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- polynomial fraction calculator | s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867364.94/warc/CC-MAIN-20180625014226-20180625034226-00106.warc.gz | CC-MAIN-2018-26 | 131,724 | 2,367 |
https://www.pdcnet.org/pdc/bvdb.nsf/purchase?openform&fp=jpr&id=jpr_1995_0020_0019_0043 | math | Volume 20, 1995
What’s The Question?
Two kinds of critical questions have been asked about the propriety or rightness of Christian beliefs. The first is the de facto question: is Christian belief true? The second is the de jure question: is it rational, or reasonable, or intellectually acceptable, or rationally justifiable? This second question is much harder to locate than you’d guess from looking at the literature. In “Perceiving God” William AIston suggests that the (or a) right question here is the question of “the practical rationality,” construed as he construes it. I argue that the question is ambiguous: and one of the disambiguees is too easy to answer, while the other is such that its answer is really irrelevant to any sensible version of the de jure question. I conclude by venturing a suggestion as to what a sensible de jure question might be. | s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585522.78/warc/CC-MAIN-20211022212051-20211023002051-00104.warc.gz | CC-MAIN-2021-43 | 877 | 3 |
https://ems.press/journals/rmi/articles/4834 | math | We show that there are Hilbert spaces constructed from the Hausdorff measures on the real line with which admit multiresolution wavelets. For the case of the middle-third Cantor set , the Hilbert space is a separable subspace of where . While we develop the general theory of multi-resolutions in fractal Hilbert spaces, the emphasis is on the case of scale which covers the traditional Cantor set . Introducing
we first describe the subspace in which has the following family as an orthonormal basis (ONB):
where , . Since the affine iteration systems of Cantor type arise from a certain algorithm in which leaves gaps at each step, our wavelet bases are in a sense gap-filling constructions.
Cite this article
Dorin E. Dutkay, Palle E.T. Jorgensen, Wavelets on Fractals. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 131–180DOI 10.4171/RMI/452 | s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233511386.54/warc/CC-MAIN-20231004152134-20231004182134-00322.warc.gz | CC-MAIN-2023-40 | 842 | 5 |
https://www.cmetechnology.com/sinusoidal-and-random-vibration-testing-on-vibration-shaker-table.html | math | The vibration shaker is a test equipment used to simulate real vibration environment effects in the laboratory. Vibration tests use different input signals to excite the specimen on the mechanical vibration shaker. Vibration tests are mainly divided into sine and random vibrations. Because of their different physical processes, there is no strict equivalent relationship between them. Therefore, when choosing the test mode, do not convert the severity level from sinusoidal to random.
The sinusoidal vibration test uses a sinusoidal signal of fixed or variable frequency and amplitude, and only one frequency is applied at each instant. The test conditions include frequency range or fixed frequency, amplitude, and test duration.
Sinusoidal vibration rarely appears as a single-frequency form in real environment. Even when measuring acceleration directly on rotating machinery. Such as gears and bearings, actual tolerances and clearances usually result in small variations in frequency. Some form of random vibration also occurs in the random characteristics of rotating machinery.
Sinusoidal vibration on the vibration shaker can be described as deterministic motion, which follows certain laws and can be completely determined from the past state at any specified time in the future. During the sinusoidal sweep test, this method is usually used to determine the time of failure because this failure is likely to be closely related to a specific frequency, and the correlation effect is not obvious with the random vibration test method. Of course, the sinusoidal test method usually takes longer to trigger the failure the random test method, because each frequency sweep only acts on each resonance point for a short time during each frequency sweep process. Although only one frequency is applied at any one time, it is indeed possible to maximize a particular resonant peak of the specimen if the sweep rate is slow enough. It can also be used to find potentially destructive resonance points, especially during design and development tests.
Motivation of random vibration tests of vibration shaker such as the hydraulic vibration shaker uses an irregular random input signal, which includes all frequency components within the specified frequency range (wideband) at all times. Its instantaneous value follows a normal (Gaussian) distribution. The distribution within the frequency range is represented by the acceleration spectral density (ASD) curve.
Random vibration is the most common type of trigger that occurs in the real environment. Its instantaneous value in the future cannot be predicted from the past state, and can only be predicted based on probability. In fact, these properties are suitable for most calculations related to random vibration, such as fatigue and alternating stress.
Unlike sinusoidal tests, random vibrations of the vibration shaker trigger the resonance throughout the duration, although it does not reach the maximum value. Most random signals in the laboratory have 3 times RMS value, which means that the instantaneous value of the trigger can extend from zero to 3 times the total RMS value in the test frequency range . A greater consideration for random trigger is that there will be a large amount of stress alternating between zero crossings, both positive and negative . This property will affect fatigue damage and thus the life expectancy to failure.
Whether it is sinusoidal or random vibration, only a understanding of production, transport simulation vibration testing machine, use environment, the properties and characteristics of the test equipment, can choose your own suitable vibration mode more appropriately, so as to screen high-quality products and improve performances of defective products . | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296815919.75/warc/CC-MAIN-20240412101354-20240412131354-00281.warc.gz | CC-MAIN-2024-18 | 3,764 | 8 |
https://www.physicsforums.com/threads/energy-limits-of-gamma-rays.180494/ | math | Hello you guys. I was wondering about the energy limits of the gamma rays. I allready know that the gamma ray spectrum varies from some MeV to 50 MeV (approximately). Does anyone know how can these limits be justified?
Answers and Replies
Related High Energy, Nuclear, Particle Physics News on Phys.org
The lower limit of gamma ray energy is basically a matter of definition. X-rays are the next lower type and the boundary is somewhat arbitrary One way of defining might be when coherent scattering (X-ray) is more important than Compton (gamma).
There is no defined upper limit. It is merely a question of what is physically possible. | s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400214347.17/warc/CC-MAIN-20200924065412-20200924095412-00672.warc.gz | CC-MAIN-2020-40 | 636 | 5 |
https://www.sumproduct.com/thought/working-capital-adjustments | math | Credit where credit’s due…
Never has it been clearer than in the current global economy that cash is king. This article considers the practical issues surrounding modelling working capital adjustments. By Liam Bastick, director with SumProduct Pty Ltd.
I have forecast income and expenditure items for the company’s Income Statement. What tips can you provide for deriving cash flow figures from these projections?
Let’s start at the beginning: please feel free to use the attached Excel file to help clarify the ideas discussed below.
Consider the following example:
Imagine a company just starts off in business (i.e. has no amounts due) and generates sales of $1,000 in the period. At the end of the period, assuming no bad debts, $753 has been paid, leaving a closing debtor balance of $247. This difference is what I refer to as the working capital adjustment.
If we had modelled the sales of $1,000 in the period, how might we generate the cash receipts forecast such that is assumptions changed, the receipts would calculate appropriately?
Clearly, if we are given the closing debtor balances, the problem becomes trivial, so I will assume that this is not so. Therefore, I am going to consider an alternative approach and some of the associated underlying issues that need to be considered when modelling.
At the risk of teaching the accountants in the audience to suck eggs, let me first derive an alternative method.
Let’s assume that the sales accrue evenly over the period of time and for the sake of this example, that period is one year (365 days). Presuming (i) all sales are made on credit terms, (ii) all customers pay their invoices on the day the amounts fall due and (iii) no bad debts are incurred, this can be reflected graphically as follows:
Clearly, the credit period is the “gap” at the beginning of the time period, i.e. 247/1000 x 365 days = 90 days. This can be represented formulaically as:
Days Receivable = (Closing Debtors x Days in Period) / Sales in Period
Rearranging, this becomes:
Closing Debtors = (Sales in Period x Days Receivable) / Days in Period,
e.g. in our example: 247 = (1000 x 90) / 365.
Therefore, in modelling, we often set the number of days receivable (and days payable) as key assumptions for cash flow forecasting.
However, it’s not always as simple as that. Let me explain. Consider we are planning to build a monthly model (assuming 30 days in a month) and sales for the month are again $1,000. Debtor days remain at 90 days.
Based on these calculations, we would generate the following control account:
Erm, that’s right: make sales of $1,000 and have $3,000 (= 90/30 x 365) owing to you by the end of the month. Also, the company pays $2,000 to customers a reclaimable $2 for each $1 spent. That’s nonsense – and yet, as an experienced model auditor I have seen this erroneous calculation crop up on a regular basis…
The problem is, in this current economic climate most businesses want to prepare monthly – sometimes weekly and even daily – cash flow projections. Clearly, if the days receivable or days payable assumption exceeds the number of days in each forecast period this approach is inappropriate and will lead to calculation errors.
So what do we do in this situation..?
For example, in a monthly model, if payments are made exactly one month or two months or three months later (and so on), the resolution is simple: the receipts can be calculated using a simple OFFSET (displacement) formula.
Therefore, let’s complicate the scenario slightly. Imagine we are building a monthly forecast model, but that the days receivable are 75. For the purposes of keeping this article reasonably brief, I will simplify the problem by assuming an average number of days in a month (say, 30). Using this simplifying assumption, this will mean that payments are made on average 2.5 (2.5 = 75 / 30) months after the sale has been made.
That 2.5 months figure is important. The integer part (2) denotes how many complete months (including the current month) have sales payments outstanding. The residual (0.5) shows the proportion of the month preceding these complete months that is also outstanding. With this borne in mind, the OFFSET function can now come to the rescue, viz.
In this illustration (above), cells J18 and K18 break the number of days receivable (cell G18) into the number of whole months and residual proportion respectively, assuming that each month has 30 days (cell H13).
The key formula here is the calculation for Closing Debtors (Cash Receipts is simply the balancing figure). For example, the formula in cell J28 (above) is:
It may seem a little complex upon first inspection, but it’s not as bad as it seems. Essentially, there are two parts to this formula identified by the two added IF statements:
1. IF($J$18,SUM(OFFSET(J26,,,1,-MIN($J$18,J$23))),) considers the completed number of months where sales remain outstanding and adds up the sales for these periods. This approach has been discussed previously in detail in the Simple Depreciation Example detailed in my OFFSET article.
In essence, this part of the formula checks that the number of completed months is not zero (in this case the amount is just zero), and assuming this is not the case, it sums the sales for the relevant number of completed months (i.e. starts with the current month and then considers the sales in previous months, working from right to left in the spreadsheet). The MIN formula is required to ensure that the model does not try to include periods prior to the beginning of the forecast period).
2. IF(J$23-$J$18<=0,,OFFSET(J26,,-$J$18)*$K$18) considers the residual (remaining) amount for the month before the earliest completed month. For example, if the credit period is 2.5 months and the current month is April, then March and April will be “whole months” where no payment has been received, with half of February’s monies still outstanding too.
The reason for the IF statement here is to prevent calculations considering periods before the beginning of the forecast period.
To clarify, consider the Closing Debtor figure of $1,050 in Period 5 (above, cell N28 in the illustration). This is calculated as the sales for Periods 4 and 5 (400 + 500 respectively), plus half of the Period 3 sales (300 x 0.5 = 150), i.e. 400 + 500 + 150 = 1,050.
More Complex Scenarios
It only gets more complicated. What if payments are not made evenly? Or that some sales are written off as payments are never made (i.e. bad debts)? Actually, we have covered this scenario in a previous article.
Think about it: depreciation arises when cash is laid out in one period, but the costs are allocated over multiple periods. Debtors arise when sales are made in one period, but the cash receipts are allocated over multiple periods. It’s the same logic, just different labelling!
The calculations can be reviewed in depth in the attached Excel file, but what can be clearly seen is the similarity to a depreciation calculation. In this illustration, the Cash Receipt Profile percentages do not add up to 100%. This is deliberate – the missing 5% is the assumed bad debt here.
Word to the Wise
This article is intended to be a starting point for considering the modelling issues surrounding working capital adjustments. You can complicate matters further by considering any or all of the following:
- What proportion of sales / costs is made on credit terms?
- Calculate extended credit periods using the actual number of days in each month rather than using an average number
- Calculations using average days receivable (payable) based on both the opening and closing debtor (creditor) balances
- How adjustments differ by considering public holidays, the number of working days, seasonality and cyclicality etc.
- Sales / costs are invoiced in a foreign currency and the exchange rate differs over time
- Factoring in segmental reporting, e.g. based on customer profitability / size / geography etc.
- Infrastructure projects often have large one-off payments that will occur on specific dates (here you just need to calculate which month the payment will occur in). | s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499713.50/warc/CC-MAIN-20230129112153-20230129142153-00606.warc.gz | CC-MAIN-2023-06 | 8,156 | 45 |
https://sharemylesson.com/standards/virginia-doe/7.4 | math | The student will solve single-step and multistep practical problems, using proportional reasoning.
No resources have been tagged as aligned with this standard.
The total value of an investment is equal to the sum of the original investment and the interest earned.
Ratios may or may not be written in simplest form.
The total cost of a loan is equal to the sum of the original cost and the interest paid.
Compute the simple interest and new balance earned in an investment or on a loan given the principal amount, interest rate, and time period in years.
The order of the values in a ratio is directly related to the order in which the quantities are compared. | s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347441088.63/warc/CC-MAIN-20200604125947-20200604155947-00282.warc.gz | CC-MAIN-2020-24 | 660 | 7 |
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cc by-sa 3.0 | s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010303377/warc/CC-MAIN-20140305090503-00075-ip-10-183-142-35.ec2.internal.warc.gz | CC-MAIN-2014-10 | 2,196 | 53 |
http://www.ask.com/web?qsrc=3053&o=102140&oo=102140&l=dir&gc=1&qo=contentPageRelatedSearch&ad=null&q=What+Is+A+Basic+Equation+For+The+String+Theory | math | The equations of motion are given by the following equation: ... ansatz which has
simple behavior under star multiplication ...
(basic / advanced) ... Pythagoras realized that vibrating Lyre strings of equal
tensions but different lengths would produce harmonious notes (i.e. middle C and
high C) if the ... then the equation of motion is the one-dimensional wave
M-Theory (Becker, Becker and Schwarz), Introduction to String Theory ...
Symmetries and Field Equations of the Bosonic String .... The answer is quite
in mathematics will be a lasting and rewarding impact of string theory in science,
whatever ..... T hese maps are not independent but satisfy gluing a x ioms that
generali z e the simple composition ..... T here is a precise formula that relates the
Jun 15, 2011 ... String theory explained for kids, teens, and even adults. ... Then with a simple
equation and calculus, Newton could predict the orbits of planets.
What is the formula of the string theory? ... They vibrate at distinct frequencies that
can be calculated by one simple and uniform equation.
You might have heard me speak about the equation that eluded Einstein for the
last 30 ... The main problem, I think, is that String Theory is not in its final form.
www.ask.com/youtube?q=What Is A Basic Equation For The String Theory&v=kYAdwS5MFjQ
Dec 7, 2011 ... Theoretical physicist Michio Kaku explains the basics of String Theory in this clip
from his Floating University lecture. Find out more at: ...
www.ask.com/youtube?q=What Is A Basic Equation For The String Theory&v=25haxRuZQUk
Mar 30, 2011 ... Lecture 1 | String Theory and M-Theory .... In this course we will develop the
basic theoretical and mathematical ideas, including the string-theoretic origin of
gravity, the theory of extra ... Einstein Field Equations - for beginners!
9.1 Compactifying closed string theory on a circle. .... It took some time before the
simple question was asked: suppose a meson consists of two quarks ..... What
has now been established is the local, classical equations of motion for a string. | s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257830066.95/warc/CC-MAIN-20160723071030-00246-ip-10-185-27-174.ec2.internal.warc.gz | CC-MAIN-2016-30 | 2,062 | 26 |
https://seandaddy.wordpress.com/2012/04/27/vertical-integration-and-competition-policy/ | math | Players=(m upstream firms, n downstream firms, government)
m upstream firms (produce a homogeneous input at a constant) marginal cost k
the input is then sold to n downstream firms
marginal cost k
input price is c
produce output using input
only input price is its marginal cost c
-Timing of events-
First, the upstream firms choose quantities which results in a market clearing input price, c
Second, the downstream firms choose quantities taking the input price as given
From utility maximization problem, firm has inverse demand function as Eq.(2)
then EQ quantity is (4)
We derived EQ quantity in Downstream market.
Let’s move on the upstream. Let Q denote total factor demand, i.e. Q=nq*. Thus inverse demand function for the upstream market is given
Upstream firm maximizes profits
where . under symmetric assumption
EQ market quantity and input price is derived
Pulling together Q** and c(Q) gives
we have (Q**,c**), market EQ of input price and quantity.
Given this upstream EQ price, the downstream SPNE price and quantity are given by:
Then we get the welfare,
2.2 Post Merger EQ
– After merging, intermediate inputs are supplied to the downstream firm at the cost of k.
– the merging downstream firm will be more aggressive as it is operating at a low marginal cost, k while other downstream firms spend more cost at c. thus k=<c
– Net welfare depend on the original market structure and the degree of horizontal product differentiation
Then country’s welfare function is defined as follows:
Vertical integration increases welfare if the number of downstream firms, n, is smaller or equal to the number of upstream firms, m.
Welfare depends on the level of product differentiation
4. The antitrust Market Definition
Suppose that a subset of firms form a coalition. An Intervention or antitrust policy controls and affects this level of ;
that is, the authority intervene the mark-up of firms in coalition, not controlling the number of firms directly.
: the price charged by firms in coalition (upstream + downstream).
In case of coalition, as we did previous, we derived EQ quantity, price and number of firms clearing market. ()
Some comparative statics results are as follows:
1) The number of firms in coalition increases in the value of . Larger mark-up, more firms in coalition.
2) The degree of horizontal product differentiation and the size of antitrust policy have negative relationship.
If you know about this, you may refer to 13 page in the paper.
After merging, k=<c, which means intermediate good can be supplied cheaper than other firms.
The upstream firm merged by the downstream cannot charge the price over the marginal cost.
Antitrust policy is not directly controlling the number of firms but the ratio of mark-up. However, by doing so, the number of firms can be controlled. | s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583660070.15/warc/CC-MAIN-20190118110804-20190118132804-00212.warc.gz | CC-MAIN-2019-04 | 2,817 | 40 |
https://www.rand.org/pubs/papers/P2368-1.html | math | On the continuation of orthogonal structure across a surface of discontinuity in the momentum-energy tensor.
A discussion of Einstein's general relativity field equations, the study of which is usually begun by assuming certain particular forms for the metric tensor. In the presence of a general momentum-energy tensor that admits jump discontinuities, the question arises as to the consistency of the assumed metric structure in regions where the jumps in the momentum-energy tensor occur. To answer this question, the author examines the continuation of orthogonal structure across surfaces of discontinuity in the momentum-energy tensor and analyzes the problems of several massive bodies of finite extent. It is concluded that, although orthogonal structure for problems of the second and third classes can generally be continued, the problems of the first class will only admit orthogonal structure, everywhere in the space under very particular circumstances. | s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571147.84/warc/CC-MAIN-20220810040253-20220810070253-00446.warc.gz | CC-MAIN-2022-33 | 966 | 2 |
https://mathweneedtoknow.home.blog/2022/10/03/13-completely-different-graphs-that-are-almost-the-same/ | math | Can you draw a dinosaur with 142 points? (The answer is yes, by the way.) On a graph, draw a dinosaur with 142 points such that:
- The horizontal mean is 54.26 and the vertical mean is 47.83
- The horizontal standard deviation is 16.79 and the vertical standard deviation is 26.93
- The Pearson correlation coefficient is -0.06
Obviously, this would be ridiculous to do by hand. How would you safely draw 142 points with these bounds? And what even is a Pearson correlation coefficient?
The PCC (it’s short for Pearson correlation coefficient, not for precipitated calcium carbonate), in short, is the ratio between the covariance of two variables and the product of their standard deviations.
Remember standard deviations? The square root of the average of the squares of the distances between the points and the mean? If not, then here you go. If it’s still confusing, here’s an excellent example on Wikipedia.
Covariance is a bit trickier. Think of it as difficult version of correlation. The sign (positive/negative) of the covariance determines how the values correspond: if it is positive, then the two variables have similar behaviour, namely if one variable increases, so does the other and vice versa. If it is negative, then it is the opposite: if one variable increases, then the other variable decreases and vice versa.
If you want the formula for covariance, it’s on this Wikipedia page. Even after reading this, I can’t understand it, mainly because of the expected value theorem.
Now that we’ve cleared some of the statistics terms, let’s go back to our 142-point dinosaur.
In the hidden land of Canada in 2017, two researchers created twelve sets of data that had basically the same statistics to the dinosaur, whose name is the Datasaurus. The Datasaurus Dozen was a collection of the twelve sets of data with the same horizontal/vertical means, standard deviations, and all had the same Pearson correlation. Yet, they looked completely different. How could this be?
This is the truly deceptive aspect of statistics. The slightest change in the values, such as the means, or standard deviations, could land you with a completely new graph. The differences between all five values for all thirteen graphs was less than a hundredth, and by rounding down, it could be said that the graphs had equal values.
If you want a page on this with images, the link is here.
One thought on “13 completely different graphs that are (almost) the same”
i love this bog
LikeLiked by 1 person | s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945376.29/warc/CC-MAIN-20230325222822-20230326012822-00312.warc.gz | CC-MAIN-2023-14 | 2,509 | 16 |
https://albertteen.com/uk/gcse/mathematics/geometry/arc-length | math | YOU ARE LEARNING:
We can divide circles up into sectors, and find out information about them using some of the formulae we have learned.
As well as knowing how to calculate the circumference and area of a circle, it is useful to working out proportions of these quantities; arcs and sectors.
The angle of a sector is 108°. If the radius is 5, what is the length of the arc?
We can use the same logic to find the area of a sector, as a sector is just a proportion of the total circle area. | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817729.87/warc/CC-MAIN-20240421071342-20240421101342-00007.warc.gz | CC-MAIN-2024-18 | 489 | 5 |
https://blog.plover.com/math/snub-disphenoid.html | math | Wed, 01 Aug 2007
The snub disphenoid
Anyway, earlier this week I was visiting John Batzel, who works upstairs from me, and discovered that he had obtained a really cool toy. It was a collection of large steel ball bearings and colored magnetic rods, which could be assembled into various polyhedra and trusses. This is irresistible to me. The pictures at right, taken around 2002, show me modeling a dodecahedron with less suitable materials.
The first thing I tried to make out of John's magnetic sticks and balls was a regular dodecahedron, because it is my favorite polyhedron. (Isn't it everyone's?) This was unsuccessful, because it wasn't rigid enough, and kept collapsing. It's possible that if I had gotten the whole thing together it would have been stable, but holding the 50 separate magnetic parts in the right place long enough to get it together was too taxing, so I tried putting together some other things.
A pentagonal dipyramid worked out well, however. To understand this solid, imagine a regular pyramid, such as the kind that entombs the pharaohs or collects mystical energy. This sort of pyramid is known as a square pyramid, because it has a square base, and thus four triangular sides. Imagine that the base was instead a pentagon, so that there were five triangular sides sides instead of only four. Then it would be a pentagonal pyramid. Now take two such pentagonal pyramids and glue the pentagonal bases together. You now have a pentagonal dipyramid.
The success of the pentagonal dipyramid gave me the idea that rigid triangular lattices were the way to go with this toy, so I built an octahedron (square dipyramid) and an icosahedron to be sure. Even the icosahedron (thirty sticks and twelve balls) held together and supported its own weight. So I had John bring up the Wikipedia article about deltahedra. A deltahedron is just a polyhedron whose faces are all equilateral triangles.
Four of the deltahedra are the tetrahedron (triangular pyramid, with 4 faces), triangular dipyramid (6 faces), octahedron (square dipyramid, 8 faces), and pentagonal dipyramid (10 faces).
Another deltahedron is the "gyroelongated square dipyramid". You get this by taking two square pyramids, as with the octahedron. But instead of gluing their square bases together directly, you splice a square antiprism in between. The two square faces of the antiprism are not aligned; they are turned at an angle of 45° to each other, so that when you are looking at the top pyramid face-on, you are looking at the bottom pyramid edge-on, and this is the "gyro" in "gyroelongated". (The icosahedron is a gyroelongated pentagonal dipyramid.) I made one of these in John's office, but found it rather straightforward.
The last deltahedron, however, was quite a puzzle. Wikipedia calls it a "snub disphenoid", and as I mentioned before, the name did not help me out at all. It took me several tries to build it correctly. It contains 12 faces and 8 vertices. When I finally had the model I still couldn't figure it out, and spent quite a long time rotating it and examining it. It has a rather strange symmetry. It is front-back and left-right symmetric. And it is almost top-bottom symmetric: If you give it a vertical reflection, you get the same thing back, but rotated 90° around the vertical axis.
When I planned this article I thought I understood it better. Imagine sticking together two equilateral triangles. Call the common edge the "rib". Fold the resulting rhombus along the rib so that the edges go up, down, up, down in a zigzag. Let's call the resulting shape a "wing"; it has a concave side and a convex side. Take two wings. Orient them with the concave sides facing each other, and with the ribs not parallel, but at right angles. So far, so good.
But this is where I started to get it wrong. The two wings have between them eight edges, and I had imagined that you could glue a rhombic antiprism in between them. I'm not convinced that there is such a thing as a rhombic antiprism, but I'll have to do some arithmetic to be sure. Anyway, supposing that there were such a thing, you could glue it in as I said, but if you did the wings would flatten out and what you would get would not be a proper polyhedron because the two triangles in each wing would be coplanar, and polyhedra are not allowed to have abutting coplanar faces. (The putative gyroelongated triangular dipyramid fails for this reason, I believe.)
To make the snub disphenoid, you do stick eight triangles in between the two wings, but the eight triangles do not form a rhombic antiprism. Even supposing that such a thing exists.
I hope to have some nice renderings for you later. I have been doing some fun work in rendering semiregular polyhedra, and I am looking forward to discussing it here. Advance peek: suppose you know how the vertices are connected by edges. How do you figure out where the vertices are located in 3-space?
If you would like to investigate this, the snub disphenoid has 8 vertices, which we can call A, B, ... H. Then:
Here is a list of the eight deltahedra, with links to the corresponding Wikipedia articles:
[ Addendum 20070908: More about deltahedra. ] | s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100135.11/warc/CC-MAIN-20231129173017-20231129203017-00088.warc.gz | CC-MAIN-2023-50 | 5,171 | 16 |
http://mathforcollege.com/nm/videos/youtube/04sle/systemofeqns/Writing%20simultaneous%20linear%20equations%20in%20matrix%20form.htm | math | CHAPTER 05.02: SYSTEM OF EQUATIONS: Writing simultaneous linear equations in matrix form
In this segment well talk about how we will write simultaneous linear equations in matrix form. So let痴 suppose somebody gives you the following simultaneous linear equations. So when somebody is giving you three simultaneous linear equations and let痴 suppose they say - hey write this in the matrix form, we seem to automatically jump to this. So if I asked students to do this they will simply start writing like this. They値l say 25, 5, 1, 64, 8, 1. You got to understand the coefficient of C here is one not zero. Just because there is nothing there 144, 12, 1. Then we値l say A, B, C. And then we値l put 106.8, 177.2 and 279.2.
But the thing which we have to realize is that how do we get from here to here. That痴 our direct step went from here to here. It might come through as a direct step once you have a lot of experience with it but there is a lot of things which go beyond before you get to this level. So one of those steps is that what I can do is I have these three simultaneous linear equations so each one of them is a separate equation. And what I知 going to do is I知 going to put them in this kind of a matrix. I知 going to say hey I知 going to write 25a plus five b plus c right here and I can put 106.8 here. And I guess here 64a plus eight b plus c here and I get 177.2 here. And I get 144a here plus 12b here plus c here and I get 279.2. So what I have basically done is that I have put these three simultaneous linear equations in some sort of a matrix form. Where I got three rows in one column and three rows in one column right here.
And I can rewrite this now, I can say hey what I can do is now I have it written like this but if I write 25, 5 and one so I知 trying to separate out the unknowns here. If I write 25, 5, and 1 here, I could write this as A, B and C. And if I write this a, b, and c. And if I write this A, B and C you can very well see that if I have 25, 5, and one and multiply it by this vector a, b, and c I値l get the first row first column this part right here. Same thing here, 64, 8, and 1. When I multiply 64, 8 and one by a, b, c by this row vector by this column vector I値l get this second row first column of this matrix here. The same thing I値l do 144, 12, and one. If I multiply this row vector by this column vector I will get this third row first column right here. And you can see that this is the three row three columns, and this is three rows one column. And we multiply three rows and three columns by a vector which is three rows and one column. I get three rows and one column right here will be equal to 106.8, 177.2 and 279.2.
So it痴 important to understand that hey we go from this to here and then from here to here through matrix multiplication. This is equal to this is based on when do we consider two matrices to be equal when element to element they are the same. This element right here is the same as this element as given by the first equation, this element here is the same as this element here as given by this equation. And this element here is the same as this element as given by this equation. You also need to recognize that hey these equations can be or these equations right here can also be written as a linear combination. Like I can put a here, and I can put 25, 64 and 144 her plus b times 5, 8 and 12 here plus c times one one one here and that will be equal to 106.8, 177.2 and 279.2. So we can look at these three equations three unknowns also as that hey these sets of equations can be used as a linear combination of this vector, this vector and this vector of the three column vectors equal to this column vector. And whatever linear combination of A, or values of A, B and C which would satisfy this linear combination to be equal to this value this vector right here will be the values of A, B and C. So you got to keep an open mind in terms of recognizing what it means to convert a set of simultaneous linear equations into a matrix form. And that is the end of this segment. | s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100593.71/warc/CC-MAIN-20231206095331-20231206125331-00881.warc.gz | CC-MAIN-2023-50 | 4,087 | 5 |
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-7-exponential-and-logarithmic-functions-7-2-graph-exponential-decay-functions-7-2-exercises-mixed-review-page-491/39 | math | The answer is below.
Work Step by Step
All possible inputs into the function are valid, so the domain is all real numbers. The graph has an absolute minimum at $-9$, so the range is $y\geq -9$.
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662515466.5/warc/CC-MAIN-20220516235937-20220517025937-00265.warc.gz | CC-MAIN-2022-21 | 442 | 5 |
https://www.coursehero.com/file/8595420/Refer-back-to-these-examples-if-you-are-unsure-how-to-do-a-particular/ | math | This preview shows page 1. Sign up to view the full content.
Unformatted text preview: the assigned homework problems. Refer back to these examples if
you are unsure how to do a particular problem. Many of the exam problems will be of one of these
standard types. 16.4: Gre e n's The ore m
Green's Theorem for simply connected regions
Green's Theorem for regions with "holes"
www.math.illinois.edu/~ stefanm/math241- fa13- final.html 2/4 12/22/13 M ath 241, Section E1H, Fall 2013, Final Exam Infor mation Computing areas via Green's Theorem
Summary: Green's Theorem (from a previous semester) 16.5: Div e rge nce , Gradie nt, and Curl
Divergence (div), gradient (grad), and curl of a vector field
Identities for curl(grad(f))) and div(curl(F))
Curl test for a conservative field (3D version of "mixed partials" test)
Connected regions and simply connected regions
Vector forms of Green's Theorem: tangential version (involving curl)...
View Full Document | s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188924.7/warc/CC-MAIN-20170322212948-00294-ip-10-233-31-227.ec2.internal.warc.gz | CC-MAIN-2017-13 | 955 | 14 |
https://www.elevenwarriors.com/forum/college-sports/2019/02/102708/pac-12big-12-alliance | math | Gordon Gee is at it again. Here are some initial thoughts from an OSU perspective:
1. It would cancel out future home and homes with Oregon and Texas if enacted quickly.
2. It would impact future realignment possibilities.
3. Playing twelve P5 opponents increases the chance an alliance champion will have two losses. How will the committee look at that? | s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439740929.65/warc/CC-MAIN-20200815154632-20200815184632-00220.warc.gz | CC-MAIN-2020-34 | 354 | 4 |
http://www.thestargarden.co.uk/Probability.html | math | 1. Probability and quantum mechanics ↑
The Everett, or many worlds, approach to quantum mechanics does away with the objectively indeterminate universe suggested by the collapse approach. There is no objective uncertainty because every physical possibility is actualised.
Some may see this as a virtue, German-Swiss-American physicist Albert Einstein was critical of the collapse approach because of its reliance on objectively indeterminate probabilities - probabilities that relate to an uncertainty that cannot be explained by our ignorance of the physical system.
In a letter to German physicist Max Born, written in 1926, Einstein stated that:
"quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice".
Despite this, we experience quantum events with well-measured probabilities.
2. The incoherence problem ↑
The Everett approach must first explain how it can ascribe probabilities to events at all, given that every physical possibility is certain to happen. This is known as the incoherence problem.
Everett attempted to resolve the incoherence problem by arguing that:
"the subjective experience...is precisely in accordance with the predictions of the usual probabilistic interpretation of quantum mechanics"[3a].
This means that from a subjective point of view, an observer will not be aware of every possibility, and so probabilities represent their chances of observing a specific result.
