов

Страница содержит ответы на “Seagull CES тест онлайн по теме Кодекс ММОГ” (Crew Evaluation Test v5), и служит в качестве базы данных вопросов и ответов, используя которую пользователь может подготовиться к экзаменам по аттестации, получению диплома или свидетельства, либо просто проверить себя в знании тех основ, которые необходимы морякам в данной области.

Воспользуйтесь поиском, чтобы найти интересующий вас вопрос. Достаточно ввести одно слово или часть его, чтобы фильтр заработал.

Количество отображаемых вопросов: 65.

Правильные ответы отмечены знаком .

According to the IMDG Code, what are “labels”?
Notices, that are fixed to the outside surfaces of small packages to indicate the hazards associated with the substances in the packages.
Large notices, that are fixed to the outside surfaces of cargo transport units to provide a warning, that the contents of the unit are dangerous goods and present risks.
The unique identifying numbers assigned to dangerous substances, materials or articles.
Large signs, identifying dangerous goods.
According to the IMDG Code, what are “placards”?
Large labels, that are fixed to the outside surfaces of cargo transport units, to indicate the hazards associated with the substances contained in the unit.
The unique identifying numbers, assigned to dangerous substances, materials or articles.
Notices, that are fixed to the outside surfaces of small packages, to provide a warning, that the contents of the package are dangerous and present risks.
Large signs, identifying dangerous goods.
According to the IMDG Code, what is the correct way to store drums, containing dangerous goods?
Upright.
On their sides.
On pallets and shrink-wrapped, in batches of eight drums.
In a pyramid pattern, not more than three tiers high.
According to the IMDG Code, where should placards be located on a freight container containing dangerous goods?
One on each end only.
One on each end and one on each side.
One on each side only.
One on each end and one visible from directly above.
According to the IMDG Code, which one of the options identifies the item or items, that road vehicles which are used for the transport of dangerous goods must have?
A yellow flashing light.
Suitable supports and securing attachments, which provide a safe and stable platform during transportation.
Motion sensors, which activate a remote alarm if the vehicle breaks loose during transportation by sea.
Suitable bracing arrangements, such that they can be attached to other vehicles, and create more rigid unit during transportation.
All substances have a unique chemical composition throughout. What chemical term is used to describe this?
Homogeneous.
Homocyclic.
Homopolymer.
Unilateral.
Besides seafarers engaged in the carriage of dangerous goods, who else does the IMDG Code specifically require should be trained in their handling?
Shore-based personnel, engaged in the handling of dangerous goods.
Pilots.
Port officials.
All persons visiting the ship while she is carrying dangerous goods.
Can a cargo transport unit used for the transport of dangerous goods, which is found to be damaged, leaking or otherwise losing its contents be accepting for loading? Select the most correct answer:
No.
Yes.
Yes, provided the area around it is regularly swept or hosed down.
Only if the shipper provides a written guarantee to be responsible for any costs, resulting from the damage.
Can the hazard, posed by a substance, arise from a dangerous reaction with other substances in the surrounding environment?
No.
Yes.
Class 2 (Gases) has three divisions. Two of these are “Flammable” and “Non-flammable, non-toxic”. Which of the following is the third division of this Class?
Corrosive.
Toxic.
Inflammable.
Reactive.
Column 16b of the Dangerous Goods List shows the requirements for segregation. What is segregation in terms of carriage of dangerous goods on board ship?
Stowage describes the location of dangerous goods horizontally on board a ship, while segregation indicates their vertical separation.
The system used in plans, showing the stowage of dangerous goods.
The way, that dangerous goods are separated by location on board.
Corrosive substances can cause severe damage to skin tissue through contact. Which of the following is also a risk with all corrosive substances shipped in packages?
That they will cause anyone entering the affected storage compartment to feel nauseous and disorientated.
That they will create a flammable atmosphere in the hold.
That they will cause toxic vapours to enter the ship’s ventilation system.
That they can damage other cargoes or the ship’s framework if they leak from damaged packaging.
For consignments of dangerous goods, the appropriate emergency information shall be available at all times. Where must it be kept?
In a place on board separate from the goods themselves and immediately accessible in the event of an incident.
With the dangerous goods themselves.
In the shore office.
Such emergency information is available by telephone or e-mail at all times directly from the IMO.
For the purposes of studying the IMDG Code, what are the three main states of matter?
Solid, liquid and gas.
Frozen, moisture and liquid.
Materials, substances and articles.
Explosive, poisonous and corrosive.
For which one of the following can guidance be found in the Supplement to the IMDG Code?
The safe use of pesticides in ships.
The safe handling of animals on board cargo ships and livestock carriers.
Reporting procedures for arrival in quarantined ports.
Anti-piracy measures.
Goods of Class 1 (Explosives) are described as “compatible” if which of the following conditions are met?
They can be safely stowed or transported together without significantly increasing the chance of an accident.
They are stowed more than 12 metres horizontally apart within the same compartment, or in a separate compartment.
They contain no live detonators or fuses.
They have been approved by the UN Committee on Classification of Dangerous Goods as mutually insensitive.
How is the technical information displayed on a portable tank?
It is stamped on a metal plate, fixed to the tank, where it can easily be seen.
It is painted on the front and back of the tank.
It is printed on both sides of the tank.
It is written on a placard fixed to the tank adjacent to the drain valve.
How many Classes of dangerous goods are there in the IMDG Code?
3.
9.
10.
12.
How must salvage packaging be specially marked?
By the word SALVAGE.
By the words EMERGENCY PACKAGING.
With the warning CAUTION – SALVAGED GOODS.
With the caution DANGER – SALVAGED GOODS.
How should marine pollutants be stowed?
They should be securely stowed in such a manner as to minimise the risk of them being lost overboard.
They should be stowed “On Deck Only”, because of the hazards they present.
They should be stowed adjacent to the living quarters.
It is permitted to stow marine pollutants anywhere within the cargo area of the ship, provided they are, at least 6 metres horizontally from the ship’s side.
If a mixture is above its upper flammable limit, how is the atmosphere described?
Too rich.
Too lean.
Saturated.
Inflammable.
If a mixture is above its lower flammable limit, how is the atmosphere described?
