text:
# Properties

 Label 245.2.e.e Level $245$ Weight $2$ Character orbit 245.e Analytic conductor $1.956$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + q^{6} + ( 3 + \beta_{3} ) q^{8} -2 \beta_{1} q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + q^{6} + ( 3 + \beta_{3} ) q^{8} -2 \beta_{1} q^{9} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{12} + ( 2 + 2 \beta_{3} ) q^{13} + ( 1 - \beta_{3} ) q^{15} + ( -3 - 3 \beta_{2} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{18} -2 \beta_{1} q^{19} + ( -1 - 2 \beta_{3} ) q^{20} -2 q^{22} + ( 1 + \beta_{1} + \beta_{2} ) q^{23} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{24} + \beta_{2} q^{25} + ( -6 + 4 \beta_{1} - 6 \beta_{2} ) q^{26} + ( -1 - \beta_{3} ) q^{27} - q^{29} + ( 1 + \beta_{2} ) q^{30} + 6 \beta_{2} q^{31} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{32} + ( 6 + 4 \beta_{1} + 6 \beta_{2} ) q^{33} + ( -6 - 4 \beta_{3} ) q^{34} + ( -8 - 2 \beta_{3} ) q^{36} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{38} + 2 \beta_{2} q^{39} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{40} + ( 5 + 2 \beta_{3} ) q^{41} + ( 5 - \beta_{3} ) q^{43} + ( 6 + 2 \beta_{1} + 6 \beta_{2} ) q^{44} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{45} + \beta_{2} q^{46} + ( 2 + 2 \beta_{2} ) q^{47} + ( -3 + 3 \beta_{3} ) q^{48} + ( 1 + \beta_{3} ) q^{50} + ( 2 + 2 \beta_{2} ) q^{51} + ( -6 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} ) q^{52} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{54} + ( -2 + 2 \beta_{3} ) q^{55} + ( -4 + 2 \beta_{3} ) q^{57} + ( 1 - \beta_{1} + \beta_{2} ) q^{58} + ( 6 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{59} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{60} + ( -3 - 6 \beta_{1} - 3 \beta_{2} ) q^{61} + ( 6 + 6 \beta_{3} ) q^{62} + ( -7 - 2 \beta_{3} ) q^{64} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{66} + ( \beta_{1} + 11 \beta_{2} + \beta_{3} ) q^{67} + ( 10 - 6 \beta_{1} + 10 \beta_{2} ) q^{68} + ( 3 - 2 \beta_{3} ) q^{69} + ( -4 + 6 \beta_{3} ) q^{71} + ( 4 - 6 \beta_{1} + 4 \beta_{2} ) q^{72} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 1 + \beta_{1} + \beta_{2} ) q^{75} + ( -8 - 2 \beta_{3} ) q^{76} + ( 2 + 2 \beta_{3} ) q^{78} + ( -12 + 2 \beta_{1} - 12 \beta_{2} ) q^{79} -3 \beta_{2} q^{80} + ( -6 \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{81} + ( -9 + 7 \beta_{1} - 9 \beta_{2} ) q^{82} + ( -1 - 9 \beta_{3} ) q^{83} + ( 2 + 2 \beta_{3} ) q^{85} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{86} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{88} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{89} + ( 4 + 2 \beta_{3} ) q^{90} + ( 3 - \beta_{3} ) q^{92} + ( 6 + 6 \beta_{1} + 6 \beta_{2} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{95} + ( -5 - 4 \beta_{1} - 5 \beta_{2} ) q^{96} + ( -6 - 4 \beta_{3} ) q^{97} + ( 8 - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 2q^{3} - 2q^{4} + 2q^{5} + 4q^{6} + 12q^{8} + O(q^{10})$$ $$4q - 2q^{2} + 2q^{3} - 2q^{4} + 2q^{5} + 4q^{6} + 12q^{8} + 2q^{10} - 4q^{11} - 6q^{12} + 8q^{13} + 4q^{15} - 6q^{16} + 4q^{17} + 8q^{18} - 4q^{20} - 8q^{22} + 2q^{23} + 2q^{24} - 2q^{25} - 12q^{26} - 4q^{27} - 4q^{29} + 2q^{30} - 12q^{31} + 6q^{32} + 12q^{33} - 24q^{34} - 32q^{36} + 8q^{38} - 4q^{39} + 6q^{40} + 20q^{41} + 20q^{43} + 12q^{44} - 2q^{46} + 4q^{47} - 12q^{48} + 4q^{50} + 4q^{51} - 20q^{52} + 8q^{53} + 6q^{54} - 8q^{55} - 16q^{57} + 2q^{58} - 8q^{59} + 6q^{60} - 6q^{61} + 24q^{62} - 28q^{64} + 4q^{65} - 4q^{66} - 22q^{67} + 20q^{68} + 12q^{69} - 16q^{71} + 8q^{72} + 4q^{73} + 2q^{75} - 32q^{76} + 8q^{78} - 24q^{79} + 6q^{80} + 2q^{81} - 18q^{82} - 4q^{83} + 8q^{85} - 6q^{86} - 2q^{87} - 4q^{88} - 6q^{89} + 16q^{90} + 12q^{92} + 12q^{93} + 4q^{94} - 10q^{96} - 24q^{97} + 32q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−1.20711 2.09077i −0.207107 + 0.358719i −1.91421 + 3.31552i 0.500000 + 0.866025i 1.00000 0 4.41421 1.41421 + 2.44949i 1.20711 2.09077i
116.2 0.207107 + 0.358719i 1.20711 2.09077i 0.914214 1.58346i 0.500000 + 0.866025i 1.00000 0 1.58579 −1.41421 2.44949i −0.207107 + 0.358719i
226.1 −1.20711 + 2.09077i −0.207107 0.358719i −1.91421 3.31552i 0.500000 0.866025i 1.00000 0 4.41421 1.41421 2.44949i 1.20711 + 2.09077i
226.2 0.207107 0.358719i 1.20711 + 2.09077i 0.914214 + 1.58346i 0.500000 0.866025i 1.00000 0 1.58579 −1.41421 + 2.44949i −0.207107 0.358719i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.e.e 4
7.b odd 2 1 35.2.e.a 4
7.c even 3 1 245.2.a.g 2
7.c even 3 1 inner 245.2.e.e 4
7.d odd 6 1 35.2.e.a 4
7.d odd 6 1 245.2.a.h 2
21.c even 2 1 315.2.j.e 4
21.g even 6 1 315.2.j.e 4
21.g even 6 1 2205.2.a.n 2
21.h odd 6 1 2205.2.a.q 2
28.d even 2 1 560.2.q.k 4
28.f even 6 1 560.2.q.k 4
28.f even 6 1 3920.2.a.bq 2
28.g odd 6 1 3920.2.a.bv 2
35.c odd 2 1 175.2.e.c 4
35.f even 4 2 175.2.k.a 8
35.i odd 6 1 175.2.e.c 4
35.i odd 6 1 1225.2.a.k 2
35.j even 6 1 1225.2.a.m 2
35.k even 12 2 175.2.k.a 8
35.k even 12 2 1225.2.b.g 4
35.l odd 12 2 1225.2.b.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 7.b odd 2 1
35.2.e.a 4 7.d odd 6 1
175.2.e.c 4 35.c odd 2 1
175.2.e.c 4 35.i odd 6 1
175.2.k.a 8 35.f even 4 2
175.2.k.a 8 35.k even 12 2
245.2.a.g 2 7.c even 3 1
245.2.a.h 2 7.d odd 6 1
245.2.e.e 4 1.a even 1 1 trivial
245.2.e.e 4 7.c even 3 1 inner
315.2.j.e 4 21.c even 2 1
315.2.j.e 4 21.g even 6 1
560.2.q.k 4 28.d even 2 1
560.2.q.k 4 28.f even 6 1
1225.2.a.k 2 35.i odd 6 1
1225.2.a.m 2 35.j even 6 1
1225.2.b.g 4 35.k even 12 2
1225.2.b.h 4 35.l odd 12 2
2205.2.a.n 2 21.g even 6 1
2205.2.a.q 2 21.h odd 6 1
3920.2.a.bq 2 28.f even 6 1
3920.2.a.bv 2 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}^{4} + 2 T_{2}^{3} + 5 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{3}^{4} - 2 T_{3}^{3} + 5 T_{3}^{2} + 2 T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$( -4 - 4 T + T^{2} )^{2}$$
$17$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$64 + 8 T^{2} + T^{4}$$
$23$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$( 1 + T )^{4}$$
$31$ $$( 36 + 6 T + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( 17 - 10 T + T^{2} )^{2}$$
$43$ $$( 23 - 10 T + T^{2} )^{2}$$
$47$ $$( 4 - 2 T + T^{2} )^{2}$$
$53$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$59$ $$3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$3969 - 378 T + 99 T^{2} + 6 T^{3} + T^{4}$$
$67$ $$14161 + 2618 T + 365 T^{2} + 22 T^{3} + T^{4}$$
$71$ $$( -56 + 8 T + T^{2} )^{2}$$
$73$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$18496 + 3264 T + 440 T^{2} + 24 T^{3} + T^{4}$$
$83$ $$( -161 + 2 T + T^{2} )^{2}$$
$89$ $$529 - 138 T + 59 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$( 4 + 12 T + T^{2} )^{2}$$