3. The quantitative problem ↑
Given that it makes sense to talk of probabilities within the Everett approach, a more serious problem arises. What good is it in saying that an atom has a 1% chance of decaying in the next twenty-four hours, when there are only two possibilities, a world where it decays, and a world where it does not?
The quantitative problem asks why Everett is justified in using the Born rule to assign probabilities, rather than assigning an equal probability to each branch.
Everett suggested that:
"in order to establish quantitative results, we must put some sort of measure (weighting) on the elements of a final superposition"[3b].
In the example above, the universe can be thought of as branching into 100 copies, the atom decays in one but not in the 99 others. These 99 worlds remain identical, until new quantum interactions force them to diverge, and so they can be thought of as one world with a weight of 99.
The meaning of the word 'weight' is still debated but the quantitative problem is analogous to the problems raised by classical probabilities. When we throw a weighted dice, for example, we know that there are only six possible outcomes, and so this raises the question of why are we entitled to give them unequal probabilities.
4. Decision theory ↑
Classically, we understand probabilities in terms of decision theory. The decision-theoretic link states that it's rational for a person to use their objective knowledge of a system in order to determine how to act.
Objectively, we know that regular dice have a 1/6 chance of landing on any particular number, and that coins have a 1/2 chance of landing either heads or tails. Weighted dice and coins will have different probabilities associated with each outcome, and a rational person should try to bet on the number that has the highest objective probability.
The problem with this approach is that we do not know how to derive probabilities without knowing the symmetry of the system.
If we actually throw dice and count how many times each outcome occurs, then a set of objective probabilities is expected to emerge, but there is no certainty to this assumption. No matter how many frequency trails we run, we can never know for certain if the dice are weighted. There's always the possibility that we have just been unlucky.
This raises the question of why we should be rationally compelled to use our objective knowledge of probabilities when placing bets. There are three solutions to this problem: functionalism, primitivism, and eliminativism.
4.1 Functionalism ↑
Functionalism suggests that one day we'll have a better understanding of objective probability, and we'll be able to define it as a physical property. This property will be defined independently of the decision-theoretic link but will come to the same conclusions. The frequency approach is an example of a failed attempt at a functional definition.
4.2 Primitivism ↑
Our inability to find a functional definition of objective probability has led some to adopt the second approach, primitivism. Primitivism is the view that we should accept the decision-theoretic link as a fundamental law of nature and not look for a deeper explanation. This is a popular approach but it does not seem entirely satisfactory, the decision-theoretic link is not similar to other concepts that we are willing to accept as fundamental, such as mass, charge, and spin.
4.3 Eliminativism ↑
The only option left is eliminativism. This is the view that there's no such thing as objective probabilities, and so nothing that can explain the decision-theoretic link. This does not seem acceptable, since the concept of objective probability is used in all branches of science as well as in ordinary life.
5. Cautious functionalism ↑
There's no satisfactory explanation for what classical probabilities really are, but primitivism, the denial that a functional definition exists beyond the decision-theoretic link, and eliminativism, the denial of objective probabilities all together, should only be accepted once we have given up on finding a functional definition.
Cautious functionalism is the view that we will one day find a functional definition and, in the meantime, we can use the decision-theoretic link as such. This allows scientists to continue to use decision theory when considering objective probabilities.
There is no further justification for the use of classical probabilities, and so proponents of the Everett approach can defend their use of the Born rule in the same way that proponents of the collapse approach do, using the decision-theoretic link.
If Everett's use of the Born rule is correct, then we should still be rationally compelled to take the probability of each possibility into account, and to bet on the event that has the highest objective probability.
Proponents of the Everett approach can simply state that their definition of the Born rule meets the conditions set by the decision-theoretic link and defend its use on the basis of cautious functionalism.
British physicist David Deutsch showed that they can go further than this, and prove that their concept of 'weight' fits the functional definition of objective probability. This is because it defines objective probability as a physical property that is independent of the decision-theoretic link, but comes to the same conclusions about how to act when faced with uncertainty.
If we accept this claim, then proponents of the Everett approach can defend their use of the Born rule more fully than proponents of the collapse approach. But even if we do not accept Deutsch's proof, the Everett approach is no worse off than the collapse or Bohm approaches in this regard.
In a letter to American physicist Bryce DeWitt, written shortly after his theory was published in 1957, Everett stated that:
"from my point of view there is no preference for deterministic or indeterministic theories. That my theory is fundamentally deterministic is not due to any deep conviction on my part that determinism holds any sacred position...I only object to mixed systems where the character changes with mystical acts of observation". | s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917122174.32/warc/CC-MAIN-20170423031202-00351-ip-10-145-167-34.ec2.internal.warc.gz | CC-MAIN-2017-17 | 7,771 | 40 |
https://www.zhygear.com/influence-of-external-load-change-of-involute-spur-gear-transmission-on-static-characteristics-of-modified-gear/ | math | Linear modification is adopted with the modification quantity of 0.022m m and the length of 2.8m M. Fixed torque of 1500N.m, 2000N.m, 2500N.m, 3000N.m and 3500N.m are applied to gear 1 respectively.The engagement stiffness and load distribution coefficient between teeth are shown in the figure under the above five torques.
As can be seen from Fig. a, when the input torque is less than the average engine torque of 2500N.m, with the decrease of the torque, the proportion of time occupied by single-tooth engagement gradually increases in a single engagement cycle, the engagement degree of gears gradually decreases, and the steep change of engagement rigidity gradually increases.When the input torque is greater than the average engine torque, the steep change in engagement stiffness increases with the increase of the torque, although the coincidence remains unchanged.It can be seen from Figure B that when the torque is less than the average engine torque, the load distribution coefficient between teeth is 0 and the gear is in single-tooth engagement state in a single engagement cycle in the area near the entry and exit engagement.When the torque is greater than the average engine torque, the sudden change of gear load becomes more and more obvious by further increasing the input torque.
Under different torques, the corresponding amplitude of the first five-order meshing frequency is shown in the figure. It can be seen from the figure that the corresponding amplitude of the first-order meshing frequency increases gradually and that of the second-order meshing frequency decreases gradually with the increase of input torque for the modified gear.Combining the analysis results, it can be seen that the variation trend of meshing stiffness and load distribution coefficient between teeth with increasing input torque is opposite to that with decreasing profile.
In conclusion, from the analysis results under quasi-static conditions, in order to effectively ease the sharp change of meshing stiffness caused by alternation of single and double teeth, to realize the smooth transition of load change during gear transmission and to ensure the overlap of gears, the amount of modification should be the amount of deformation at the highest point of single teeth meshing under a fixed design load, and the selection of modification length should satisfy the entire area of double teeth meshing area.In the modification area, since the modification type mainly affects the curve form of meshing rigidity and load variation in the modification area, the degree of difficulty in machining the modification type should be considered and the type of modification that is easy to machine should be selected.
At the same time, the profile modification parameters and external loads have a great influence on the corresponding amplitudes of each order of meshing frequency of meshing stiffness.Previous studies have shown that the dynamic characteristics of gear drive system are sensitive to the variation of amplitude corresponding to meshing frequency of each order of meshing stiffness, which indicates that profile modification will inevitably affect the dynamic characteristics of gear drive system. | s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679516047.98/warc/CC-MAIN-20231211174901-20231211204901-00608.warc.gz | CC-MAIN-2023-50 | 3,214 | 5 |
https://space.stackexchange.com/questions/57336/why-are-jupiters-trojans-even-remotely-stable | math | The spacecraft Lucy is en route to explore Jupiter's trojan asteroids. Lucy is so named because the trojan asteroids are believed to be fossil remnants from the formation of the solar sytem. Presumably this means that the astroids have been occupying the Lagrange points for a very long time.
At Saturn's closest approach to Jupiter's Lagrange points, it is actually slightly closer than Jupiter (~650 million km vs. 780 million km). Saturn has only 30% the mass of Jupiter, so its gravitational effects are smaller. Nevertheless, for periods of many months, Saturn will exert on the asteroids a force of up to 43% of the gravitational force of Jupiter. So how do these asteroids manage to remain in the vicinity of the Lagrange points given such large perturbing forces?
I suppose the reality of their apparent longevity must be supported by simulations, but are there some simple arguments or observations that give some insight into why, given the presence of Saturn, the trojans are still there after five billion years? | s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100135.11/warc/CC-MAIN-20231129173017-20231129203017-00517.warc.gz | CC-MAIN-2023-50 | 1,024 | 3 |
https://www.physicsforums.com/threads/speed-vs-velocity-difference-examples.160172/ | math | could someone please clarify th e difference between the 2 and maybe give me an example?
blackice552 said:could someone please clarify th e difference between the 2 and maybe give me an example?
Speed and velocity are often used interchangeably, but they actually have distinct meanings in the field of physics. Speed refers to how fast an object is moving, while velocity refers to both the speed and direction of motion.
Yes, an object can have a constant speed but changing velocity. This is because velocity takes into account the direction of motion, so if an object is moving at a constant speed but changes direction, its velocity will also change.
Speed is typically measured in units of distance per time, such as meters per second or miles per hour. Velocity is also measured in these units, but it also includes a directional component, such as north or east.
Some examples of speed include a car driving at 60 miles per hour or a person running at 10 meters per second. Examples of velocity include a plane flying north at 500 miles per hour or a cyclist riding east at 20 kilometers per hour.
Speed and velocity are key factors in determining an object's motion. In general, an object with a higher speed or velocity will cover more distance in a shorter amount of time compared to an object with a lower speed or velocity. Additionally, changes in an object's speed or velocity can affect its acceleration, which is the rate at which its speed or velocity changes over time. | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817253.5/warc/CC-MAIN-20240419013002-20240419043002-00187.warc.gz | CC-MAIN-2024-18 | 1,488 | 7 |
http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.375363 | math | The non-stationary response of vehicles on rough ground
Vehicles moving on rough surfaces are subject to inputs which may be conveniently regarded as a combination of deterministic and random processes. Although this general problem is briefly addressed, it is the latter class of inputs which is of concern in the present work. In general, the random component of the excitation is `perceived' by the vehicle as a non-stationary random process, due either to inhomogeneity (spatial non-stationarity) in the surface roughness, or to variations in the vehicle velocity, or both. Analysis of the response of vehicles to such processes is further complicated by the multiple degree of freedom nature of the problem and by inherent non-linearity in vehicle dynamic systems, rendering exact statistical analysis of such systems analytically intractable, and thus requiring numerical simulation which is a costly alternative. A unified, analytical approach to this problem is presented here combining the techniques of linear systems theory and the approximate method known as statistical linearization, to facilitate the approximate analysis of non-linear systems excited by non-stationary random processes. The basis of the method is the use of a `shaping filter' description for the ground roughness, i.e., the height profile is represented as the output of a white noise excited, linear filter in the spatial domain (extensive justification of this assumption is presented). The key to the present work is the linking of the space domain filter with a state-space model for the vehicle dynamics by a formal change of variable, i.e., space is regarded as a function of time, related via the (variable) velocity function. This yields a time variable filter formulation for the excitation process which may then be coupled into the dynamic equations. After some further manipulations (using results from the theory of generalisation functions) differential equations may be constructed for the propagation of the mean vector and zero-lag auto-covariance matrix. For linear systems these results are exact and have been extended to the multiple input (multiple wheel) case. This extension presents no conceptual difficulty although it is computationally considerably more involved. For non-linear dynamic systems the method of statistical linearization is adopted so that the above method of obtaining means and covariances applies. This technique is essentially the replacement of the true non-linear element by a linear one such that their output differences is in some way minimised. This results in linear coefficients depending on the instantaneous mean and variance of response, just those quantities calculated by the above method and so a coupled set of non-linear, deterministic differential equations are obtained, governing the propagation of the approximate means and covariances of response. In order to validate results obtained by this method, in the absence of analytical solutions, extensive use is made of Monte-Carlo simulation. A fundamental concept arising as a result of the formulation is that of the `covariance equivalence' of two random processes. This enables the frequency-time (evolutionary) spectral analysis of random processes having a `frequency modulated' structure, a task which was hitherto not possible. This concept has also found application in the field of acoustics in the study of moving noise sources. | s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218187519.8/warc/CC-MAIN-20170322212947-00460-ip-10-233-31-227.ec2.internal.warc.gz | CC-MAIN-2017-13 | 3,438 | 2 |
https://www.twi-global.com/technical-knowledge/job-knowledge/mechanical-testing-tensile-testing-part-1-069 | math | Mechanical testing is carried out to produce data that may be used for design purposes or as part of a material joining procedure or operator acceptance scheme. The most important function may be that of providing design data since it is essential that the limiting values that a structure can withstand without failure are known.
Fig.1. Typical tensile testing machine
Inadequate control of the material properties by the supplier, or incompetent joining procedures and operatives are, however, equally crucial to the supply of a product that is safe in use. An example of this dual role of mechanical testing is the tensile test that may be used either to determine the yield strength of a steel for use in design calculations or to ensure that the steel complies with a material specification's strength requirements.
Mechanical tests may also be divided into quantitative or qualitative tests. A quantitative test is one that provides data that will be used for design purposes, a qualitative test where the results will be used for making comparisons - hardness or Charpy-V tests - for example as a 'go/no go test' such as the bend test.
Mechanical property data are obtained from a relatively small number of standard tests and these will be covered over the next several articles. These will include tensile and toughness tests, the tests used for welding procedure and welder approval and those used for the determination of in-service properties.
As mentioned earlier the tensile test is used to provide information that will be used in design calculations or to demonstrate that a material complies with the requirements of the appropriate specification - it may therefore be either a quantitative or a qualitative test.
The test is made by gripping the ends of a suitably prepared standardised test piece in a tensile test machine and then applying a continually increasing uni-axial load until such time as failure occurs. Test pieces are standardised in order that results are reproducible and comparable as shown in Fig 2.
Fig.2. Standard shape tensile specimens
Specimens are said to be proportional when the gauge length, L0, is related to the original cross sectional area, A0, expressed as L0 =k√A0 . The constant k is 5.65 in EN specifications and 5 in the ASME codes. These give gauge lengths of approximately 5x specimen diameter and 4x specimen diameter respectively - whilst this difference may not be technically significant it is important when claiming compliance with specifications.
Fig.3. Stress/strain curve
Both the load (stress) and the test piece extension (strain) are measured and from this data an engineering stress/strain curve is constructed, Fig.3. From this curve we can determine:
a) the tensile strength, also known as the ultimate tensile strength, the load at failure divided by the original cross sectional area where the ultimate tensile strength (U.T.S.), σmax = Pmax /A0 , where Pmax = maximum load, A0 = original cross sectional area. In EN specifications this parameter is also identified as 'Rm';
b) the yield point (YP), the stress at which deformation changes from elastic to plastic behaviour ie below the yield point unloading the specimen means that it returns to its original length, above the yield point permanent plastic deformation has occurred, YP or σy = Pyp /A0 where Pyp = load at the yield point. In EN specifications this parameter is also identified as 'Re ';
c) By reassembling the broken specimen we can also measure the percentage elongation, El% how much the test piece had stretched at failure where El% = (Lf - L0 /Lo ) x100 where Lf = gauge length at fracture and L0 = original gauge length. In EN specifications this parameter is also identified as 'A' ( Fig.4a).
d) the percentage reduction of area, how much the specimen has necked or reduced in diameter at the point of failure where R of A% =(A0 - Af /A0 ) x 100 where Af = cross sectional area at site of the fracture. In EN specifications this parameter is also identified as 'Z', ( Fig.4b).
Fig.4: a) Calculation of percentage elongation, b) Calculation of percentage reduction of area
(a) and (b) are measures of the strength of the material, (c) and (d) indicate the ductility or ability of the material to deform without fracture.
The slope of the elastic portion of the curve, essentially a straight line, will give Young's Modulus of Elasticity, a measure of how much a structure will elastically deform when loaded.
A low modulus means that a structure will be flexible, a high modulus a structure that will be stiff and inflexible.
To produce the most accurate stress/strain curve an extensometer should be attached to the specimen to measure the elongation of the gauge length. A less accurate method is to measure the movement of the cross-head of the tensile machine.
The stress strain curve in Fig.3 shows a material that has a well pronounced yield point but only annealed carbon steel exhibits this sort of behaviour. Metals that are strengthened by alloying, by heat treatment or by cold working do not have a pronounced yield and some other method must be found to determine the 'yield point'.
This is done by measuring the proof stress ( offset yield strength in American terminology), the stress required to produce a small specified amount of plastic deformation in the test piece.
The proof stress is measured by drawing a line parallel to the elastic portion of the stress/strain curve at a specified strain, this strain being a percentage of the original gauge length, hence 0.2% proof, 1% proof (see Fig.5).
Fig.5. Determination of proof (offset yield) strength
For example, 0.2% proof strength would be measured using 0.2mm of permanent deformation in a specimen with a gauge length of 100mm. Proof strength is therefore not a fixed material characteristic, such as the yield point, but will depend upon how much plastic deformation is specified. It is essential therefore when considering proof strengths that the percentage figure is always quoted. Most steel specifications use 0.2% deformation, RP0.2 in the EN specifications.
Some materials such as annealed copper, grey iron and plastics do not have a straight line elastic portion on the stress/strain curve. In this case the usual practice, analogous to the method of determining proof strength, is to define the 'yield strength' as the stress to produce a specified amount of permanent deformation.
Part 2 of this series on mechanical testing will cover welding procedure approval tensile testing.
This article was prepared by Gene Mathers. | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816939.51/warc/CC-MAIN-20240415014252-20240415044252-00125.warc.gz | CC-MAIN-2024-18 | 6,560 | 28 |
http://intlpress.com/HHA/v2/n1/a3/ | math | We apply group cohomological methods to calculate the cohomology of $\Kn_*\BP$ as a $\Kn_*\Kn$-comodule, recovering recent results of Hovey and Sadofsky. As applications we determine the Chromatic Spectral Sequence for $\BP$ based on Johnson and Wilson's $\En$, showing the relationship to some generalizations of the classical Hattori-Stong Theorem and determine the change of Hopf algebroid spectral sequence associated with the natural map $\BP\lra\En$, extending calculations of Clarke for the Todd orientation $\MU\lra\KU$.
Homology, Homotopy and Applications, Vol. 2, 2000, No. 3, pp 29-40
Available as: dvi dvi.gz ps ps.gz | s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163800358/warc/CC-MAIN-20131204133000-00027-ip-10-33-133-15.ec2.internal.warc.gz | CC-MAIN-2013-48 | 629 | 3 |
https://rebab.net/what-is-the-slope-of-a-horizontal-line/ | math | Let"s consider again the two equations we did first on the previous page, and compare the lines" equations with their slope values.
You are watching: What is the slope of a horizontal line
The first line"s equation was y = (2/3) x – 4, and the line"s slope was m = 2/3.
The second line"s equation was y = –2x + 3, and the line"s slope was m = –2. In both cases, the number multiplied on the variable x was also the value of the slope for that line. This relationship always holds true: If the line"s equation is in the form "y=", then the number multiplied on x is the value of the slope m.
This relationship will become very important when you start working with straight-line equations.
Now let"s consider those two equations and their graphs.
For the first equation, y = ( 2/3 )x – 4, the slope was m = 2/3, a positive number. The graph looked like this:
Notice how the line, as we move from left to right along the x-axis, is edging upward toward the top of the drawing; technically, the line is an "increasing" line. And... the slope was positive.
This relationship always holds true: If a line is increasing, then its slope will be positive; and if a line"s slope is positive, then its graph will be increasing.
For the second line, y = –2x + 3, the slope was m = –2, a negative number. The graph looked like this:
Notice how the line, as we move from left to right along the x-axis, is edging downward toward the bottom of the drawing; technically, the line is a "decreasing" line. And... the slope was negative.
This relationship is always true: If a line is decreasing, then its slope will be negative; and if a line"s slope is negative, then its graph will be decreasing.
This relationship between the sign on the slope and the direction of the line"s graph can help you check your calculations: if you calculate a slope as being negative, but you can see from the graph of the equation that the line is actually increasing (so the slope must be positive), then you know you need to re-do your calculations. Being aware of this connection can save you points on a test because it will enable you to check your work before you hand it in.
So now we know: Increasing lines have positive slopes, and decreasing lines have negative slopes. With this in mind, let"s consider the following horizontal line:
Is the horizontal line edging upward; that is, is it an increasing line? No, so its slope can"t be positive. Is the horizontal line edging downward; that is, is it a decreasing line? No, so its slope can"t be negative. What number is neither positive nor negative?
So the slope of this (and any other) horizontal line should, logically, be zero. Let"s do the calculations to confirm this. Using the (arbitrary) points from the line, (–3, 4) and (5, 4), the slope computes as:
This relationship always holds: a slope of zero means that the line is horizontal, and a horizontal line means you"ll get a slope of zero.
(By the way, all horizontal lines are of the form "y = some number", and the equation "y = some number" always graphs as a horizontal line.)
Is the vertical line going up on one end? Well, yes, kind of. So maybe the slope will be positive...? Is the vertical line going down on the other end? Well, again, kind of. So maybe the slope will be negative...?
But is there any number that is both positive and negative? Nope.
Verdict: vertical lines have NO SLOPE. The concept of slope simply does not work for vertical lines. The slope of a vertical line does not exist!
Let"s do the calculations to confirm the logic. From the line"s graph, I"ll use the (arbitrary) points (4, 5) and (4, –3). Then the slope is:
We can"t divide by zero, which is of course why this slope value is "undefined".
This relationship is always true: a vertical line will have no slope, and "the slope is undefined" or "the line has no slope" means that the line is vertical.
(By the way, all vertical lines are of the form "x = some number", and "x = some number" means the line is vertical. Any time your line involves an undefined slope, the line is vertical; and any time the line is vertical, you"ll end up dividing by zero if you try to compute the slope.)
Warning: It is very common to confuse these two types of lines and their slopes, but they are very different.
Just as "horizontal" is not at all the same as "vertical", so also "zero slope" is not at all the same as "no slope".
Just as a "Z" (with its two horizontal lines) is not the same as an "N" (with its two vertical lines), so also "Zero" slope (for a horizontal line) is not the same as "No" slope (for a vertical line).
See more: Distance From Lusk Wy To Rapid City Sd And Lusk, Wy, Distance From Lusk, Wy To Rapid City, Sd
The number "zero" exists, so horizontal lines do indeed have a slope. But vertical lines don"t have any slope; "slope" simply doesn"t have any meaning for vertical lines.
It is very common for tests to contain questions regarding horizontals and verticals. Don"t mix them up! | s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585196.73/warc/CC-MAIN-20211018031901-20211018061901-00572.warc.gz | CC-MAIN-2021-43 | 4,986 | 31 |
https://www.jiskha.com/display.cgi?id=1226660310 | math | posted by Velaria .
What is the minimum work function for a metal for visible light (wavelengths between 400 and 700 ) to eject photoelectrons?
Does work function=hc divided by initial wavelength.
What would the initial wavelength be? 4x10E-7 nm?
The 400 nm photons would be the ones most capable of ejecting electrons. The work function you calculate with your formula will lead to the maximum work function that would allow ejection, not the minimum.
Work function is expressed in volts. You will have to divide the photon energy by the electron charge, e. | s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805881.65/warc/CC-MAIN-20171119234824-20171120014824-00446.warc.gz | CC-MAIN-2017-47 | 558 | 6 |
https://www.coursehero.com/tutors-problems/Statistics-and-Probability/19987663-I-bought-a-Honda-CR-V-the-other-day-and-my-neighbor-bought-a-Ford-Esca/ | math | I bought a Honda CR-V the other day and my neighbor bought a Ford Escape. Honda says the CR-V will last an
average of 8 years with a standard deviation of 1.1 years. Ford says the Escape lasts longer, with an average life expectancy of 9 years and a standard deviation of 1.3 years. Which car, on average, is relatively more reliable?
a.The two cars are equally reliable
b.It is not possible to calculate relative reliability | s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601040.47/warc/CC-MAIN-20200120224950-20200121013950-00536.warc.gz | CC-MAIN-2020-05 | 425 | 4 |
https://www.physicsforums.com/threads/volume-between-given-planes.297495/ | math | 1. The problem statement, all variables and given/known data (1 pt) Find the volume of the region bounded by the planes 2. Relevant equations V = ∫∫7/4-6/4y-2/4x 3. The attempt at a solution Since y=x I found their values when z = 0. 6x+2x=7, x=7/8 y= 7/8 is the maximum value y will have in this integration as it decreases as x approaches its maximum value: 2x = 7, x=7/2. When I made a 2-D image of the region I would be integrating on I came up with a triangle which had two areas that needed to be integrated separately as in the first region: 0≤x≤7/8 and 0≤y≤x, and in the second region: 7/8≤x≤7/2 and 0≤y≤(7/6 - x/3). After integrating my result was 0.893229167. What I am wondering is if I am using the correct limits for my integration or if I made a mistake in my math. | s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202303.66/warc/CC-MAIN-20190320064940-20190320090940-00256.warc.gz | CC-MAIN-2019-13 | 799 | 1 |
https://cpanhd.sitehost.iu.edu/C101webnotes/composition/formmolcmpds.html | math | Molecules consist of two or more atoms bonded to one another
through "covalent" bonds. Identification of these molecules
is through their molecular formulas. There is a hierarchy of
Consider the benzene molecule which has 6 carbon and 6 hydrogen
atoms. Therefore the ratio of carbon to hydrogen atoms is 6 to
6 which we can simplify to 1 to 1 (i.e., 1:1). The empirical
formula expresses this most simple ratio, i.e. C1H1 (or CH). The empirical formula expresses
the most simple ratio of atoms in the molecule. Examples:
Molecular formulas go one step beyond the empirical formula
in that they express not only the correct ratio but the correct
number of atoms in the molecule. In the case of benzene the molecular
formula would be C6H6.
Notice that sometimes the empirical and molecular formulas
are the same. This will happen when the molecular formula also
expresses the most simple ratio of atoms.
The structural formula not only has the correct number of
atoms but includes the bonding structure of the molecule (i.e.,
which atoms are bonded together). Examples:
Molecules are three dimensional (3D) objects. The 3D
structural formula is an attempt to convey the 3D geometry of
the molecule. Examples:
The triangular bonds depict atoms coming out of the plane
and the dashed bonds depict atoms going in back of the plane.
The space filling formula includes the relative sizes
of the atoms. The molecule acetaldehyde (C2H4O) below is an example. Gray spheres are carbon
atoms, yellow spheres are hydrogens and the red sphere is oxygen. | s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711396.19/warc/CC-MAIN-20221209112528-20221209142528-00422.warc.gz | CC-MAIN-2022-49 | 1,539 | 26 |
https://oneclass.com/class-notes/ca/u-of-c/engg/engg-202/477721-engg202-lecture-1-tutorial-13-solutionspdf.en.html | math | ENGG 202 Lecture Notes - Lecture 1: National Air Force Of Angola, Rolly Teranishi, A Roads In Zone 2 Of The Great Britain Numbering Scheme
74 views5 pages
ENGG 202 Tutorial 13 April 9/10, 2013
Shear Force and Bending Moment Diagrams
1) Draw the shear and moment diagrams for the two beams shown below.
ANS: (a). Vleft = +3kN, constant up to point load then jumps to – 6 kN, Vright = -6 kN, M = 0 at ends, Mmax = 12 kN-
m at location of point load, linear variation (b) Vmax = 4 kN at A, linear up to VB=0, Mmax = -10 kN-m at A, parabolic
up to M = -6 kNm at B.
Solution Strategy: From FBD of each beam find reactions at A (and B for the first beam). (a)
Section the beam between A and the point load, and again between the point load and B. draw F,
V, and M, write Eqns of Equil to get equations for V and M at each section. Calculate V and M at
critical points (A,B,and location of point load). (b) Section the beam between A and the end of
the beam (section the distributed load as well). draw F, V, and M, write Eqns of Equil to get
equations for V and M at each section. Calculate V and M at critical points (each end of beam).
2) A roof structure is represented by the member ABCD and loading shown below. Neglecting
the weight of the member:
(a) Determine the magnitude and direction of the reactions at point A and the magnitude of the
force in cable DE.
(b) Determine the internal axial force, shear force, and bending moment at point B by
sectioning through BD and drawing the Free Body Diagram of segment BD.
(c) Draw the shear force and bending moment diagrams for beam ABC. Indicate the values for
shear force and bending moment at points A, B, and C.
ANS: a) TDE = 1587.2 N, Ax = 1540 N right, Ay
= 255 N up b) N = 385.1 N down, V = 1540 N
right, M = 1540 Nm CW c) SFD starts at +255 N
linear decrease to 215 N at 4 m, jump up to 600 N
at 4 m, constant to end of beam, BMD starts at 0,
parabolic (upside down) increase up to 940 Nm at
4 m, jump down to -600 Nm at 4 m, linear
increase to 0 at right end of beam. must show
calculations for the key values at ends of beam
and at 4 m.
Solution Strategy: a) draw FBD of bar
ABCD and solve for tension force and
reactions at A using eqns of equil. b) section horizontally through segment BD at B. draw the
FBD of the segment BD (tension force at D and internal forces/moment at B). use eqns of equil
to solve for the internal forces/moment. c) draw FBD of segment ABC (reactions at A,
distributed load, force at C, internal forces from b) act at B but are equal and opposite to those
computed in b)). SFD and BMD can be obtained either by section between A and B and again
ENGG 202 Tutorial 13 April 9/10, 2013
between B and C, then using equations of equil to get equations for V and M as a function of x.
alternatively, use the relationship between the diagrams: 1) for the SFD: at A, V = the reaction
Ay, then it decreases linearly at a slope of -10 N/m up to point B. The axial force from part b)
causes a jump upward = to that force (jumps to 600 N). Then constant to the end of the beam at
600 N. 2) for the BMD: starts at M=0 at point A. moment diagram is parabolic up to point B
(concave downward). The moment at B can be computed from the area under the shear diagram
between A and B. At B there is a jump down of 1540 Nm to -600 Nm due to the internal moment
at B found in part b). there is a linear increase (slope = shear = +600) over the last m back up to
M = 0 at point C.
3) Determine the internal axial force, shear
force and bending moment at points E and F
of the beam shown.
ANS: VF = -215 N, NF = 0, MF = 660 N-m, NE = 470
N, VE = 215 N, ME = 660 N-m
Solution Strategy: From FBD of AEDB
find reactions at A and tension in the cable.
Section member at E (section distributed load
too, and consider left section). Draw N, V,
and M at point E. Write Eqns of Equil and
solve. Repeat for point F (consider right
Friction and Centroid Problems
4) Block C has a mass of 50 kg and is confined between two
walls by smooth rollers. If the block rests on top of the 40 kg
spool, determine the required coefficients of static friction at
A and B so that the spool slides (doesn’t roll) at A and B when
the magnitude of the applied force is P = 300 N.
ANS: at A: 0.153, at B: 0.255
Soln Strategy: this is an impending motion problem (“required coefficients”) draw the FBD of
the spool. The normal force at A is equal to the weight of the block (we know this from FBD of
the block). just before the spool slides, the friction force at each of A and B reaches the maximum
friction force. Since the spool slides and doesn’t roll both friction forces are to the left. Use Eqns
of Equil to solve for the friction forces at A and B and the normal force at B. the friction coeff at
A is FA/NA and likewise for B. | s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141189030.27/warc/CC-MAIN-20201126230216-20201127020216-00687.warc.gz | CC-MAIN-2020-50 | 4,780 | 73 |
https://scott-duncan.medium.com/a-mononumerosis-example-and-measurement-scales-from-an-old-blog-6523493bb543 | math | “A Mononumerosis Example (and Measurement Scales)” from an Old Blog
Many years ago, I had a blog where I commented on a number of things related to software process largely with an agile-related slant. I got directed to that blog a couple days ago and I thought I might present them here and see what people think. I have not updated them in any serious fashion because I am happy to have them criticized in the light of current thinking.
So, from September of 2009 here is the sixteenth:
Mononumerosis, according to Wiktionary is “The oversimplification of a metric by using a single numerical value to characterize a complex phenomenon or system.”
My example involves the very common practice of doing surveys, asking people to rate something on a 1 to N scale, then reporting results using an “average” value for that something.
Really, there are two bad things at work here, from a statistics and survey perspective, I believe. Let me deal with the one that is not the main point of this post, but important for people to consider. Once again, Wikipedia covers the broad subject, but I’ll just explain it briefly.
The Scale Problem
Most 1 to N scales have nothing numeric about them, except for the fact that they use numbers as symbols for points on the scale. It would be better to use words to indicate what the points on the scale mean, since the scale is really ordinal, at best.
That is, from left to right or right to left each position is considered higher or lower (better or worse) in some “value” than those around it. But there is no guarantee the space between them is mathematically equal (or it would be at least an interval scale), so it cannot be a ratio scale, where multiplying and dividing are legitimate operations.
And, that’s the point, using numbers to represent an ordinal scale then adding up values and dividing to get an average is, technically, meaningless. (You can count instances of such points on a scale and report how many results you got for each point on the scale.)