Too rich.
Too lean.
Saturated.
Inflammable.
If it is safe to do so, how should a spillage of a substance listed as a marine pollutant preferably be dealt with?
It should be collected for safe disposal and absorbent material should be used to collect spilt liquid.
It should be washed directly overboard.
It should be sprayed with caustic soda and then washed overboard.
It should be pumped down to the engine-room and held in a spare storage tank until it can be landed to shore.
If you discover a leaking tank container on the weather deck, which of the following actions should you take first?
Report it immediately to the Duty Officer.
Wash it away immediately.
Examine the tank container and try and close off any leaking valves.
Make sure the Chief Officer knows about it before arrival, so that no vehicles drive through the spillage.
In addition to the IMDG Code, which other book should be referred to for guidance on security matters?
MARPOL 73/78 (as amended).
The International Maritime Security (IMS) Code.
The Seafarers Security Handbook.
The International Ship and Port Facility Security (ISPS) Code.
In which of these can you find the Dangerous Goods List?
Volume 1 of the IMDG Code.
Volume 2 of the IMDG Code.
The Supplement to the IMDG Code.
MARPOL Annex II.
In which part of the diamond-shaped shipping label for dangerous goods would you find the class and division numbers?
The upper part.
The lower part.
The left-hand part.
The right-hand part.
Packaging’s shall be so constructed and sealed as to prevent any loss of contents. Which two of the following are among those circumstances, which the IMDG Code says should be considered in this respect?
Vibration.
Changes in pressure.
Being inverted.
Damage through an impact equal to a force of 300 kg point load.
Ships are divided into two groups for the purpose of making appropriate stowage recommendations in accordance with the IMDG Code. Which are they?
Cargo ships or passenger ships carrying a limited number of passengers, being a maximum of 25 or 1 per 3 metres of length.
Other passenger ships in which a larger number of passengers is carried.
Oil tankers, chemical tankers and gas carriers, which are not subject to the provisions outlined in volume 2 of the IMDG Code.
Cargo ships equipped for the carriage of dangerous goods in packaging’s, IBC’s, portable tanks and MEGCs.
The MFAG is written around the substances, material and articles covered by the IMDG Code and which other Code?
The IMSBC Code.
The IGC Code.
The ISPS Code.
The IBC Code.
The Medical First Aid Guide for Use in Accidents Involving Dangerous Goods provides advice intended for which of the following?
The initial management of chemical poisoning and diagnosis within the limits of the facilities available at sea.
The comprehensive and total management of chemical poisoning and diagnosis to full recovery, with assistance from shore as required.
The transfer of a casualty into shore side care with full and appropriate medical documentations.
The limited care of casualties in exposure incidents involving liquid chemicals.
The classes drawn up by the UN Committee of Experts for the safe handling of hazardous substances are based on which of the following?
Alphabetical order.
UN Number.
The type of hazard they present.
Random laboratory tests carried out under the UN guidelines.
The proper shipping name and UN Number of a substance must be listed material or article on the Dangerous Goods Form. This is so that the Master has the necessary information to hand, if he has to report a pollution incident involving this substance. But which of the following is also a reason?
So that the substance can be readily identified during transport, especially in the event of an emergency.
To assist in stowage of the package.
To avoid customs fines for improper documentation when passing through controlled waterways.
To demonstrate compliance with Part B of the ISPS Code.
The proper shipping name and UN Number of a substance must be listed on the Dangerous Goods Form. This is so that the substance can be readily identified during transport, especially in the event of an emergency. But which one of the following is also a reason for this?
So that the Master has the necessary information to hand if he has to report a pollution incident involving this substance.
To assist in stowage of the package.
To avoid being detailed by the authorities for improper documentation when entering port.
To demonstrate compliance with section 6.5 of the ISPS Code.
The reporting of incidents involving harmful substances and/or marine pollutants is regulated under which international regulations?
MARPOL 73/78.
Part A of SOLAS 1974.
Appendix B of the Supplement to the IMDG Code.
The EmS Guide.
What do the letters “IMDG”stand for when we refer to the IMDG Code?
International Maritime Dangerous Goods.
Internal Marine Dangerous Goods.
International Marine Danger Group.
International Monitoring Department (Geopolitics).
What does the EmS number refer to?
The relevant emergency schedules for FIRE and SPILLAGE in the Emergency Response Procedures for Ships Carrying Dangerous Goods.
The emergency response procedures as outlined in 46 CFR 32 of the US Federal Code of Regulations and expanded in the USCG Guidelines on the Transportation of Dangerous Goods by Sea.
The relevant guidelines listed in the International Maritime Organization Emergency Anti-Pollution Measures (1974).
The appropriate entry in the Emergency Procedures (Shipping).
What does the abbreviation “IBC” stand for?
Intermediate Bulk Container.
Intermediate Bulk Code.
Internal Bulk Construction.
International Booking Code.
What is a “UN Number”?
An international fire-fighting reference number.
An emergency contact number for use in the event of an injury or pollution incident involving a dangerous substance as listed in the Dangerous Goods List in the IMDG Code.
A unique international chemical symbol assigned to a dangerous substance or material.
A unique identifying number assigned to a dangerous substance, material or article by the United Nations Committee of Experts on the Transport of Dangerous Goods.
What is an “article”?
A device, that contains a dangerous substance or mixture of substances.
A form, that contains important information about a dangerous substance.
A device, that is responsible for initiating a dangerous reaction.
Something, that is packed within a freight container.
What is meant by the term salvage packaging in relation to the IMDG Code?
Special packaging into which damaged, defective, leaking or non-conforming packages or dangerous goods, that have spilled or leaked, are placed for the purposes of transport for recovery or disposal.
Packaging, used only at the direction of a contracted environmental protection response company.
Packaging, which is manufactured from recycled material.
Packaging, in which goods recovered from the sea are held, pending investigation by the appropriate authority.
What special mark must a plastic packaging bear if it is recycled?
REC.
R.
RECYCLED.
Caution! Recycled Packaging!
When should all placards, orange panels, marks and signs be removed from a cargo transport unit?