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# “Is Algebra Necessary?” Are You High?

“This room smells of mathematics!
Go out and fetch a disinfectant spray!”

A.H. Trelawney Ross, Alan Turing’s form master

It’s been a while since I’ve felt riled enough to blog. But now, the spirit moves within me once more.

First, I encourage you to read Andrew Hacker’s op-ed in The New York Times,Is Algebra Necessary?” Then, sample a few reactions:

I will try to gather a few observations here which I haven’t seen made elsewhere, for the most part.

Towards the end, Hacker’s reasoning gets just bizarre. He keeps emphasising how important “citizen statistics” is. I’m baffled as to how one could teach statistics in any useful way without the material he wants to throw out! Prerequisites: we needz them. “Is Algebra Necessary?” If you want to do statistics or economics, yes, it is.

Quantitative literacy clearly is useful in weighing all manner of public policies, from the Affordable Care Act, to the costs and benefits of environmental regulation, to the impact of climate change. Being able to detect and identify ideology at work behind the numbers is of obvious use. Ours is fast becoming a statistical age, which raises the bar for informed citizenship. What is needed is not textbook formulas but greater understanding of where various numbers come from, and what they actually convey.

OK, so how are we supposed to teach where “numbers come from” and “what they actually convey” when the students can’t manipulate algebraic formulas? That’s like expecting to raise a generation of musicians who’ve never practised a scale.

It’s also nice how he skips past the serious problems we have with infrastructure. You know, schools not being able to buy books any more, and teachers who feel they aren’t capable of teaching even middle-school mathematics—because they aren’t. But saying that teachers need to be trained and books must be bought doesn’t make you a daring iconoclast worthy of a New York Times soapbox, now, does it?

Hacker’s standard for what counts as advanced—and, therefore, unnecessary—mathematics seems to drift from algebra to calculus:

Medical schools like Harvard and Johns Hopkins demand calculus of all their applicants, even if it doesn’t figure in the clinical curriculum, let alone in subsequent practice.

Maybe because doctors need to understand clinical trials, which requires understanding statistics, which in turn requires calculus? (I’m looking at The Cartoon Guide to Statistics, on page 66. Yup, integral sign.) Hey, here’s a radical thought: maybe not enough doctors really get what’s going on with clinical trials, and we’d be better off if more of them knew slightly higher mathematics, instead of fewer?

Of course, in Hacker’s world, everyone knows what they want to be when they grow up, and nobody needs training for anything else in case they want or need to switch to a new job.

It’s not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it’s not easy to see why potential poets and philosophers face a lofty mathematics bar.

Philosophers don’t need to know mathematics? Not even mathematics which is hundreds or thousands of years old? What. The. Fuck?

(by Oliver Byrne)

Oh, poor Euclid. Such a fate, and for you who alone have looked on beauty bare….

I’m guessing Hacker is referring to Fermat’s Last Theorem, but I don’t really know. (Nor does anyone else I know who read Hacker’s piece.) It doesn’t really fit the sense of “dilemma.” Fermat stated a problem and claimed that he had a solution. Perhaps the “dilemma” was that he felt compelled to write it down even though he didn’t have space to include his proof? Come on, I’m trying to be charitable here.

I might as well close with this bit from Hacker:

Yes, young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions.