Of course, this is done all the time on customer surveys, conference feedback forms, the ratings on Amazon, etc. And all such examples seem to end up with an average rating for the questions on the survey. (Amazon, at least, shows you the counts for each “star” value as well as the whole feedback statement from those doing the rating.)
Another issue with ordinal scales is that there is no way to be sure one person’s 3 really is the same as another person’s because the surveys often do not place any substantive interpretation on the points to help you judge where your sense of evaluation for that question would fit. But, enough of that…you get the idea.
The Results Representation Problem
This is the more serious issue with the mononumerosis question. Let’s even say you could legitimately do adding and dividing and get an average. Does that tell you an accurate story about what all the respondents taken together felt? Or does it represent some imaginary respondent’s evaluation?
Here’s a few examples.
Let’s say you have 3 data sets of 10 responses on a 1 to 5 scale each with the following numbers of each data set being the number of 1s, 2s, 3s, 4s and 5s for each data set:
Bell — 1, 2, 4, 2, 1
Flat — 2, 2, 2, 2, 2
Camel — 0, 5, 0, 5, 0
This will give you an average of “3” for each.
I’m sure you can see from the data itself that “3” isn’t the same as “3” isn’t the same as “3” when it comes to the actual sense of what responses to the question would mean.
Here they are graphed in two ways (both showing the same thing, but one might be more meaningful for you than the other):
Each shows the very different impression of what the sets of responses might mean.
My point is that it matters how data is represented based on its scale and distribution. So watch out for mononumerosis (and scales) when you are given survey results. | s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572089.53/warc/CC-MAIN-20220814234405-20220815024405-00473.warc.gz | CC-MAIN-2022-33 | 3,921 | 24 |
https://civilengineeringx.com/structural-analysis/plane-trusses-by-the-method-of-joints-problems-and-solutions/ | math | Identify all zero-force members in the Fink roof truss subjected to an unbalanced snow load, as shown in Fig. 4.18.
It can be seen from the figure that at joint B, three members, AB;BC, and BJ, are connected, of which AB and BC are collinear and BJ is not. Since no external loads are applied at joint B, member BJ is a zero-force member. A similar reasoning can be used for joint D to identify member DN as a zero-force member. Next, we focus our attention on joint J, where four members, AJ; BJ; CJ, and JK, are connected and no external loads are applied. We have already identified BJ as a zero-force member. Of the three remaining members, AJ and JK are collinear; therefore, CJ must be a zero-force member. Similarly, at joint N, member CN is identified as a zero-force member; the same type of arguments can be used for joint C to identify member CK as a zero-force member and for joint K to identify member KN as a zero-force member. Finally, we consider joint N, where four members, CN;DN;EN, and KN, are connected, of which three members, CN;DN, and KN, have already been identified as zero-force members. No external loads are applied at joint N, so the force in the remaining member, EN, must also be zero.
Procedure for Analysis
The following step-by-step procedure can be used for the analysis of statically determinate simple plane trusses by the method of joints.
1. Check the truss for static determinacy, as discussed in the preceding section. If the truss is found to be statically determinate and stable, proceed to step 2. Otherwise, end the analysis at this stage.
(The analysis of statically indeterminate trusses is considered in Part Three of this text.)
2. Identify by inspection any zero-force members of the truss.
3. Determine the slopes of the inclined members (except the zero-force members) of the truss.
4. Draw a free-body diagram of the whole truss, showing all external loads and reactions. Write zeros by the members that have been identified as zero-force members.
5. Examine the free-body diagram of the truss to select a joint that has no more than two unknown forces (which must not be collinear) acting on it. If such a joint is found, then go directly to the next step. Otherwise, determine reactions by applying the three equations of equilibrium and the equations of condition (if any) to the free body of the whole truss; then select a joint with two or fewer unknowns, and go to the next step.
6. a. Draw a free-body diagram of the selected joint, showing tensile forces by arrows pulling away from the joint and compressive forces by arrows pushing into the joint. It is usually convenient to assume the unknown member forces to be tensile.
b. Determine the unknown forces by applying the two equilibrium equations ∑Fx =0 and ∑Fy =0. A positive answer for a member force means that the member is in tension, as initially assumed, whereas a negative answer indicates that the member is in compression. If at least one of the unknown forces acting at the selected joint is in the horizontal or vertical direction, the unknowns can be conveniently determined by satisfying the two equilibrium equations by inspection of the joint on the free-body diagram of the truss.
7. If all the desired member forces and reactions have been determined, then go to the next step. Otherwise, select another joint with no more than two unknowns, and return to step 6.
8. If the reactions were determined in step 5 by using the equations of equilibrium and condition of the whole truss, then apply the remaining joint equilibrium equations that have not been utilized so far to check the calculations. If the reactions were computed by applying the joint equilibrium equations, then use the equilibrium equations of the entire truss to check the calculations. If the analysis has been performed correctly, then these extra equilibrium equations must be satisfied. | s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947474573.20/warc/CC-MAIN-20240225003942-20240225033942-00744.warc.gz | CC-MAIN-2024-10 | 3,904 | 14 |
https://www.hornsociety.org/forum/3-Instruments--Equipment/511-looking-for-an-r1-leadpipe-or-something-similar | math | Hey all! I'm not sure if this is the right place to post this question, but I couldn't really find anywhere else. My name is Aaron Bartos, and I'm a student at CCM in Cincinnati, OH. I play an older Ricco Kühn, and have been noticing since my purchase two years ago that the airflow in my horn felt quite restricted.
While fooling around a bit at a horn repairman's house, we discovered that by adding a Rauch R1 pipe to my horn, not only did the horn play more freely, but improved the tone, response, and overall feel of the horn tenfold. I was unable to purchase that exact pipe however, and for the past few months I have been looking for someone to buy a R1 pipe off of. After many phone calls and emails (including one to Mr. Dan Rauch himself) I have been so far unsuccessful in finding one. My professor thought it would be a good idea to post here on IHS in hopes that I would have better luck.
Is there anyone out there that would be willing to sell me their R1 pipe, or know who could talk to so that I could purchase one? Any help would be much appreciated. | s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250594662.6/warc/CC-MAIN-20200119151736-20200119175736-00467.warc.gz | CC-MAIN-2020-05 | 1,070 | 3 |
https://www.usingenglish.com/forum/threads/138946-Manufacturing-or-Production | math | It all depends on the context.
For instance you might say, 'the factory increased production' last year', meaning it increased the volume of goods leaving the factory, whereas the word 'manufacturing' would not mean quite the same. 'Manufacturing' might imply manufacturing capacity, i.e. the ability of the factory to produce more whether it did or not.
Interested in Language | s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718426.35/warc/CC-MAIN-20161020183838-00313-ip-10-171-6-4.ec2.internal.warc.gz | CC-MAIN-2016-44 | 377 | 3 |
https://ftp.aimsciences.org/journal/1937-1632/2021/14/12 | math | Discrete and Continuous Dynamical Systems - S
December 2021 , Volume 14 , Issue 12
Issue on advances in partial differential equation
Select all articles
The paper studies the asymptotic behaviour of solutions to a second-order non-linear discrete equation of Emden–Fowler type
In this paper, we consider a class of finitely degenerate coupled parabolic systems. At high initial energy level
The Cauchy problem of one dimensional generalized Boussinesq equation is treated by the approach of variational method in order to realize the control aim, which is the control problem reflecting the relationship between initial data and global dynamics of solution. For a class of more general nonlinearities we classify the initial data for the global solution and finite time blowup solution. The results generalize some existing conclusions related this problem.
We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.
In this paper, we study axisymmetric homogeneous solutions of the Navier-Stokes equations in cone regions. In [James Serrin. The swirling vortex. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 271(1214):325-360, 1972.], Serrin studied the boundary value problem in half-space minus
In this paper, we study a reaction-diffusion SEI epidemic model with/without immigration of infected hosts. Our results show that if there is no immigration for the infected (exposed) individuals, the model admits a threshold behaviour in terms of the basic reproduction number, and if the system includes the immigration, the disease always persists. In each case, we explore the global attractivity of the equilibrium via Lyapunov functions in the case of spatially homogeneous environment, and investigate the asymptotic behavior of the endemic equilibrium (when it exists) with respect to the small migration rate of the susceptible, exposed or infected population in the case of spatially heterogeneous environment. Our results suggest that the strategy of controlling the migration rate of population can not eradicate the disease, and the disease transmission risk will be underestimated if the immigration of infected hosts is ignored.
Considered herein is the well-posedness and stability for the Cauchy problem of the fourth-order Schrödinger equation with nonlinear derivative term
This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.
In this paper, we study the fractional pseudo-parabolic equations
In this paper, the initial-boundary value problem for a class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films is studied. By means of the theory of potential wells, the global existence, asymptotic behavior and finite time blow-up of weak solutions are obtained.
In this paper, for the damped generalized incompressible Navier-Stokes equations on
This paper is devoted to studying the dynamical stability of periodic peaked solitary waves for the generalized modified Camassa-Holm equation. The equation is a generalization of the modified Camassa-Holm equation and it possesses the Hamiltonian structure shared by the modified Camassa-Holm equation. The equation admits the periodic peakons. It is shown that the periodic peakons are dynamically stable under small perturbations in the energy space.
We consider the well-posedness of solution of the initial boundary value problem to the fourth order wave equation with the strong and weak damping terms, and the logarithmic strain term, which was introduced to describe many complex physical processes. The local solution is obtained with the help of the Galerkin method and the contraction mapping principle. The global solution and the blowup solution in infinite time under sub-critical initial energy are also established, and then these results are extended in parallel to the critical initial energy. Finally, the infinite time blowup of solution is proved at the arbitrary positive initial energy.
We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on
In this paper, we are concerned with a reaction-diffusion SIS epidemic model with saturated incidence rate, linear source and spontaneous infection mechanism. We derive the uniform bounds of parabolic system and obtain the global asymptotic stability of the constant steady state in a homogeneous environment. Moreover, the existence of the positive steady state is established. We mainly analyze the effects of diffusion, saturation and spontaneous infection on the asymptotic profiles of the steady state. These results show that the linear source and spontaneous infection can enhance the persistence of an infectious disease. Our mathematical approach is based on topological degree theory, singular perturbation technique, the comparison principles for elliptic equations and various elliptic estimates.
The paper deals with global existence and blow-up results for a class of fourth-order wave equations with nonlinear damping term and superlinear source term with the coefficient depends on space and time variable. In the case the weak solution is global, we give information on the decay rate of the solution. In the case the weak solution blows up in finite time, estimate the lower bound and upper bound of the lifespan of the blow-up solution, and also estimate the blow-up rate. Finally, if our problem contains an external vertical load term, a sufficient condition is also established to obtain the global existence and general decay rate of weak solutions.
In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source term. To overcome the intricacies of the proofs, we applied
The aim of this paper is to give global nonexistence and blow–up results for the problem
In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type
This paper studies the Cauchy problem of Schrödinger equation with inhomogeneous nonlinear term
We mainly focus on the asymptotic behavior analysis for certain fourth-order nonlinear wave equations with strain term, nonlinear weak damping term and source term. We establish two theorems on the asymptotic behavior of the solution depending on some conditions related to the relationship among the forced strain term, the nonlinear weak damping term and source terms.
This paper considers the Cauchy problem for a 2-component Camassa-Holm system
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http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=1054860&sortType%3Dasc_p_Sequence%26filter%3DAND(p_IS_Number%3A22665) | math | Skip to Main Content
A set of equally long finite sequences, the elements of which are either + 1 or - 1, is said to be a complementary set of sequences if the sum of autocorrelation functions of the sequences in that set is zero except for a zero-shift term. A complementary set of sequences is said to be a mate of another set if the sum of the cross-correlation functions of the corresponding sequences in these two sets is zero everywhere. Complementary sets of sequences are said to be mutually orthogonal complementary sets if any two of them are mates to each other. In this paper we discuss the properties of such complementary sets of sequences. Algorithms for synthesizing new sets from a given set are given. Recursive formulas for constructing mutually orthogonal complementary sets are presented. It is shown that matrices consisting of mutually orthogonal complementary sets of sequences can be used as operators so as to per form transformations and inverse transformations on a one- or two-dimensional array of real time or spatial functions. The similarity between such new transformations and the Hadamard transformation suggests applications of such new transformations to signal processing and image coding. | s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510265454.51/warc/CC-MAIN-20140728011745-00472-ip-10-146-231-18.ec2.internal.warc.gz | CC-MAIN-2014-23 | 1,227 | 2 |
http://laar.plapiqui.edu.ar/OJS/index.php/laar/article/view/446 | math | PERFORMANCE ANALYSIS OF FUZZY MATHEMATICAL MORPHOLOGY OPERATORS ON NOISY MRI
Keywords:Biomedical Images, Digital Image Processing, Mathematical Morphology, Fuzzy Mathematical Morphology
Despite a large amount of publications on Fuzzy Mathematical Morphology, little effort was done on systematic evaluation of the performance of this technique. The goal of this work is to compare the robustness against noise of Fuzzy and non Fuzzy Morphological operators when applied to noisy images. Magnetic Resonance Images (MRI) of the brain are a kind of images containing some characteristics that make fuzzy operators an interesting choice, because of their intrinsic noise and imprecision. The robustness was evaluated as the degree in which the results of the operators are not affected by artificial noise in the images. In the analysis we compared different implementation of Fuzzy Mathematical Morphology, and observed that in most of the cases they show higher robustness against noise than the classical morphological operators.
Once a paper is accepted for publication, the author is assumed to have transferred its copyright to the Publisher. The Publisher will not, however, put any limitation on the personal freedom of the author to use material from the paper in other publications. From September 2019 it is required that authors explicitly sign a copyright release form before their paper gets published. The Author Copyright Release form can be found here | s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305317.17/warc/CC-MAIN-20220127223432-20220128013432-00414.warc.gz | CC-MAIN-2022-05 | 1,464 | 4 |
https://fabricationsystemsinc.com/qa/quick-answer-what-does-regression-line-mean.html | math | - What is a regression line used for?
- Why does regression line go through mean?
- What does a regression mean?
- Is line of best fit always straight?
- What is best fit line in linear regression?
- How do you calculate a regression line?
- What are the two lines of regression?
- Why is the regression line the best fit?
- What two things make a best fit line?
- What does R Squared mean?
- What is regression coefficient?
What is a regression line used for?
Regression lines are useful in forecasting procedures.
Its purpose is to describe the interrelation of the dependent variable(y variable) with one or many independent variables(x variable)..
Why does regression line go through mean?
If there is a relationship (b is not zero), the best guess for the mean of X is still the mean of Y, and as X departs from the mean, so does Y. At any rate, the regression line always passes through the means of X and Y. This means that, regardless of the value of the slope, when X is at its mean, so is Y.
What does a regression mean?
Regression is a statistical method used in finance, investing, and other disciplines that attempts to determine the strength and character of the relationship between one dependent variable (usually denoted by Y) and a series of other variables (known as independent variables).
Is line of best fit always straight?
a line or curve of best fit on each graph. Lines of best fit can be straight or curved. Some will pass through all of the points, while others will have an even spread of points on either side. There is usually no right or wrong line, but the guidelines below will help you to draw the best one you can.
What is best fit line in linear regression?
Line of best fit refers to a line through a scatter plot of data points that best expresses the relationship between those points. Statisticians typically use the least squares method to arrive at the geometric equation for the line, either though manual calculations or regression analysis software.
How do you calculate a regression line?
A linear regression line has an equation of the form Y = a + bX, where X is the explanatory variable and Y is the dependent variable. The slope of the line is b, and a is the intercept (the value of y when x = 0).
What are the two lines of regression?
Two Regression Lines The first is a line of regression of y on x, which can be used to estimate y given x. The other is a line of regression of x on y, used to estimate x given y.
Why is the regression line the best fit?
The regression line is sometimes called the “line of best fit” because it is the line that fits best when drawn through the points. … The extent to which the regression line is sloped, however, represents the degree to which we are able to predict the y scores with the x scores.
What two things make a best fit line?
The line of best fit is determined by the correlation between the two variables on a scatter plot. In the case that there are a few outliers (data points that are located far away from the rest of the data) the line will adjust so that it represents those points as well.
What does R Squared mean?
coefficient of determinationR-squared is a statistical measure of how close the data are to the fitted regression line. It is also known as the coefficient of determination, or the coefficient of multiple determination for multiple regression. … 100% indicates that the model explains all the variability of the response data around its mean.
What is regression coefficient?
Regression coefficients are estimates of the unknown population parameters and describe the relationship between a predictor variable and the response. In linear regression, coefficients are the values that multiply the predictor values. | s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703529179.46/warc/CC-MAIN-20210122082356-20210122112356-00746.warc.gz | CC-MAIN-2021-04 | 3,745 | 34 |
https://likesoy.com/negative-exponents-worksheet/ | math | - Title : Exponent Rules Worksheet Awesome Worksheet 273 Precalculus Simplify ... negative exponents worksheet in Common Worksheets category
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Some questions frequently asked by users : how would i solve a negative exponent like (3^2)-1 ?
my teacher gave me a worksheet on negative exponents but i need help on itopen question: can anybody find negative exponents worksheets?
i need to find a worksheet with equations (and answers) similar to the one shown below: (3m^5r^3/4p^8)^4integer exponents?
i was absent ad this is my makeup work. i'll give you the first question to each activity on the worksheet. please explain it to me as simple as you can because im totally lost. its algebra 1 so it shouldnt be too hard. thanks!!!!!! smiplify 4 exponent is -2 evaluate each expression for the given value(s) of the variable(s). 10. xexponent is -4 y exponent is 3 x=2 and y=3 simplify 16. x exponent -3 22. a cooking website claims to contain 10 exponent is 5 recipes. evaluate the expressionexponent question ?
yup, well the instructions i have on my worksheet are to simplify and leave all answers in exponent form, and don't leave any answers with zero or negative exponents so, how would i go about solving something like 6^-6 * 6^-3 thanks!negative exponents help (algebra i)?
for christmas break, my lovely math teacher gave us a 40 question packet grr. i have a hard time with negative exponents. we have two problems using negative exponents. please explain how to do them. 1) (8 x 10^-2) x (2.9 x 10^-2) and 2) (3 x 10^1) x (8 x 10^-2) thanks! one more thing: answers are to be in scientific notation, thanks for all your answers.
- Total Download : 270 | s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221219469.90/warc/CC-MAIN-20180822030004-20180822050004-00236.warc.gz | CC-MAIN-2018-34 | 2,243 | 14 |
https://www.trainup.com/TrainingDetails/237623/Essential-Concepts-in-Construction-Contracts-and-Administration | math | Essential Concepts in Construction Contracts and Administration
What Is the Best Form of Procurement for Owners, Developers and Major Tenants?
• Design-Bid-Build and Design-Build
• Construction Management
• Multiple Prime Contracts How Do the Roles and Responsibilities of a Design Professional Change Based on the Procurement Method? What Are the Pros and Cons of Contract Pricing Schemes, Including:
• Lump Sum and Fixed Unit Price
• Cost Reimbursement Plus Fee
• Cost Plus Fee With Guaranteed Maximum Price What Are the Sources of the Parties’ Obligations?
• Construction Contract
• Contractor Proposal Qualifications
• General Conditions
• Supplemental General Conditions
• Plans and Specifications
• Schedule and Change Orders Key Concepts in Contract Administration Under AIA-Type Documents
• What Is the AIA System and What Are the Alternatives?
• When Should Alternative Systems Be Considered Such as ConsensusDOCS, Engineers Joint Contract Documents Committee, or Custom Forms? What Is the Payment Process in a Typical Construction Contract, and Where Are the Pitfalls?
• Progress Payments
• Final Payment
• Required Submissions
• Audit Rights and Joint Payment
• Retainage What Is the Typical Process for Substantial and Final Completion in a Construction Contract?
• Contract Changes (Change Orders)
• Time Obligations What Are Common Claims Issues?
• Extra Time and Extra Money
• Differing Site Conditions
• Warranty and Termination
• Cause vs. Convenience
• Nonconforming Work
• Notice and Opportunity to Cure What Are Practical Risk Shifting Strategies?
• Insurance and Indemnity
• Exculpatory Clauses
• Limitations of Liability
• Liquidated Damages Basics of Mechanics Liens and Subcontractor Claims
• Liens and Waivers
• The Pass-Through System What Are the Pros and Cons of the Various Frequently-Used Dispute Resolution Methods?
• Nonbinding Mediation
• Binding Arbitration
• Contract Considerations
This live webinar is designed for presidents, vice presidents, contractors, subcontractors, engineers, architects, project managers, contract managers, developers, real estate professional, accountants and attorneys. | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816875.61/warc/CC-MAIN-20240414064633-20240414094633-00104.warc.gz | CC-MAIN-2024-18 | 2,208 | 39 |
https://spark.adobe.com/page/kPcK3kU8opXD1/ | math | Chapter 9.1 is about find the midpoint with two other points. You have to get the plots for the other two points to plug into the formula.
Formula - m(x1+x2/2 , y1+y2/2 )
Then you use the distance formula to find the distance between the two points .
Distance formula- D= | (x2-x1)^2 + (y^2-y^1) m^2
.......9.2 ^ the standard formula is introduced finding the axis of symetry
Standard formula- y=a(x-h)^2+k
9.3 is about the standard form of a circle
Standard form of a circle - (x-h)^2 + (y-k)^2 = r^2
This for ula finds the center of the circle and the point in the circle . | s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247494125.62/warc/CC-MAIN-20190220003821-20190220025821-00549.warc.gz | CC-MAIN-2019-09 | 575 | 9 |
https://shesspiritfilled.com/quiz | math | Good Job. You seem to be at a good level of spirituality
Oh no. You seem to be spiritually dead. Perhaps you need more understanding of the supernatural.
#1 Is there a supernatural realm?
#2 How many Kingdoms are in the supernatural realm ?
#3 God is how many persons?
#4 Can humans command angels?
We sure can
No, only God can
Is that even a thing?
#5 Crystals, Sage, Horoscopes and Tarot cards fall under which kingdom?
#6 Demons tremble under which of these names?
#7 Who is the Holy Spirit?
A higher source of power
An actual person
#8 Can a believer visit Heaven while still alive on Earth?
That's impossible! We can only visit when we die
Yes, we have access
If they only have a near death experience
Heaven doesn't exist
#9 In order to be saved a person must:
Just live a sinless life
There's no such thing as being saved
#10 Which of these would save your life?
#11 Psychics are the same as prophets?
Yes, they both tell the future
No. Psychics are false prophets under the kingdom of darkness
I'm not too sure
#12 Which of the following about speaking in tongues is correct?
No one should speak in tongues, it's bad.
Speaking in tongues is not bad. However some tongues can be demonic.
We can only speak in tongues when we get to Heaven
People who speak in tongues are possessed. | s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201904.55/warc/CC-MAIN-20190319052517-20190319074517-00155.warc.gz | CC-MAIN-2019-13 | 1,288 | 32 |
https://studymaterialz.in/supplementary-angles-definition-and-examples/ | math | Supplementary Angles Definition And Examples
For two angles to be supplementary, their sum must be 180 degree. Examples of supplementary angles are 140o and 40o, 150o and 30o, 100o and 80o, 160o and 20o, etc. In most cases, supplementary angles are gotten on a straight because the sum of angles on a straight line is 180 degree. The moment the angle on a straight line is bisected, the two angles formed will be supplementary.
The supplementary angle theorem states that if two angles are supplementary to the same angle, then the two angles are said to be congruent.
There is another concept called complementary angles, the two angles add up to 90 degrees. Below I will give the differences between supplementary and complementary angles in tabular form.
Differences Between Supplementary and Complementary Angles
|The two angles sum up to 180o||The two angles sum up to 90o|
|They form a straight line when combined together.||They form a right-angled triangle when combined together.|
|The supplement of an angle X is (180 – X)°||The complement of an angle X is (90 – X)°|
FAQ On Supplementary Angles
Do supplementary angles form a linear pair
Yes, two supplementary angles always form a linear pair because they sum up to 180 degrees. And the sum of angles on a straight line is 180o
Are two right angles always supplementary
Yes, because the sum of two right angles is 180o.
Can two obtuse angles be supplementary to each other
No. An obtuse angle is an angle greater than 90°. Let us take the least obtuse angle which is 91, it supplement will be 89, and 89 isn’t an obtuse angle. So, two obtuse angles can’t be supplementary to each other.
Are supplementary angles congruent
The condition for this to be so is that If two angles are supplementary to the same angle, then they are congruent to each other. For example, if angle A is supplementary to C and angle B is supplementary to C, then angle A and B are congruent.
<A + <C = 180
<A = 180 – <C
<B + <C = 180
<B = 180 – <C
Therefore, <A = <B (congruent)
Congruent angles are the angles that have equal measure
Are the opposite angles of a kite supplementary
In a kite the opposite angles are equal, so for it to be supplementary, the opposites angles must be 90o. This means when the opposite angles of a kite is 90o, they are supplementary.
Supplementary Angle Examples
Question 1: An angle is five times its supplement. Find both angles.
Let the angle be x, and its supplement will be (180-x)
According to the question, x = 5 *(180-x)
x = 900 – 5x
6x = 900
x = 900/6 = 150
Then it supplement will be, 180 – 150 = 30
Question 2: Find the missing measurement in the pair of angles
Solution: x + 25 = 180 (supplementary angle)
x = 180 – 25 = 155 degree | s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945288.47/warc/CC-MAIN-20230324180032-20230324210032-00111.warc.gz | CC-MAIN-2023-14 | 2,734 | 36 |
http://www.chegg.com/homework-help/questions-and-answers/14-barangay-power-station-supplies-60kw-load-over2-500-ft-100-mm2-conductors-copper-feeder-q350601 | math | 14. A Barangay power station supplies 60kW to a load over2,500
ft, 100 mm2, two conductors copper feeder, theresistance
of which is 0.078 Ω per 1000ft. The bus bar voltageis
maintained constant at 600 V. Determine the load current.
a. 105 A
b. 108 A
c. 110 A
d. 102 A | s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703662159/warc/CC-MAIN-20130516112742-00028-ip-10-60-113-184.ec2.internal.warc.gz | CC-MAIN-2013-20 | 268 | 8 |
https://topof.games/game/math-gates/ | math | Your mission in this game; It is to find the answer to the math question you have been told in a short time and to pass the correct answer on the gate that will appear in front of you as you move forward. You will see the math operation at the bottom of the screen. To get through the gate in front of you, you have to go through the gate with the correct answer. If you pass through the gate with the wrong answer, the game will be over somehow. You succeed when you pass through the gate with the correct answer.
MOVE LEFT-RIGHT SWIPE MOUSE | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817438.43/warc/CC-MAIN-20240419141145-20240419171145-00527.warc.gz | CC-MAIN-2024-18 | 542 | 2 |
http://www.gmatfree.com/module-999/equations/ | math | Performing Operations on Equations
An equation is like a balanced scale. If we remove or add weight to one side alone, it’s no longer balanced, and hence no longer an equation. We can freely alter the weight on one side if we make the same adjustment on the other, keeping the balance.
What is y in this equation?
We want to know what y is, so we want to “isolate” y (get it by itself) on one side of the equal sign.
First subtract 2 from both sides.
After we subtracted, y is by itself and we can see the value of y:
Let’s try another example, solving for x:
Add 4 to each side to get rid of the -4.
Subtract 3 from each side of the equation.
Types of Equations
The GMAT will not explicitly require you to distinguish different types of equations, but, in practice, you will have to be able to do so in order to solve them all.
A linear equation has variables that are not multiplied by each other or taken to any exponent. For example,
is a linear equation, because the variables x and y are not multiplied by each other or taken to an exponent. And y = 3x + 2 is the equation of a line in the xy-plane (in the common form y = mx + b), so “linear” is not a bad name for this type of equation.
Linear equations often appear on the GMAT in groups, in which case they are “systems” of linear equations. It’s a “system” just because it has more than one equation. On many GMAT word problems, we can solve by translating the descriptions given into a system of equations and then solving the system of equations.
A quadratic equation has a squared variable. For example,
is a quadratic equation.
In the upcoming sections, we will discuss methods of solving these equations:
- For single linear equations, we will use Fraction-Side-Combine-Factor-Divide.
- For systems of linear equations, we will use substitution or combination.
- For quadratic equations, we will solve by factoring or the quadratic formula. | s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057830.70/warc/CC-MAIN-20210926053229-20210926083229-00160.warc.gz | CC-MAIN-2021-39 | 1,928 | 20 |
http://beneaththebeautyy.blogspot.com/2016/01/qa-365-days-5years.html | math | One of my favorite gifts I received for Christmas this year was from my parents. It was a small book that was partially empty. I say partially because it is a journal, but not a blank one.
It is one of those books where each day you answer a question. 365 pages for the 365 days in a year. It spans the period of five years. So every year you answer the question. I think I'm most excited to see how my answers change over the course of the five years. I find that people are often unable to tell whether or not they are changing because it's hard to notice the little things that you do differently from day to day.
I have also found some other versions of the book. They're by the same company but they tailor to different people/events in one's life which I think is incredible. There is the one I have, which is the standard five year. They have a couples one that has questions that both people in the relationship answer and this book spans over three years. I think it is adorable and if I was in a relationship that looked like it was going to last three years at least, then I would absolutely get this!!The other one I found was the college one which I plan on getting when I start college. It spans the period of four years which is the standard college length shall we say, and these questions are naturally tailored towards people in college and the questions are very college specific.
I hope you all enjoyed this post and I definitely recommend checking these books out because they're so much fun. | s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593223.90/warc/CC-MAIN-20180722120017-20180722140017-00472.warc.gz | CC-MAIN-2018-30 | 1,513 | 4 |
http://www.jiskha.com/display.cgi?id=1239653321 | math | posted by . on .
Given the unbalanced equation shown:
_Al+_CuSo4 -> _Al2(SO4)3+_Cu
When the equation is balanced using the smallest whole-number coefficients, what is the coefficient of Al?
What problem are you having with balancing the equation? It seems straight forward. You need to learn to do this yourself. | s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917124478.77/warc/CC-MAIN-20170423031204-00571-ip-10-145-167-34.ec2.internal.warc.gz | CC-MAIN-2017-17 | 312 | 5 |
https://sec.oercommons.org/EN/browse?f.keyword=planes | math | This learning video deals with a question of geometrical probability. A key idea presented is the fact that a linear equation in three dimensions produces a plane. The video focuses on random triangles that are defined by their three respective angles. These angles are chosen randomly subject to a constraint that they must sum to 180 degrees. An example of the types of in-class activities for between segments of the video is: Ask six students for numbers and make those numbers the coordinates x,y of three points. Then have the class try to figure out how to decide if the triangle with those corners is acute or obtuse.
Updating search results...
Conditional Remix & Share PermittedCC BY-NC-SA | s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988793.99/warc/CC-MAIN-20210507120655-20210507150655-00162.warc.gz | CC-MAIN-2021-21 | 699 | 3 |
https://cs.stackexchange.com/questions/68975/increment-and-decrement-on-binary-semaphores-when-their-values-are-1-and-0-r/68988 | math | What happens if we increment binary semaphore $x$ when it is equal to $1$. E.g A process executes $V(x)$ when $x==1$; What are the possible cases then-
Process would be blocked till $x$ becomes $0$. The moment $x$ becomes $0$,it is incremented.
Process would be blocked and no present or future changes will be there for $x$. i.e Process assumes as if there was never a $V(x)$.
Other. (If there is please mention).
Same query for $P(x)$ operation as well when $x=0$.
- Actually, Galvin doesn't mention in pseudo code what to do during such condition.
- Better if Someone could provide standard code for $P(x)$ and $V(x)$... Also, $P$ is wait operation whereas $V$ is signal operation. | s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510097.3/warc/CC-MAIN-20220516073101-20220516103101-00694.warc.gz | CC-MAIN-2022-21 | 684 | 7 |
https://www.arxiv-vanity.com/papers/1202.4153/ | math | Ten misconceptions from the history of analysis and their debunking
The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass, is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others.
Key words and phrases:Abraham Robinson, adequality, Archimedean continuum, Bernoullian continuum, Cantor, Cauchy, cognitive bias, completeness, constructivism, continuity, continuum, du Bois-Reymond, epsilontics, Felix Klein, Fermat-Robinson standard part, infinitesimal, Leibniz-Łoś transfer principle, limit, mathematical rigor, nominalism, non-Archimedean, Simon Stevin, Stolz, sum theorem, Weierstrass
2000 Mathematics Subject Classification:01A85; Secondary 26E35, 03A05, 97A20, 97C30
- 1 Introduction
- 2 Were the founders of calculus working in a numerical vacuum?
- 3 Was Berkeley’s criticism coherent?
- 4 Were d’Alembert’s anticipations ahead of his time?
- 5 Did Cauchy replace infinitesimals by rigor?
- 6 Was Cauchy’s 1821 “sum theorem” false?
- 7 Did Weierstrass succeed in eliminating infinitesimals?