As soon as the dangerous goods or their residues are discharged.
As soon as the cargo transport unit is discharged from the ship.
Once the cargo transport unit has been loaded and sealed.
Placards, orange panels, marks and signs should never be removed from a cargo transport unit.
Where dangerous goods have been loaded into a road vehicle, what is the name of the form, that must be supplied by the person, responsible for that operation (assuming the information has NOT been incorporated into the Dangerous Goods Form)?
Container/vehicle packing certificate.
Vehicle manifest.
Packing note.
Modular operation transport certificate (MOTC).
Which abbreviation is used to identify a “Marine Pollutant” in the Dangerous Goods List?
SMP.
MARPOL.
P.
Which group of dangerous goods is generally understood to be the most dangerous for marine transportation?
Toxic gases.
Infectious substances.
Flammable liquids.
Organic peroxides.
Which of the following is a design feature of all intermediate bulk containers?
They must be fitted with a minimum of three pressure-relief valves.
They must be capable of withstanding a drop test of 12 metres.
They must be individually approved by the competent authority.
They must be suitable for handling by mechanical means.
Which of the following is a general principle of segregation?
Incompatible goods must be stowed in separate compartments, or at least 15 metres apart horizontally within the same compartment.
Incompatible goods must be stowed on separate decks.
Compatible goods shall be segregated from one another.
Incompatible goods shall be segregated from one another.
Which of the following is an example of an artificial mixture?
Glass.
Air.
Milk.
Petroleum.
Which of the following is another name for “smell”?
Odour.
Odonate.
Odeum.
Odorant.
Which one of the following is a correct statement?
The hazard posed by a substance can arise from a dangerous reaction of the substance or article with other substances in the surrounding environment.
Substances in the surrounding environment can have no dangerous impact on packaged dangerous goods.
A dangerous reaction between a substance or article and other substances in the surrounding environment is only possible, if those goods are packaged in a container constructed of soft, non-water-resistant material.
The hazard posed by a substance is no higher on board ship than it is ashore.
Which one of the following is the most accurate definition of the term “incompatible goods” in relation to the IMDG Code?
Those, which when stowed together result in a hazardous situation in the event of spillage, leakage or any other accident.
This is just another way of saying “dangerous goods”.
Those, which when stowed together with foodstuffs present a risk to health through the leakage of poisonous vapours.
Goods of different classes, as listed in the Dangerous Goods List.
Which one of the following would you consult for guidance on the fire-fighting procedures to follow in the event of an incident involving dangerous goods?
The MFAG.
The EmS Guide.
MARPOL 73/78.
The International Marine Fire-Fighting Directory.
Which one of the options best describes a multi-element gas container in relation to the IMDG Code?
A portable tank, which is divided by a number of internal bulkheads and is capable of being emptied via a single drain valve.
A special arrangement of cylinders and tubes, which has a manifold and is assembled within a framework and fitted with service and structural equipment.
A number of independent gas cylinders being transported under a single shipment and with the same UN Number.
A gas container, which is divided into individual compartments and is capable of simultaneous storage of different gases.
Which one of the options describes a requirement of the IMDG Code in respect of the testing of packaging?
All packaging’s must be of a successfully tested design.
Each individual packaging must undergo testing by the manufacturer before use.
Packaging will be subject to testing in the event of leakage or other failure.
The marking on every package must indicate when it was last tested.
Which one of the options identifies the first part of any marking on a packaging as required by the IMDG Code?
The UN packaging symbol.
The UN Number.
The mark of the authorizing State.
The manufacturer’s mark.
Which one of the options identifies where empty, uncleaned cylinders, which have previously contained gas of a toxic nature, be safely stowed, according to the IMDG Code?
Only on deck.
Only under deck.
Either on deck or under deck – there is no restriction on empty cylinders.
Empty gas cylinders must be cleaned before they can be accepted for carriage by sea.
Which one of these statements defines a “marine pollutant”?
A substance, which is subject to the provisions of Annex III of MARPOL.
A substance, which, because of its tendency to degrade in seafood, or because of its low hazard potential to the aquatic environment is subject to the provisions of SOLAS and transported accordingly.
A substance, which requires the highest form of protection during transportation and is regulated accordingly by the International Transportation Authority.
A substance, that discolours the water.
Which organization today is responsible for maintaining the IMDG Code?
The International Maritime Organization.
The International Hazardous Goods Committee.
The Governments of Member States.
Which packing group is assigned to substances presenting a low level of danger?
Packing Group I.
Packing Group II.
Packing Group III.
Packing Group L.
Which two kinds of “schedules” are found in the EmS Guide?
Fire.
Spillage.
Atmospheric pollution.
Medical emergencies.
Which two of the following are standard tests performed on representative samples of packagings?
Drop test.
Stacking test.
Rigidity test.
Crash test.
Which two of the following can the term “cargo transport unit” be applied to?
Aviation transport containers.
International livestock container.
Portable tanks.
Who is primarily responsible for completing the documents, relating to the shipping of dangerous goods (unless the Code specifically requires otherwise)?
The Shipper.
The Master.
Why should dangerous cargo be stowed clear of living quarters on the ship?
Because of the possibility, that dangerous vapours may leak and be drawn into the air intakes and ventilation ducts, entering the accommodation or machinery space.
Because they will be off-loaded first and so should be at the other end of the ship.
Because the area next to the accommodation is designated for the carriage of foodstuffs.
Because there are no hook-up points for the power to drive electrical cooling units; these must all be close to machinery space according to the Code.
Within the IMDG Code, the term “marking” refers to which one of the following?
Identification of the substance contained in a package.
Identification of the risks and hazards presented by substances contained in a package.
Identification of the risks and hazards presented by substances contained in cargo transport units.
Damage done to a package of dangerous goods while on board ship.
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# Tag Info