So: the line is drawn wherever Hacker feels it should. Good to know. (I wonder: if long division is mandatory, what about, say, extracting square roots by hand? My mother learned how to do that in grade school, but I didn’t pick it up until I read the Feynman Lectures.) Incidentally, as a workaday scientist, I work with vectors and discontinuities far more often than I have to use long division. Honestly, I think “the line is broken in two pieces” is a significantly easier concept to get across than the rigmarole of pulling numbers down and multiplying them up and subtracting them down. But, of course, “vectorial angles” and “discontinuous functions” sound awfully intimidating to the proletariat, so let’s just hold them up as examples of what we don’t need to teach!

(by Eugene)

UPDATE (30 July 2012): I notice some people have been finding this blog entry by websearching for “Fermat’s dilemma,” so I figured I should say a bit more about that, following thoughts I developed on Twitter. As I said above, you could, if you really tried, interpret this odd choice of words as Hacker’s trying to be artful. But even if you try to read it that way, it’s sorry writing. In aiming to be evocative, Hacker just comes across ambiguous. You could say that Fermat’s little theorem—not to be confused with his Last Theorem!—embodies a dilemma. The little theorem gives us a test for saying whether a number is prime, but that test doesn’t always work: it can be fooled by a “pseudoprime.” Do we use a fallible test or not? We have a dilemma on our hands! Or, if that sounds like too much of a stretch, take the subject of probability theory, which Fermat and Blaise Pascal jumpstarted back in 1654. They wanted to answer a puzzle posed by Antoine Gombaud, Chevalier de Méré: how should two gamblers divvy up a pot of money if their game-playing is interrupted? Sorting out this kind of dilemma led Pascal and Fermat to probability theory.

See, we give things names in order to identify them, and if you make up your own names for things which already have them, you’re apt to confuse people. Sometimes, when the old terminology is muddled or unclear, a new turn of phrase is just what we need, but this is not one of those times. Hacker throws around big words like “polynomial functions and parametric equations” in order to make algebra sound intimidating, and when he makes up his own terms, he fumbles.

UPDATE (11 August 2012): Another item in the “oh, this is too fun” category. Hacker mocks the notion that being able to prove

$(x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2$

can lead to “more credible political opinions or social analysis.” I didn’t say anything about that until now, because I figured the rejoinder was implicit in what I said up top: if you can’t say that

$(x^2 + y^2)^2 = (x^2)^2 + (y^2)^2 + 2(xy)^2$

and

$(x^2 – y^2)^2 = (x^2)^2 + (y^2)^2 – 2(xy)^2,$

and then combine the two and so deduce

$(x^2 + y^2)^2 = (x^2 – y^2)^2 + 4(xy)^2,$

then actually learning or doing anything with “citizen statistics” will be beyond you. Them’s the breaks. However, there’s something else interesting here, which goes back to Hacker’s low expectations for poets and philosophers. To make things more clear, I’m going to replace $x^2$ with $a$ and $y^2$ with $b$. Then the Equation Mocked by Hacker becomes

$(a + b)^2 = (a – b)^2 + 4ab.$

If we divide both sides of this equation by 4, we get a restatement of it which is just as true:

$\frac{(a + b)^2}{4} = \frac{(a – b)^2}{4} + ab.$

And this is the algebraic statement of what Euclid said geometrically in Proposition II.5 of The Elements—which happens to be one of the oldest fragments of Euclid we have a surviving copy of. In fact, since this papyrus fragment was probably written between 75 and 125 CE, it’s likely older than any copy of any part of the Gospels. Rylands Library Papyrus P52, a scrap with a few verses from John 18, is the oldest one of those we’ve got, and it dates to about 125 CE itself.

It’s easy to say that not being willing to learn mathematics cuts one off from intellectual history. But damn, does Hacker ever put a point on it.

Congratulations, Professor Hacker. Your op-ed has earned a negative score on the Pritchard scale.

“Take the mathematical developments out of the history of science, and you suppress the skeleton which supported and kept together all the rest. Mathematics gives to science its innermost unity and cohesion, which can never be entirely replaced with props and buttresses or with roundabout connections, no matter how many of these may be introduced.”
—George Sarton, science historian

format: plain


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## Statement

difficulty

On a given instant of time, a rocket has a velocity of  . Knowing that its mass is 2kg. What is the momentum on that instant?

## Solution

Data

Resolution

Applying the definition of quantity of motion:

## Formulas worksheet

These are the main formulas that you must know to solve this exercise. If you are not clear about their meaning, we recommend you to check the theory in the corresponding sections. Furthermore, you will find in them, under the Formulas tab, the codes that will allow you to integrate these equations in external programs like Word or Mathematica.

Formulas
Related sections
$\stackrel{\to }{p}=m·\stackrel{\to }{v}$


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# SPSS Tutorials: Weighting Cases

This SPSS tutorial shows how to use Weight Cases to apply a weighting variable, especially when your data measures counts.

## Weighting Cases

In SPSS, weighting cases allows you to assign "importance" or "weight" to the cases in your dataset. Some situations where this can be useful include:

• Your data is in the form of counts (the number of occurrences) of factors or events. The "weight" is the number of occurrences.
• Your data requires adjustments to correct for over- or under-representation of certain characteristics in your sample. (This often happens with large surveys: a "weighting" variable is developed to adjust a sample's composition to be reflective of the population's composition, or to control for over- or under-reporting from a certain group.)
• The Pew Research Center often makes their raw survey data available online to the public. Many of these surveys include weighting as a part of the study methodology.

Weighting cases in SPSS works the same way for both situations.

To turn on case weights, click Data > Weight Cases.

• To enable a weighting variable, click Weight cases by, then double-click on the name of the weighting variable in the left-hand column to move it to the Frequency Variable field. Click OK.
• To turn off an enabled weighting variable, open Weight Cases window again, and click Do not weight cases. Click OK.

## Example: Reproducing an Existing 3x2 Table

### Problem Statement

Suppose you are helping a friend with their statistics homework, and see that they have included the following write-up in their report:

You immediately notice several things wrong with this report so far:

1. You can never prove anything with a hypothesis test--not even if the p-value is really, really small.
2. You do not see any test statistics anywhere, and it is unclear what test he has run.

You get the feeling that they may have used the Chi-square test of independence, but want to verify this for yourself.