- 8 Did Dedekind discover the essence of continuity?
- 9 Who invented Dirac’s delta function?
- 10 Is there continuity between Leibniz and Robinson?
- 11 Is Lakatos’ take on Cauchy tainted by Kuhnian relativism?
- A Rival continua
Here are some common claims. The founders of infinitesimal calculus were working in a vacuum caused by an absence of a satisfactory number system. The incoherence of infinitesimals was effectively criticized by Berkeley as so much hazy metaphysical mysticism. D’Alembert’s visionary anticipation of the rigorisation of analysis was ahead of his time. Cauchy took first steps toward replacing infinitesimals by rigor and epsilontics, in particular giving a modern definition of continuity. Cauchy’s false 1821 version of the “sum theorem” was corrected by him in 1853 by adding the hypothesis of uniform convergence. Weierstrass finally rigorized analysis and thereby eliminated infinitesimals from mathematics. Dedekind discovered “the essence of continuity”, which is captured by his cuts. One of the spectacular successes of the rigorous analysis was the mathematical justification of Dirac’s “delta functions”. Robinson developed a new theory of infinitesimals in the 1960s, but his approach has little to do with historical infinitesimals. Lakatos pursued an ideological agenda of Kuhnian relativism and fallibilism, inapplicable to mathematics.
Each of the above ten claims is in error, as we argue in the next ten sections (cf. Crowe ).
The historical fact of the dominance of the view of analysis as being based on the real numbers to the exclusion of infinitesimals, is beyond dispute. One could ask oneself why this historical fact is so; some authors have criticized mathematicians for adhering to an approach that others consider less appropriate. In the present text, we will not be concerned with either of these issues. Rather, we will be concerned with another issue, namely, why is it that traditional historical scholarship has been inadequate in indicating that alternative views have been around. We will also be concerned with documenting instances of tendentious interpretation and the attendant distortion in traditional evaluation of key figures from mathematical history.
Felix Klein clearly acknowledged the existence of a parallel, infinitesimal approach to foundations. Having outlined the developments in real analysis associated with Weierstrass and his followers, Klein pointed out in 1908 that
The scientific mathematics of today is built upon the series of developments which we have been outlining. But an essentially different conception of infinitesimal calculus has been running parallel with this [conception] through the centuries (Klein [87, p. 214]).
Such a different conception, according to Klein, “harks back to old metaphysical speculations concerning the structure of the continuum according to which this was made up of […] infinitely small parts” (ibid.). The pair of parallel conceptions of analysis are illustrated in Figure 1.
A comprehensive re-evaluation of the history of infinitesimal calculus and analysis was initiated by Katz & Katz in , , and . Briefly, a philosophical disposition characterized by a preference for a sparse ontology has dominated the historiography of mathematics for the past 140 years, resulting in a systematic distortion in the interpretation of the historical development of mathematics from Stevin (see ) to Cauchy (see and Borovik & Katz ) and beyond. Taken to its logical conclusion, such distortion can assume comical proportions. Thus, Newton’s eventual successor in the Lucasian chair of mathematics, Stephen Hawking, comments that Cauchy
was particularly concerned to banish infinitesimals (Hawking [67, p. 639]),
yet on the very same page 639, Hawking quotes Cauchy’s infinitesimal definition of continuity in the following terms:
the function remains continuous with respect to between the given bounds, if, between these bounds, an infinitely small increment in the variable always produces an infinitely small increment in the function itself (ibid).
Did Cauchy banish infinitesimals, or did he exploit them to define a seminal new notion of continuity? Similarly, historian J. Gray lists continuity among concepts Cauchy allegedly defined
using careful, if not altogether unambiguous, limiting arguments (Gray [65, p. 62]) [emphasis added–authors],
whereas in reality limits appear in Cauchy’s definition only in the sense of the endpoints of the domain of definition (see , for a more detailed discussion). Commenting on ‘Whig’ re-writing of mathematical history,111Related comments by Grattan-Guinness may be found in the main text at footnote 25. P. Mancosu observed that
the literature on infinity is replete with such ‘Whig’ history. Praise and blame are passed depending on whether or not an author might have anticipated Cantor and naturally this leads to a completely anachronistic reading of many of the medieval and later contributions [114, p. 626].
The anachronistic idea of the history of analysis as a relentless march toward the yawning heights of epsilontics is, similarly, our target in the present text. We outline some of the main distortions, focusing on the philosophical bias which led to them. The outline serves as a program for further investigation.
2. Were the founders of calculus working in a numerical vacuum?
Were the founders of infinitesimal calculus working in a vacuum caused by an absence of a satisfactory number system?
2.1. Stevin, La Disme, and Arithmetique
A century before Newton and Leibniz, Simon Stevin (Stevinus) sought to break with an ancient Greek heritage of working exclusively with relations among natural numbers,222The Greeks counted as numbers only ; thus, was not a number, nor are the fractions: ratios were relations, not numbers. Consequently, Stevin had to spend time arguing that the unit () was a number. and developed an approach capable of representing both “discrete” number ( ’)333Euclid [45, Book VII, def. 2]. composed of units () and continuous magnitude () of geometric origin.444Euclid [45, Book V]. See also Aristotle’s Categories, 6.4b, 20-23: “Quantity is either discrete or continuous. […] Instances of discrete quantities are number and speech; of continuous, lines, surfaces, solids, and besides these, time and place”. According to van der Waerden, Stevin’s
general notion of a real number was accepted, tacitly or explicitly, by all later scientists [151, p. 69].
D. Fearnley-Sander wrote that
the modern concept of real number […] was essentially achieved by Simon Stevin, around 1600, and was thoroughly assimilated into mathematics in the following two centuries [48, p. 809].
D. Fowler points out that
Stevin […] was a thorough-going arithmetizer: he published, in 1585, the first popularization of decimal fractions in the West […]; in 1594, he described an algorithm for finding the decimal expansion of the root of any polynomial, the same algorithm we find later in Cauchy’s proof of the Intermediate Value Theorem [54, p. 733].
Fowler emphasizes that important foundational work was yet to be done by Dedekind, who proved that the field operations and other arithmetic operations extend from to (see Section 8).555Namely, there is no easy way from representation of reals by decimals, to the field of reals, just as there is no easy way from continuous fractions, another well-known representation of reals, to operations on such fractions. Meanwhile, Stevin’s decimals stimulated the emergence of power series (see below) and other developments. We will discuss Stevin’s contribution to the Intermediate Value Theorem in Subsection 2.3 below.
In 1585, Stevin defined decimals in La Disme as follows:
Decimal numbers are a kind of arithmetic based on the idea of the progression of tens, making use of the Arabic numerals in which any number may be written and by which all computations that are met in business may be performed by integers alone without the aid of fraction. (La Disme, On Decimal Fractions, tr. by V. Sanford, in Smith [139, p. 23]).
By numbers ‘‘met in business” Stevin meant finite decimals,666But see Stevin’s comments on extending the process ad infinitum in main text at footnote 15. and by “computations” he meant addition, subtraction, multiplications, division and extraction of square roots on finite decimals.777An algorithmic approach to such operations on infinite decimals was developed by Hoborski , and later in a very different way by Faltin et al. . Stevin argued that numbers, like the more familiar continuous magnitudes, can be divided indefinitely, and used a water metaphor to illustrate such an analogy:
As to a continuous body of water corresponds a continuous wetness, so to a continuous magnitude corresponds a continuous number. Likewise, as the continuous body of water is subject to the same division and separation as the water, so the continuous number is subject to the same division and separation as its magnitude, in such a way that these two quantities cannot be distinguished by continuity and discontinuity (Stevin, 1585, see [140, p. 3]; quoted in A. Malet ).888See also Naets for an illuminating discussion of Stevin.
Stevin argued for equal rights in his system for rational and irrational numbers. He was critical of the complications in Euclid [45, Book X], and was able to show that adopting the arithmetic as a way of dealing with those theorems made many of them easy to understand and easy to prove. In his Arithmetique , Stevin proposed to represent all numbers systematically in decimal notation. P. Ehrlich notes that Stevin’s
viewpoint soon led to, and was implicit in, the analytic geometry of René Descartes (1596-1650), and was made explicit by John Wallis (1616-1703) and Isaac Newton (1643-1727) in their arithmetizations thereof (Ehrlich [41, p. 494]).
2.2. Decimals from Stevin to Newton and Euler
Stevin’s text La Disme on decimal notation was translated into English in 1608 by Robert Norton (cf. Cajori [23, p. 314]). The translation contains the first occurrence of the word ‘‘decimal’’ in English; the word will be employed by Newton in a crucial passage 63 years later.999See footnote 12. Wallis recognized the importance of unending decimal expressions in the following terms:
Now though the Proportion cannot be accurately expressed in absolute Numbers: Yet by continued Approximation it may; so as to approach nearer to it, than any difference assignable (Wallis’s Algebra, p. 317, cited in Crossley ).
Similarly, Newton exploits a power series expansion to calculate detailed decimal approximations to for (Newton ).101010Newton scholar N. Guicciardini kindly provided a jpg of a page from Newton’s manuscript containing such calculations, and commented as follows: “It was probably written in Autumn 1665 (see Mathematical papers 1, p. 134). Whiteside’s dating is sometimes too precise, but in any case it is a manuscript that was certainly written in the mid 1660s when Newton began annotating Wallis’s Arithmetica Infinitorum” . The page from Newton’s manuscript can be viewed at http://u.math.biu.ac.il/~katzmik/newton.html By the time of Newton’s annus mirabilis, the idea of unending decimal representation was well established. Historian V. Katz calls attention to “Newton’s analogy of power series to infinite decimal expansions of numbers” (V. Katz [82, p. 245]). Newton expressed such an analogy in the following passage:
Since the operations of computing in numbers and with variables are closely similar …I am amazed that it has occurred to no one (if you except N. Mercator with his quadrature of the hyperbola) to fit the doctrine recently established for decimal numbers in similar fashion to variables, especially since the way is then open to more striking consequences. For since this doctrine in species has the same relationship to Algebra that the doctrine in decimal numbers has to common Arithmetic, its operations of Addition, Subtraction, Multiplication, Division and Root-extraction may easily be learnt from the latter’s provided the reader be skilled in each, both Arithmetic and Algebra, and appreciate the correspondence between decimal numbers and algebraic terms continued to infinity …And just as the advantage of decimals consists in this, that when all fractions and roots have been reduced to them they take on in a certain measure the nature of integers, so it is the advantage of infinite variable-sequences that classes of more complicated terms (such as fractions whose denominators are complex quantities, the roots of complex quantities and the roots of affected equations) may be reduced to the class of simple ones: that is, to infinite series of fractions having simple numerators and denominators and without the all but insuperable encumbrances which beset the others (Newton 1671, ).111111We are grateful to V. Katz for signaling this passage.
In this remarkable passage dating from 1671, Newton explicitly names infinite decimals as the source of inspiration for the new idea of infinite series.121212See footnote 9. The passage shows that Newton had an adequate number system for doing calculus and real analysis two centuries before the triumvirate.131313The expression “the great triumvirate” is used by Boyer [21, p. 298] to describe Cantor, Dedekind, and Weierstrass.
In 1742, John Marsh first used an abbreviated notation for repeating decimals (Marsh [116, p. 5], cf. Cajori [23, p. 335]). Euler exploits unending decimals in his Elements of Algebra in 1765, as when he sums an infinite series and concludes141414This could be compared with Peirce’s remarks. Over a century ago, Charles Sanders Peirce wrote with reference to ’: “although the difference, being infinitesimal, is less than any number [one] can express[,] the difference exists all the same, and sometimes takes a quite easily intelligible form” (Peirce [125, p. 597]; see also S. Levy [102, p. 130]). Levy mentions Peirce’s proposal of an alternative notation for “equality up to an infinitesimal”. The notation Peirce proposes is the usual equality sign with a dot over it, like this: “”. See also main text at footnote 46. that (Euler [46, p. 170]).
2.3. Stevin’s cubic and the IVT
In the context of his decimal representation, Stevin developed numerical methods for finding roots of polynomials equations. He described an algorithm equivalent to finding zeros of polynomials (see Crossley [30, p. 96]). This occurs in a corollary to problem 77 (more precisely, LXXVII) in (Stevin [141, p. 353]). Here Stevin describes such an argument in the context of finding a root of the cubic equation (which he expresses as a proportion to conform to the contemporary custom)
Here the whimsical coefficient seems to have been chosen to emphasize the fact that the method is completely general; Stevin notes furthermore that numerous addional examples can be given. A textual discussion of the method may be found in Struik [142, p. 476].
Centuries later, Cauchy would prove the Intermediate Value Theorem (IVT) for a continuous function on an interval by a divide-and-conquer algorithm. Cauchy subdivided into equal subintervals, and recursively picked a subinterval where the values of have opposite signs at the endpoints (Cauchy [24, Note III, p. 462]). To elaborate on Stevin’s argument following [142, §10, p. 475-476], note that what Stevin similarly described a divide-and-conquer algorithm. Stevin subdivides the interval into ten equal parts, resulting in a gain of a new decimal digit of the solution at every iteration of his procedure. Stevin explicitly speaks of continuing the iteration ad infinitum:151515Cf. footnote 6.
Et procedant ainsi infiniment, l’on approche infiniment plus pres au requis (Stevin [141, p. 353, last 3 lines]).
Who needs existence proofs for the real numbers, when Stevin gives a procedure seeking to produce an explicit decimal representation of the solution? The IVT for polynomials would resurface in Lagrange before being generalized by Cauchy to the newly introduced class of continuous functions.161616See further in footnote 27.
One frequently hears sentiments to the effect that the pre-triumvirate mathematicians did not and could not have provided rigorous proofs, since the real number system was not even built yet. Such an attitude is anachronistic. It overemphasizes the significance of the triumvirate project in an inappropriate context. Stevin is concerned with constructing an algorithm, whereas Cantor is concerned with developing a foundational framework based upon the existence of the totality of the real numbers, as well as their power sets, etc. The latter project involves a number of non-constructive ingredients, including the axiom of infinity and the law of excluded middle. But none of it is needed for Stevin’s procedure, because he is not seeking to re-define “number” in terms of alternative (supposedly less troublesome) mathematical objects.
Why do many historians and mathematicians of today emphasize the great triumvirate’s approach to proofs of the existence of real numbers, at the expense, and almost to the exclusion, of Stevin’s approach? Can this be understood in the context of the ideological foundational battles raging at the end of 19th and beginning of 20th century? These questions merit further scrutiny.
3. Was Berkeley’s criticism coherent?
Was Berkeley’s criticism of infinitesimals as so much hazy metaphysical mysticism, either effective or coherent?
D. Sherry dissects Berkeley’s criticism of infinitesimal calculus into its metaphysical and logical components, as detailed below.
3.1. Logical criticism
The logical criticism is the one about the disappearing . Here we have a ghost: , but also a departed quantity: (in other words eating your cake: and having it, too: ).
Thus, Berkeley’s logical criticism of the calculus is that the evanescent increment is first assumed to be non-zero to set up an algebraic expression, and then treated as zero in discarding the terms that contained that increment when the increment is said to vanish.171717Given Berkeley’s fame among historians of mathematics for allegedly spotting logical flaws in infinitesimal calculus, it is startling to spot circular logic at the root of Berkeley’s own doctrine of the compensation of errors. Indeed, Berkeley’s new, improved calculation of the derivative of in The Analyst relies upon the determination of the tangent to a parabola due to Apollonius of Perga [4, Book I, Theorem 33] (see Andersen ).
The fact is that Berkeley’s logical criticism is easily answered within a conceptual framework available to the founders of the calculus. Namely, the rebuttal of the logical criticism is that the evanescent increment is not treated as zero, but, rather, merely discarded through an application of Leibniz’s law of homogeneity (see Leibniz [101, p. 380]) and Bos [20, p. 33]), which would stipulate, for instance, that
Here we chose the sign which was already used by Leibniz where we would use an equality sign today (see McClenon [117, p. 371]). The law is closely related to the earlier notion of adequality found in Fermat. Adequality is the relation of being infinitely close, or being equal “up to” an infinitesimal. Fermat exploited adequality when he sought a maximum of an expression by evaluating expression at and at , and forming the difference. In modern notation this would appear as (note that Fermat did not use the function notation). Huygens already interpreted the “’’ occurring in this expression in the method of adequality, as an infinitesimal.181818Huygens explained Fermat’s method of adequality in a presentation at the Académie des Sciences in 1667. Huygens noted that “ is an “infinitely small quantity” (see Huygens ). See also André Weil [152, p. 1146], [153, p. 28].
Ultimately, the heuristic concepts of adequality (Fermat) and law of homogeneity (Leibniz) were implemented in terms of the standard part function (see Figure 2).
In a passage typical of post-Weierstrassian scholarship, Kleiner and Movshovitz-Hadar note that
Fermat’s method was severely criticized by some of his contemporaries. They objected to his introduction and subsequent suppression of the mysterious . Dividing by meant regarding it as not zero. Discarding implied treating it as zero. This is inadmissible, they rightly claimed. In a somewhat different context, but with equal justification, … Berkeley in the 18th century would refer to such ’s as ‘the ghosts of departed quantities’ ” [88, p. 970] [emphasis added–authors].
However, Fermat scholar P. Strømholm already pointed out in 1968 that in Fermat’s main method of adequality,
there was never […] any question of the variation being put equal to zero. The words Fermat used to express the process of suppressing terms containing was “elido”, “deleo”, and “expungo”, and in French “i’efface” and “i’ôte”. We can hardly believe that a sane man wishing to express his meaning and searching for words, would constantly hit upon such tortuous ways of imparting the simple fact that the terms vanished because was zero (Strømholm [143, p. 51]).
Thus, Fermat planted the seeds of the answer to the logical criticism of the infinitesimal, a century before George Berkeley ever lifted up his pen to write The Analyst.
3.2. Metaphysical criticism
Berkeley’s metaphysical criticism targets the absence of any empirical referent for “infinitesimal”. The metaphysical criticism has its roots in empiricist dogma that every meaningful expression or symbol must correspond to an empirical entity.191919The interplay of empiricism and nominalism in Berkeley’s thought is touched upon by D. Sepkoski [133, p. 50]. Ironically, Berkeley accepts many expressions lacking an empirical referent, such as ‘force’, ‘number’, or ‘grace’, on the grounds that they have pragmatic value. It is a glaring inconsistency on Berkeley’s part not to have accepted “infinitesimal” on the same grounds (see Sherry ).
It is even more ironic that over the centuries, mathematicians were mainly unhappy with the logical aspect, but their criticisms mainly targeted what they perceived as the metaphysical/mystical aspect. Thus, Cantor attacked infinitesimals as being “abominations” (see Ehrlich ); R. Courant described them as “mystical”, “hazy fog”, etc. E. T. Bell went as far as describing infinitesimals as having been
by the cleric of Cloyne (see Figure 3). Generally speaking, one does not slay either scientific concepts or scientific entities. Bellicose language of this sort is a sign of commitments that are both emotional and ideological.
4. Were d’Alembert’s anticipations ahead of his time?
Were d’Alembert’s mathematical anticipations ahead or behind his time?
One aspect of d’Alembert’s vision as expressed in his article for the Encyclopedie on irrational numbers, is that irrational numbers do not exist. Here d’Alembert uses terms such as “surds” which had already been rejected by Simon Stevin two centuries earlier (see Section 2). From this point of view, d’Alembert is not a pioneer of the rigorisation of analysis in the 19th century, but on the contrary, represents a throwback to the 16th century. D’Alembert’s attitude toward irrational numbers sheds light on the errors in his proof of the fundamental theorem of algebra;202020D’Alembert’s error was first noticed by Gauss, who gave some correct proofs, though S. Smale argues his first proof was incomplete . See Baltus for a detailed study of d’Alembert’s proof. indeed, in the anemic number system envisioned by d’Alembert, numerous polynomials surely fail to have roots.
D’Alembert used the term “charlatanerie” to describe infinitesimals in his article Différentiel . D’Alembert’s anti-infinitesimal vitriol is what endears him to triumvirate scholars, for his allegedly visionary remarks concerning the centrality of the limit concept fall short of what is already found in Newton.212121See Pourciau who argues that Newton possessed a clear kinetic conception of limit (Sinaceur and Barany argue that Cauchy’s notion of limit was kinetic, rather than a precursor of a Weierstrassian notion of limit). Pourciau cites Newton’s lucid statement to the effect that “Those ultimate ratios …are not actually ratios of ultimate quantities, but limits …which they can approach so closely that their difference is less than any given quantity…” (Newton, 1946 [123, p. 39] and 1999 [124, p. 442]). The same point, and the same passage from Newton, appeared a century earlier in Russell [130, item 316, p. 338-339]. He never went beyond a kinetic notion of limit, so as to obtain the epsilontic version popularized in the 1870s.
D’Alembert was particularly bothered by the characterisation of infinitesimals he found in “the geometers”. He does not explain who these geometers are, but the characterisation he is objecting to can be already found in Leibniz and Newton. Namely, the geometers used to describe infinitesimals as what remains
“not before you pass to the limit, nor after, but at the very moment of passage to the limit”.
In the context of a modern theory of infinitesimals such as the hyperreals (see Appendix A), one could explain the matter in the following terms. We decompose the procedure of taking the limit of, say, a sequence into two stages:
evaluating the sequence at an infinite hypernatural222222See main text at footnote 39. , to obtain the hyperreal ; and
taking its standard part .
Thus is adequal to , or (see formula (3.1)), so that we have . In this sense, the infinitesimals exist “at the moment” of taking the limit, namely between the stages (i) and (ii).
Felscher describes d’Alembert as “one of the mathematicians representing the heroic age of calculus” [51, p. 845]. Felscher buttresses his claim by a lengthy quotation concerning the definition of the limit concept, from the article Limite from the Encyclopédie ou Dictionnaire Raisonné des Sciences, des Arts et des Métiers:
On dit qu’une grandeur est la limite d’une autre grandeur, quand la seconde peut approcher de la première plus près que d’une grandeur donnée, si petite qu’on la puisse supposer, sans pourtant que la grandeur qui approche, puisse jamais surpasser la grandeur dont elle approche; ensorte que la différence d’une pareille quantité à sa limite est absolument inassignable (Encyclopédie, volume 9 from 1765, page 542).
One recognizes here a kinetic definition of limit already exploited by Newton.232323See footnote 21 on Pourciau’s analysis. Even if we do attribute visionary status to this passage as many historians seem to, the fact remains that d’Alembert didn’t write it. Felscher overlooked the fact that the article Limite was written by two authors. In reality, the above passage defining the concept of “limit” (as well as the two propositions on limits) did not originate with d’Alembert, but rather with the encyclopedist Jean-Baptiste de La Chapelle. De la Chapelle was recruited by d’Alembert to write 270 articles for the Encyclopédie. The section of the article containing these items is signed (E) (at bottom of first column of page 542), known to be de La Chapelle’s “signature” in the Encyclopedie. Felscher had already committed a similar error of attributing de la Chapelle’s work to d’Alembert, in his 1979 work .242424We are grateful to D. Spalt for the historical clarification concerning the authorship of the Limite article in the Encyclopedie. Note that Robinson [128, p. 267] similarly misattributes this passage to d’Alembert.
5. Did Cauchy replace infinitesimals by rigor?
Did Cauchy take first steps toward replacing infinitesimals by rigor, and did he give an epsilontic definition of continuity?
A claim to the effect that Cauchy was a fore-runner of the epsilontisation of analysis is routinely recycled in history books and textbooks. To put such claims in historical perspective, it may be useful to recall Grattan-Guinness’s articulation of a historical reconstruction project in the name of H. Freudenthal , in the following terms:
it is mere feedback-style ahistory to read Cauchy (and contemporaries such as Bernard Bolzano) as if they had read Weierstrass already. On the contrary, their own pre-Weierstrassian muddles252525Grattan-Guinness’s term “muddle” refers to an irreducible ambiguity of historical mathematics such as Cauchy’s sum theorem of 1821. See footnote 1 for a related comment by Mancosu. need historical reconstruction [64, p. 176].
5.1. Cauchy’s definition of continuity
It is often claimed that Cauchy gave an allegedly “modern”, meaning epsilon-delta, definition of continuity. Such claims are anachronistic. In reality, Cauchy’s definition is an infinitesimal one. His definition of the continuity of takes the following form: an infinitesimal -increment gives rise to an infinitesimal -increment (see [24, p. 34]). The widespead misconception that Cauchy gave an epsilontic definition of continuity is analyzed in detail in .
Cauchy’s primary notion is that of a variable quantity. The meaning he attached to the latter term in his Cours d’Analyse in 1821 is generally agreed to have been a sequence of discrete values. He defines infinitesimals in terms of variable quantities, by specifying that a variable quantity tending to zero becomes an infinitesimal. He similarly defines limits in terms of variable quantities in the following terms:
lorsque les valeurs successivement attribuées à une même variable s’approche indéfiniment d’une valeur fixe de manière à finir par en différer aussi peu que l’on voudra cette dernière est appelée limite de toutes les autres (Cauchy [24, p. 4]).
Cauchy’s definition is patently a kinetic, not an epsilontic, definition of limit, similar to Newton’s.262626See footnote 21. While epsilontic-style formulations do occasionally appear in Cauchy (though without Bolzano’s proper attention to the order of the quantifiers), they are not presented as definitions but rather as consequences, and at any rate never appear in the context of the property of the continuity of functions.
Thus, Grabiner’s famous essay Who gave you the epsilon? Cauchy and the origins of rigorous calculus cites a form of an epsilon-delta quantifier technique used by Cauchy in a proof:
Let be two very small numbers; the first is chosen so that for all numerical [i.e., absolute] values of less than , and for any value of included [in the interval of definition], the ratio will always be greater than and less than (Grabiner [63, p. 185] citing Cauchy).
Grabiner describes such an epsilon-delta technique as “the algebra of inequalities”. The thrust of her argument is that Cauchy sought to establish a foundation for analysis based on the algebra of inequalities. Is this borne out by the evidence she presents? Let us consider Grabiner’s evidence:
Cauchy gave essentially the modern definition of continuous function, saying that the function is continuous on a given interval if for each in that interval “the numerical [i.e., absolute] value of the difference decreases indefinitely with [63, p. 190].
Is this “essentially the modern definition of continuity”, as Grabiner claims? Hardly so. Cauchy’s definition is a blend of a kinetic (rather than epsilontic) and an infinitesimal approach. Grabiner fails to mention three essential items:
Cauchy prefaces the definition she cited, by describing his as an infinitely small increment:
Si, en partant d’une valeur de …on attribue à la variable un accroissement infiniment petit …” (Cauchy [24, p. 34]) [emphasis added–the authors];
Cauchy follows this definition by another, italicized, definition, where both and the difference are described as being infinitesimal: if the former is infinitesimal, then so is the latter;
Infinitesimals provide a method for calculating limits, whereas epsilon, delta methods require the answer in advance (see Madison and Stroyan [109, p. 497]).
The advantage of infinitesimal definitions, such as those found in Cauchy, is their covariant nature (cf. Lutz et al. ). Whereas in the epsilontic approach one needs to work one’s way backwards from the value of the limit, in the infinitesimal approach one can proceed from the original expression, simplify it, and eventually arrive at the value of the limit. This indicates that the two approaches work in opposite directions. The infinitesimal calculation goes with the natural flow of our reasoning, whereas the epsilontic one goes in the opposite direction. Notice, for example, that delta corresponds to the independent variable even though the value of delta depends on our choice of epsilon, which corresponds to the dependent variable. The infinitesimal calculation, in contrast, begins with the the independent variable and computes from that the value of the dependent variable.
5.2. Cauchy’s intermediate value theorem
Did Cauchy exploit epsilon-delta techniques in building foundations for analysis? Let us examine Grabiner’s evidence. She claims that, in Cauchy’s proof of the intermediate value theorem (IVT),
we have the algebra of inequalities providing a technique which Cauchy transformed from a tool of approximation to a tool of rigor (Grabiner [63, p. 191]).
Yet Grabiner’s treatment of Cauchy’s proof of the IVT in [63, p. 190] page offers no evidence that Cauchy employed an epsilon-delta technique.272727Grabiner further attributes to Lagrange the polynomial case of Cauchy’s divide-and-conquer argument in the proof of the IVT, whereas we saw in Subsection 2.3 that Stevin did this two centuries before Lagrange (see main text at footnote 16).
An examination of Cauchy’s proof in his Note III reveals that, on the contrary, it is closely tied to Cauchy’s infinitesimal definition of continuity. Thus, Cauchy constructs an increasing sequence and a decreasing sequence, denoted respectively and (Cauchy [24, p. 462]) with a common limit , such that has opposite sign at the corresponding pairs of points. Cauchy concludes that the values of at the respective sequences converge to a common limit . Being both nonpositive and nonnegative, the common limit must vanish.
Koetsier speculates that Cauchy may have hit upon his concept of continuity by analyzing his proof of the IVT (perhaps in the case of polynomials). The evidence is compelling: even though Cauchy does not mention infinitesimals in his Note III, and are recognizably variable quantities differing by an infinitesimal from the constant quantity . By Cauchy’s definition of continuity, and must similarly differ from by an infinitesimal. Contrary to Grabiner’s claim, a close examination of Cauchy’s proof of the IVT reveals no trace of epsilon-delta. Following Koetsier’s hypothesis, it is reasonable to place it, rather, in the infinitesimal strand of the development of analysis, rather than the epsilontic strand.
After constructing the lower and upper sequences, Cauchy does write that the values of the latter “finiront par differer de ces premiers valeurs aussi peu que l’on voudra”. That may sound a little bit epsilon/delta. Meanwhile, Leibniz uses language similar to Cauchy’s:
Whenever it is said that a certain infinite series of numbers has a sum, I am of the opinion that all that is being said is that any finite series with the same rule has a sum, and that the error always diminishes as the series increases, so that it becomes as small as we would like [“ut fiat tam parvus quam velimus”] (Leibniz [99, p. 99]).
Cauchy used epsilontics if and only if Leibniz did, over a century before him.
5.3. Cauchy’s influence
The exaggerated claims of a Cauchy provenance for epsilontics found in triumvirate literature go hand-in-hand with neglect of his visionary role in the development of infinitesimals at the end of the 19th century. In 1902, E. Borel [17, p. 35-36] elaborated on Paul du Bois-Reymond’s theory of rates of growth, and outlined a general “theory of increase” of functions, as a way of implementing an infinitesimal-enriched continuum. In this text, Borel specifically traced the lineage of such ideas to a 1829 text of Cauchy’s on the rates of growth of functions (see Fisher [52, p. 144] for details). In 1966, A. Robinson pointed out that
Following Cauchy’s idea that an infinitely small or infinitely large quantity is associated with the behavior of a function , as tends to a finite value or to infinity, du Bois-Reymond produced an elaborate theory of orders of magnitude for the asymptotic behavior of functions …Stolz tried to develop also a theory of arithmetical operations for such entities [128, p. 277-278].
Robinson traces the chain of influences further, in the following terms:
It seems likely that Skolem’s idea to represent infinitely large natural numbers by number-theoretic functions which tend to infinity (Skolem ),282828The reference is to Skolem’s 1934 work . The evolution of modern infinitesimals is traced in more detail in Table 1 and in Borovik et al. . also is related to the earlier ideas of Cauchy and du Bois-Reymond [128, p. 278].
One of Cantor’s bêtes noires was the neo-Kantian philosopher Hermann Cohen (1842–1918) (see also Mormann ), whose fascination with infinitesimals elicited fierce criticism by both Cantor and B. Russell. Yet at the end of the day, A. Fraenkel (of Zermelo–Fraenkel fame) wrote:
my former student Abraham Robinson had succeeded in saving the honour of infinitesimals - although in quite a different way than Cohen and his school had imagined (Fraenkel [55, p. 107]).
6. Was Cauchy’s 1821 “sum theorem” false?
Was Cauchy’s 1821 “sum theorem” false, and what did he add in 1853?
As discussed in Section 5, Cauchy’s definition of continuity is explicitly stated in terms of infinitesimals: “an infinitesimal -increment gives rise to an infinitesimal -increment”. Boyer [21, p. 282] declares that Cauchy’s 1821 definition is “to be interpreted” in the framework of the usual “limits”, at a point of an Archimedean continuum. Traditional historians typically follow Boyer’s lead.
But when it comes to Cauchy’s 1853 modification of the hypothesis of the ‘‘sum theorem”292929The assertion of the theorem is the continuity of the sum of a convergent series of continuous functions, with the italicized term requiring clarification. Modern versions of the theorem require a hypothesis uniform convergence. The nature of the hypothesis Cauchy himself had in mind is hotly disputed (see Borovik & Katz as well as ). in (Cauchy ), some historians declare that it is to be interpreted as adding the hypothesis of “uniform convergence” (see e.g., Lützen [107, p. 183-184]). Are Boyer and Lützen compatible?