0

The other answer's suggestion of encrypting the file-encryption key twice is good, so I will only answer the part about backup key size. Now while using a 53bit key as a backup key may seem as a crypto joke, when we actually consider that an average password would be [...] If you go with the average password complexity, those users who do choose a ...

1

This kind of key management problem is a big reason why encryption is not more widely used for everyday data. I see a single easy way to do this, I think it is similar to the way disk encryption systems work, and it may not work for your uses. For each file, a random key $kf$ is generated. 2 copies of the key are encrypted, one with a key derived from the ...

1

I see no obvious security flaws, but it seems inefficient and is nonstandard. You would be better off always using the root key as the HMAC key and basing the HMAC message/data on only the key index (or other identity) rather than previous keys. That way you can create any of the keys independently. That is essentially what HKDF does in HKDF-Expand. If you ...

0

So is $2$ the private key here ? If it's a private key then both Alice and Bob know it even though the eavesdroppers don't know. Please explain how public key/private key pair is generated from this shared secret $2$. As the others said, $2$ is a shared secret, rather than a private key. It is usually used to derive one or more symmetric keys (e.g. for ...

2

$s$ is a shared secret key. It's known to both Alice and Bob. You could call is a private key, but the usual terminology is “secret key” here, for no deep reason. Alice has a private/public key pair: $a$ is her private key, $A$ is her public key. Ditto with $b$ and $B$ for Bob. These values are not useful in isolation though; in normal use, the only point ...

4

So is 2 the private key here ? No, it's referred to as a "shared secret" (because it is shared between Alice and Bob, and is secret to everyone else). If there were 'private' and 'public' keys (which is not the standard terminology with DH), then Alice's private key would be $a=6$, and the public key would be $g^a = 8$. In this case, the 'private key' ...

5

An important principle in cryptography is "key separation" which holds that one should "use distinct keys for distinct algorithms and distinct modes of operation". Violating key separation often opens up avenues of attacks that may break confidentiality, integrity, or even recover the key. You can use a KDF to derive cryptographically independent keys from ...

3

You don't actually need 384 bits of key material. The IV for GCM does not need to be secret, and may be chosen deterministically, e.g. as an incremental counter. Thus, you only need 256 bits for the AES key, which you already have. That said, if you did actually need more key material, you could use any standard KDF to expand your 256 bits. Since you ...

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# zbMATH — the first resource for mathematics

Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes. (English) Zbl 1176.76084
A number of nonconservative hyperbolic models have been introduced in fluid dynamics to serve as (simplified) models of two-phase or two-layer flows. The authors’ aim, in the present paper, is to address the fundamental question whether finite difference schemes for nonconservative systems converge toward correct weak solutions containing shock waves. Addressing this important issue requires detailed numerical computations, which they carry out here.
For the theory and numerical analysis of nonconservative products, the authors refer to Berthon, Coquel, and LeFloch, who connected the theory of nonconservative products with the concept of a kinetic relation. They introduced a general framework to handle nonconservative systems, which encompasses a large number of examples arising in applications. In particular, they rigorously analyzed a typical model of turbulent fluid dynamics by establishing the existence and properties of a physically relevant family of traveling waves and deriving the corresponding kinetic function.
The use of a numerical strategy based on a direct discretization of the nonconservative system by means of a finite difference scheme, which is formally path-consistent, is advisable and may have the following advantages:
1. The numerical solution is formally consistent with the definition of nonconservative product in the sense of Dal Maso, LeFloch and Murat and, in turn, in the special case when the system admits a conservative subsystem, the numerical scheme is conservative for this subsystem in the sense of Lax.
2. The approximations of the shock provided by the schemes are consistent with a regularization of the system with higher-order terms that vanish as $$\Delta x$$ tends to 0. (Obviously, the main drawback is that this regularization depends on the chosen family of paths and on the numerical scheme itself. This issue is dealt with in the present paper.)
3. As originally pointed out by T. Y. Hou, Ph. Rosakis and Ph. LeFloch [ibid. 150, No. 2, 302–331 (1999; Zbl 0936.74052)], in the (simpler) case of scalar hyperbolic equations, the convergence error, measured in terms of convergence error measure or in terms of Hugoniot curves, is noticeable for very fine meshes, for discontinuities of large amplitude, and/or for large-time simulations, only.
4. This strategy is extendable to high-order methods or to multidimensional problems, as developed, together with collaborators, by Coquel and Parés.
The convergence error should also be compared with the experimental error. In the case of the two-layer shallow water system, the shocks captured by Roe scheme and the family of straight lines have been found to be in good agreement with the experimental measurements of internal bores in the Strait of Gibraltar, despite of the simplicity of the family of paths.
In certain special situations, the convergence error measure is found to vanish identically. This is the case of systems whose nonconservative product is associated with a linearly degenerate characteristic field. For more general models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration; as the authors demonstrate, plotting the shock curves provides a convenient approach for evaluating the validity of a given scheme.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 76L05 Shock waves and blast waves in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text:
##### References:
 [1] Berthon, C.; Coquel, F., Nonlinear projection methods for multi-entropies navier – stokes systems, (), 278-304 · Zbl 1078.76573 [2] Berthon, C.; Coquel, F., Nonlinear projection methods for multi-entropies navier – stokes systems, Math. comput., 76, 1163-1194, (2007) · Zbl 1110.76035 [3] C. Berthon, F. Coquel, P.G. LeFloch, Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems, unpublished notes. · Zbl 1234.35190 [4] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Math., Birkhäuser Verlag, Bäsel, 2004. · Zbl 1086.65091 [5] Castro, M.J.; Gallardo, J.M.; Parés, C., High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems, Math. comput., 75, 1103-1134, (2006) · Zbl 1096.65082 [6] Castro, M.J.; Gallardo, J.M.; Parés, C., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. comput. phys., 227, 574-601, (2007) · Zbl 1126.76036 [7] Castro, M.J.; García, J.A.; González, J.M.; Macías, J.; Parés, C.; Vázquez, M.E., Numerical simulation of two layer shallow water flows through channels with irregular geometry, J. comput. phys., 195, 202-235, (2004) · Zbl 1087.76077 [8] Castro, M.J.; Macías, J.; Parés, C., A Q-scheme for a class of systems of coupled conservation laws with source term. application to a two-layer 1-D shallow water system, Math. mod. numer. anal., 35, 107-127, (2001) · Zbl 1094.76046 [9] Castro, M.J.; Pardo, A.; Parés, C., Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique, Math. mod. meth. appl. sci., 17, 2055-2113, (2007) · Zbl 1137.76038 [10] M.J. Castro, A. Pardo, C. Parés, E.F. Toro, in preparation. [11] C. Chalons, F. Coquel, Numerical capture of shock solutions of nonconservative hyperbolic systems via kinetic functions. Analysis and simulation of fluid dynamics, Adv. Math. Fluid Mech., Birkhäuser, Bäsel, 2007, pp. 45-68. · Zbl 1291.35125 [12] Dal Maso, G.; LeFloch, P.G.; Murat, F., Definition and weak stability of nonconservative products, J. math. pures appl., 74, 483-548, (1995) · Zbl 0853.35068 [13] Hayes, B.T.; LeFloch, P.G., Nonclassical shocks and kinetic relations: scalar conservation laws, Arch. rational mech. anal., 139, 1-56, (1997) · Zbl 0902.76053 [14] Hayes, B.T.; LeFloch, P.G., Nonclassical shocks and kinetic relations: finite difference schemes, SIAM J. numer. anal., 35, 2169-2194, (1998) · Zbl 0938.35096 [15] Hou, T.Y.; LeFloch, P.G., Why nonconservative schemes converge to wrong solutions: error analysis, Math. comput., 62, 497-530, (1994) · Zbl 0809.65102 [16] Lax, P.D.; Wendroff, B., Systems of conservation laws, Comm. pure appl. math., 13, 217-237, (1960) · Zbl 0152.44802 [17] LeFloch, P.G., Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form, Comm. partial differen. equat., 13, 669-727, (1988) · Zbl 0683.35049 [18] P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Mathematics and its Application, Minneapolis, Preprint # 593, 1989. [19] P.G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems in conservative form, in: J. Ballmann, R. Jeltsch (Eds.), Proceedings of the International Conference on Hyperbolic problems, Note on Num. Fluid Mech., vol. 24, Viewieg, Braunschweig, 1989, pp. 362-373. [20] P.G. LeFloch, On some nonlinear hyperbolic problems (in English), Habilitation à Diriger des Recherches, Université de Paris, vol. 6, 1990. [21] P.G. LeFloch, Hyperbolic systems of conservation laws: The theory of classical and nonclassical shock waves, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2002. · Zbl 1019.35001 [22] LeFloch, P.G., Graph solutions of nonlinear hyperbolic systems, J. hyper. differen. equat., 1, 643-689, (2004) · Zbl 1071.35078 [23] LeFloch, P.G.; Liu, T.-P., Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum math., 5, 261-280, (1993) · Zbl 0804.35086 [24] LeFloch, P.G.; Mohamadian, M., Why many shock wave theories are necessary. fourth-order models, kinetic functions, and equivalent equations, J. comput. phys., 227, 4162-4189, (2008) · Zbl 1264.76055 [25] LeFloch, P.G.; Thanh, M.D., The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Comm. math. sci., 1, 763-796, (2003) · Zbl 1091.35044 [26] LeFloch, P.G.; Thanh, M.D., The Riemann problem for the shallow water equations with discontinuous topography, Comm. math. sci., 5, 865-885, (2007) · Zbl 1145.35082 [27] Muñoz-Ruiz, M.L.; Parés, C., Godunov method for nonconservative hyperbolic systems, Math. method. anal. numer., 41, 169-185, (2007) · Zbl 1124.65077 [28] Noelle, S.; Xing, Y.; Shu, C.W., High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. comput. phys., 226, 29-58, (2007) · Zbl 1120.76046 [29] Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. numer. anal., 44, 300-321, (2006) · Zbl 1130.65089 [30] Parés, C.; Castro, M.J., On the well-balance property of roe’s method for nonconservative hyperbolic systems. applications to shallow-water systems, Math. model. numer. anal., 38, 821-852, (2004) · Zbl 1130.76325 [31] J.B. Schijf, J.C. Schonfeld, Theoretical considerations on the motion of salt and fresh water, in: Proceedings of the Minn. Int. Hydraulics Conv., Joint Meeting IAHR Hydro. Div. ASCE. (September 1953) 1953, pp. 321-333. [32] Xing, Y.; Shu, C.-W., High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, J. comput. phys., 214, 567-598, (2006) · Zbl 1089.65091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# How did the ancients view *infinitesimals*?

With some category/topos theory we can now put infinitesimals on a rigorous ground, as in Bell's A Primer of Infinitesimal Analysis, where the author introduces $\epsilon$ satisfying $$\epsilon\ne 0, \epsilon^2=0.$$

However, he also points out that this version of infinitesimal is not compatible with the law of excluded middle.

Meanwhile, the author seems convinced that this $\epsilon$ is the infinitesimal in the eyes of Newton and Leibniz among many others, when they were attacking problems like instantaneous speed and area under a curve.

I wonder whether this is true. I know people like Newton and Leibniz did not use limiting argument. But this does not mean they think of infinitesimals as nilsquare elements as described by Bell, because there are still other models of infinitesimals available.

Thanks very much.