### Running the Procedure

Whenever you want to re-create a frequency table or crosstab, you first need to figure out how many unique combinations of the factors there are, and how many observations there were for each factor combination. In this situation, we have two variables: ClassRank (which has three levels) and PickedAMajor (which has two levels). So there are 3*2 = 6 unique factor combinations. They are:

 Freshman No 212 responses Freshman Yes 114 responses Sophomore No 171 responses Sophomore Yes 168 responses Junior No 92 responses Junior Yes 198 responses

When we go to enter our data in SPSS, we will need to create three new variables: ClassRank, PickedAMajor, and a frequency variable (let's name it "Freq"). After entering the data, your Data View window should look like this:

Now we need to weight the cases with respect to Freq. Click Data > Weight Cases.

Click Weight cases by, then double-click Freq to move it to the Frequency Variable field. Click OK.

Now we can run our crosstab and verify your friend's results. Click Analyze > Descriptive Statistics > Crosstabs.

When the Crosstabs window opens, select the variable ClassRank in the left column and move it to the Row(s) field using the first arrow button, then select variable PickedAMajor in the left column and transfer it to the Column(s) field using the second arrow button. Doing this will reproduce the 3x2 table that your friend made.

To produce a Chi-square test of independence, click Statistics. This will open the Crosstabs: Statistics window. Select the Chi-square check box in the upper left-hand corner, then click Continue.

Click OK to run the crosstab.

#### Syntax

WEIGHT BY Freq.

CROSSTABS
/TABLES=ClassRank BY PickedAMajor
/FORMAT=AVALUE TABLES
/STATISTICS=CHISQ
/CELLS=COUNT
/COUNT ROUND CELL.

### Output

Within your output, you should see the following two tables:

The 3x2 table matches your friend's output. From the Chi-square Tests table, we see that this test result was significant at the 5% level (χ(2)=68.207, p < .001). From this result, we infer that there is a significant association between a student's class rank and whether or not they have picked a major.


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• Create Account

#Actualjohnmarinelli

Posted 30 August 2013 - 08:13 PM

Hi everyone,

I know this question has been asked a few times before, but I still can't get my program to work.

I'm working on an Asteroids clone in OpenGL, and I'm starting off by simply getting my spaceship to move around.  I also have 4 quads set up in each respective corner so I know if the movement is working or not.

However, it seems like I'm moving the quads around, and not my spaceship.  Here's the code:

void renderScene()
{
glClear(GL_COLOR_BUFFER_BIT);

glMatrixMode(GL_MODELVIEW);

glPushMatrix();

gSpaceship.render();

glPopMatrix();

glPushMatrix();
//move to bottom right corner
glTranslatef(SPACE_WIDTH / 2.f, SPACE_HEIGHT / 2.f, 0.f);

glColor3f(0.f, 1.f, 0.f);
glVertex2f(0.f, 0.f);
glVertex2f(50.f, 0.f);
glVertex2f(50.f, 50.f);
glVertex2f(0.f, 50.f);
glEnd();

//move to bottom left corner
glTranslatef(-SPACE_WIDTH, 0.f, 0.f);

glColor3f(0.f, 0.f, 1.f);
glVertex2f(0.f, 0.f);
glVertex2f(50.f, 0.f);
glVertex2f(50.f, 50.f);
glVertex2f(0.f, 50.f);
glEnd();

//move to top left corner
glTranslatef(0.f, -SPACE_HEIGHT, 0.f);

glColor3f(1.f, 1.f, 1.f);
glVertex2f(0.f, 0.f);
glVertex2f(50.f, 0.f);
glVertex2f(50.f, 50.f);
glVertex2f(0.f, 50.f);
glEnd();

//move to top right corner
glTranslatef(SPACE_WIDTH, 0.f, 0.f);

glColor3f(1.f, 0.f, 1.f);
glVertex2f(0.f, 0.f);
glVertex2f(50.f, 0.f);
glVertex2f(50.f, 50.f);
glVertex2f(0.f, 50.f);
glEnd();
glPopMatrix();

glFlush();

glutSwapBuffers();
}

and heres' gSpaceship.render(), and consequently, JTexture.render():

void Spaceship::render()
{
//mGraphics is a JTexture
//m_xPos and m_yPos start out in the center of the 4 quads
mGraphics.render(m_xPos, m_yPos);
}

void JTexture::render(GLfloat x, GLfloat y)
{
if (mTextureID != 0)
{
glTranslatef(x, y, 0.f);

glBindTexture(GL_TEXTURE_2D, mTextureID);

glTexCoord2f(0.f, 0.f); glVertex2f(0.f, 0.f );
glTexCoord2f(1.f, 0.f); glVertex2f(mTextureWidth, 0.f );
glTexCoord2f(1.f, 1.f); glVertex2f(mTextureWidth, mTextureHeight);
glTexCoord2f(0.f, 1.f); glVertex2f(0.f, mTextureHeight);
glEnd();
}
}

also, I'd like to eventually be able to rotate the texture and move in the direction its facing; I know how to rotate a texture, but I know I will be very lost when it comes to moving in a direction thats not axis-aligned.  How does one go about this?

Thanks a lot!

#1johnmarinelli

Posted 30 August 2013 - 08:01 PM

Hi everyone,

I know this question has been asked a few times before, but I still can't get my program to work.

I'm working on an Asteroids clone in OpenGL, and I'm starting off by simply getting my spaceship to move around.  I also have 4 quads set up in each respective corner so I know if the movement is working or not.

However, it seems like I'm moving the quads around, and not my spaceship.  Here's the code:

void renderScene()
{
glClear(GL_COLOR_BUFFER_BIT);

glMatrixMode(GL_MODELVIEW);

glPushMatrix();

gSpaceship.render();

glPopMatrix();

glPushMatrix();
//move to bottom right corner
glTranslatef(SPACE_WIDTH / 2.f, SPACE_HEIGHT / 2.f, 0.f);

glColor3f(0.f, 1.f, 0.f);
glVertex2f(0.f, 0.f);
glVertex2f(50.f, 0.f);
glVertex2f(50.f, 50.f);
glVertex2f(0.f, 50.f);
glEnd();

//move to bottom left corner
glTranslatef(-SPACE_WIDTH, 0.f, 0.f);

glColor3f(0.f, 0.f, 1.f);
glVertex2f(0.f, 0.f);
glVertex2f(50.f, 0.f);
glVertex2f(50.f, 50.f);
glVertex2f(0.f, 50.f);
glEnd();