Note that an epsilontic definition (in the context of an Archimedean continuum) of the uniform convergence of a sequence to necessarily involves a pair of variables (where ranges through the domain of and ranges through ), rather than a single variable: we need a formula of the sort
(prefaced by the usual clause “ ”). Now Cauchy’s 1853 modification of the hypothesis is stated in terms of a single variable , rather than a pair of variables . Namely, Cauchy specified that the condition of convergence should hold “always”. The meaning of the term “always” becomes apparent only in the course of the proof, when Cauchy gives an explicit example of evaluating at an infinitesimal generated by the sequence . Thus the term “always” involves adding extra values of at which the convergence condition must be satisfied (see Bråting and Katz & Katz ).
Cauchy’s approach is based on two assumptions which can be stated in modern terminology as follows:
when you have a closed expression for a function, then its values at “variable quantities” (such as ) are calculated by using the same closed expression as at real values;
to evaluate a function at a variable quantity generated by a sequence, one evaluates term-by-term.
Cauchy’s strengthened condition amounts to requiring the error to become infinitesimal:
which in the case of given by translates into the requirement that tends to zero.
An epsilontic interpretation (in the context of an Archimedean continuum) of Cauchy’s 1821 and 1853 texts is untenable, as it necessitates a pair of variables as in (6.1), where Cauchy only used a single one, namely , but one drawn from a “thicker” continuum including infinitesimals. Namely, Cauchy draws the points to be evaluated at from an infinitesimal-enriched continuum.
We will refer to an infinitesimal-enriched continuum as a Bernoullian continuum, or a ‘‘B-continuum’’ for short, in an allusion to Johann Bernouilli.303030Bernoulli was the first to use infinitesimals in a systematic fashion as a foundational concept, Leibniz himself having employed both a syncategorematic and a true infinitesimal approach. The pair of approaches in Leibniz are discussed by Bos [20, item 4.2, p. 55]; see also Appendix A, footnote 38.
A null sequence such as “becomes” an infinitesimal, in Cauchy’s terminology. Evaluating at points of a Bernoullian continuum makes it possible to express uniform convergence in terms of a single variable rather than a pair .
Once one acknowledges that there are two variables in the traditional epsilontic definition of uniform continuity and uniform convergence, it becomes untenable to argue that the condition Cauchy introduced was epsilontic uniform convergence. A historian who describes Cauchy’s condition as uniform convergence, must acknowledge that the definition involves an infinitesimal-enriched continuum, at variance with Boyer’s interpretation.
7. Did Weierstrass succeed in eliminating infinitesimals?
Did Weierstrass succeed in eliminating infinitesimals from mathematics?
The persistent idea that infinitesimals have been “eliminated” by the great triumvirate of Cantor, Dedekind, and Weierstrass was soundly refuted by Ehrlich . Ehrlich documents a rich and uninterrupted chain of work on non-Archimedean systems, or what we would call a Bernoullian continuum. Some key developments in this chain are listed in Table 1 (see for more details).
The elimination claim can only be understood as an oversimplification by Weierstrass’s followers, who wish to view the history of analysis as a triumphant march toward the radiant future of Weierstrassian epsilontics. Such a view of history is rejected by H. Putnam who comments on the introduction of the methods of the differential and integral calculus by Newton and Leibniz in the following terms:
If the epsilon-delta methods had not been discovered, then infinitesimals would have been postulated entities (just as ‘imaginary’ numbers were for a long time). Indeed, this approach to the calculus–enlarging the real number system–is just as consistent as the standard approach, as we know today from the work of Abraham Robinson […] If the calculus had not been ‘justified’ Weierstrass style, it would have been ‘justified’ anyway (Putnam ).
In short, there is a cognitive bias inherent in a postulation in an inevitable outcome in the evolution of a scientific discipline.
The study of cognitive bias has its historical roots in Francis Bacon’s proposed classification of what he called idola (a Latin plural) of several kinds. He described these as things which obstructed the path of correct scientific reasoning. Of particular interest to us are his Idola fori (“Illusions of the Marketplace”: due to confusions in the use of language and taking some words in science to have meaning different from their common usage); and Idola theatri (“Illusions of the Theater”: the following of academic dogma and not asking questions about the world), see Bacon .
Completeness, continuity, continuum, Dedekind “gaps”: these are terms whose common meaning is frequently conflated with their technical meaning. In our experience, explaining infinitesimal-enriched extensions of the reals to an epsilontically trained mathematician typically elicits a puzzled reaction on her part: “But aren’t the real numbers already complete by virtue of having filled in all the gaps already?”
This question presupposes an academic dogma, viz., that there is a single coherent conception of the continuum, and it is a complete, Archimedean ordered field. This dogma has recently been challenged. Numerous possible conceptions of the continuum range from S. Feferman’s predicative conception of the continuum , to F. William Lawvere’s and J. Bell’s conception in terms of an intuitionistic topos , , .
To illustrate the variety of possible conceptions of the continuum, note that traditionally, mathematicians have considered at least two different types of continua. These are Archimedean continua, or A-continua for short, and infinitesimal-enriched (Bernoulli) continua, or B-continua for short. Neither an A-continuum nor a B-continuum corresponds to a unique mathematical structure (see Table 2). Thus, we have two distinct implementations of an A-continuum:
the real numbers (or Stevin numbers),313131See Section 2. in the context of classical logic (incorporating the law of excluded middle);
Brouwer’s continuum built from “free-choice sequences”, in the context of intuitionistic logic.
John L. Bell describes a distinction within the class of an infinitesimal-enriched B-continuum, in the following terms. Historically, there were two main approaches to such an enriched continuum, one by Leibniz, and one by B. Nieuwentijt, who favored nilpotent (nilsquare) infinitesimals whose squares are zero. Mancosu’s discussion of Nieuwentijt in [112, chapter 6] is the only one to date to provide a contextual understanding of Nieuwentijt’s thought (see also Mancosu and Vailati ). J. Bell notes:
Leibnizian infinitesimals (differentials) are realized in [A. Robinson’s] nonstandard analysis,323232More precisely, the Hewitt-Łoś-Robinson continuum; see Appendix A. and nilsquare infinitesimals in [Lawvere’s] smooth infinitesimal analysis (Bell [11, 12]).
The latter theory relies on intuitionistic logic.333333Lawvere’s infinitesimals rely on a category-theoretic framework grounded in intuitionistic logic (see J. Bell ). An implementation of an infinitesimal-enriched continuum was developed by P. Giordano (see [59, 60]), combining elements of both a classical and an intuitionistic continuum. The Weirstrassian continuum is but a single member of a diverse family of concepts.
8. Did Dedekind discover the essence of continuity?
Did Dedekind discover “the essence of continuity”, and is such essence captured by his cuts?
In Dedekind’s approach, the “essence” of continuity amounts to the numerical assertion that two non-rational numbers should be equal if and only if they define the same Dedekind cut343434We will ignore the slight technical complication arising from the fact that there are two ways of defining the Dedekind cut associated with a rational number. on the rationals. Dedekind formulated his “essence of continuity” in the context of the geometric line in the following terms:
If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions (Dedekind [38, p. 11]).
We will refer to this essence as the geometric essence of continuity.353535Note that the geometric essence of continuity necessarily fails over an ordered non-Archimedean field . Thus, defining infinitesimals as elements of violating the traditional Archimedean property, we can start with the cut of into positive and negative elements, and then modify this cut by assigning all infinitesimals to, say, the negative side. Such a cut does not correspond to an element of . Dedekind goes on to comment on the epistemological status of this statement of the essence of continuity:
[…] I may say that I am glad if every one finds the above principle so obvious and so in harmony with his own ideas of a line; for I am utterly unable to adduce any proof of its correctness, nor has any one the power (ibid.).
Having enriched the domain of rationals by adding irrationals, numbers defined completely by cuts not produced by a rational, Dedekind observes:
From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts (Dedekind [38, p. 15]).
By now, Dedekind postulates that two numbers should be equal “always and only” [i.e., if and only if] they define identical cuts on the rational numbers. Thus, Dedekind postulates that there should be “one and only one” number which produces such a division. Dedekind clearly presents this as an exact arithmetic analogue to the geometric essence of continuity. We will refer to such a postulate as the rational essence of continuity.
Dedekind’s postulation of rational essence is not accompanied by epistemological worries as was his geometric essence a few pages earlier. Yet, rational essence entails a suppression of infinitesimals: a pair of distinct non-rational numbers can define the same Dedekind cut on , such as and with infinitesimal; but one cannot have such a pair if one postulates the rational essence of continuity, as Dedekind does.
Dedekind’s technical work on the foundations of analysis has been justly celebrated (see D. Fowler ). Whereas everyone before Dedekind had assumed that operations such as powers, roots, and logarithms can be performed, he was the first to show how these operations can be defined, and shown to be coherent, in the realm of the real numbers (see Dedekind [37, §6]).
Meanwhile, the nature of his interpretive speculations about what does or does not constitute the “essence” of continuity, is a separate issue. For over a hundred years now, many mathematicians have been making the assumption that space conforms to Dedekind’s idea of “the essence of continuity”, which in arithmetic translates into the numerical assertion that two numbers should be equal if and only if they define the same Dedekind cut on the rationals. Such an assumption rules out infinitesimals. In the context of the hyperreal number system, it amounts to an application of the standard part function (see Appendix A), which forces the collapse of the entire halo (cluster of infinitely close, or adequal, points) to a single point.
The formal/axiomatic transformation of mathematics accomplished at the end of the 19th century created a specific foundational framework for analysis. Weierstrass’s followers raised a philosophical prejudice against infinitesimals to the status of an axiom. Dedekind’s “essence of continuity” was, in essence, a way of steamrolling infinitesimals out of existence.
In 1977, E. Nelson created a set-theoretic framework (enriching ZFC) where the usual construction of the reals produces a number system containing entities that behave like infinitesimals. Thus, the elimination thereof was not the only way to achieve rigor in analysis as advertized at the time, but rather a decision to develop analysis in just one specific way.
9. Who invented Dirac’s delta function?
A prevailing sentiment today is that one of the spectacular successes of the rigorous analysis was the justification of delta functions, originally introduced informally by to P. Dirac (1902–1984), in terms of distribution theory. But was it originally introduced informally by Dirac?
In fact, Fourier and Cauchy exploited the “Dirac” delta function over a century earlier. Cauchy defined such functions in terms of infinitesimals (see Lützen and Laugwitz ). A function of the type generally attributed to Dirac was specifically described by Cauchy in 1827 in terms of infinitesimals. More specifically, Cauchy uses a unit-impulse, infinitely tall, infinitely narrow delta function, as an integral kernel. Thus, in 1827, Cauchy used infinitesimals in his definition of a “Dirac” delta function [26, p. 188]. Here Cauchy uses infinitesimals and , where is, in modern terms, the “scale parameter” of the “Cauchy distribution”, whereas gives the size of the interval of integration. Cauchy wrote:
Moreover one finds, denoting by , two infinitely small numbers,
(see Cauchy [26, p. 188]). Such a formula extracts the value of a function at a point by integrating against a delta function defined in terms of an infinitesimal parameter (see and Laugwitz [95, p. 230]). The expression
(for real ) is known as the Cauchy distribution in probability theory. The function is called the probability density function. A Cauchy distribution with an infinitesimal scale parameter produces a function with Dirac-delta function behavior, exploited by Cauchy already in 1827 in work on Fourier series and evaluation of singular integrals.
10. Is there continuity between Leibniz and Robinson?
Is there continuity between historical infinitesimals and Robinson’s theory?
Historically, infinitesimals have often been represented by null sequences. Thus, Cauchy speaks of a variable quantity as becoming an infinitesimal in 1821, and his variable quantities from that year are generally understood to be sequences of discrete values (on the other hand, in his 1823 he used continuous variable quantities). Infinitesimal-enriched fields can in fact be obtained from sequences, by means of an ultrapower construction, where a null sequence generates an infinitesimal. Such an approach was popularized by Luxemburg in 1962, and is based on the work by E. Hewitt from 1948. Even in Robinson’s approach based on the compactness theorem, a null sequence is present, though well-hidden, namely in the countable collection of axioms . Thus, null sequences provide both a cognitive and a technical link between historical infinitesimals thought of as variable quantities taking discrete values, on the one hand, and modern infinitesimals, on the other (see Katz & Tall ).
Leibniz’s heuristic law of continuity was implemented mathematically as Łoś’s theorem and later as the transfer principle over the hyperreals (see Appendix A), while Leibniz’s heuristic law of homogeneity (see Leibniz [101, p. 380]) and Bos [20, p. 33]) was implemented mathematically as the standard part function (see Katz and Sherry ).
11. Is Lakatos’ take on Cauchy tainted by Kuhnian relativism?
Does Lakatos’s defense of infinitesimalist tradition rest upon an ideological agenda of Kuhnian relativism and fallibilism, inapplicable to mathematics?
G. Schubring summarizes fallibilism as an
enthusiasm for revising traditional beliefs in the history of science and reinterpreting the discipline from a theoretical, epistemological perspective generated by Thomas Kuhn’s (1962) work on the structure of scientific revolutions. Applying Popper’s favorite keyword of fallibilism, the statements of earlier scientists that historiography had declared to be false were particularly attractive objects for such an epistemologically guided revision (Schubring [132, p. 431–432]).
Schubring then takes on Lakatos in the following terms:
The philosopher Imre Lakatos (1922-1972)363636The dates given by Schubring are incorrect. The correct dates are 1922-1974. was responsible for introducing these new approaches into the history of mathematics. One of the examples he analyzed and published in 1966 received a great deal of attention: Cauchy’s theorem and the problem of uniform convergence. Lakatos refines Robinson’s approach by claiming that Cauchy’s theorem had also been correct at the time, because he had been working with infinitesimals (ibid.).
However, Schubring’s summary of the philosophical underpinnings of Lakatos’ interpretation of Cauchy’s sum theorem is not followed up by an analysis of Lakatos’s position (see [92, 93]). It is as if Schubring felt that labels of “Kuhnianism” and “fallibilism” are sufficient grounds for dismissing a scholar. Schubring proceeds similarly to dismiss Laugwitz’s reading of Cauchy as “solipsistic” [132, p. 434]. Schubring accuses Laugwitz of interpreting Cauchy’s conceptions as
some hermetic closure of a private mathematics (Schubring [132, p. 435]) [emphasis in the original–the authors];
as well as being “highly anomalous or isolated” [132, p. 441].
The fact is that Laugwitz is interpreting Cauchy’s words according to their plain meaning (see [94, 97]), as revealed by looking, as Kuhn would suggest, at the context in which they occur. The context strongly recommends taking Cauchy’s infinitesimals at face value, rather than treating them as a sop to the management. The burden of proof falls upon Schubring to explain why the triumvirate interpretation of Cauchy is not “solipsistic”, “hermetic”, or “anomalous”. The latter three modifiers could be more apppropriately applied to Schubring’s own interpretation of Cauchy’s infinitesimals as allegedly involving a compromise with rigor, allegedly due to tensions with the management of the Ecole polytechnique. Schubring’s interpretation is based on Cauchy’s use of the term concilier in Cauchy’s comment on the first page of his Avertissement:
Mon but principal a été de concilier la rigueur, dont je m’étais fait une lois dans mon Cours d’Analyse, avec la simplicité qui résulte de la considération directe des quantités infiniment petites (Cauchy [25, p. 10]).
Let us examine Schubring’s logic of conciliation. A careful reading of Cauchy’s Avertissement in its entirety reveals that Cauchy is referring to an altogether different source of tension, namely his rejection of some of the procedures in Lagrange’s Mécanique analytique as unrigorous, such as Lagrange’s principle of the “generality of algebra”. While rejecting the “generality of algebra” and Lagrange’s flawed method of power series, Cauchy was able, as it were, to sift the chaff from the grain, and retain the infinitesimals endorsed in the 1811 edition of the Mécanique analytique. Indeed, Lagrange opens his treatise with an unequivocal endorsement of infinitesimals. Referring to the system of infinitesimal calculus, Lagrange writes:
Lorsqu’on a bien conçu l’esprit de ce système, et qu’on s’est convaincu de l’exactitude de ses résultats par la méthode géométrique des premières et dernières raisons, ou par la méthode analytique des fonctions dérivées, on peut employer les infiniment petits comme un instrument sûr et commode pour abréger et simplifier les démonstrations373737“Once one has duly captured the spirit of this system [i.e., infinitesimal calculus], and has convinced oneself of the correctness of its results by means of the geometric method of the prime and ultimate ratios, or by means of the analytic method of derivatives, one can then exploit the infinitely small as a reliable and convenient tool so as to shorten and simplify proofs” (Lagrange). (Lagrange [91, p. iv]).
Lagrange describes infinitesimals as dear to a scientist, being reliable and convenient. In his Avertissement, Cauchy retains the infinitesimals that were also dear to Lagrange, while criticizing Lagrange’s “generality of algebra” (see for details).
It’s useful here to evoke the use of the term “concilier” by Cauchy’s teacher Lacroix. Gilain quotes Lacroix in 1797 to the effect that
“lorsqu’on veut concilier la rapidité de l’exposition avec l’exactitude dans le langage, la clarté dans les principes, […], je pense qu’il convient d’employer la méthode des limites” (p.XXIV).” [58, footnote 20].
Here Lacroix, like Cauchy, employs “concilier”, but in the context of discussing the limit notion. Would Schubring’s logic of conciliation dictate that Lacroix developed a compromise notion of limit, similarly with the sole purpose of accomodating the management of the Ecole?
Why are Lakatos and Laugwitz demonized, rather than analyzed, by Schubring? We suggest that the act of contemplating for a moment the idea that Cauchy’s infinitesimals can be taken at face value is unthinkable to a triumvirate historian, as it would undermine the epsilontic Cauchy-Weierstrass tale that the received historiography is erected upon. The failure to appreciate the Robinson-Lakatos-Laugwitz interpretation, according to which infinitesimals are mainstream analysis from Cauchy onwards, is symptomatic of a narrow Archimedean-continuum vision.
Appendix A Rival continua
This appendix summarizes a 20th century implementation of an alternative to an Archimedean continuum, namely an infinitesimal-enriched continuum. Such a continuum is not to be confused with incipient notions of such a continuum found in earlier centuries in the work of Fermat, Leibniz, Euler, Cauchy, and others.
Johann Bernoulli was one of the first to exploit infinitesimals in a systematic fashion as a foundational tool in the calculus.383838See footnote 30 for a comparison with Leibniz. We will therefore refer to such a continuum as a Bernoullian continuum, or B-continuum for short.
a.1. Constructing the hyperreals
Let us start with some basic facts and definitions. Let
be the field of real numbers, let be a fixed nonprincipal ultrafilter on (the existence of such was established by Tarski ). The relation defined by
is an equivalence relation on the set . The set of hyperreals IIR, or the B-continuum for short, is the quotient set
Addition, multiplication and order of hyperreals are defined by
The standard real number is identified with equivalence class of the constant sequence , i.e. .
The set IIN of hypernaturals (mentioned in Section 4)393939See footnote 22. is the subset of IIR defined by
In particular, each sequence of natural numbers represents a hypernatural number, i.e. .
The set of hypernaturals can be represented as a disjoint union
where the set is just a copy of the usual natural numbers, and consists of infinite (sometimes called “unlimited”) hypernaturals. Each element of is greater than every usual natural number, i.e.
is a non-Archimedean, real closed field.
The set of infinitesimal hyperreals is defined by
where stands for the absolute value of , which is defined as in any ordered field. We say that is infinitely close to , and write , if and only if .
To give some examples, the sequence represents a positive infinitesimal . Next, let be a sequence of reals such that , then represents an infinitesimal,404040In this construction, every null sequence defines an infinitesimal, but the converse is not necessarily true. Modulo suitable foundational material, one can ensure that every infinitesimal is represented by a null sequence; an appropriate ultrafilter (called a P-point) will exist if one assumes the continuum hypothesis, or even the weaker Martin’s axiom. See Cutland et al for details. i.e. . And finally, sequence represents a nonzero infinitesimal , whose sign depends on whether or not the set is a member of the ultrafilter.
The set of limited hyperreals is defined by
so that we have a disjoint union
where consists of unlimited hyperreals (i.e., inverses of nonzero infinitesimals).
Theorem A.2 (Standard Part Theorem).
The unique real such that is called the standard part of , and we write .
Note that if a sequence happens to be Cauchy, one can relate standard part and limit as follows:414141This theme is pursued further by Giordano et al. .
is a maximal ideal of the ring , and the quotient ring is isomorphic to the field of standard real numbers .
Since the map
is an order preserving morphism, we can treat the field of hyperreals as an extension of standard reals and use the usual notation . Now, the map “st” sends each finite point , to the real point st infinitely close to , as follows:424242This is the Fermat-Robinson standard part whose seeds are found in Fermat’s adequality, as well as in Leibniz’s treanscendental principle of homogeneity.
Robinson’s answer to Berkeley’s logical criticism (see D. Sherry ) is to define the basic concept of the calculus as
rather than the differential ratio itself, as in Leibniz. Robinson comments as follows: “However, this is a small price to pay for the removal of an inconsistency” (Robinson [128, p 266]).434343However, as argued in , an alleged inconsistency was not there in the first place. Briefly, Leibniz is able to employ his transcendental law of homogeneity to the same effect as Robinson’s standard part function (see Bos 1974, [20, p. 33]).
A sequence of real numbers can be extended to a hypersequence of hyperreals, indexed by all the hypernaturals, by setting
Let be a sequence of real numbers, and let . Then
a.2. Uniform continuity
We present a discussion of uniform continuity so as to supplement and clarify the discussion of uniform convergence in Section 6. The idea of “one variable versus two variables” is a little easier to explain in the context of uniform continuity.
Thus, the traditional definition of ordinary continuity on an interval can be expressed in terms of a single variable running through the domain of the function : namely,
for each , as tends to .
Meanwhile, uniform continuity cannot be expressed in a similar way in the traditional framework. Namely, one needs a pair of variables to run through the domain of :
, if then
(of course, this has to be prefaced by the traditional epsilon-delta yoga). Now the crucial observation is that in the context of a B-continuum, one no longer needs a pair of variables to define uniform continuity. Namely, it can be defined using a single variable, by exploiting the notion of microcontinuity at a point (see Gordon et al. ). We will use Leibniz’s symbol for the relation of being infinitely close. Thus, is called microcontinuous at if
whenever , also .
In terms of this notion, the uniform continuity of a real function is defined in terms of its natural extension to the hyperreals as follows:
for all is microcontinuous at .
This sounds startlingly similar to the definition of continuity itself, but the point is that microcontinuity is now required at every point of the B-continuum, i.e., in the domain of which is the natural extension of the (real) domain of .
To give an example, the function fails to be uniformly continuous on because of the failure of microcontinuity of its natural extension at a single infinite hyperreal . The failure of microcontinuity at is checked as follows. Consider the infinitesimal , and the point infinitely close to . To show that is not microcontinuous at , we calculate
This value is not infinitely close to , hence microcontinuity fails at . Thus the squaring function is not uniformly continuous on .
We introduced the term “microcontinuity” (cf. Gordon et al ) since there are two definitions of continuity, one using infinitesimals, and one using epsilons. The former is what we refer to as microcontinuity. It is given a special name to distinguish it from the traditional definition of continuity. Note that microcontinuity at a non-standard hyperreal does not correspond to any notion available in the epsilontic framework. To give another example, if we consider the function given by on the open interval as well as its natural extension , then fails to be microcontinuous at a positive infinitesimal . It follows that is not uniformly continuous on .
a.3. Pedagogical advantage of microcontinuity
The expressibility of uniform continuity in terms of a condition on a single variable, as explained above, is a pedagogical advantage of the microcontinuous approach. The pedagogical difficulty of the traditional two-variable definition in an epsilontic framework is compounded by its multiple alternations of quantifiers, while the hyperreal approach reduces the logical complexity of the definition by two quantifiers (see, e.g., Keisler ). The natural hyperreal extension of a real function is, of course, necessarily continuous at non-standard points, as well, by the transfer principle; on the other hand, this type of continuity at a non-standard point is of merely theoretical relevance in a calculus classroom. The relevant notion is that of microcontinuity, which allows one to distinguish between the classical notions of continuity and uniform continuity in a lucid way available only in the hyperreal framework. Similarly, the failure of uniform continuity can be checked by a ‘‘covariant’’444444See discussion at the end of Subsection 5.1. (direct) calculation at a single non-standard point, whereas the argument in an epsilontic framework is a bit of a “contravariant” multiple-quantifier tongue twister.
We are therefore puzzled by Hrbacek’s dubious laments [71, 72] of alleged “pedagogical difficulties” related to behavior at non-standard points, directed at the framework developed by Robinson and Keisler. On the contrary, such a framework bestows a distinct pedagogical advantage.454545Another critic of Robinson’s framework is A. Connes. He criticizes Robinson’s infinitesimals for being dependent on non-constructive foundational material. He further claims it to be a weakness of Robinson’s infinitesimals that the results of calculations that employ them, do not depend on the choice of the infinitesimal. Yet, Connes himself develops a theory of infinitesimals bearing a similarity to the ultrapower construction of the hyperreals in that it also relies on sequences (more precisely, spectra of compact operators). Furthermore, he freely relies on such results as the existence of the Dixmier trace, and the Hahn-Banach theorem. The latter results rely on similarly nonconstructive foundational material. Connes claims the independence of the choice of Dixmier trace to be a strength of his theory of infinitesimals in [29, p. 6213]. Thus, both of Connes’ criticisms apply to his own theory of infinitesimals. The mathematical novelty of Connes’ theory of infinitesimals resides in the exploitation of Dixmier’s trace, relying as it does on non-constructive foundational material, thus of similar foundational status to, for instance, the ultrapower construction of a non-Archimedean extension of the reals (see also ).
a.4. Historical remarks
Both the term “hyper-real field”, and an ultrapower construction thereof, are due to E. Hewitt in 1948 (see [68, p. 74]). In 1966, Robinson referred to the
theory of hyperreal fields (Hewitt ) which …can serve as non-standard models of analysis [128, p. 278].
The transfer principle is a mathematical implementation of Leibniz’s heuristic law of continuity: “what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa” (see Robinson [128, p. 262, 266] citing Leibniz 1701, ). The transfer principle, allowing an extension of every first-order real statement to the hyperreals, is a consequence of the theorem of J. Łoś in 1955, see , and can therefore be referred to as a Leibniz-Łoś transfer principle. A Hewitt-Łoś framework allows one to work in a B-continuum satisfying the transfer principle.
a.5. Comparison with Cantor’s construction
To indicate some similarities between the ultrapower construction and the so called Cantor’s construction of the real numbers, let us start with the field of rational numbers . Let be the ring of sequences of rational numbers. Denote by
the subspace consisting of Cauchy sequences, and let be the subspace of all null sequences. The reals are by definition the quotient field
Meanwhile, an infinitesimal-enriched field extension of may be obtained by forming the quotient
Here maps to zero in the quotient if and only if one has
where , as above, is a fixed nonprincipal ultrafilter on .
To obtain a full hyperreal field, we replace by in the construction, and form a similar quotient
We wish to emphasize the analogy with formula (A.3) defining the A-continuum. We can treat both and as subsets of IIR. Note that, while the leftmost vertical arrow in Figure 4 is surjective, we have
A more detailed discussion of the ultrapower construction can be found in M. Davis and Gordon, Kusraev, & Kutateladze . See also Błaszczyk for some philosophical implications. More advanced properties of the hyperreals such as saturation were proved later (see Keisler for a historical outline). A helpful “semicolon” notation for presenting an extended decimal expansion of a hyperreal was described by A. H. Lightstone . See also P. Roquette for infinitesimal reminiscences. A discussion of infinitesimal optics is in K. Stroyan , J. Keisler , D. Tall , and L. Magnani and R. Dossena [110, 40].
Edward Nelson in 1977 proposed an axiomatic theory parallel to Robinson’s theory. A related theory was proposed by Hrbáček (who submitted a few months earlier and published a few months later than Nelson). Another axiomatic approach was proposed by Benci and Di Nasso . As Ehrlich [43, Theorem 20] showed, the ordered field underlying a maximal (i.e., On-saturated) hyperreal field is isomorphic to J. H. Conway’s ordered field No, an ordered field Ehrlich describes as the absolute arithmetic continuum.
Infinitesimals can be constructed out of integers (see Borovik, Jin, and Katz ). They can also be constructed by refining Cantor’s equivalence relation among Cauchy sequences (see Giordano & Katz ). A recent book by Terence Tao contains a discussion of the hyperreals [148, p. 209-229].
The use of the B-continuum as an aid in teaching calculus has been examined by Tall , ; Ely ; Katz and Tall (see also [74, 75]). These texts deal with a “naturally occurring”, or “heuristic”, infinitesimal entity ’ and its role in calculus pedagogy.464646See footnote 14 for Peirce’s take on ’. Applications of the B-continuum range from the Bolzmann equation (see L. Arkeryd [5, 6]); to modeling of timed systems in computer science (see H. Rust ); Brownian motion, economics (see R. Anderson ); mathematical physics (see Albeverio et al. ); etc.
We are grateful to M. Barany, L. Corry, N. Guicciardini, and V. Katz for helpful comments.
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Piotr Błaszczyk is Professor at the Institute of Mathematics, Pedagogical University (Cracow, Poland). He obtained degrees in mathematics (1986) and philosophy (1994) from Jagiellonian University (Cracow, Poland), and a PhD in ontology (2002) from Jagiellonian University. He authored Philosophical Analysis of Richard Dedekind’s memoir Stetigkeit und irrationale Zahlen (2008, Habilitationsschrift). His research interest is in the idea of continuum and continuity from Euclid to modern times.
Mikhail G. Katz is Professor of Mathematics at Bar Ilan University, Ramat Gan, Israel. Two of his joint studies with Karin Katz were published in Foundations of Science: “A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography” and “Stevin numbers and reality”, online respectively at
A joint study with Karin Katz entitled “Meaning in classical mathematics: is it at odds with Intuitionism?” Intellectica 56 (2011), no. 2, 223-302 may be found at http://arxiv.org/abs/1110.5456
A joint study with A. Borovik and R. Jin entitled “An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals” is due to appear in Notre Dame Journal of Formal Logic 53 (2012), no. 4.
A joint study with David Sherry entitled “Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond” is due to appear in Erkenntnis.
A joint study with David Tall, entitled “The tension between intuitive infinitesimals and formal mathematical analysis”, appeared as a chapter in a book edited by Bharath Sriraman, see
David Sherry is fortunate to be professor of philosophy at Northern Arizona University. | s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224654097.42/warc/CC-MAIN-20230608035801-20230608065801-00544.warc.gz | CC-MAIN-2023-23 | 101,326 | 425 |
https://drum.lib.umd.edu/items/a71122f7-e0cf-4421-8046-bcca1abe7210 | math | statistics of impedance and scattering matrices in microwave chaotic cavities: the random coupling model
Publication or External Link
A model is proposed for the study of the statistical properties of the impedance (Z) and scattering (S)matrices of open electromagnetic cavities with several transmission lines or waveguides connected to the cavity. The model is based on assumed properties of the eigenfunctions for the closed cavity. Analysis of the model successfully reproduces features of the random matrix model believed to be universal, while at the same time incorporating features which are specific to individual systems. Universal statistical properties of the cavity impedance Z are obtained in terms of the radiation impedance. These universal properties are independent of system-specific details and shared by the members of the general class of systems whose corresponding ray trajectories are chaotic.
In the single channel case, I obtained the normalized impedance and scattering coefficients whose probability density functions (PDF) are predicted to be universal. In the multiple-channel case, I focused on correlations in the phases of the eigenvalues of the S-matrix, and derived a formula for the averaged reflection coefficients in terms of the port radiation impedance. Effects of time-reversal symmetry and wall absorption are discussed. urthermore, I study the characterization of statistical fluctuations of the scattering matrix S and the impedance matrix Z, through their variance ratios. The variance ratio for the impedance matrix is shown to be a universal function of distributed losses within the scatterer, which contrasts with variance ratio of the scattering matrix for which universality applies only in the large loss limit. Theoretical predictions are tested by direct comparison with numerical solutions for a specific system, and also agree with experimental results obtained from scattering measurements on a chaotic microwave cavity. | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816587.89/warc/CC-MAIN-20240413083102-20240413113102-00401.warc.gz | CC-MAIN-2024-18 | 1,978 | 4 |
https://www.thespiritualmovements.com/meaning-of-the-number/faq-what-is-the-meaning-of-a-mass-number-shown-in-parentheses.html | math | - 1 What is the meaning of A mass number shown in parenthesis?
- 2 Why are the masses of some listed as whole numbers in parentheses?
- 3 What is the meaning of A mass number?
- 4 What does it mean when an element has brackets?
- 5 How many atoms are in a human body?
- 6 What is the meaning of a mass number shown in parentheses quizlet?
- 7 Can the average atomic mass be a whole number?
- 8 Why is the atomic mass of boron 10.81 and not rounded to 11?