-
I'm not sure Newton and Leibniz count as ancients! –  Peter Smith Feb 17 '13 at 15:56
@PeterSmith Pardon my English. But I am comparing their approach with the modern categorical one :) –  Hui Yu Feb 17 '13 at 15:57
@K.Stm. This is not that surprising. There are different logic systems. Please search intuitionist logic by yourself because I do not know how to insert a link in comments. –  Hui Yu Feb 17 '13 at 16:05
@HuiYu To insert link in a comment just [put text here](and your link here) ;-) –  dtldarek Feb 17 '13 at 16:17
@AsafKaragila, yes Robinson in the 1960s en.wikipedia.org/wiki/Non-standard_analysis –  alancalvitti Feb 17 '13 at 17:56

I think (but am very open to correction by those who know the history better than I do) that the situation is basically this. Newton, Leibniz and other early practitioners made a bunch of assumptions (both overt and implicit) about infinitesimals that can't coherently all be held true together. That's why from the earliest days, starting with Bishop Berkeley's mockery of the very idea, there was an unresolved anxiety about infinitesimals that lasted right into the twentieth century. They "worked" wonderfully well, but seemingly shouldn't have.

So the name of the game isn't to find the coherent account of infinitesimals that Newton and Leibniz had in view. Rather the game is one of optimal rational reconstruction. What coherent view gives us a theory which satisfies the most important features that Newton and Leibniz wanted in a calculus of infinitesimals, and does so in the neatest, most mathematically fruitful way?

But note there needn't be a fact of the matter about what are the most important features, about what's neatest, about what's most fruitful. These are going to be judgement calls, and maybe there will be different ways to go, with rather different costs and benefits. Even if we decide that the kind of theory which Bell presents is overall a "best buy", it would certainly be pushing the principle of charitable interpretation to breaking point to say that it was therefore what Newton and Leibniz "really" meant all along. Almost certainly, any coherent theory will have to deny things they would have taken for granted.

But then is Bell's smooth infinitesimal analysis the best buy? As the OP indicates, there are other and more popular modern versions of nonstandard analysis which stick closer to Robinson's original. For an extended development of one, see Nader Vakil's impressive and illuminating Real Analysis Through Modern Infinitesmials (in the Cambridge Encyclopedia of Mathematics series). But I wouldn't want to say either that such a theory rather than Bell's -- attractive though it is -- is what Newton and Leibniz really were talking about! Being the best buy in the neighbourhood is recommendation enough.

-
I upvoted your answer. Can you give an example of two implicit beliefs held by Newton or Leibniz which were contradictory? –  k.stm Feb 18 '13 at 12:41

Your question got me reading (googling). This paper: Leibniz's Infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russel and beyond (Katz & Sherry, 2012), in section 4, explicitly addresses your question, with a quote by Leibniz himself even. In his calculus, $\epsilon^2 \neq 0$. I'm only 1/3 in, but this is a nice paper. Here is the quote:

"being driven to fall back on assumptions that are admitted by no one; such as that something diﬀerent is obtained by multiplying 2 by m and by multiplying m by 2; that the latter was impossible in any case in which the former was possible; also that the square or cube of a quantity is not a quantity or Zero (Leibniz translated by Child [28, p. 146])." (emphasis added)

-

The Archimedean property of the real numbers says that there is no $\varepsilon>0$ such that $$\underbrace{\varepsilon+\cdots\cdots+\varepsilon}_{n\text{ terms}}$$ remains less than $1$ regardless of how small $\varepsilon$ is, as long as the finite cardinal number $n$ is large enough.

One explanation of why it's called "Archimedean" is that Archimedes of Syracuse (287 BC – c. 212 BC), a geometer, physicist, engineer, and inventor, said such quantities don't exist. Nonetheless, he used them brilliantly. He said his arguments using infinitesimals fall short of being complete proofs because infinitesimals don't exist. All of his arguments using infinitesimals are in one work, sometimes titled The Method.

Here is an example:

• Draw a secant line to a parabola with endpoints $A$ and $B$. (This need not be orthogonal to the axis, as one might think from some illustrations.)
• Through $B$, draw the tangent line; through $A$ draw a line parallel to the axis. These meet at $C$.

Archimedes claimed: One-third of the area of triangle $ABC$ is inside the curve.

To demonstrate this, he relied on the concept of center of gravity, which had first been introduced by him. He also relied on the concept of torque on a lever, also first introduced by him.

• Let $D$ be the midpoint between $A$ and $C$. Consider $D$ to be the fulcrum of a lever, which is the line $DB$.
• Let $E$ be on the line $DB$, just as far from the fulcrum $D$ as $B$ is, but in the opposite direction.

Archimedes showed that the center of gravity of the interior of the triangle is on this line $DB$, one-third of the way from $D$ to $B$. If one could let the whole weight of the triangle rest at that center of gravity, and a weight equal to that bounded by the curve and the secant line at $E$, then the lever is in equilibrium precisely of the proposition to be demonstrated is true.

To show that, he considered cross-sections parallel to the axis. Let the infinitesimal weight of each such cross section (proportional to its length) rest on the lever at the point where the cross-section intersects the lever. Archimedes claimed this would exert just as much torque on the lever as if the whole weight of the triangle rests at the center of gravity.

So let the infinitesimal weight one such cross-section rest on the lever $DB$ at the point where it crosses $DB$. And let a weight equal to that of the part of that cross-section that is inside the curve rest at $E$. If the lever is then in equilibrium, then we're done. But that is just what in modern language we would call the equation of the parabola. QUOD ERAT DEMONSTANDUM

Archimedes used the same method to show that the center of gravity of the interior of a hemisphere (i.e. half a sphere) is five-eights of the way from the pole to the center. And maybe more than a dozen other propositions; I don't remember exactly how many.

-

The reals admit to axiomatization that allows for infinitesimals. Look at http://en.wikipedia.org/wiki/Hyperreal_number

Edit: Excuse me. I misread the question. No one can say with certainty. The ancients must have thought of infinitesimals as "very small" or "infinitely small" change, as in the differential change "dx." Newton and Leibniz did not have a formal definition of the reals and were not aware of their completeness. They had superb intuition. I doubt anyone will give you a more satisfactory explanation.

p.s. This question feels a lot like the high school:

What did Dumas mean when he wrote such and such in his "blah-blah?"