//move to top left corner
glTranslatef(0.f, -SPACE_HEIGHT, 0.f);

glColor3f(1.f, 1.f, 1.f);
glVertex2f(0.f, 0.f);
glVertex2f(50.f, 0.f);
glVertex2f(50.f, 50.f);
glVertex2f(0.f, 50.f);
glEnd();

//move to top right corner
glTranslatef(SPACE_WIDTH, 0.f, 0.f);

glColor3f(1.f, 0.f, 1.f);
glVertex2f(0.f, 0.f);
glVertex2f(50.f, 0.f);
glVertex2f(50.f, 50.f);
glVertex2f(0.f, 50.f);
glEnd();
glPopMatrix();

glFlush();

glutSwapBuffers();
}

and heres' gSpaceship.render(), and consequently, JTexture.render():

void Spaceship::render()
{
//mGraphics is a JTexture
mGraphics.render(m_xPos, m_yPos);
}

void JTexture::render(GLfloat x, GLfloat y)
{
if (mTextureID != 0)
{
glTranslatef(x, y, 0.f);

glBindTexture(GL_TEXTURE_2D, mTextureID);

glTexCoord2f(0.f, 0.f); glVertex2f(0.f, 0.f );
glTexCoord2f(1.f, 0.f); glVertex2f(mTextureWidth, 0.f );
glTexCoord2f(1.f, 1.f); glVertex2f(mTextureWidth, mTextureHeight);
glTexCoord2f(0.f, 1.f); glVertex2f(0.f, mTextureHeight);
glEnd();
}
}

also, I'd like to eventually be able to rotate the texture and move in the direction its facing; I know how to rotate a texture, but I know I will be very lost when it comes to moving in a direction thats not axis-aligned.  How does one go about this?

Thanks a lot!

PARTNERS


text:
# Where does Tris finish in the first stage of initiation?

Where does Tris finish in the first stage of initiation?


text:
## Basic College Mathematics (9th Edition)

The tally is 5 "sticks" and the class frequency is $5$.
There are 5 numbers from 120 and 129 in the data list. Thus, the tally is 5 "sticks" and the class frequency is $5$.


text:
The following terminology is used throughout this Design Section:

Warning: REQUIRED - Indicates design standards stipulated by the MPCA Construction General Permit (CGP)(or other consistently applicable regulations)

HIGHLY RECOMMENDED - Indicates design guidance that is extremely beneficial or necessary for proper functioning of theinfiltration practice, but not specifically required by the MPCA CGP.

RECOMMENDED - Indicates design guidance that is helpful for infiltration practice performance but not critical to the design.

## Major design elements

### Physical feasibility initial check

Drainage Area: It is HIGHLY RECOMMENDED that the following infiltration practices be designed with the indicated maximum drainage areas.

• Dry well – 1 acre.
• Infiltration Trench – 5 acres.
• Underground Infiltration System – 10 acres.
• Infiltration Basin – between 5 and 50 acres.

Site Topography and Slopes: Unless slope stability calculations demonstrate otherwise, it is HIGHLY RECOMMENDED that infiltration practices be located a minimum horizontal distance of 200 feet from down-gradient slopes greater than 20 percent, and that slopes in contributing drainage areas be limited to 15 percent.

Soils: It is HIGHLY RECOMMENDED that native soils in proposed infiltration areas have a minimum infiltration rate of 0.2 inches per hour (typically Hydrologic Soil Group A, B and C soils). Initially, soil infiltration rates can be estimated from NRCS soil data, and confirmed with an on-site infiltration evaluation or geotechnical investigation (see Step 6 of the Design procedures section for investigation procedures). It is HIGHLY RECOMMENDED that native soils have silt/clay contents less than 40 percent and clay content less than 20 percent, and that infiltration practices not be situated in fill soils.

Warning: It is REQUIRED that impervious area construction be completed and pervious areas established with dense and healthy vegetation prior to introduction of stormwater into an infiltration practice.

Depth to groundwater table and bedrock:

Warning: A separation distance of 3 feet is REQUIRED between the bottom of the infiltration practice and the elevation of the seasonally high water table (saturated soil) or top of bedrock (i.e. there must be a minimum of 3 feet of undisturbed soil beneath the infiltration practice and the seasonally high water table or top of bedrock). See also Step 8 under the Design procedures section.

Local authorities may require greater separation depths. Site Location / Minimum Setbacks: It is HIGHLY RECOMMENDED that infiltration practices not be hydraulically connected to structure foundations or pavement, to avoid seepage and frost heave concerns, respectively. If groundwater contamination is a concern, it is RECOMMENDED that groundwater mapping be conducted to determine possible connections to adjacent groundwater wells.

Warning: The minimum setback distance from a stormwater infiltration system to a community public water-supply well is 50 feet as REQUIRED by the Minnesota Department of Health. The setback is 35 feet to all other water-supply wells.

Recommended minimum setback requirements. This represents the minimum distance from the infiltration practice to the structure of concern. If the structure is aboveground, the distance is measured from the edge of the permeable pavement to the structure. If the structure is underground, the setback distance represents the distance from the point of infiltration through the bottom of the permeable pavement system to the structure.

Setback from Minimum Distance [feet]
Property Line 10
Building Foundation* 10
Private Well 50
Septic System Tank/Leach Field 35
* Minimum with slopes directed away from the building.

Karst: It is HIGHLY RECOMMENDED that infiltration practices not be used in active karst formations without adequate geotechnical testing.

Wellhead Protection Areas: It is HIGHLY RECOMMENDED to review the Minnesota Department of Health guidance on stormwater infiltration in Wellhead Protection Areas.

### Conveyance

It is HIGHLY RECOMMENDED that a flow splitter or diversion structure be provided to divert the Vwq to the infiltration practice and allow larger flows to bypass the practice, unless the infiltration practice is sized to retain Vcp, Vp10 or Vp100. Where a flow splitter is not used, it is HIGHLY RECOMMENDED that contributing drainage areas be limited to the appropriate size given the BMP and an overflow be provided within the practice to pass part of the Vwq to a stabilized watercourse or storm drain. It is also HIGHLY RECOMMENDED that overflow associated with the Vp10 or Vp100 storm (depending on local drainage criteria) be controlled such that velocities are non-erosive at the outlet point (to prevent downstream slope erosion), and that when discharge flows exceed 3 cubic feet per second, the designer evaluate the potential for erosion to stabilized areas and infiltration facilities.