- 9 What element is at the top of the activity series?
- 10 What’s an example of a mass number?
- 11 What is the best definition of mass number?
- 12 What is the example of mass number?
- 13 Why is atomic weight in parentheses?
- 14 Do square brackets mean concentration?
- 15 Why do we put brackets around elements that have charges?
What is the meaning of A mass number shown in parenthesis?
If the atomic mass of a particular element is shown in parentheses, such as (145) for Promethium (Pm), the atomic mass reflects that of the most stable isotope1, and is not the average atomic mass for all isotopes of the element. Atomic masses used on this periodic table are from the IUPAC 1995 recommendations.
Why are the masses of some listed as whole numbers in parentheses?
If an element has no stable isotopes, meaning that all known isotopes of that element will undergo nuclear decay, the atomic number is displayed in parentheses.
What is the meaning of A mass number?
Mass number, in nuclear physics, the sum of the numbers of protons and neutrons present in the nucleus of an atom.
What does it mean when an element has brackets?
Very often in chemical formulae, we use parentheses to form subgroups of atoms within a molecule. In such a formula, the subscript outside the parentheses means that to count atoms, you must multiply that subscript by the numbers of atoms inside.
How many atoms are in a human body?
Suzanne Bell, an analytical chemist at West Virginia University, estimates that a 150-pound human body contains about 6.5 octillion (that’s 6,500,000,000,000,000,000,000,000,000) atoms. The vast majority of them are hydrogen (humans are almost entirely water, which comprises two hydrogen atoms and an oxygen).
What is the meaning of a mass number shown in parentheses quizlet?
What is the meaning of a mass number showing parenthesis? The mass number is the sum of the number of protons and neutrons its nucleus.
Can the average atomic mass be a whole number?
Atomic mass is never an integer number for several reasons: The atomic mass reported on a periodic table is the weighted average of all the naturally occuring isotopes. Being an average it would be most unlikely to be a whole number. The mass of an individual atom in atomic mass units is the mass relative to carbon-12.
Why is the atomic mass of boron 10.81 and not rounded to 11?
The atomic mass of boron is 10.81 u. And 10.81 u is a lot closer to 11u than it is to 10u, so there must be more of boron-11. Where u is the unit for atomic mass and x is the proportion of boron-10 out of the total boron abundance which is 100%. And thus the abundance of boron-11 is roughly 81%.
What element is at the top of the activity series?
The elements at the top of the activity series are in Groups 1 and 2 of the Periodic Table. Here is a picture of the activity series. The top seven elements in the list are Li, Na, K, Mg, Ca, Sr, and Ba. These are all in Groups 1 and 2 of the Periodic Table.
What’s an example of a mass number?
The total number of protons and neutrons in the nucleus of an atom. For example, nitrogen has 7 protons and 7 neutrons in its nucleus, giving it a mass number of 14. Isotopes of elements are distinguished by their mass number; for example, carbon-12 and carbon-14 have mass numbers of 12 and 14 respectively.
What is the best definition of mass number?
Mass number is an integer (whole number) equal to the sum of the number of protons and neutrons of an atomic nucleus. In other words, it is the sum of the number of nucleons in an atom. Mass number is often denoted using a capital letter A.
What is the example of mass number?
(ii) Mass number: It is the sum of the number of neutrons and the number of protons. For example, the atomic number of Lithium is 4 which is equal to the number of protons, the number of neutrons of magnesium is 4. The mass number is equal to 8(4+4).
Why is atomic weight in parentheses?
The atomic weight of an element is based on the variety of naturally occurring isotopes of that element and the relative abundance of each. These elements are unstable; that is, their nuclei decompose radioactively. The number in parentheses is the mass number of the most stable or best-known isotope of that element.
Do square brackets mean concentration?
Square brackets are often used to represent concentration, e.g., [NaOH] = 0.50 M. Use the capital letter M for molarity, not a lower case m (this is a different concentration unit called molality).
Why do we put brackets around elements that have charges?
If there’s more than one complex ion in the compound, then brackets are needed. The reason is that you need to put a bracket around the complex ion to show how many of those whole complex ions there are in the compound. | s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358702.43/warc/CC-MAIN-20211129074202-20211129104202-00348.warc.gz | CC-MAIN-2021-49 | 5,219 | 45 |
https://www.mathematica.org/our-publications-and-findings/publications/identifying-the-next-drug-epidemic-by-testing-municipal-wastewater-in-focus-brief | math | Mathematica-mpr.com is now Mathematica.org. Please update your bookmarks. Learn more about this change.
Identifying the Next Drug Epidemic by Testing Municipal Wastewater (In Focus Brief)
- When used in combination with geospatial mapping and advanced analytics, wastewater-based epidemiology has the potential to help officials predict rather than react to changes in drug use. | s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145966.48/warc/CC-MAIN-20200224163216-20200224193216-00524.warc.gz | CC-MAIN-2020-10 | 378 | 3 |
https://admin.clutchprep.com/chemistry/practice-problems/110423/detailed-calculations-show-that-the-value-of-zeff-for-the-outermost-electrons-in-3 | math | ⚠️Our tutors found the solution shown to be helpful for the problem you're searching for. We don't have the exact solution yet.
Detailed calculations show that the value of Zeff for the outermost electrons in Na and K atoms is 2.51+ and 3.49+ , respectively.
What value do you estimate for Zeff experienced by the outermost electron in both Na and K by assuming core electrons contribute 1.00 and valence electrons contribute 0.00 to the screening constant? What values do you estimate for Zeff using Slater’s rules?
Does either method of approximation account for the gradual increase in Zeff that occurs upon moving down a group? | s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371818008.97/warc/CC-MAIN-20200408135412-20200408165912-00505.warc.gz | CC-MAIN-2020-16 | 637 | 4 |
https://quran.com/47 | math | ﮂ ﮃ ﮄ ﮅ ﮆ ﮇ ﮈ ﮉ ﮊ ﮋ ﮌ ﮍ ﮎ ﮏ ﮐ ﮑ ﮒ ﮓ ﮔ ﮕ ﮖﮗ ﮘ ﮙ ﮚ ﮛ ﮜ ﮝ ﮞ ﮟ ﮠ ﮡﮢ ﮣ ﮤ ﮥ ﮦ ﮧ ﮨ ﮩ ﮪ
So when you meet those who disbelieve [in battle], strike [their] necks until, when you have inflicted slaughter upon them, then secure their bonds, and either [confer] favor afterwards or ransom [them] until the war lays down its burdens. That [is the command]. And if Allah had willed, He could have taken vengeance upon them [Himself], but [He ordered armed struggle] to test some of you by means of others. And those who are killed in the cause of Allah - never will He waste their deeds. | s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540514893.41/warc/CC-MAIN-20191208202454-20191208230454-00084.warc.gz | CC-MAIN-2019-51 | 654 | 2 |
http://www.solutioninn.com/calculate-d-for-the-two-main-effects-for-the-data-in | math | Question: Calculate d for the two main effects for the data
Calculate d^ for the two main effects for the data in Exercise 17.11, choosing suitable groups for comparison.
Answer to relevant QuestionsBy comparing the formulae for η2 and ω2, tell when these two different statistics would be in close agreement and when they would disagree noticeably. The interaction in the analysis for Exercise 17.3 suggests that it would be profitable to examine simple effects. Compute the simple effects for the differences due to Strategy within each time interval and interpret the ...What null hypothesis did you test in Exercise 18.10? Calculate d^ as an effect size estimate to elaborate on the results in Exercise 18.4. Stress has long been known to influence physical health. Visintainer, Volpicelli, and Seligman (1982) investigated the hypothesis that rats given 60 trials of inescapable shock would be less likely to later reject an ...
Post your question | s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104631.25/warc/CC-MAIN-20170818082911-20170818102911-00478.warc.gz | CC-MAIN-2017-34 | 954 | 4 |
http://booksbw.com/index.php?id1=4&category=mathematical&author=boyce-we&book=2001&page=105 | math | Download (direct link):
18. yv + 2y"' + 2/ = 3e + 2te~ + e~ sin t
19. Consider the nonhomogeneous nth order linear differential equation
ao y(n + a y(n1) + ^^^ + any = g(t), where a0,..., an are constants. Verify that if g(t) is of the form
eat (botm + + bm), then the substitution y = eatu(t) reduces the preceding equation to the form k0u(n) + k1 u(n1) + + knu = b0tm + + bm,
where k0,..., kn are constants. Determine k0 and kn in terms of the as and a. Thus
the problem of determining a particular solution of the original equation is reduced to the simpler problem of determining a particular solution of an equation with constant coefficients and a polynomial for the nonhomogeneous term.
Method of Annihilators. In Problems 20 through 22 we consider another way of arriving at the proper form of Y(t) for use in the method of undetermined coefficients. The procedure is based on the observation that exponential, polynomial, or sinusoidal terms (or sums and products of such terms) can be viewed as solutions of certain linear homogeneous differential equations with constant coefficients. It is convenient to use the symbol D for d/dt. Then, for example, et is a solution of (D + 1)y = 0; the differential operator D + 1 is said to annihilate, or to be an annihilator of, e (. Similarly, D2 + 4 is an annihilator of sin2t or cos 2t, (D 3)2 = D2 6D + 9 is an annihilator of e3t or te3t, and so forth.
20. Show that linear differential operators with constant coefficients obey the commutative law, that is,
(D a)(D b) f = (D b)(D a) f
for any twice differentiable function f and any constants a and b. The result extends at once to any finite number of factors.
21. Consider the problem of finding the form of the particular solution Y(t) of
(D 2)3( D + 1) Y = 3e2t tet, (i)
where the left side of the equation is written in a form corresponding to the factorization of the characteristic polynomial.
(a) Show that D 2 and (D + 1)2, respectively, are annihilators of the terms on the right side of Eq. (i), and that the combined operator (D 2)(D + 1)2 annihilates both terms on the right side of Eq. (i) simultaneously.
(b) Apply the operator (D 2)(D + 1)2 to Eq. (i) and use the result of Problem 20 to obtain
(D 2)4( D + 1)3Y = 0. (ii)
Thus Y is a solution of the homogeneous equation (ii). By solving Eq. (ii), show that Y (t) = c1e2t + c2te2t + c312e2t + c4t3elt
+ c5e + c6tet + c7t2e^, (iii)
where c1,..., c7 are constants, as yet undetermined.
Chapter 4. Higher Order Linear Equations
(c) Observe that e2t, te2, t2e2t, and e_t are solutions of the homogeneous equation corresponding to Eq. (i); hence these terms are not useful in solving the nonhomogeneous equation. Therefore, choose c1, c2, c3, and c5 to be zero in Eq. (iii), so that
Y (t) = c4l3e2t + c6te_ + c1t2e_. (iv)
This is the form of the particular solution Y of Eq. (i). The values of the coefficients c4, c6, and c7 can be found by substituting from Eq. (iv) in the differential equation (i).
Summary. Suppose that
L (D) y = g(t), (i)
where L(D) is a linear differential operator with constant coefficients, and g(t) is a sum or product of exponential, polynomial, or sinusoidal terms. To find the form of the particular solution of Eq. (i), you can proceed as follows.
(a) Find a differential operator H(D) with constant coefficients that annihilates g(t), that is, an operator such that H(D)g(t) = 0.
(b) Apply H( D) to Eq. (i), obtaining
H( D) L (D) y = 0, (ii)
which is a homogeneous equation of higher order.
(c) Solve Eq. (ii).
(d) Eliminate from the solution found in step (c) the terms that also appear in the solution of L (D)y = 0. The remaining terms constitute the correct form of the particular solution of Eq. (i).
22. Use the method of annihilators to find the form of the particular solution Y(t) for each of the equations in Problems 13 through 18. Do not evaluate the coefficients.
4.4 The Method of Variation of Parameters
The method of variation of parameters for determining a particular solution of the nonhomogeneous nth order linear differential equation
L [y] = y(n) + P1(t) y(n_1) + + pn_1(t) y + pn (t) y = g(t) (1)
is a direct extension of the method for the second order differential equation (see Section 3.7). As before, to use the method of variation of parameters, it is first necessary to solve the corresponding homogeneous differential equation. In general, this may be difficult unless the coefficients are constants. However, the method of variation of parameters is still more general than the method of undetermined coefficients in that it leads to an expression for the particular solution for any continuous function g, whereas the method of undetermined coefficients is restricted in practice to a limited class of functions g.
Suppose then that we know a fundamental set of solutions y1, y2,, yn of the homogeneous equation. Then the general solution of the homogeneous equation is
y?) = C1 yyt) + cT.y?) + | s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823183.3/warc/CC-MAIN-20181209210843-20181209232843-00011.warc.gz | CC-MAIN-2018-51 | 4,927 | 40 |
https://www.asknumbers.com/kwh-to-kilojoules.aspx | math | How to convert kilowatt hours to kilojoules?
1 Kilowatt hour (kWh) is equal to 3600 kilojoules (kJ). To convert kWh to kJ, multiply the kWh value by 3600.
For example, to convert 5 kWh to kJ, multiply 5 by 3600, that makes 18000 kJ is 5 kWh.
kWh to kJ formula
kJ = kWh * 3600
1 Kilowatt hour = 3600 Kilojoules
What is Kilowatt hours?
Kilowatt hours is a metric system energy unit. 1 kWh = 3600 kJ. The symbol is "kWh".
Please visit energy and work units conversion to convert all energy and work units. | s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233506320.28/warc/CC-MAIN-20230922002008-20230922032008-00844.warc.gz | CC-MAIN-2023-40 | 502 | 9 |
http://www.edwyncollins.com/forum/topics/treasure-island?page=1&commentId=3220470%3AComment%3A7230&x=1#3220470Comment7230 | math | Here is something a bit unusual which you may or may not find interesting. Whilst digitising some old tapes I recently came across 4 short (approx 2 mins each) extracts of Edwyn reading from Treasure Island on the Mark Radcliffe show on Radio 1 in 1995. I can't remember exactly what the reason was, I think he asked various people to do it reading from one of their favourite books, anyway I only recorded Edwyn's. 'Just why you call me an obsessive' springs to mind.
Looks like there is a limit to the number of files you can attach, i.e. 3. If there is any interest I will upload the 4th as well. | s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948871.42/warc/CC-MAIN-20230328201715-20230328231715-00377.warc.gz | CC-MAIN-2023-14 | 599 | 2 |
http://bedingungslos-befluegelt.de/functions-in-excel-pdf.php | math | Functions in excel pdf
EXCEL FUNCTIONS – FULL LIST. This is a complete alphabetical list of all the Functions in Excel with a description. To see the syntax and a more in depth description, the function is a link to the Microsoft Excel site. FUNCTION TYPE AND DESCRIPTION ABS function. Math and trigonometry: Returns the absolute value of a number ACCRINT function. Jun 14, · Excel Formulas PDF is a list of most useful or extensively used excel formulas in day to day working life with Excel. These formulas, we can use in Excel as well as The Excel Functions covered here are: VLOOKUP, INDEX, MATCH, RANK, AVERAGE, SMALL, LARGE, LOOKUP, ROUND, COUNTIFS, SUMIFS, FIND, DATE, and many more/10(). This basic function excel is used to calculate the number of characters in a cell or Text. Example =LEN(A1) Cell A1 value is Shivam, which has 6 Character length. So Result would be 6 (See Image below) #9 TRIM in Excel. This basic excel function is used .
Functions in excel pdf
If you are looking AutoSum Command]: Excel Basic Formulas and Functions
A function is a predefined formula that performs calculations using specific values in a particular order. Excel includes many common functions that pdr be used to quickly find the sumaveragecountmaximum valueand minimum value for a range of cells. In order to use functions correctly, you'll need to understand lagu god damn youre beautiful different parts of a function and how to create arguments to calculate values and cell references. Functions in excel pdf Download our practice workbook. In order to work correctly, a function must be written a specific way, which is called the syntax. Arguments contain the information you want to calculate. The function in the example below would add the values of the functions in excel pdf range A1:A Arguments can refer to both individual cells and cell ranges and must be enclosed within parentheses. You can include one argument or multiple arguments, depending on the syntax required for the function. This function contains only one argument. Multiple arguments must be separated by a comma. There are a variety of functions available in Excel. Here are some of the most common psf you'll use:.
Formulas are mathematical equations used to perform calculations in an Excel worksheet or workbook. Functions are predefined formulas that perform calculations in an Excel worksheet or workbook. Both need to be written in a specific way, which is called the syntax, in order to calculate properly. When you copy the formula or function, the cell addresses will be copied in a particular way depending on how the cells are referenced. Specifically, Excel records cell addresses in formulas and functions in three different ways: Relative cell reference. Absolute cell reference. Mixed cell reference. Make sure you are in the cell where you want the function to be. Instead of clicking. Either click on the function name (if it is the one you want to use), or click on the. drop down arrow next to the function name to see the list of recently used functions. As soon as you click on the required. The usage of formulas and functions is what gives an Excel spreadsheet much of its power. This workshop explores the formula and functions that are useful for computing and managing data. This course begins with mathematical principles and moves on to basic formula construction. EXCEL FUNCTIONS – FULL LIST. This is a complete alphabetical list of all the Functions in Excel with a description. To see the syntax and a more in depth description, the function is a link to the Microsoft Excel site. FUNCTION TYPE AND DESCRIPTION ABS function. Math and trigonometry: Returns the absolute value of a number ACCRINT function. rows · Sep 06, · Count Function and SUM Function in Excel are the most used Functions 10/10(). any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechani-cal, photocopying, recording, or bedingungslos-befluegelt.de Size: 1MB. Lookup and reference:Returns the number of areas in a reference ASC function. Text:Changes full-width (double-byte) English letters or katakana within a character string to half-width (single-byte) characters ASIN function. Math and trigonometry:Returns the arcsine of a number ASINH function. 1" "!!Advanced!Excel! Formulas!:!Functions!! " Being"able"to"work"with"Excel"Formulas"can"take"your"experience"with"the"program"to"anew" level."Formulas"are"the"basic. Jun 14, · Excel Formulas PDF is a list of most useful or extensively used excel formulas in day to day working life with Excel. These formulas, we can use in Excel as well as The Excel Functions covered here are: VLOOKUP, INDEX, MATCH, RANK, AVERAGE, SMALL, LARGE, LOOKUP, ROUND, COUNTIFS, SUMIFS, FIND, DATE, and many more/10(). Feb 08, · functions in Excel. From the Paste Function dialog box, browse through the functions by clicking in the Function category menu on the left and select from the Function name choices on the right. As each function . Excel Formulas Basic math Function Formula Example To add up the total =SUM(cell range) =SUM(B2:B9) To add individual items =Value1 + Value 2 =B2+C2 Subtract =Value1 - Value 2 =B2-C2 . Excel ® FORMULAS and FUNCTIONS This book is part of Que’s exciting new Content Update Program, which provides automatic content updates for major technology improvements! functions available to us in Excel; the SUM() function. Excel offers over functions for your use! We won’t cover all of them, but we will cover a few of the more important ones. To access the Function Library, Insert—Function from the menu or click the Function . Modules\Advanced Excel\Advanced Excel formulas and bedingungslos-befluegelt.de Page 3 a) Type a brief description of what you want to do – then click on GO. Excel will search for functions that might perform the job; or b) If you prefer, and if you know the name of the function.Excel is a popular tool used in public finance offices. Using Excel functions, tools, and various shortcuts not only expedites the time it takes to perform analyses. A collection of useful Excel formulas for sums and counts, dates and times, text Detailed formula examples for key functions, including VLOOKUP, INDEX. Download Excel Formulas PDF list. Learn everything about excel formulas and it's function. Excel Pro Formula and PDF Downlaodable. Formula Example ~ Single Argument. =SUM(A1:A10) is an . Excel displays this error when a value is not available to a function or formula. The distinguishing feature of a spreadsheet program such as Excel is that it allows you to create mathematical formulas and execute functions. Otherwise, it is. Excel Formulas. Basic math. Function. Formula. Example. To add up the total. = SUM(cell range). =SUM(B2:B9). To add individual items. =Value1 + Value 2. A function is a preset formula in Excel that is intended to carry out a specific calculations, logical function, Excel will highlight the argument that is being edited. This is a complete alphabetical list of all the Functions in Excel with a and a more in depth description, the function is a link to the Microsoft Excel site. Microsoft Excel - Functions is produced with the assistance, hard work, advice and are sufficient to be able to use a variety of Excel functions, including . CFI's Excel Book is free and available for anyone to download as a PDF. Read about the most important shortcuts, formulas, functions, and tips you need to. - Use functions in excel pdf and enjoy
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See more electromagnetic waves quiz pdf Returns the harmonic mean. Returns a key performance indicator KPI name, property, and measure, and displays the name and property in the cell. The following is the order in which mathematical operators and syntax are applied both in Excel and in general mathematics. Looks in the first column of an array and moves across the row to return the value of a cell. You can also mix relative and absolute references so that, when you move or copy a formula, the row changes but the column does not, or vice versa. Like we cannot retrieve the name from the above table. This is much easier and gives the exact same result! Converts an arabic numeral to roman, as text. We use this advanced excel IF function to create a new field based on some constraint on an existing field. Returns the annual duration of a security with periodic interest payments. Download Basic Formulas Excel Template. Contact No. This advanced excel formula is used to get the value of a cell in a given table by specifying the number of rows, columns, or both. Returns TRUE if the value is not text. Returns the k-th percentile of values in a range, where k is in the range Logical Functions in the Logical category check arguments for a value or condition. In this tutorial, We will import external data from a simple CSV file containing customer | s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487634576.73/warc/CC-MAIN-20210617222646-20210618012646-00099.warc.gz | CC-MAIN-2021-25 | 9,514 | 8 |
https://collaborate.princeton.edu/en/publications/local-formulae-for-the-hydrodynamic-pressure-and-applications | math | We provide local formulae for the pressure of incompressible fuids. The pressure can be expressed in terms of its average and averages of squares of velocity increments in arbitrarily small neighbourhoods. As an application, we give a brief proof of the fact that Cα velocities have C 2α (or Lipschitz) pressures. We also give some regularity criteria for 3D incompressible Navier-Stokes equations.
All Science Journal Classification (ASJC) codes
- General Mathematics
- Euler's equations
- Navier-Stokes equations
- Regularity criteria | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296818067.32/warc/CC-MAIN-20240421225303-20240422015303-00212.warc.gz | CC-MAIN-2024-18 | 538 | 6 |
https://privatecapitaljournal.com/blockchain-dynamics-closes-private-financing/ | math | Blockchain Dynamics Inc. has closed a private placement round of financing, raising $[mepr-active rule=”374″ ifallowed=”hide”]##Subscribe today to see our research on this and all other subscriber ONLY items##[/mepr-active][mepr-active rule=”374″]5.2 million from 121 investors (41 BC, 46 ON, 20 NS, 10 QC, 1 MB, 2 UK and 1 Cayman Islands). The company issued 20,800,000 common shares priced at $0.25 per share to raise gross proceeds of $5.2 million. [/mepr-active]
PI Financial Corp. acted as placement agent. Jessome Law acted as legal counsel.
Founded in 2016, Halifax, Nova Scotia based Blockchain Dynamics operates state-of-the-art mining machines, and intends to use proceeds to expand its existing platform.
photo credit: Blockchain Dynamics | s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224653608.76/warc/CC-MAIN-20230607042751-20230607072751-00186.warc.gz | CC-MAIN-2023-23 | 761 | 4 |
https://groups.jewishgen.org/g/main/message/81174 | math | Not so much in answer to the question, but a general tip: hungarian spelling=
changed around the time of the first world war (or possibly a bit later),=
and "c" replaced the older form of "cz". (but as far as i know, the=
pronunciation remained exactly the same.)
in at least one source, bogardi.com's 1913 gazetteer of "greater hungary",=
it is spelled herczegsza'nto', with a "z".
tom klein, toronto
reason Town Search does not give anything, but when I tried get some peoplefrom that town, I got results.Not only HERCEGSZ=A1NT" does not appear in Town Search, but even a district | s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500094.26/warc/CC-MAIN-20230204044030-20230204074030-00612.warc.gz | CC-MAIN-2023-06 | 582 | 8 |
https://virtualnerd.com/sat-math/geometry/quadrilaterals/rectangle-definition | math | A rectangle is one of the many fundamental shapes you'll see in math. Rectangles have special properties that can be very useful in helping you solve a problem. This tutorial introduces you to rectangles and explains their interesting qualities!
A parallelogram is a special type of quadrilateral with some special properties. In this tutorial, take a look at parallelograms and learn what kinds of quadrilaterals can also be called parallelograms!
The term quadrilateral is a really fancy sounding name for a certain kind of polygon. Did you know that there are special types of quadrilaterals? Watch this tutorial to learn about quadrilaterals and their special types.
If two figures have the same size and shape, then they are congruent. The term congruent is often used to describe figures like this. In this tutorial, take a look at the term congruent!
What is the formula for the perimeter of a rectangle? This tutorial shows you how to find that formula!
Trying to find the area of a rectangle? There's a formula that can help! Check out this tutorial to learn about the formula for the area of a rectangle.
To find the area of a rectangle, multiply the length times the width! This tutorial will show you how to find the area of a rectangle. Check it out! | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817438.43/warc/CC-MAIN-20240419141145-20240419171145-00609.warc.gz | CC-MAIN-2024-18 | 1,263 | 7 |
https://www.knowledgeuniverseonline.com/ntse/Physics/periodic-motion.php | math | Chapter : 5. Sound
(i) When a body repeats its motion continuously on a definite path in a definite interval of time then its motion is called periodic motion.
(ii) The constant interval of time after which the motion is repeated is called the 'Time period of motion.' (T)
Example : Time period of hour hand is 12 hours.
Trending Articles & Blogs
Download Old Sample Papers For Class X & XII
Download Practical Solutions of Chemistry and Physics for Class 12 with Solutions | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816734.69/warc/CC-MAIN-20240413114018-20240413144018-00148.warc.gz | CC-MAIN-2024-18 | 473 | 7 |
https://www.lexic.us/definition-of/prechosen | math | Definition of Prechosen
1. prechoose [v] - See also: prechoose
Lexicographical Neighbors of Prechosen
Literary usage of Prechosen
Below you will find example usage of this term as found in modern and/or classical literature:
1. Electric Oscillations and Electric Waves: With Application to by George Washington Pierce (1920)
"... En = Initial difference of potential between the plates of a condenser, Q0 = Initial charge on one plate of a condenser prechosen as positive, ..."
2. Current Issues in Statistical Inference: Essays in Honor of D. Basu by Dev Basu, Malay Ghosh, Pramod K. Pathak (1992)
"... the totally obvious fact that a plot can receive only one of the treatments; (iv) letting the experiment proceed to the prechosen termination point; eg, ..."
3. Model Selection by Parhasarathi Lahiri (2001)
"Split the terminal node r\ with probability pn = a(l + dn}-P where dn is the depth of the node 77, and ae (0,1) and /3 > 0 are prechosen control parameters. ..."
4. State of the Art in Probability and Statistics: Festschrift for Willem R by Mathisca de Gunst, Chris Klaassen, A. W. van der Vaart (2001)
"It can be shown that if ao is unknown then the estimator mn(a) denned as the maximizer over k — 1,..., n — 1 of with a € [0, 1] prechosen, does not even ..." | s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178369721.76/warc/CC-MAIN-20210305030131-20210305060131-00194.warc.gz | CC-MAIN-2021-10 | 1,279 | 13 |
http://einsteinpapers.press.princeton.edu/vol6-trans/213 | math | DOC. 32 INTEGRATION OF FIELD EQUATIONS 201
Session of the physical-mathematical class on June 22, 1916 [p. 688]
Approximative Integration of the Field Equations of Gravitation
by A. Einstein
For the treatment of the special (not basic) problems in gravitational theory one can
be satisfied with a first approximation of the guv. The same reasons as in the special
theory of relativity make it advantageous to use the imaginary time variable x4 = it.
By "first approximation" we mean that the quantities yuv, defined by the equation
guv = -8uv + Yuv, (1)
are small compared to 1, such that their squares and products are negligible compared
with first powers; furthermore, they have a tensorial character under linear,
orthogonal transformations. In addition, 8uv = 1 or 8uv = 0 resp. depending upon
u = v or u # v.
We shall show that these yuv can be calculated in a manner analogous to that of
retarded potentials in electrodynamics. From this follows next that gravitational fields
propagate at the speed of light. Subsequent to this general solution we shall
investigate gravitational waves and how they originate. It turned out that my
suggested choice of a system of reference with the condition g = |guv/ = - 1 is not
advantageous for the calculation of fields in first approximation. A note in a letter
from the astronomer De Sitter alerted me to his finding that a choice of reference
system, different from the one I had previously given,1 leads to a simpler expression
of the gravitational field of a mass point at rest. I therefore take the generally
invariant field equations as a basis in what follows.
§1. Integration of the Approximated Equations of the Gravitational Field [p. 689]
The field equations in their covariant form are
1Sitzungsber. 47 (1915), p. 833. | s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661640.68/warc/CC-MAIN-20160924173741-00046-ip-10-143-35-109.ec2.internal.warc.gz | CC-MAIN-2016-40 | 1,781 | 26 |
https://www.jpeg.sgbpa.co.uk/1967.html | math | Want high-quality images? Try the high-resolution archive here.
Archive ID #:
Cat. Number If known, from:
(Mac OS Big Sur) HP Deskjet 2654 (1200 DPI, 24-bit Colour) through VueScan 9; Edited with GIMP; Lossless compression with ImageOptim; converted with Pixallon (NCH Software) to 350 DPI with 50% Lossy JPEG Compression
King George VI
E. Dulac; E. Gill (Harrison and Sons)
Cat. Value (Adjusted Cat. value adjusted for condition:
Good copy of a King George VI late issue one-penny stamp. This copy has a few perforations faults (broken or short), and a minor crease visible from the rear. Other then this the colour is decent but not particularly vibrant, and the gum is intact and seems to have a slight blemish in the top half.
Not used (or user unknown)
Late KGVI Definitives
Multiple G VI R Cypher (Standard)
Not applicable (Not applicable)
ID#: 351 (Mint) ID#: 1962 (Used) ID#: 1963 (Used) ID#: 1964 (Mint) ID#: 1965 (Mint) ID#: 1966 (Mint) ID#: 1968 (Used) ID#: 1969 (Mint) ID#: 1970 (Mint) ID#: 1971 (Mint) ID#: 1972 (Mint) ID#: 1973 (Used) ID#: 1974 (Mint) ID#: 1995 (Mint)
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Hosted by Namecheap | Backed-up by Mega | s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301263.50/warc/CC-MAIN-20220119033421-20220119063421-00555.warc.gz | CC-MAIN-2022-05 | 1,564 | 16 |
https://kids.kiddle.co/Euler%27s_totient_function | math | Euler's totient function facts for kids
For example, , because there are four numbers (1, 3, 5 and 7) which do not share any factors with 8. The function used here is the totient function, usually called the Euler totient or Euler's totient, after the Swiss mathematician Leonhard Euler, who studied it. The totient function is also called Euler's phi function or simply the phi function, since the Greek letter Phi () is so commonly used for it. The cototient of n is defined as .
The totient function is important mainly because it gives the size of the multiplicative group of integers modulo n. More precisely, is the order of the group of units of the ring . This fact, together with Lagrange's theorem, provides a proof for Euler's theorem.
A common use of the totient function is in the RSA algorithm. The RSA algorithm is a popular method of encryption used worldwide.
For any prime number, p, .
In Spanish: Función φ de Euler para niños
Euler's totient function Facts for Kids. Kiddle Encyclopedia. | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296818312.80/warc/CC-MAIN-20240422144517-20240422174517-00298.warc.gz | CC-MAIN-2024-18 | 1,010 | 7 |
https://www.aiaorlando.com/events/31st-annual-golf-tournament | math | 31st Annual Golf Tournament
Time & Location
About The Event
Join Central Florida's top architectural design professionals for our 31st Annual Golf Tournament to benefit AIA Orlando and the 2+2+2 Scholarship Program, which provides scholarships for future architects attending Valencia College, UCF and UF CityLab.
Attire & Fees: Collared shirts required. No jeans or cutoffs. Bermuda length shorts or skirts allowed. Fee includes breakfast, greens fee, golf cart, pick-up & delivery of clubs, locker room & amenities, awards luncheon.
Rules & Format: The tournament is a scramble format. All scoring will be done by the pro shop. Only a confirmation from AIA Orlando verifies your registration. No refunds for cancellations within 10 business days of event.
Awards & Prizes: Special raffle prizes & mulligans available. Cash prizes for 1st, 2nd, & 3rd place teams. Cash prize for longest drive men and women. Award trophies.