-
Surreals seem even less likely than nilpotent infinitesimals, to be honest. –  Zhen Lin Feb 17 '13 at 16:16
Did you perhaps mean Hyperreal numbers? –  Asaf Karagila Feb 17 '13 at 17:34
Any of the super, hyper, and surreals have infinitesimals, @Asaf. –  Student Feb 17 '13 at 17:38
I am aware of that. But this is not "an axiomatization of the reals". –  Asaf Karagila Feb 17 '13 at 17:51
Thank you, for pointing that out. –  Student Feb 17 '13 at 18:17


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What is the digit on the units place in the expanded value of 97^275 –

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02 Jul 2019, 21:20
1
The units place in the expanded value of 97^275–32^44?
7 raise to any power has cyclicity of 7, 9 , 3, 1 as its unit digit
97^275 = 97^ (272+3) the unit digit for the result would be 3

Similarly, 2 raise to any power has cyclicity of 2, 4 , 8, 6 as its unit digit
32^44 the unit digit for the result would be 6

3-6 --> 13-6--> 7

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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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02 Jul 2019, 22:01
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We are only concerned about the digit in the units place.
So, even in the above problem $$97^{275}– 32^{44}$$ we can only be concerned about $$...7^{275} – ...2^{44}$$

The unit’s digit of powers of 7 are in order of 7, 9, 3, 1
Remainder of $$\frac{275}{4}$$ = 3
The unit’s digit of the third power of 7 is 3. Thus, $$7^{275}$$ will also have units digit of 3.

Similarly,
The unit’s digit of powers of 2 are in order of 2, 4, 8, 6
Remainder $$\frac{44}{4}$$ = 0 i.e 4
The unit’s digit of the fourth power of 2 is 6. Thus $$2^{44}$$ will also have the unit’s digit of 6.

Hence,
Unit’s digit of $$7^{275} – 2^{44}$$ = Unit’s digit of $$97^{275} – 32^{44}$$ = ….3 - ….6 = …7

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02 Jul 2019, 22:03
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Numbers ending with 7 will end with 7,9,3,1 based on the power. --> Power cycle for 7 is 4.
97^275 will end with 3 since 275 when divided by 4 remainder is 3.

Similarly, any number ending with 2 will have 2/4/8/6 in its unit;s digit based on the power.
32^44 will end with 6, as 44 when divided by 4 remainder is 0.

Also, notice that 97^275 > 32^44. Hence, 97^275 -32^44= .........3 -...........6=............7

OA is D
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02 Jul 2019, 22:03
A)1

I just tried to get the units digit of 7^5 -2^4
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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02 Jul 2019, 22:04
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To solve this question we will use the pattern method.

We know that we can find the unit digit of any number from 0 to 9 by finding out if the power is of 4k+1, 4k+2, 4k+3 or 4k.

Using this we get that power 275 is of form 4k+3 and 44 is of form 4k.

Unit digit of 7 for power of form 4k+3 is 3 and unit of 2 for power of the form 4k is 6.

The difference between a number of X3 (where X any interger > 0) and 6 will be 7.

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02 Jul 2019, 22:16
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Unit digit of 97^275 would be same as 7^275, which would be same as 7^3 (the unit digits of 7^x repeat after x = 4), which is 3.
Similarly, unit digit of 32^44 would be same as 2^44, which would be same as 2^4 (the unit digits of 2^x repeat after x = 4), which is 6.

Now, ____3-____6 will be same as 13-6 = 7 (where "____x" represents some number string ending with x)
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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02 Jul 2019, 22:29
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Quote:
What is the digit on the units place in the expanded value of $$97^{275}–32^{44}$$?

A. 1
B. 3
C. 5
D. 7
E. 9

Unit digit of $$97^{275}–32^{44}$$ = Unit digit of $$7^{275}–2^{44}$$

CONCEPT: The unit digit of any number depends on the unit digit of the number so any digit other than unit digit is irrelevant such as $$97^{275}$$ will have same unit digit as $$387^{275}$$ or $$7^{275}$$

Now, Unit digit of $$7^{275}–2^{44}$$ = Unit digit of $$7^3–2^4$$

CONCEPT: The cyclicity of unit digit of 7 and 2 is 4 i.e.e unit digits repeat after every 4th power. therefore Remainder (275/4) = 3 hence unit digit of $$7^275$$ = unit digit of $$7^3$$

Unit digit of $$7^3–2^4$$ = 3-6 = 7

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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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02 Jul 2019, 22:31
1
Unit digits for the powers of 97: 7, 9, 3, 1, 7, 9, 3........ and so on. Cyclicity = 4

Unit digits for the powers of 32: 2, 4, 8, 6, 2, 4, 8........ and so on. Cyclicity = 4

Now, for the 275th power of 97, unit digit will be "3". (275=68*4+3. So, 3rd term in the series [7,9,3,1])

For the 44th power of 32, unit digit will be "6". (44=11*4, 4th term in the series [2,4,8,6])

So, difference for any number with these unit digits will be "7". (X...X3 - X..X6 = X..X7)

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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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02 Jul 2019, 22:52
1
Last digit of the first no is 7 which a cyclicity of 4 . so last digit will be 3.
Similarly last digit of second no is 4. So last digit will be 6.
So effective last digit will be 3-6 =7
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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02 Jul 2019, 23:16
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1
Check for the cyclicity of 7- 3M which means unit digit is going to be 3 and then cyclicity of 2 is 4M which means that the unit digit is going to be 6.

So 3 - 6 which means 13 - 6 = 7 Answer is D
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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02 Jul 2019, 23:21
1
97^1, 97^5, 97^9, etc. have a units digit of 7
97^2, 97^6, 97^10, etc. have a units digit of 9
97^3, 97^7, 97^11, etc. have a units digit of 3
97^4, 97^8, 97^12, etc. have a units digit of 1

From the above-mentioned pattern, one can deduce that 97^275 has a units digit of 3.