### Pre-treatment

Warning: It is REQUIRED that some form of pre-treatment, such as a plunge pool, sump pit, filter strip, sedimentation basin, grass channel, or a combination of these practices be installed upstream of the infiltration practice.

It is HIGHLY RECOMMENDED that the following pre-treatment sizing guidelines be followed:

• Before entering an infiltration practice, stormwater should first enter a pre-treatment practice sized to treat a minimum volume of 25 percent of the Vwq.
• If the infiltration rate of the native soils exceeds 2 inches per hour a pre-treatment practice capable of treating a minimum volume of 50 percent of the Vwq should be installed.
• If the infiltration rate of the native soils exceeds 5 inches per hour a pre-treatment practice capable of treating a minimum volume of 100 percent of the Vwq should be installed.

It is HIGHLY RECOMMENDED that pre-treatment practices be designed such that exit velocities from the pre-treatment systems are non-erosive (less than 3 feet per second) and flows are evenly distributed across the width of the practice (e.g., by using a level spreader).

Caution: - Use low-impact earth moving equipment - DO NOT Overexcavate

### Treatment

Space Occupied: Space varies depending on the depth of the practice. Typically, infiltration trenches are 3 to 12 feet deep with a width less than 25 feet. A dry well is essentially a smaller version of an infiltration trench, consistent with the fact that the drainage area to an infiltration trench is typically five times greater (or larger) than that of a dry well. Underground infiltration systems are larger practices that range in depth from approximately 2 to 12 feet. The surface area of all infiltration practices is a function of MPCA’s 48-hour drawdown requirement and the infiltration capacity of the underlying soils.

Practice Slope: It is RECOMMENDED that the bottom of all infiltration practices be flat, in order to enable even distribution and infiltration of stormwater. It is RECOMMENDED that the longitudinal slope range only from the ideal 0 percent up to 1 percent, and that lateral slopes be held at 0 percent.

Side Slopes: It is HIGHLY RECOMMENDED that the maximum side slopes for an infiltration practice be 1:3 (V:H).

Schematic showing the effective infiltration area for side slopes less than 1V:3H.

Depth: The depth of an infiltration practice is a function of the maximum drawdown time and the design infiltration rate.

Warning: The REQUIRED drawdown time for infiltration practices is 48 hours or less, and so the depth of the practice should be determined accordingly.
Warning: Groundwater Protection: It is REQUIRED that runoff from potential stormwater hotspots (PSHs) not be infiltrated unless adequate pre-treatment has been provided. Infiltration of runoff from confirmed hotspot areas, industrial areas with exposed significant materials, or vehicle fueling and maintenance areas is PROHIBITED.

Aesthetics: infiltration basins can be effectively integrated into the site planning process, and aesthetically designed as attractive green spaces planted with native vegetation. Infiltration trenches are less conducive to site aesthetics, but the surface of trenches can be designed with turf cover crops if desired.

### Landscaping

Warning: It is REQUIRED that impervious area construction be completed and pervious areas established with dense and healthy vegetation prior to introduction of stormwater into an infiltration practice.

It is RECOMMENDED that vegetation associated with infiltration practices be established to blend into the surrounding area, that native species be used wherever possible. It is HIGHLY RECOMMENDED that deep rooted plants such as prairie grass be used, because they increase the infiltration capacity of the underlying soils. Dry wells and infiltration trenches can be covered with permeable topsoil and planted with grass to match the surrounding landscape.

Due to soil compaction concerns, it is HIGHLY RECOMMENDED that infiltration areas not be used for recreational purposes unless a soil amendment is used to off-set compaction.

It is HIGHLY RECOMMENDED that vegetation associated with infiltration practices be regularly maintained and bare areas seeded. Mowing practices can be used to maintain native vegetation.

It is RECOMMENDED that soil testing be conducted in infiltration practices, to determine if fertilizer application is warranted. Incorporating mulch or compost into the soil or planting with salt tolerant grasses can counter soil fertility problems caused by high chloride concentrations.

It is HIGHLY RECOMMENDED that designs include an observation well consisting of an anchored six-inch diameter perforated PVC pipe fitted with a cap to facilitate periodic inspection and maintenance.

### Safety

Dry wells, infiltration trenches and subsurface infiltration systems do not pose any major safety hazards. Infiltration basins should have similar side slope considerations as ponds and wetlands.

Warning: If a dry well or infiltration trench is greater than five feet deep, it is REQUIRED that OSHA health and safety guidelines be followed for safe construction practices.

Additional information on safety for construction sites is available from OSHA.

## Design procedure

The following steps outline a recommended design procedure for infiltration practices in compliance with the MPCA Permit for new construction. Design recommendations beyond those specifically required by the permit are also included and marked accordingly.

When riser pipe outlets are used in infiltration basins, it is HIGHLY RECOMMENDED that they be constructed with manholes that either have locks or are sufficiently heavy to prevent easy removal.

Fencing of dry wells and infiltration trenches is neither necessary nor desirable. Infiltration basins may warrant fencing in some situations.

### Design steps

#### Step 1. Make a preliminary judgement

Make a preliminary judgment as to whether site conditions are appropriate for the use of an infiltration practice, and identify the function of the practice in the overall treatment system.

A. Consider basic issues for initial suitability screening, including the following (see physical feasibility for detailed information).

• site drainage area (See the Summary of infiltration practices for given drainage areas table below);
• site topography and slopes;
• soil infiltration capacity;
• regional or local depth to groundwater and bedrock;
• site location/ minimum setbacks; and
• presence of active karst.

B. Determine how the infiltration practice will fit into the overall stormwater treatment system.

• Decide whether the infiltration practice is the only BMP to be employed, or if are there other BMPs addressing some of the treatment requirements.
• Decide where on the site the infiltration practice is most likely to be located.

Summary of infiltration practices for given drainage areas.

Infiltration Practice Drainage Area
Infiltration Basin 5 to 50 acres
Infiltration Trench less than 5 acres
Dry Wells less than one acre
Underground Infiltration Systems less than 10 acres

#### Step 2. Confirm design criteria and applicability

A. Determine whether the infiltration practice must comply with the MPCA Permit.

B. Check with local officials, watershed organizations, and other agencies to determine if there are any additional restrictions and/or surface water or watershed requirements that may apply.