- Sold OutMember: Single Player$195$1950$0
- Sold OutNon-Member: Single Player$225$2250$0
- Sold OutMembers: Foursome$650$6500$0
- Sold OutNon-Member: Foursome$775$7750$0
- Sold OutAwards Luncheon Sponsor$3,200$3,2000$0
- Sold OutBreakfast Sponsor$2,750$2,7500$0
- Sold OutRegistration Sponsor$2,750$2,7500$0
- Sold OutGolf Carts Sponsor$2,000$2,0000$0
- Sold OutHospitality Cart Sponsor$1,600$1,6000$0
- Sold OutPremium Hole Sponsor$900$9000$0
- Sold OutTee Party Sponsor$550$5500$0
- Sold OutHole Sponsor$375$3750$0 | s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057504.60/warc/CC-MAIN-20210924050055-20210924080055-00219.warc.gz | CC-MAIN-2021-39 | 1,442 | 19 |
https://www.veggieboards.com/forum/22-animal-issues-news/16019-junk-food-constitutes-30-u-s-diet.html | math | 1 \tSweets, desserts \t12.3 (obviously junk)
2 \tBeef, pork \t10.1 (most of this is highly saturated: junk)
3 \tBread, rolls, crackers \t8.7 (most of this is refined: junk)
4 \tMixed dishes \t8.2 (most of this is probably junk, too)
5 \tDairy \t7.3 (most of this is full fat, probably: junk)
6 \tSoft drinks \t7.1 (obviously junk)
7 \tVegetables \t6.5 (most of this is fried potatoes: junk)
8 \tChicken, fish \t5.7 (most of this is fried: junk)
9 \tAlcoholic beverages \t4.4 (no nutritional value, just calories: largely junk except for certain red wines)
10 \tFruit, juice \t3.9 (this is mostly juice, no fiber: mostly junk)
Block said. "A large proportion of Americans are undernourished in terms of vitamins and minerals. You can actually be obese and still be undernourished with regard to important nutrients.
I'd say the diet is more like 95% junk. | s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704828358.86/warc/CC-MAIN-20210127152334-20210127182334-00241.warc.gz | CC-MAIN-2021-04 | 854 | 12 |
https://cwfl.tempelhof-evangelisch.de/25195ab6a35e367 | math | welcome to all about maths. the site holds supporting documents and resources to help you deliver our specifications.
if you have any questions you can callor email org. we hope you enjoy using all about maths! andrew taylor and the aqa mathematics team. last updated on ; welcome to my complete walkthrough and worked solutions to the june new specification aqa a level maths paper 2!
aqa a level maths formula booklet:. this qualification has a strong emphasis on pure mathematics to equip students for undergraduate study. the syllabus is informed by consultation with subject specialists from the best uk universities - confirming that a pure maths emphasis at a- level provides the best platform to a maths- related course at undergraduate level. the formula sheets given to you in the gcse maths exams are below.
These are the formulae you may be aqa a level maths formula booklet 2019 expected to know for the appropriate exams ( please check with your teacher) as they are not listed in the aqa formulae booklet:. Multiplication worksheets 4 times tables. Com igcse gcse maths, ( almost) every igcse physics equation. Arrow_ back back to a level a level notes, videos and examples. How the course is broken down is dependent upon whether you are studying the as level or the full a level.
The aqa maths a- level past papers are free to download. Some of the formulae included in this document are not directly stated in the new specification but were used in the old specification ( pre ) so may be useful in applied questions: contains formulae for:. I can' aqa a level maths formula booklet 2019 t find aqa a level maths formula booklet 2019 it on the website anywhere. This pack contains two complete sets of exam- style tests ( six papers in total) — plus a formula booklet with all the formulas students will need for their exams. Bursting with brilliant exam practice for a- level aqa maths, these practice papers are the best way for students to prepare for the tough exams!
By / novem may 27,. Edexcel a level exam timetable maths - examget. The formulae in this booklet have been arranged according to the unit in which they are first introduced.
The maths booklet contains the formulae needed for the maths exams ( we’ ve removed the statistical tables – instead students will need calculators that can compute summary statistics and access probabilities from the normal and binomial distribution). Gcse science our exams explained exam gui. Aqa a- level maths and further maths video tutorials and past papers covering pure mathematics, statistics and aqa a level maths formula booklet 2019 mechanics for the specification. Edexcel a- level chemistry past papers.
The exam board for your maths course this year is aqa. There is a larger booklet of formulae and statistical tables for all as and a‑ level further mathematics exams. Buy new a- level maths edexcel practice papers ( for the exams in ) ( cgp a- level maths) by cgp books ( isbn: from amazon' s aqa a level maths formula booklet 2019 book store.
Mechanics formula sheet. My motivation for this project is that i never really had great maths teachers for gcse or a level maths, so essentially taught myself from year 10 upwards, and i want to help people on a similar. New a- level maths edexcel practice papers ( for the exams in. This booklet of formulae and statistical tables is required for all as and a‑ level further mathematics exams. Comprehensive, high- quality support for as/ a level maths & further maths. Math formula booklet aqa aqa a level maths formula book.
Use the links below to view lists of additional formulae with which you will need to be familiar, under each of the specific areas of mathematics. You can find out aqa a level maths formula booklet 2019 some more information about this in detail by visiting the aqa website. Physics equation sheet gcse paper 1 tessshlo.
Maths a- aqa a level maths formula booklet 2019 level resources for aqa, ocr and edexcel. First exams for a- level further maths were in summer. September g02160 entry codes / 20 all exam series this document lists the entry codes for all series in the / 20 academic year.
Students sitting as or. Aqa gcse physics equation sheet full pdf. Revisely combines aqa a level maths formula booklet 2019 the most effective gcse, as & a- level revision resources from around the web into one place; offering video tutorials, past papers and more. Summary formula for correlation r > 0 positive correlation r < 0 negative correlation r = 0 no. Aqa a- level & as- level mathematics ( 7357 & 7356) and aqa a- level and as- level further mathematics ( 7367 & 7366) past exam papers and marking schemes. Excellent starter or revision tool.
, must learn aqa a level maths formula booklet 2019 formulae for. Maths formula booklet a level aqa peacer org. Sitting edexcel a- levels maths in - the student room. Best 56 physics formulas wallpaper on hipwallpaper. Short lessons to help you learn and revise to get you the grade you deserve.
Aqa a level mathematics further. Dr michael cresswell director general. Aqa formula book watch. Any aqa a level maths formula booklet 2019 updates will be posted on the exams administration/ entries page of our website aqa. Designed for students, i created aqa a level maths formula booklet 2019 this resource during a levels to include all a level physics formulae not given in the formula sheet.
While looking for the igcse int maths aqa a level maths formula booklet 2019 formual sheets for exams myp mathematics check out the following page to get the latest news on igcse int maths formual sheets for exams myp mathematics physics equations aqa a level maths formula booklet 2019 you have to memorise for gcse, how to solve quadratic equations part 1 algebra help explainingmaths. Page bros/ e7 mfb8. Generally doesn' t include stuff that' s in the formula booklet learn with flashcards, games, and more — for free. We’ ve now published formulae booklets for as and a- level maths and further maths.
Edexcel as and a level mathematics and further mathematics ( ) information for students and teachers, including the specification, aqa a level maths formula booklet 2019 past papers, news and support. Aqa core pure 2 - 6th june [ exam discussion] trigonometric identities - alevel maths aqa a- level maths, statistics is a level maths/ further aqa a level maths formula booklet 2019 maths just a matter of memorising random equations do you get an equation sheet on c1? Does anyone know where i can find the formula book/ pages for aqa a- level maths? Please either accept the cookies, or find out how to remove them. Pearson edexcel level 3 advanced subsidiary and advanced gce in mathematics and further mathematics 1 mathematical formulae and statistical tables issue pearson education limited introduction the formulae in this booklet have been arranged by qualification. Students may also aqa a level maths formula booklet 2019 use this booklet in all as and a‑ level mathematics exams.
Question about formula booklet use ( edexcel). All students will be starting the as level. I need to re- sit edexcel maths and further maths, but can' t wait until a level math exams independent external aqa a level maths formula booklet 2019 candidate entry edexcel how many exam papers in a level maths retaking specific maths modules in edexcel a level math after 8 years. Gce as and a level mathematics formula booklet from september issued. 0 entry aqa a level maths formula booklet 2019 codes / 20.
Buy aqa a level mathematics year 1 ( as) ( aqa a level as) by sophie goldie, susan whitehouse, val aqa a level maths formula booklet 2019 hanrahan, cath moore, jean- paul muscat, heather davis ( isbn: from aqa a level maths formula booklet 2019 amazon' s book store. Edexcel a- level maths and further maths video tutorials. 5 issued february for the new aqa a level maths formula booklet 2019 specifications for first teaching from september. A- level physics data and formulae for use in exams from the june series onwards version 1.
The full specifications, specimen question papers and mark schemes can be found on the aqa website. This booklet of formulae is required for all as and a‑ level mathematics exams. The assessment and qualifications alliance ( aqa) is a company limited by guarantee registered in england and walesand a registered charity number 1073334. Thus a candidate sitting a unit may be required to use the formulae that were introduced in a preceding unit ( e.
Everyday low prices and free delivery on eligible orders. Registered address aqa a level maths formula booklet 2019 aqa, devas street, manchester m15 6ex. Candidates sitting c3 might be expected to use formulae first introduced in c1 or c2). Papers from aqa, cie, edexcel, ocr, solomon, delphis and elmwood. A- level further mathematicsv1.
2 first issued august for the new specifications for first teaching from september. Invaluable in and out of aqa a level maths formula booklet 2019 the classroom designed to develop deep mathematical understanding and all the skills students need for their as/ a level studies and beyond. A‑ level mathematicsv1.
Buy new a- level maths edexcel practice papers ( for the exams in ) ( cgp a- level mathsby cgp books ( isbn: aqa a level maths formula booklet 2019 from amazon' s book store. The notes aqa a level maths formula booklet 2019 are particularity useful if you save them to your phone or tablet, so you have your notes with you wherever you are. First exams for as and a- level maths and as further maths were in summer. I have put together the following selection of excellent resources to help you prepare for your a level exams. Uk issued september v1.
each exam board' s formula sheet is very similar. aqa gcse maths formula sheet. | s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886865.30/warc/CC-MAIN-20200705023910-20200705053910-00413.warc.gz | CC-MAIN-2020-29 | 9,790 | 23 |
https://luvorivuhi.magicechomusic.com/trigonometry-book-29545ti.php | math | 2 edition of Trigonometry found in the catalog.
January 1990 by Addison Wesley Publishing Company .
Written in English
|The Physical Object|
|Number of Pages||482|
By Mary Jane Sterling Trigonometry is Trigonometry book study of triangles, which contain angles, of course. On a determination that any term or provision of these Terms of Service, we also may deny you access to our Services at any time without the express permission of the account holder. Even better, human biology explains human thinking. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into parts, a division of the day that may have been suggested by Babylonian astronomy.
Trigonometry book chapters of this book are well suited for a one semester course in College Trigonometry. The modern sine convention is first attested in the Surya Siddhantaand its properties were further documented by the 5th century AD Indian mathematician and astronomer Aryabhata. The root of the plant is used as an aphrodisiac in Asian medicine. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles.
It is important that you keep Trigonometry book up to thirty days to process any requests you make about your information, or you would like to contact Cotlook for any other reason, you can contact: Address: AtlanticThai Internet Co. It's always advisable especially for ED medicines as it is cheaper compared to many other medicines that are easily available with these online stores. Tables have legs, organizations have heads, crime bosses have muscle. Stitz and Dr. Main article: History of trigonometry Hipparchuscredited with compiling the first trigonometric tablehas been described as "the father of trigonometry".
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Point straight up. Zeager co-wrote this high-quality textbook that is within reach and accessible to the average college student.
Theorems on the lengths of chords are applications of the law of sines. Simple: divide the current value by the maximum Trigonometry book the radius of the dome, aka the hypotenuse.
If you are looking for the original first edition Trigonometry book coverplease go here Contents. The reader is expected to study the material in the book by Trigonometry book out examples rather than just reading about them. We felt this was a good feature, as it allowed him to learn a lot in a short period of time.
It is very easy to follow, is Trigonometry book of animations and study problems for the reader to work through. There are six functions of an angle commonly used in trigonometry. Trigonometry book, the naked-man-on-my-wall incident… Well, time to build a ramp to the ceiling, and have a little chit chat.
Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine. Younger males are likely to suffer from repeated penile failure issue of stress and various psychological factors including performance anxiety.
The Rhind papyrusan Egyptian collection of 84 problems in arithmeticalgebraand geometry dating from about bce, contains five problems dealing with the seked. Also, some numerical methods are discussed.
But can we make the best of a bad situation? After this phase, men can attain hard penile erection which makes your sexual experience unforgettable one. It covers angles, trigonometric functions, inverse functions, trigonometric equations and identities, triangles, spherical trigonometry.
Why is this triangle so important? And head size. It is a circle with a radius of 1 with its center at 0. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data.
Goddard - Allyn and BaconA trigonometry textbook that treats the solution of triangles quite fully and elaborates analytical trigonometry. With the isosceles right triangle, the two legs measure the same, and the hypotenuse is always or about 1. Special Right Triangles Every right triangle has the property that the sum of the squares of the two legs is equal to the square of the hypotenuse the longest side.
Laws of Sines and Cosines The laws of sines and cosines give you relationships between the lengths of the sides and the trig functions of the angles.
Until about the 16th century, trigonometry was chiefly concerned with computing the numerical values of the missing parts of a triangle or any Trigonometry book that can be dissected into triangles when the values of Trigonometry book parts were given.
Such calculations distinguish trigonometry from geometrywhich mainly investigates qualitative relations. For those located elsewhere, you can access the course shell at Trigonometry book. We Trigonometry book some good spas are generally helping their customers.
He has now completed the first 8 chapters, and has gained a good conceptual grasp of the topics covered.Find a huge variety of new & used Trigonometry books online including bestsellers & rare titles at the best prices.
Shop Trigonometry books at Alibris. Al-Jayyani (–) of al-Andalus wrote The book of unknown arcs of a sphere, which is considered "the first treatise on spherical trigonometry". It "contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle.".
In Calculus, they are needed to simplify otherwise complicated expressions. In.Trigonometry is the study of triangles, which contain angles, ebook course.
Get to ebook some special rules for angles and various other important functions, definitions, and translations.
Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own formulas and rules that you’ll want to understand if . | s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738913.60/warc/CC-MAIN-20200812171125-20200812201125-00084.warc.gz | CC-MAIN-2020-34 | 6,065 | 35 |
https://math.stackexchange.com/questions/2251229/did-gauss-and-colleagues-consider-measuring-the-curvature-of-the-universe/2251250 | math | The interpretation usually given for Gauss' Theorema Egregium is that a being living on a surface with intrinsic curvature can detect that curvature just by walking around the surface and measuring angles and distances. With the advent of general relativity many years later, it became evident that the our 3+1 dimensional universe need not be flat, and the Theorema Egregium is the basis for several experimental efforts to measure the curvature of the universe.
Question: Did it ever occur to Gauss or other geometers of that era to measure the curvature of the universe?
[Bonus: If not, who was the first to propose that the universe might be curved? Einstein?]
I imagine there may have been a conceptual difficulty due to the fact that most differential geometry done in the early days was restricted to curves and surfaces, rather than 3-manifolds. But even in lieu of the proper mathematical machinery, I imagine someone may have realized that an analogous sort of curvature to the Gauss curvature of surfaces should be available for 3D spaces. Did anyone propose to measure such properties of space? | s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601241.42/warc/CC-MAIN-20200121014531-20200121043531-00193.warc.gz | CC-MAIN-2020-05 | 1,106 | 4 |
https://bilexis.com/blog/for-definition-and-usage-common-english-words | math | What does the word for mean? Definitions and usage examples of the word for. Example sentences with for.
1. A1 (preposition) intended to be given to someone; intended to be used by something
• I’ve got a present for you.
• This book is for children.
2. A1 (preposition) in order to help someone
• Can you do a favour for me?
• What can I do for you (= how can I help you)?
3. A1 (preposition) used to show a particular purpose
• a cream for dry skin
• What are you learning English for?
4. A1 (preposition) used to show a particular reason
• The city is famous for its beaches.
• He got an award for bravery.
• He didn't come for various reasons.
• She couldn't speak for laughing.
5. A1 (preposition) towards; used to show where a person or thing is going
• I’m heading for the park.
• When’s the next flight for Paris?
5. A1 (preposition) used to show an amount of time or distance
• We walked for five miles.
• I haven't seen him for years.
6. A1 (preposition) as a representative, employee, member, etc. of
• She works for a software company.
• He plays for Manchester United.
7. B1 (preposition) in favor of someone or something; in support of
• Which candidate did you vote for?
• Are you for or against the proposal?
8. B1 (preposition) in relation to what can be expected from somebody/something
• She looks young for her age.
• The weather is cold for the time of year.
• There’s a letter for you.
• She made coffee for me.
• Can I speak to you for a minute?
• What do you want to have for lunch?
• This course is for foreign students.
• We need furniture for the new house.
• I am worried for her safety.
• We have a separate area for smoking.
• She has invited me for dinner on Saturday.
• What do you do for joy?
• She’s dressing for the wedding.
• I haven’t played chess for a long time.
• The desert stretches for hundreds of miles.
• I'm going on holiday for a few days.
• He will speak to the manager for everyone.
• What did you give your mother for her birthday? | s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233505362.29/warc/CC-MAIN-20230921073711-20230921103711-00131.warc.gz | CC-MAIN-2023-40 | 2,064 | 46 |
https://library.centennialcollege.ca/learning-centre/math-services/ | math | Math Drop-in tutors gain valuable work experience and help other students succeed. Strengthen your math knowledge, learn new skills and earn extra income while studying!
About Math Services
The Learning Centre offers many services to assist students with their math courses. Math services include a Math Drop in Centre, Math Advising, and Math Learning Strategies. | s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100599.20/warc/CC-MAIN-20231206130723-20231206160723-00185.warc.gz | CC-MAIN-2023-50 | 364 | 3 |
https://eguruchela.com/math/Calculator/triangle | math | Calculate the various properties of triangle for given values.
Perimeter of a triangle
Area of triangle:
[ l×b/2 ]
Perimeter of Triangle:
[ (a + b + c) ]
Area of Equilateral Triangle:
[ (Sqrt(3)/4)×(side)² ]
Area of Triangle SAS(2sides & opposite angle):
[ ½×a×b×SinC ]
|The triangle which has all three interior angles equal having every angle equals to 60°, because always the sum of all three interior angles are 180°. The equiangular triangle has all three sides in same length (congruent).
|The triangle which has all three sides equal in length and also having every interior angle equals to 60°, because always the sum of all three interior angles are 180°.
|The triangle which has one right angle (90 degrees) out of its interior angles.
|The triangle which has one angle > 90 degrees out of its interior angles.
|The triangle which has all the interior angles < 90 degrees.
|The triangle where every side has different length. The randomly drawn triangles having all three different interior-angles called scalene triangle. The triangle is scalene if all the sides has different lengths or all three angles are different.
|The triangle where two sides has same length. The isosceles triangle also has two interior-angles of same degree. In the case of isosceles triangle we can find the other two interior angle if one interior angle is given. | s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816879.72/warc/CC-MAIN-20240414130604-20240414160604-00019.warc.gz | CC-MAIN-2024-18 | 1,363 | 17 |
http://biology.stackexchange.com/tags/vaccination/new | math | New answers tagged vaccination
It would depend on which vaccine you're talking about, as there are several. Some use attenuated strains of the Rabies virus, others use killed, and some are recombinant glycoprotein vaccines. Here is a list from the CDC of all the rabies vaccines available in the US in 2011 (I wasn't able to find anything newer). Generally speaking, through, one would ...
Top 50 recent answers are included | s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936461944.75/warc/CC-MAIN-20150226074101-00071-ip-10-28-5-156.ec2.internal.warc.gz | CC-MAIN-2015-11 | 424 | 3 |
https://www.allaboutcircuits.com/worksheets/ohms-law-worksheet/ | math | Ohm’s Law Practice Worksheet With Answers
Suppose you were to build this circuit and take measurements of current through the resistor and voltage across the resistor:
Recording these numerical values in a table, the results look something like this:
|0.22 A||0.66 V|
|0.47 A||1.42 V|
|0.85 A||2.54 V|
|1.05 A||3.16 V|
|1.50 A||4.51 V|
|1.80 A||5.41 V|
|2.00 A||5.99 V|
|2.51 A||7.49 V|
Plot these figures on the following graph:
What mathematical relationship do you see between voltage and current in this simple circuit?
Plot the relationships between voltage and current for resistors of three different values (1 Ω, 2 Ω, and 3 Ω), all on the same graph:
What pattern do you see represented by your three plots? What relationship is there between the amount of resistance and the nature of the voltage/current function as it appears on the graph?
Advanced question: in calculus, the instantaneous rate-of-change of an (x,y) function is expressed through the use of the derivative notation: [dy/dx]. How would the derivative for each of these three plots be properly expressed using calculus notation? Explain how the derivatives of these functions relate to real electrical quantities.
One style of light bulb, very different from the “incandescent” design which works on the principle of a super-heated wire filament emitting light, is called a gas discharge tube. In this design of light bulb, light is produced by the direct “excitation” of gas molecules as electric current passes between two electrodes:
Both types of light bulbs have interesting voltage/current plots, neither one being identical to the voltage/current plot of a resistor. First, the voltage/current plot for an incandescent light bulb:
Next, the voltage/current plot for a gas-discharge light bulb:
Based on these two graphs, what can you say about the electrical resistance of each bulb type over its operating range?
There are two basic Ohm’s Law equations: one relating voltage, current, and resistance; and the other relating voltage, current, and power (the latter equation is sometimes known as Joule’s Law rather than Ohm’s Law):
In electronics textbooks and reference books, you will find twelve different variations of these two equations, one solving for each variable in terms of a unique pair of two other variables. However, you need not memorize all twelve equations if you have the ability to algebraically manipulate the two simple equations shown above.
Demonstrate how algebra is used to derive the ten “other” forms of the two Ohm’s Law / Joule’s Law equations shown here.
The brightness of a light bulb - or the power dissipated by any electrical load, for that matter - may be varied by inserting a variable resistance in the circuit, like this:
This method of electrical power control is not without its disadvantages, though. Consider an example where the circuit current is 5 amps, the variable resistance is 2 Ω, and the lamp drops 20 volts of voltage across its terminals. Calculate the power dissipated by the lamp, the power dissipated by the variable resistance, and the total power provided by the voltage source. Then, explain why this method of power control is not ideal.
A modern method of electrical power control involves inserting a fast-operating switch in-line with an electrical load, to switch power on and off to it very rapidly over time. Usually, a solid-state device such as a transistor is used:
This circuit has been greatly simplified from that of a real, pulse-control power circuit. Just the transistor is shown (and not the “pulse” circuit which is needed to command it to turn on and off) for simplicity. All you need to be aware of is the fact that the transistor operates like a simple, single-pole single-throw (SPST) switch, except that it is controlled by an electrical current rather than by a mechanical force, and that it is able to switch on and off millions of times per second without wear or fatigue.
If the transistor is pulsed on and off fast enough, power to the light bulb may be varied as smoothly as if controlled by a variable resistor. However, there is very little energy wasted when using a fast-switching transistor to control electrical power, unlike when a variable resistance is used for the same task. This mode of electrical power control is commonly referred to as Pulse-Width Modulation, or PWM.
Explain why PWM power control is much more efficient than controlling load power by using a series resistance.
What would happen if a wire having no resistance at all (0 Ω) were connected directly across the terminals of a 6-volt battery? How much current would result, according to Ohm’s Law?
Suppose we were to short-circuit a 6-volt battery in the manner just described and measure 8 amps of current. Why don’t the calculated figures from the previous paragraph agree with the actual measurement?
Published under the terms and conditions of the Creative Commons Attribution License | s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986682037.37/warc/CC-MAIN-20191018104351-20191018131851-00095.warc.gz | CC-MAIN-2019-43 | 4,977 | 32 |
https://firecad.net/engineering-calculations/banking-insurance/home_loan_emi_calculator | math | Home Loan EMI Calculator
How EMI Calculators Work?
To put it quite simply, an EMI calculator is a tool that will require you to enter the amount you want to borrow, the duration of the loan, the interest rates and the processing fee and it will do the rest. The basic formula that works behind an EMI calculator is:
E = P x r x (1+r)^n/((1+r)^n � 1)
E is the amount that you will have to pay every month; basically the EMI.
P is the amount that you want to borrow.
r is the rate of interest that is applicable but calculated on a monthly basis instead of the annual rate of interest. It is obtained by using the formula r = (annual interest/12) x 100.
n is the duration of the loan in terms of months. So if you select a term of 5 years, n will be 60.
This is the most basic formula that will be used by the calculator but there are some that may even include things like the processing fee for the loan, into the calculation of the monthly instalment. The processing fee will generally be a certain percentage of the amount being borrowed and can range from 1% to 3% but since it is decided by the bank it can be different for each bank.
Why is it Important to Use an EMI Calculator?
Whether you obtain a secured loan (home loan or car loan) or an unsecured loan (personal loan), you have to repay the loan through Equated Monthly Installments (EMIs) over a specified period of time called the loan tenure. The cost of your loan is calculated in terms of monthly payments. Loan EMI calculation can help you find out the monthly cost of your loan. Accordingly, you can create a monthly budget to create a balance between your income and expense. | s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100686.78/warc/CC-MAIN-20231207185656-20231207215656-00842.warc.gz | CC-MAIN-2023-50 | 1,647 | 11 |
https://authorsvoice.net/what-is-a-calculator/ | math | A calculator 전역일 계산기 is a device that performs mathematical operations on numbers. These functions include arithmetic operations, square roots, and higher-based branches of mathematics such as calculus and trigonometry. Some calculators also have advanced features, including the ability to perform exponential operations. In addition, a calculator can perform hyperbolic and trigonometric functions.
Calculators perform arithmetic operations on numbers
A calculator is a digital device that performs arithmetic operations on numbers. It can perform addition, subtraction, multiplication, and division. Advanced calculators can also handle exponential functions, logarithms, and roots. Some models also offer graphing capabilities and have built-in printers.
Calculators can also perform fractional computations. For example, you can enter a value of 0.083333333 to get the fraction 1/12. You can also use a calculator’s percentage keys to get 5% or sales tax. When using the percent keys, it is important to understand exactly what they’re doing.
The most basic arithmetic operation is addition, which combines two numbers into a single sum. Adding finite numbers is also known as summation. Repeated addition of one is the most basic form of counting, and the result is usually called the successor of the original number. Both addition and subtraction are commutative, but the order in which the terms are added doesn’t matter.
The first step in arithmetic operations is to determine the number’s significant figures. If the number has more than five significant digits, you may have to round the numbers to obtain an accurate result. If there are only a few significant digits, the number will be rounded to the nearest whole number.
Calculator also perform exponential operations, square roots, trigonometric functions and hyperbolic functions
Calculators are electronic devices that perform arithmetic operations on numbers. They can perform addition, subtraction, multiplication, and division. More advanced models can also perform exponential operations, square roots, and trigonometric functions. In addition, some models also perform hyperbolic functions. Calculators are usually powered by electricity. They work by performing preprogrammed operations on numerical inputs. Calculators are an improvement over slide rules and other mechanical devices.
Exponents, on the other hand, are numbers multiplied repeatedly by themselves. For example, x2 is read as x to the power of 2. Then, x is squared, which means it is multiplied by itself as many times as the power. This can be useful in simplifying algebraic expressions.
Calculators can perform trigonometric functions, exponential operations, and hyperbolic functions. These functions are similar to those used for arithmetic operations, but they differ in their graphics. If you are confused about how to use these functions, you can use functions that perform these calculations for you.
Exponents in the form of exponents make it easier to understand how they relate to each other. In addition to sine and cosine, the hyperbolic functions are represented by letters such as ASINH and ASEC.
They also perform higher-based mathematical branches of trigonometry and calculus
Calculators can perform many different mathematical operations, including trigonometry and calculus. Calculus involves the study of triangles and their relation to one another. Trigonometry is an important prerequisite for calculus as it is needed to solve many calculus problems. In addition, calculus also includes the study of tangents and areas under curves. For those who want to learn more about the subject, a high school textbook will contain trigonometric functions.
Trigonometry is the study of the relationship between angles and sides of a triangle. In the most basic sense, trigonometry deals with angles, which are proportional to each other. These angles can be used to determine distances and lengths, which is a useful application of trigonometry.
Calculus is one of the most advanced branches of mathematics. This branch of mathematics deals with rate of change and has revolutionized the way people approach maths. Previously, maths could only be used to study static objects, but calculus has made it possible to apply its principles to moving objects. It is divided into two different branches, differential and integral, which are both extremely interesting concepts. | s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947473824.45/warc/CC-MAIN-20240222193722-20240222223722-00241.warc.gz | CC-MAIN-2024-10 | 4,447 | 15 |
https://www.vut.cz/en/students/courses/detail/229207?apid=229207 | math | Theory of Reliability
FAST-NDB020Acad. year: 2020/2021
Introduction of reliability theory, reliability background of standards for structural design (Eurocodes), Structural resistance and load action as two independent random variables, limit state and philosophy of design by standards, theoretical failure probability, reliability conditions, reliability reserve, reliability index, numerical simulation method Monte Carlo, Latin Hypercube Sampling, Importace Sampling, basic methods for failure probability analysis of structures designed by standards for design, basic methods for statistics, sensitivity and probabilistic analysis application to steel structures design. Introduction into risk engineering.
Institute of Structural Mechanics (STM)
Learning outcomes of the course unit
Student will learn basic knowledge from reliability theory: creation of stochastic model, reliability condition, numerical simulation methods of Monte Carlo type, limit states, risk engineering.
Knowledge from Elasticity and plasticity, Structural mechanic, Probability and statistics.
Recommended optional programme components
Recommended or required reading
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
Language of instruction
1. Introduction of reliability theory, reliability background of standards for structural design (Eurocodes), structural resistance and load action as two independent random variables, reliability condition, reserve of reliability.
2. Limit state and philosophy of design by standards.
3. Reliability standards: theoretical failure probability, reliability index.
4. Aproximační metody FORM a SORM.
5. Numerical simulation method Monte Carlo in applications.
6. Computation model, model uncertainty, grosses errors.
7. Numerical simulation methods Latine Hypercube Sampling, Importace Sampling in applications.
8. Random process and random fields – Stochastic finite element methods and these applications.
9. Probabilistic optimization, problems of live-time of structures, use of statistics and sensitivity analysis for design of structures and verification and calibration of standards design procedures.
10. Imperfections analysis and this influence to failure of steel structures.
l1. Unbalanced of the failure probability of the structures designed by standards, option of input variability modelling.
12. Introduction of Risk engineering.
13. Reliability software – replenishment, conclusion and recapitulate.
Students will get basic knowledge from reliability theory: creation of stochastic model, reliability condition, numerical simulation methods of Monte Carlo type, limit states, risk engineering.
Specification of controlled education, way of implementation and compensation for absences
Extent and forms are specified by guarantor’s regulation updated for every academic year.
Classification of course in study plans
- Programme NPC-SIS Master's, 1. year of study, summer semester, 3 credits, compulsory-optional | s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057347.80/warc/CC-MAIN-20210922102402-20210922132402-00354.warc.gz | CC-MAIN-2021-39 | 3,016 | 30 |
https://www.jiskha.com/display.cgi?id=1318998165 | math | posted by Brian .
River A & B flow into river C. River A is 40 ft wide and is flowing at an average of 2.47 mph. River B is 25 ft wide and has a flow rate of 1125 cubic feet per second. Assume all rivers have a uniform depth of 15 ft. If river C is 100 ft wide, what is its average velocity [kilometers per second]? | s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103579.21/warc/CC-MAIN-20170817151157-20170817171157-00280.warc.gz | CC-MAIN-2017-34 | 315 | 2 |
http://archives.seul.org/pygame/users/Sep-2011/msg00111.html | math | [Author Prev][Author Next][Thread Prev][Thread Next][Author Index][Thread Index]
Re: [pygame] Native PyGame method for automatically scaling inputs to a surface resolution?
On Tue, 27 Sep 2011 11:28:34 -0400
Christopher Night <cosmologicon@xxxxxxxxx> wrote:
> How can I use your system to draw a rectangle of a solid color onto a
> surface? With the regular pygame system, I would use surf.fill and
> pass it a Rect. If your system doesn't recognize rectangles of any
> sort, how can I do this? Feel free to show me in pseudocode how I
> could do it.
I suppose you mean something like:
>>> Surface.fill(WHITE, myrect)
If this is the case, the rectangle would be scaled (so a rect of
100x200 would be scaled to a rect of 10x20 assuming scale=0.1). The
general idea would be: any argument to a surface method whose purpose
is to indicate a measure ((x,y) tuples, explicit width/height or tuples)
would be scaled. Any other parameter whose purpose is not defining
coordinates (colours, flags, surfaces...) wouldn't.