32^1, 32^5, 32^9, etc. have a units digit of 2
32^2, 32^6, 32^10,etc. have a units digit of 4
32^3, 32^7, 32^11,etc. have a units digit of 8
32^4, 32^8, 32^12,etc. have a units digit of 6

Similarly, one can deduce that 32^44 has a units digit of 6.

Therefore, the digit on the units place 97^275–32^44 is: ...3 - ...6 = ...7

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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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03 Jul 2019, 00:52
7 follows a cycle of 4 and 2 follows a cycle of 4
The unit digit is of the two number is (3-2) =1
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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03 Jul 2019, 01:23
1
97^275:Since the cyclicity is of 4 numbers(7,9,3,1), we can divide the power, 275, by 4 to get a remainder of 3. so the last digit will be 3
32^44:Since the cyclicity is of 4 numbers(2,4,8,6), we can divide the power, 44, by 4 to get a remainder of 0. so the last digit will be 6.
so the answer should be 7
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03 Jul 2019, 01:25
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What is the digit on the units place in the expanded value of 97275–3244
97
275

32
44
?

A. 1
B. 3
C. 5
D. 7
E. 9

Since we are looking for the unit's place, let's figure out the unit's places of the numbers involved
97^275 will have 3 at the unit's place (formula: cyclicity of 4)

32^44 will have 6 at the unit's place (formula: cyclicity of 4)

Difference: _ _ _ 3 - _ _ _ 6 = _ _ _ 7

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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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03 Jul 2019, 01:29
1
97275–3244

The last digit of the number 97^275 is defined by the the power of 7. The power of 7 has the cycle of 4 numbers (7, 9, 3, 1). => 275 = 4*68 + 3 => the last digit of the number 97^275 is 3.
The last digit of the number 32^44 is defined by the the power of 2. The power of 2 has the cycle of 4 numbers (2, 4, 8, 6). => 44 = 4*11 => the last digit of the number 32^44 is 6.

XXX3 - XXX6 = XXX7 = > Answer is D.
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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03 Jul 2019, 01:37
$$7^5$$ = digit in the unit place would be 7(for square- 9, for cube- 3 and so on and so forth)
$$2^4$$ = digit in the unit place would be 6 (for square- 4, for cube- 8 and so on and so forth)
therefore, 7-6(subtraction, since the last digit can be subtracted directly) would yield answer 1.
Option A
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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03 Jul 2019, 02:27
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the first part is having units digit as 3, as 7 is having a cyclicity of 4,the power 275 when divided by 4, remainder is 3, so 7^3, last digit is 3;

similarly the second part is having units digit as 6, as cyclicity of 2 is 5, & 44 divided by 5 gives remainder as 4, 2^4, last digit will be 6;

subtraction of XXXX..X3-YYY.YY6; last digit will be 7
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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03 Jul 2019, 02:51
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Using the cyclical nature of unit digits raised to a power, we can easily calculate the answer.
For units digit of 97^275,
7 is the units digit of 97.
We know 7^1=7
7^2=9 (the units digit)
7^3=3
7^4=1
The units digits then repeat in 7,9,3,1 order. the power of 97, 275 is in the form of 4n+3.
So 97^275 would have 3 as units digit.

For units digit of 32^44,
2 is the units digit of 32.
We know 2^1=2
2^2=4
2^3=8
2^4=6(the units digit)
The units digits then repeat in 2,4,8,6 order. the power of 32, 44 is in the form of 4n.
So 32^44 would have 6 as units digit.

So the units digit of the substraction of these numbers would be = 3-6
Since 6 is bigger here than 3, 1 would carry over from the tens side to 3 making it 13.
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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03 Jul 2019, 02:56
It's 7^275-2^44

275/4 will.give a remainder of 3 , so units digit will 3

44/4 will.give a remainder of 0 , unit digit will be 2

3-2=1

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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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03 Jul 2019, 03:25
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imo 7,

question can be written as 7^3 - 2^4 so 3-6 gives ans 7
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Re: What is the digit on the units place in the expanded value of 97^275 –   [#permalink] 03 Jul 2019, 03:25

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# How to create a color of the same brightness and saturation?

Let's consider three colors: Red, Green and Blue. For blue I created a specific shade, something like #F2F5F7. If you see this color, it has a certain level of shade, brightness and saturation value of blue.

I want to create the same equivalent color but for Green and Red, how to get that?

• Did you mean to say "but from Green and Red"? – JohnB Apr 17 '15 at 10:58
• Hi Faraz, welcome to GDSE and thanks for your question. If you want to know more about the site, please see the help center or ping one of us in chat once your reputation is sufficient (20). Keep contributing and enjoy the site! – Vincent Apr 17 '15 at 11:05
• use adobe kuler for a fast way. – Lunaticbrain Apr 17 '15 at 12:40
• I'd like to say that with your example, the difference between the three colours is going to be nigh invisible--all three of them are almost white. – Vincent Apr 17 '15 at 13:54
• Consider using the Lab color space: "Unlike the RGB and CMYK color models, Lab color is designed to approximate human vision." There's a color calculator at easyrgb.com and conversion formulas RGB->XYZ and XYZ->Lab. One way to use this for your case would be to pick any green color, convert to Lab, adjust L to equal the L of the blue, and then convert back to RGB. – Supr Apr 17 '15 at 14:53

## 1 Answer

This is exactly what the HSB colour model is for. You have almost exactly named the model's variables in your question:

• Hue is the 'kind' of colour: red, blue, orange, yellow;
• Saturation is inverse with the amount of white you add to the hue;
• Brightness is inverse with the amount of black you add.

So, to take your example colour of #F2F5F7. you can convert this into HSB in Photoshop or Illustrator's colour picker, yielding H=203, S=1, B=96 as HSB values.

All you have to do now is change the H to the hue you want. For a green, try H=83 and leave the S and B values intact. This is converted by the colour picker right away and yields #f3f4f2.