#### Step 3. Perform field verification of site suitability

If the initial evaluation indicates that an infiltration practice would be a good BMP for the site, it is RECOMMENDED that a minimum of 3 soil borings or pits be dug (in the same location as the proposed infiltration practice) to verify soil types and infiltration capacity characteristics and to determine the depth to groundwater and bedrock.

It is RECOMMENDED that the minimum depth of the soil borings or pits be 5 feet below the bottom elevation of the proposed infiltration practice.

It is HIGHLY RECOMMENDED that soil profile descriptions be recorded and include the following information for each soil horizon or layer (Source: Site Evaluation for Stormwater Infiltration, Wisconsin Department of Natural Resources Conservation Practice Standards, 2004):

• thickness, in inches or decimal feet;
• Munsell soil color notation;
• soil mottle or redoximorphic feature color, abundance, size and contrast;
• USDA soil textural class with rock fragment modifiers;
• soil structure, grade size and shape;
• soil consistency, root abundance and size;
• soil boundary; and
• occurrence of saturated soil, impermeable layers/lenses, groundwater, bedrock or disturbed soil.

It is HIGHLY RECOMMENDED that the field verification be conducted by a qualified geotechnical professional.

#### Step 4. Compute runoff control volumes

Calculate the Water Quality Volume (Vwq), Channel Protection Volume (Vcp), Overbank Flood Protection Volume (Vp10), and the Extreme Flood Volume (Vp100) (see Unified sizing criteria).

Warning: If the infiltration practice is being designed to meet the requirements of the MPCA Permit, the REQUIRED treatment volume is the water quality volume of 1 inch of runoff from the new impervious surfaces created from the project.

If part of the overall Vwq is to be treated by other BMPs, subtract that portion from the Vwq to determine the part of the Vwq to be treated by the infiltration practice.

The design techniques in this section are meant to maximize the volume of stormwater being infiltrated. If the site layout and underlying soil conditions permit, a portion of the Channel Protection Volume (Vcp), Overbank Flood Protection Volume (Vp10), and the Extreme Flood Volume (Vp100) may also be managed in the infiltration practice (see Step 7).

#### Step 5. Select design variant based on physical suitability evaluation

Once the Physical Suitability Evaluation is complete, it is HIGHLY RECOMMENDED that the designer apply the better site design principles in sizing and locating the infiltration practice(s) on the development site. Given the water quality volume and the drainage area, select the appropriate infiltration practice for the first iteration of the design process.

Note: Information collected during the site suitability evaluation (see Steps 1 and 3) should be used to explore the potential for multiple infiltration practices versus relying on a single infiltration facility. The use of smaller infiltration practices dispersed around a development is usually more sustainable than a single regional facility that is more likely to have maintenance and groundwater mounding problems (Source: Site Evaluation for Stormwater Infiltration, Wisconsin Department of Natural Resources Conservation Practice Standards, 2004).

#### Step 6. Determine infiltration rate and size infiltration practice (Note: Steps 6, 7, 8 and 9 are iterative)

After following the steps outlined above, the designer will presumably know the location of naturally occurring permeable soils, the depth to the water table, bedrock or other impermeable layer, and the contributing drainage area. While the first step in sizing an infiltration practice is selecting the type of infiltration practice for the site, the basic design procedures are very similar.

Infiltration Location: Given the steps performed in the Physical Suitability Evaluation, identify the most suitable location for the infiltration practice.
Warning: Pre-treatment prior to infiltration is REQUIRED to remove total suspended solids and other pollutants associated with stormwater (see Step 9)
Determine infiltration rate: For design purposes, there are two ways of determining the soil infiltration rate. The first, and preferred method, is to field-test the soil infiltration rate using appropriate methods described below. The other method uses the typical infiltration rate of the most restrictive underlying soil (determined during soil borings).

If infiltration rate measurements are made, a minimum of one infiltration test in a soil pit must be completed at the elevation from which exfiltration would occur (i.e. interface of gravel drainage layer and in situ soil). When the SCM surface area is between 1000 and 5000 square feet, two soil pit measurements are needed. Between 5000 and 10000 square feet of surface area, a total of three soil pit infiltration measurements should be made. Each additional 5000 square feet of surface area triggers an additional soil pit.

Recommended number of soil borings, pits or permeameter tests for bioretention design. Designers select one of these methods.

Surface area of stormwater control measure (BMP)(ft2) Borings Pits Permeameter tests
< 1000 1 1 5
1000 to 5000 2 2 10
5000 to 10000 3 3 15
>10000 41 41 202

1an additional soil boring or pit should be completed for each additional 2,500 ft2 above 12,500 ft2
2an additional five permeameter tests should be completed for each additional 5,000 ft2 above 15,000 ft2

The median measured infiltration rate should be utilized for design. Soil pits should be dug during the design phase and should be a minimum of two feet in diameter for measurement of infiltration rate. Infiltration testing in the soil pit can be completed with a double-ring infiltrometer or by filling the pit with water and measuring stage versus time. If the infiltration rate in the first pit is greater than 2 inches per hour, no additional pits shall be needed.

Alternatively, a Modified Philip-Dunne permeameter can be used to field test infiltration rate. Modified Philip-Dunne permeameter tests may be made in conjunction with soil borings or may be completed using a handheld soil auger. Borings should be lined with a plastic sleeve to prevent infiltration from the sides of the borehole (i.e. restrict flow to vertical infiltration). Soil borings should be filled with water. The time for the borehole to drain should be recorded and divided by the initial ponding depth in the borehole to provide an infiltration rate measurement. The design infiltration rate should be the lower of the median soil pit infiltration rate or the median borehole method infiltration rate.

Illustration of how to determine the appropriate number of permeameter samples. When the standard deviation for all measurements flattens out with successive measurements, collection of additional permeameter tests may be halted, provided a minimum of 5 samples have been collected.

NOTE: In the table above, the recommended number of permeameter tests increases by 5 tests per each additional 5000 square feet of surface area. For larger sites, this can result in a very large number of samples. There may be situations where fewer permeameter tests may be used (5 is the minimum) . For example, in situations where the variability in saturated hydraulic conductivity between measurements is not great, fewer samples may be taken. One method for determining the number of samples is to plot standard deviation versus number of samples. Measurements may be halted when the standard deviation becomes relatively constant from one sample to the next. In the example to the right the standard deviation flattens at about 7 to 10 samples. Therefore, 7 to 10 samples would be an appropriate number of samples for this situation.