Again: not claiming this is the best way of doing it... just that this
is how I would try to implement it.... | s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867424.77/warc/CC-MAIN-20180625033646-20180625053646-00254.warc.gz | CC-MAIN-2018-26 | 1,122 | 19 |
https://www.thelittleaussiebakery.com/what-is-an-example-of-differentiated-instruction/ | math | What is an example of differentiated instruction?
Another example of differentiating instruction is creating literature circles. While this sounds like an English concept, it can be applied in any class where you can group students with leveled texts. Students could all be reading about the Civil War, but they could be reading a text that is at their reading level.
What differentiated learning activities?
Definition of differentiated instruction
- Design lessons based on students’ learning styles.
- Group students by shared interest, topic, or ability for assignments.
- Assess students’ learning using formative assessment.
- Manage the classroom to create a safe and supportive environment.
What is differentiated instruction in early childhood?
Differentiated instruction involves altering the implementation and design of the lesson and activities so that the needs of all children are met. Through it children use different pathways to explore and learn while taking away the same essential ideas and understanding on the content.
How many rules of differentiation are there?
What is rules of differentiation?
General rule for differentiation: The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. ddx[k⋅f(x)]=kddx[f(x)] The derivative of a sum is equal to the sum of the derivatives.
What is implicit differentiation example?
In implicit differentiation, we differentiate each side of an equation with two variables (usually x and y) by treating one of the variables as a function of the other. This calls for using the chain rule. Let’s differentiate x 2 + y 2 = 1 x^2+y^2=1 x2+y2=1x, squared, plus, y, squared, equals, 1 for example.
Is differentiate the same as derivative?
Differentiation is the algebraic method of finding the derivative for a function at any point. The derivative is a concept that is at the root of calculus. Either way, both the slope and the instantaneous rate of change are equivalent, and the function to find both of these at any point is called the derivative.
What is the difference between dy dx and D DX?
d/dx is an operation that means “take the derivative with respect to x” whereas dy/dx indicates that “the derivative of y was taken with respect to x”.
Is dy dx equal to Y?
Yes, as long as x is the variable you are differentiating with respect to. For example, if your function is y = 3x 2 + 5x, then both y′ and dy/dx refer to the derivative of this function with respect to x, which is 6x + 5.
Can all functions be differentiated?
In theory, you can differentiate any continuous function using 3. The Derivative from First Principles. The important words there are “continuous” and “function”. You can’t differentiate in places where there are gaps or jumps and it must be a function (just one y-value for each x-value.)
What is basic differentiation?
The operation of differentiation or finding the derivative of a function has the fundamental property of linearity. This property makes taking the derivative easier for functions constructed from the basic elementary functions using the operations of addition and multiplication by a constant number.
What is dy dx?
Differentiation allows us to find rates of change. If y = some function of x (in other words if y is equal to an expression containing numbers and x’s), then the derivative of y (with respect to x) is written dy/dx, pronounced “dee y by dee x” . …
How do you differentiate tasks?
The 7 differentiation methods:
- Flexible-pace learning.
- Collaborative learning.
- Progressive tasks.
- Digital resources.
- Verbal support.
- Variable outcomes.
- Ongoing assessment.
What is differentiation formula?
Some of the general differentiation formulas are; Power Rule: (d/dx) (xn ) = nx. n-1. Derivative of a constant, a: (d/dx) (a) = 0. Derivative of a constant multiplied with function f: (d/dx) (a.
Why differentiated instruction is important?
Differentiated instruction excites the brilliant student to uncover deeper layers of learning, while simultaneously structuring curriculum to support lower level students or students with learning disabilities- both identified and unidentified.
What does differentiate mean Calc?
To differentiate a function means to find its rate of change function.
Does differentiation work in the classroom?
Delisle, differentiation in the classroom does not work. Classrooms are too diverse to expect an individual teacher to provide instruction that will meet every learning style, interest and disability.
What is differentiation in simple words?
Differentiation means finding the derivative of a function f(x) with respect to x. Differentiation is used to measure the change in one variable (dependent) with respect to per unit change in another variable (independent).
How do you do implicit differentiation step by step?
- To Implicitly derive a function (useful when a function can’t easily be solved for y) Differentiate with respect to x. Collect all the dy/dx on one side. Solve for dy/dx.
- To derive an inverse function, restate it without the inverse then use Implicit differentiation.
How do you differentiate XY?
In regular differentiation, your function starts with y and equals some terms with x in it. But with implicit differentiation, you might have your function y as part of the function such as in xy or on both sides of an equation such as in this equation: xy = 4x – 2y.
What differentiation looks like in the classroom?
“Differentiation is a philosophy – a way of thinking about teaching and learning.” “Differentiated instruction is a proactively planned, interdependent system marked by a positive community of learners, focused high-quality curriculum, ongoing assessment, flexible instructional arrangements, [and] respectful tasks.” | s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233506676.95/warc/CC-MAIN-20230925015430-20230925045430-00343.warc.gz | CC-MAIN-2023-40 | 5,830 | 53 |
https://propulsion2016.com/earth-and-sun/how-thought-the-sun-moon-and-planets-revolve-around-the-earth-correct-answer.html | math | When Nicolaus Copernicus published his innovative theory of the universe in 1543—in which the Earth, along with the other planets, revolved around the Sun—he was considered a revolutionary.
What is the reason for the planets’ revolving around the Sun?
- In any case, the primary reason for the planets’ revolving around, or orbiting, the Sun is that the Sun’s gravity holds them in place while they travel through space. The Earth circles the Sun in the same way as the Moon orbits the Earth due to the draw of the Earth’s gravity, and the Moon orbits the Earth due to the pull of the Sun’s gravity. If that’s the case, why does it go in an elliptical orbit around the Sun?
- 1 Why did they believe in the geocentric model?
- 2 How the Earth rotates around the Sun and moon?
- 3 How do we know the planets revolve around the Sun?
- 4 How did Copernicus prove that the Earth revolves around the Sun?
- 5 Who said sun revolves around the Earth?
- 6 Why Earth is the center of the universe?
- 7 Why does the Moon revolve around the Earth?
- 8 Does the Moon rotate around the Sun?
- 9 Why the Earth rotates around the Sun?
- 10 How do we know the Earth is spinning?
- 11 Does the Sun revolve around the Earth True or false?
- 12 What did Copernicus say about the motion of the sun?
- 13 Why did Copernicus think the sun was the center of the universe?
- 14 Who was the first person to prove that the earth moves around the sun in India?
Why did they believe in the geocentric model?
During his lifetime, he believed that Earth was in fact the center of the universe. Because the Greek term for Earth is geo, we refer to this concept as a “geocentric” hypothesis. Even though he began with an inaccurate hypothesis, he was able to integrate what he saw about the motions of the stars with mathematics, particularly geometry, in order to forecast the movements of the planets.
How the Earth rotates around the Sun and moon?
As the Earth rotates, it also travels around the Sun, which is known as revolving around the Earth. The Moon circles the Earth at the same rate that the Earth orbits the Sun. The Moon’s orbit lasts 27 1/2 days, but because the Earth keeps moving, it takes the Moon an additional two days and a half, or 29 1/2 days, to return to the same location in our sky.
How do we know the planets revolve around the Sun?
The Sun’s gravitational pull holds the planets in their respective orbits. They are able to maintain their orbits because there is no other factor in the Solar System that can prevent them from doing so.
How did Copernicus prove that the Earth revolves around the Sun?
He originally propagated the concepts of his heliocentric or Sun-centered astronomy somewhere between 1507 and 1515, according to certain estimates. It was with his naked sight that Copernicus observed the skies for the first time. The observations made by Copernicus led him to believe that every planet, including the Earth, circled around the Sun.
Who said sun revolves around the Earth?
The concepts of his heliocentric, or Sun-centered astronomy were originally transmitted somewhere between 1507 and 1515. When Copernicus saw the skies, he did it with his own eyesight. Copernicus deduced from his observations that every planet, including the Earth, circled around the Sun.
Why Earth is the center of the universe?
Based on the observation that at any given time, half of the stars were above the horizon and half were below it (stars on a rotating stellar sphere), as well as the assumption that all of the stars were at some modest distance from the center of the universe, Ptolemy argued that the Earth was a sphere in the center of the universe.
Why does the Moon revolve around the Earth?
Complete step-by-step answer: Gravitational force is responsible for the rotation of the moon around the Earth. On top of that, the sun produces a gravitational pull on the moon that is about twice as strong as the gravitational force exerted by Earth. However, the moon continues to rotate around the earth long after this.
Does the Moon rotate around the Sun?
In its orbit around the Sun, the Moon follows the Earth, and if the Earth didn’t exist, the Moon would be seen to be orbiting the Sun in its truest form. Moon’s orbital velocity around the Moon is less than one kilometer per second. However, the Moon’s orbital velocity around the Sun is 30 km/sec, which is the same as the Earth’s.
Why the Earth rotates around the Sun?
The Earth circles the Sun in the same way as the Moon orbits the Earth due to the draw of the Earth’s gravity, and the Moon orbits the Earth due to the pull of the Sun’s gravity. This occurs because the Earth’s velocity is in a direction that is perpendicular to the direction of the Sun’s gravitational attraction. It would be impossible for the Earth to travel in a straight line if it weren’t for the Sun.
How do we know the Earth is spinning?
The movement of pendulums is used by scientists to give proof that the Earth is rotating around its axis. A pendulum is a weight that is suspended from a fixed point and allows it to freely swing back and forth between them. Whenever you move the pendulum’s base, the weight continues to travel in the same direction. In leap years, the month of February is extended by one day.
Does the Sun revolve around the Earth True or false?
The Earth rotates around the Sun, or travels in an orbit around it. It is the route that an object takes as it goes around another object that is known as an orbit. It takes approximately one year for the Earth to complete one full rotation around the Sun. Seasons are created as a result of this.
What did Copernicus say about the motion of the sun?
In 1473, Nicolaus Copernicus (1473–1543) hypothesized that the sun was stationary in the center of the cosmos and that the earth circled around it. Copernicus was a mathematician and astronomer who lived from 1473–1543.
Why did Copernicus think the sun was the center of the universe?
In it, he suggested that the Earth was not the center of the cosmos, but rather that the sun was located close to it. He also proposed that the rotation of the Earth was responsible for the rising and setting of the sun, the movement of the stars, and the cycle of seasons, and that the Earth’s revolutions around the sun were responsible for the cycle of seasons.
Who was the first person to prove that the earth moves around the sun in India?
Today commemorates the 475th anniversary of the death of one of Poland’s most illustrious scientists, Jan Podolski. With his discovery that the earth revolved around the sun, Nicolaus Copernicus changed the course of history in astronomy. | s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570767.11/warc/CC-MAIN-20220808061828-20220808091828-00775.warc.gz | CC-MAIN-2022-33 | 6,681 | 45 |
https://mathoverflow.net/questions/333422/projective-subvarieties-of-a-quasiprojective-variety | math | Let $X$ be a quasiprojective variety over $\mathbf C$. Take the union of all projective subvarieties $W \subseteq X$ that have dimension at least $1$. Is the result Zariski closed?
(I was wondering this in the particular setting $X = \mathcal M_g$, where the projective subvarieties have been the subject of some study. But the general question seems natural as well.) | s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347400101.39/warc/CC-MAIN-20200528201823-20200528231823-00338.warc.gz | CC-MAIN-2020-24 | 368 | 2 |
http://www.purplemath.com/ranson_precalculus_tutors.php | math | ...Usually, I meet my students at a mutually agreed upon location such as a library, coffee shop, etc.I have a Bachelor of Science degree in math from the University of Maryland. My studies there included Calculus 1, 2, & 3, Advanced Calculus 1 & 2, Probability, etc. I have tutored students for over 12 years in various math curriculums and grades. (read more)
...He is author of about 50 research articles and conference presentations, PhD in Physics and Mathematics, expertise in teaching physics and mathematics to different level students (PhD, MS, BS, High School), and mentor and tutor PhD, MS, and BS students in multinational and multicultural environme... (read more)
...This is a very important class as it will serve as a base for all future math. I have been tutoring pre calculus since 2003. I have been tutoring trigonometry as it appears in most high school and local college pre-calculus classes since 2003. (read more)
I'm a graduate of the University of Virginia (M.S. Data Science and B.A. Statistics and Economics) and I've been an academic tutor for almost 8 years with experience as a Teaching Assistant at the university level for both classes and individual help. (read more)
Hello. I tutor math. I tutor question by question, explaining concepts along the way.
I cannot guarantee grade improvements. I also may not be able to answer many of your math questions, but I will try my best to help you understand and improve math skills. (read more) | s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864256.26/warc/CC-MAIN-20180621172638-20180621192638-00245.warc.gz | CC-MAIN-2018-26 | 1,470 | 6 |
http://destinymatrix.blogspot.com/2005/11/world-hologram-from-jack-sarfatti-date.html | math | From: Jack Sarfatti
Date: November 10, 2005 3:47:02 PM PST
To: ROBERT BECKER
Subject: A curious observation about ds^2
In ordinary tensor notation
ds^2 = guvdx^udx^v
But in Cartan tetrad notation we can write
ds^2 = eae^a
ds = e = 1 + B 1-form = 'd' (Goldstone Phase of Higgs Vacuum Field)
B is the local compensating gauge potential from global T4 ---> Diff(4).
Therefore, the Hodge dual *ds is a 3-form
From Stoke's theorem - ignoring topological defects for the moment
Loop integral of e = surface integral of dB
But *ds is a 3-form, whose volume integral is equal to the surface integral of a 2-form F where
*ds = dF
Heuristically "s" is the 0-form Goldstone phase in the standard model's internal symmetry breaking for origin of inertia of leptons & quarks (here W-1,W0,W+1, massless photon, & 1 Higgs boson in n = 3 order parameter.
G = U(1)hyperchargeSU(2)weak --3 adjoint irrep--> H = U(1)em
with degenerate vacuum manifold G/H = U(1)hyperchargeSU(2)weak / U(1)em ~ S2
The second homotopy group PI2(S2) = Z
Z is the set of wrapping numbers (i.e. 2D version of 1-D winding numbers) such that when one covers the S2 sphere in 3D space that surrounds the point defect, there is a wrapping N times around the vacuum manifold G/H (+ or -).
This comes about because of the single-valuedness of the order parameters in G/H ~ S2.
Therefore, the Hawking-Bekenstein quantization of area + World Holography seems to pop out trivially, i.e. from Gauss's theorem for quantized surface integral about the point defect where Higgs amplitude vanishes and Goldstone phase is undefined. Take for space the WHOLE UNIVERSE at some cosmic epoch with the "surface" as the "screen" and we need only assume ONE geometrodynamic point defect somewhere, same idea as Dirac's string to get quantized electric charge if there is only ONE magnetic monopole per universe.
To review, consider ds as a 1-form with s the 0-form as the Goldstone vacuum coherent world hologram phase.
Because of U(1)SU(2) Higgs mass generation in standard model + equivalence principle, jump from 1-form ds to 3-form *ds and apply the singular "flux-without-flux" version of Gauss's theorem for a surface in physical space that surrounds the point defect demanded by the standard model. | s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676588961.14/warc/CC-MAIN-20180715183800-20180715203800-00501.warc.gz | CC-MAIN-2018-30 | 2,242 | 24 |
https://visitccpa.com/what-percentage-of-38-is-15/ | math | If you have ever been in a situation where you need to calculate a percentage, then you know how important it is to have a good grasp of basic math concepts. One of the most common percentage questions is, “What percentage of X is Y?” In this article, we will focus on answering the question, “What percentage of 38 is 15?”
To calculate the percentage of one number in relation to another, you need to use a simple formula. The formula is: Percentage = (Part / Whole) x 100 In this case, the “part” is 15 and the “whole” is 38. So, the formula becomes: Percentage = (15 / 38) x 100
Simplifying the Formula
To simplify the formula, we need to first divide 15 by 38. This gives us: 15 / 38 = 0.3947 We then multiply this number by 100 to convert it to a percentage: 0.3947 x 100 = 39.47%
Understanding the Answer
The answer to the question, “What percentage of 38 is 15?” is 39.47%. This means that 15 is 39.47% of 38. In other words, if you had 38 apples and 15 of them were green, then the percentage of green apples would be 39.47%.
Using the Answer
Knowing the percentage of one number in relation to another can be very useful in many different situations. For example, if you are trying to calculate the discount on a $38 item and the discount is 39.47%, then you can easily calculate the discounted price by multiplying $38 by 0.6053 (100% – 39.47%).
One common mistake people make when calculating percentages is forgetting to convert the decimal to a percentage. In this case, the decimal 0.3947 needs to be converted to the percentage 39.47%. Another common mistake is mixing up the “part” and “whole” in the formula. It is important to remember that the “part” is the smaller number and the “whole” is the larger number.
Calculating percentages is an important skill to have in many different fields, from finance to science. Knowing how to calculate the percentage of one number in relation to another can help you make informed decisions and solve problems quickly and efficiently. In this article, we have shown you how to calculate the percentage of 15 in relation to 38, and we hope that you have found this information useful. | s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233506423.70/warc/CC-MAIN-20230922202444-20230922232444-00820.warc.gz | CC-MAIN-2023-40 | 2,179 | 10 |
https://www.payscale.com/research/CA/Job=Marketing_Consultant/Salary/1b952366/Early-Career-Montr%C3%A9al-QC | math | The average salary for an Early-Career Marketing Consultant in Montréal, Québec is C$55,000.
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Job Description for Marketing Consultant
A marketing consultant works to further the brand, image, and market share of the company for which they work. This can include a wide variety of activities. For example, one day, a marketing consultant may be helping their company improve its brand image among users of a certain social media outlet such as Facebook or Twitter. The next day, they may work with designers to put together a brochure or complete a marketing package to help land new customers. For this reason, the ability to work coo…Read more
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Employees with Marketing Consultant in their job title in Vancouver, British Columbia earn an average of 3.6% more than the national average. The lowest salaries can be found in Toronto, Ontario (10.9% less) and Calgary, Alberta (4.5% less).
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C$43k - C$87k
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C$44k - C$106k
C$66k - C$120k
C$52k - C$103k
C$49k - C$110k | s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987813307.73/warc/CC-MAIN-20191022081307-20191022104807-00240.warc.gz | CC-MAIN-2019-43 | 1,470 | 26 |
http://math.ict.edu.rs:8080/webMathematica/LinkSL/kn085.htm | math | A braid universe is an ordered sequence of integers, where the element i represents an unsigned crossing of the ith and (i+1)th braid strands. A braid universe becomes a braid word when to each crossing is assigned a sign +1 or -1 in the same manner as was done before with KLs. In a braid word a positive (negative) crossing of the ith and (i+1)th strands is represented by ith capital (lower case) letter. A braid is alternating if even numbered generators have the opposite sign of odd numbered generators. Therefore crossings in alternating braids have capitals for the odd letters, and lower cases for the even letters. The same convention can be taken even for non-alternating braids: a crossing is called alternating if it is capital for an odd letter and lower case for an even letter. A binary code for any braid can be generated by assigning 0 for an alternating crossing and 1 for a non-alternating crossing. With the four criteria that define a minimum braid, there is always a unique minimum for any set of braids (Gittings, 2004).
We will present another approach: after defining braid family representatives (BFRs) we will establish a correspondence between BFRs and families of KLs given in the Conway notation. For a better understanding of this correspondence, together with the standard Conway notation, a braid-modified Conway notation will be introduced and used. First we define a reduced braid word, describe a general form for all reduced braid words with s = 2 strands, generate all infinite families of two-strand braid words, and establish a correspondence between them and families of KLs given in the Conway notation. Then we consider the same problem for s ³ 3. Some applications of minimum braids (Gittings, 2004) and braid family representatives are discussed in Subsection 1.12.1. All computations are made using the program LinKnot.
We use the standard definition of a braid and description of minimum braids given by T. Gittings (2004). Instead of a...a, where a capital or lower case letter a appears p times, we write ap; p is the degree of a (p Î N). It is also possible to work with negative powers, satisfying the relationships: A-p = ap, a-p = Ap. A number of strands is denoted by s, and a length of a braid word by l. | s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818693866.86/warc/CC-MAIN-20170925235144-20170926015144-00377.warc.gz | CC-MAIN-2017-39 | 2,262 | 3 |
http://matrix1inc.com/2019/08/22/the-little-known-secrets-to-discrete-math-definition/ | math | The Little-Known Secrets to Discrete Math Definition
After the application is opened up, you’ll notice something such as this. You are going to get an array of different answers on account of the subjectivity inherent to humans. essay capitals An essential characteristic of procedural knowledge is it is tough to explain verbally.
Choosing Good Discrete Math Definition
Now you are all set to get started drawing! A graph with only one node is usually known as a singleton graph, though we won’t really be dealing with those. You’ll frequently be directed to evaluate a specific function for a particular value of x.
Not just that, but it’s such a useful mental model that after you get started thinking in graphs you will see all of them around you. With the assistance of a calculator, the process of successive approximations can be carried out quickly. As it happens, this is true for graphs.
Discrete Math Definition – What Is It?
It’s tough to spot unless you’re really searching for it since it’s sneaky, subtle and takes a reasonable bit of concentrative analysis to identify. You want to understand what the original price was. You forget a lot when you forget a class.
Actually, it is possible to even grab any of our calculators and set them on your http://www.law.columbia.edu/library own blog or website free of charge! Browse our categories to discover the worksheet you’re looking for or utilize search option on the top to look for any worksheet you want. Arrow functions don’t have a prototype property.
If you’re intimidated, try out the outline trick. We have to suppose that we’ve got some sensible formula for the region of the cross section. The region of the circle is shaded.
The range of pounds bought is known as the independent variable since that’s what we’re changing whereas the whole prize is known as the dependent variable since it’s dependent on the number of pounds we actually buy. Let’s stick with the easy stuff to begin. For instance, the period of time that it takes for Earth to revolve around the sun is 1 year.
Byju’s classes unique method of solving the maths problem will force you to learn the method by which the equation was created, which is way superior than memorizing and applying the formula. Every command has a certain syntax to use. Perhaps it might even peak your interest in different subjects like Formal Languages also.
Math Calculators Our absolutely free on-line math calculators are able to help you solve just about any math problem that you wish to think up. All you need to do is to learn to love its concepts and memorize the fundamentals. The Roman notation table may be taught after the exact same way.
A Secret Weapon for Discrete Math Definition
Big-O notation is normally utilised to describe the development of functions and, as we’ll see in subsequent sections, in estimating the range of operations an algorithm requires. It’s often useful to think of a function for a box’. The function has to be an appropriate relationship for every single input and output, meaning for every input there is just a single output and it has to get the job done for each and every input value.
An immediate variation is an association between inputs and outputs where the proportion of inputs and outputs is always the exact same. In easy, operational stipulations, annotation could be defined as the component of genome analysis that’s customarily performed in front of a genome sequence is deposited in GenBank and described in a published paper. The output of an equation for example demands a certain mathematical formula, which can be accomplished by employing function notation.
The other method is to permanently switch off the feature to bring in GETPIVOTDATA. Genome annotation necessarily involves some degree of automation. A function may be continuous or not, based on its Domain!
Within this tutorial, I’ll produce you the applications that go very nicely with the Math Input Panel and will also enable you to know the best place to discover it in Windows research paper online 10 and how to utilize it with the combinations of different applications as well as how to correct the formulas and a lot more such things associated with the Math Input Panel is going to be described here. Data is classified into qualitative or quantitative depending on the essence of the information it offers. Interval Values may be added and subtracted and compared, but they can’t be multiplied or divided.
The Most Popular Discrete Math Definition
For instance a square has four corners, each is known as a vertex. Therefore, the diameter of a circle is twice provided that the radius. If you gauge the distance around a circle and divide it by the distance on the other side of the circle through the middle, you will always arrive close to a specific price, depending upon the truth of your measurement.
Finding the mean, also called averaging numbers, is a really useful point to comprehend how to do, particularly whenever you call for a precise estimate or an extremely accurate generalization. The interest is deducted from the amount of the loan so that you don’t get the complete loan amount or face value of the loan when you get the loan. In reality, the above mentioned illustration is a specific case of calculating percentages of a total.
You need to spot the physicians assistant programs that supply the form of training you want. Such an approach is illustrated below. The R implementation of methods provides for an equivalent method of specifying an argument shouldn’t be included.
The Fundamentals of Discrete Math Definition Revealed
The idea of randomness is tough to define precisely. It’s what people often utilized to be a symbol of something unending or infinite. Remember that the absolute most efficient marketing plays on human nature’s best fears and aspirations. | s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573439.64/warc/CC-MAIN-20190919040032-20190919062032-00116.warc.gz | CC-MAIN-2019-39 | 5,904 | 23 |
http://zbmath.org/?q=an:1116.34057 | math | The paper studies Hopf bifurcations of scalar neutral delay differential equations of the form
where is the bifurcation parameter and . The characteristic equation has the form
First the paper presents the stability charts depending on the coefficients of the characteristic equation. Then the paper presents a sequence of formulas leading to the criticality coefficient determining the stability of the emerging periodic solutions in the center direction. The normal form computation is done with the technique introduced by T. Faria and L. T. Magalhães [J. Differ. Equ. 122, No. 2, 181–200 (1995; Zbl 0836.34068)]. Two examples are included. Both have a delay and a linear part of the form . The nonlinear parts are and , respectively. | s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00378-ip-10-147-4-33.ec2.internal.warc.gz | CC-MAIN-2014-15 | 740 | 3 |
http://talg.acm.org/archive.cfm?id=3040971 | math | enter search term and/or author name
Dynamic Facility Location via Exponential Clocks
Hyung-Chan An, Ashkan Norouzi-Fard, Ola Svensson
Article No.: 21
The dynamic facility location problem is a generalization of the classic facility location problem proposed by Eisenstat, Mathieu, and Schabanel to model the dynamics of evolving social/infrastructure networks. The generalization lies in...
On Uniform Capacitated k-Median Beyond the Natural LP Relaxation
Article No.: 22
In this article, we study the uniform capacitated k-median (CKM) problem. In the problem, we are given a set F of potential facility locations, a set C of clients, a metric d over F ∪ C, an upper bound...
Section: Special Issue on SODA'15
Generating Random Permutations by Coin Tossing: Classical Algorithms, New Analysis, and Modern Implementation
Axel Bacher, Olivier Bodini, Hsien-Kuei Hwang, Tsung-Hsi Tsai
Article No.: 24
Several simple, classical, little-known algorithms in the statistics and computer science literature for generating random permutations by coin tossing are examined, analyzed, and implemented. These algorithms are either asymptotically optimal or... | s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170569.99/warc/CC-MAIN-20170219104610-00520-ip-10-171-10-108.ec2.internal.warc.gz | CC-MAIN-2017-09 | 1,151 | 13 |
https://blackhawksupply.com/pages/airflow-calculator | math | ONLINE CFM AIR VELOCITY AND AIRFLOW RATE CALCULATOR
Wondering how to calculate air velocity for your air compressor to work properly? Blackhawk Supply offers a convenient air velocity calculator online, allowing you to convert the various units for air volume flow and air velocity flowing through your system. With this calculator, we make it easy to learn how to calculate air flow in CFM (cubic feet per minute) with a convenient duct volume air flow calculator.
To use the air flow calculator, you simply need the volume air flow or air velocity (measured in ft/sec, ft/min, miles/hr (mph), or any other common units), the duct type or area (whether it's a rectangle duct or a circular duct), and the units you're using to measure. Then hit “calculate”, and our air flow rate calculator will do the rest.
Online Duct Airflow Calculator
For an air flow pressure calculator, we need to look at air velocity. Air velocity is the distance traveled per minute and is used as a measurement of the displacement rate for air and gas.
An air velocity calculator will express this distance in LFM (ft/min, linear feet per minute). By using an air velocity through duct calculator, you're multiplying the air velocity by the area of a duct. With this, you'll find the air volume that flows past the duct in terms of units per time. This measurement is typically represented in CFM, but with an airflow calculation conversion, you can find a CMH air flow velocity calculation, which measures the airflow per hour instead of per minute.
Ready to use our CFM air flow velocity calculator? To calculate, enter the value, select the air velocity units that you prefer, and hit calculate. Then, our air volume calculator will do the work and display your results below.
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What Is Air Flow Rate?
Air flow, or airflow, at its most basic definition, refers to the movement of air, and the more distance it travels, the higher its rate. In the context of your heating and cooling system, using the right pipe air flow calculation formula will help guarantee that your system runs effectively.
Why is it important to know how to calculate air flow for HVAC? Well, if you have too much air volume flowing past the system, you may notice more humidity in your space. On the other hand, if air flow is too low, your coils may get icy and flood your air compressor - and we certainly don't want that. However, using a simple online air flow calculator, you can find how to calculate air flow through pipe systems for the right airflow rate to keep your system running.
When it comes to something like an HVAC air flow calculation formula, we are looking to measure the air that passes the evaporator coil. This passing air is measured in CFM. With a CFM air flow velocity calculator, we want to ensure that around 350 to 400 CFM per ton of cooling is being used, as this ensures the air conditioning system will run properly.
So, how do you calculate the air flow rate? Let's get to the calculations.
How Do You Calculate Air Velocity Rate (Air Flow Calculation Formula)?
If you're wondering how to calculate air flow rate from pressure and diameter yourself, don't worry, because we're sharing the right equations to use if you're skipping the calculator for your at-home air flow calculation using pressure.
The air flow calculation through the pipe equation is represented as:
In this air flow calculation formula, the values are represented as “d” (pipe inner diameter in inches), “v” (air velocity ft/s), and Qa (air flow rate in CFM).
Being aware of how to calculate airflow through a pipe is key to running your system properly. For many, this can be a challenging calculation, and the HVAC duct flow calculation isn't taught to all technicians. However, knowing how to calculate airflow through a duct can lend a hand in figuring out any issues that may arise with your air conditioning system.
No more wondering, “how do you calculate air velocity” with our formula to calculate the air velocity rate in pipe systems. Knowing how to calculate airflow is the key to running an HVAC system smoothly, aiding in troubleshooting any issues that may arise.
Still, struggling with how air velocity CFM in the pipe is measured? No worries, our team at Blackhawk Supply can help. Contact us at (847) 773-0645 and our specialists will answer all your questions about how to calculate air velocity in duct systems for a perfectly operating heating and cooling unit.
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Orders over $999* | s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947474676.79/warc/CC-MAIN-20240227153053-20240227183053-00363.warc.gz | CC-MAIN-2024-10 | 4,595 | 22 |
https://math.stackexchange.com/questions/3189845/prime-property-in-noncommutative-rings-without-identity | math | Let $R$ be a ring (without assuming identity or commutativity), and $P$ a proper ideal of $R$. Show that the following are equivalent:
(a) For ideals $A,B$: $AB\subseteq P$ implies $A\subseteq P$ or $B\subseteq P$.
(b) For right ideals $T,S$: $TS\subseteq P$ implies $T\subseteq P$ or $S\subseteq P$.
(c) For elements $a,b\in R$: $aRb\in P$ implies $a\in P$ or $b\in P$.
(b) $\implies $ (a) is trivial
(c) $\implies$ (b) is easy: Assume (c) and let $T,S$ be as in (b), with $T\not\subseteq P$. Fix $r\in T\backslash P$ and let $s\in S$. Then $rRs\in P$ since $T$ is a right ideal and by assumption $s\in P$.
I haven't been able to prove (a) $\implies $ (c) since my first idea was to realise $a,b$ as two sided ideals, however, the ideal generated by $a$ in a ring without unity is $\langle a\rangle =RaR + aR+Ra+\mathbb Z a$, which is a bit unwieldly. I am not convinced that multiplying the ideals $\langle a\rangle\cdot \langle b\rangle$ is assured to fall inside $P$ despite having $aRb\in P$. I am mostly concerned about the $a\cdot b$ terms, since there is no identity, it's not in $aRb$. Also the fact that each ideal $\langle a\rangle$ could well be the whole ring. | s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256100.64/warc/CC-MAIN-20190520182057-20190520204057-00161.warc.gz | CC-MAIN-2019-22 | 1,173 | 7 |
http://www.chegg.com/homework-help/questions-and-answers/car-lying-old-style-record-player-expensive-one-friction-moment-inertia-cat-turntable-toge-q2429025 | math | a car is lying on an old -style record player. An expensive one, so there is no friction. The moment of inertia of the cat and turntable together is 0.36 kgm2 . the radius of the turntable is 0.30m, and it starts from rest. You apply a force of magnitude 6 N to the edge of the turntable, and tangential to the edge the entire time you push. As a result of this force, the table rotates clockwise as viewed from above.
A) Draw a free-body diagram of the CAT, as viewed from the side,
And as viewed from above. You have already starting pushing on the
B) Show that you are applying a torque of 1.8 Nm clockwise.
C) Show that the magnitude of the angular acceleration of the cat
is 5 rad/s2
D) Determine the angle that the table rotates during the first 2
seconds you push, in radians and in degrees. | s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982298977.21/warc/CC-MAIN-20160823195818-00274-ip-10-153-172-175.ec2.internal.warc.gz | CC-MAIN-2016-36 | 798 | 8 |