For information on conducting soil infiltration rate measurements, see Determining soil infiltration rates.

If the infiltration rate is not measured, use the table below to estimate an infiltration rate for the design of infiltration practices. These infiltration rates represent the long-term infiltration capacity of a practice and are not meant to exhibit the capacity of the soils in the natural state.

Caution: Select the design infiltration rate from the table based on the least permeable soil horizon within the first 5 feet below the bottom elevation of the proposed infiltration practice

Caution: The table for design infiltration rates has been modified. Field testing is recommended for gravelly soils (HSG A; GW and GP soils; gravel and sandy gravel soils). If field-measured soil infiltration rates exceed 8.3 inches per hour, the Construction Stormwater permit requires the soils be amended. Guidance on amending these soils can be found here.

Design infiltration rates, in inches per hour, for A, B, C, and D soil groups. Corresponding USDA soil classification and Unified soil Classifications are included. Note that A and B soils have two infiltration rates that are a function of soil texture.*
The values shown in this table are for uncompacted soils. This table can be used as a guide to determine if a soil is compacted. For information on alleviating compacted soils, link here. If a soil is compacted, reduce the soil infiltration rate by one level (e.g. for a compacted B(SM) use the infiltration rate for a B(MH) soil).

Hydrologic soil group Infiltration rate (inches/hour) Infiltration rate (centimeters/hour) Soil textures Corresponding Unified Soil Classification
A
Although a value of 1.63 inches per hour (4.14 centimeters per hour) may be used, it is Highly recommended that you conduct field infiltration tests or amend soils.b See Guidance for amending soils with rapid or high infiltration rates and Determining soil infiltration rates.

gravel
sandy gravel

GW - well-graded gravels, sandy gravels
GP - gap-graded or uniform gravels, sandy gravels
1.63a 4.14

silty gravels
gravelly sands
sand

GM - silty gravels, silty sandy gravels

0.8 2.03

sand
loamy sand
sandy loam

B
0.45 1.14 SM - silty sands, silty gravelly sands
0.3 0.76 loam, silt loam MH - micaceous silts, diatomaceous silts, volcanic ash
C
0.2 0.51 Sandy clay loam ML - silts, very fine sands, silty or clayey fine sands
D
0.06 0.15

clay loam
silty clay loam
sandy clay
silty clay
clay

GC - clayey gravels, clayey sandy gravels
SC - clayey sands, clayey gravelly sands
CL - low plasticity clays, sandy or silty clays
OL - organic silts and clays of low plasticity
CH - highly plastic clays and sandy clays
OH - organic silts and clays of high plasticity

*NOTE that this table has been updated from Version 2.X of the Minnesota Stormwater Manual. The higher infiltration rate for B soils was decreased from 0.6 inches per hour to 0.45 inches per hour and a value of 0.06 is used for D soils (instead of < 0.2 in/hr).
Source: Thirty guidance manuals and many other stormwater references were reviewed to compile recommended infiltration rates. All of these sources use the following studies as the basis for their recommended infiltration rates: (1) Rawls, Brakensiek and Saxton (1982); (2) Rawls, Gimenez and Grossman (1998); (3) Bouwer and Rice (1984); and (4) Urban Hydrology for Small Watersheds (NRCS). SWWD, 2005, provides field documented data that supports the proposed infiltration rates. (view reference list)
aThis rate is consistent with the infiltration rate provided for the lower end of the Hydrologic Soil Group A soils in the Stormwater post-construction technical standards, Wisconsin Department of Natural Resources Conservation Practice Standards.
bThe infiltration rates in this table are recommended values for sizing stormwater practices based on information collected from soil borings or pits. A group of technical experts developed the table for the original Minnesota Stormwater Manual in 2005. Additional technical review resulted in an update to the table in 2011. Over the past 5 to 7 years, several government agencies revised or developed guidance for designing infiltration practices. Several states now require or strongly recommend field infiltration tests. Examples include North Carolina, New York, Georgia, and the City of Philadelphia. The states of Washington and Maine strongly recommend field testing for infiltration rates, but both states allow grain size analyses in the determination of infiltration rates. The Minnesota Stormwater Manual strongly recommends field testing for infiltration rate, but allows information from soil borings or pits to be used in determining infiltration rate. A literature review suggests the values in the design infiltration rate table are not appropriate for soils with very high infiltration rates. This includes gravels, sandy gravels, and uniformly graded sands. Infiltration rates for these geologic materials are higher than indicated in the table.
References: Clapp, R. B., and George M. Hornberger. 1978. Empirical equations for some soil hydraulic properties. Water Resources Research. 14:4:601–604; Moynihan, K., and Vasconcelos, J. 2014. SWMM Modeling of a Rural Watershed in the Lower Coastal Plains of the United States. Journal of Water Management Modeling. C372; Rawls, W.J., D. Gimenez, and R. Grossman. 1998. Use of soil texture, bulk density and slope of the water retention curve to predict saturated hydraulic conductivity Transactions of the ASAE. VOL. 41(4): 983-988; Saxton, K.E., and W. J. Rawls. 2005. Soil Water Characteristic Estimates by Texture and Organic Matter for Hydrologic Solutions. Soil Science Society of America Journal. 70:5:1569-1578.

The infiltration capacity and existing hydrologic regime of natural basins are inherently different than constructed practices and may not meet MPCA Permit requirements for constructed practices. In the event that a natural depression is being proposed to be used as an infiltration system, the design engineer must demonstrate the following information:

• infiltration capacity of the system under existing conditions (inches per hour)
• existing drawdown time for the high water level (HWL) and a natural overflow elevation.

The design engineer should also demonstrate that operation of the natural depression under post-development conditions mimics the hydrology of the system under pre-development conditions.

If the infiltration rates are measured, the tests shall be conducted at the proposed bottom elevation of the infiltration practice. If the infiltration rate is measured with a double-ring infiltrometer the requirements of ASTM D3385 (Standard test method for infiltration rate of soils in field using double-ring infiltrometer) should be used for the field test.

Warning: The measured infiltration rate shall be divided by a safety factor of 2.

The safety factor of 2 adjusts the measured infiltration rates for the occurrence of less permeable soil horizons below the surface and the potential variability in the subsurface soil horizons throughout the infiltration site. This safety factor also accounts for the long-term infiltration capacity of the stormwater management facility.

To meet requirements of the Stormwater General Permit (CSW permit), the surface area (As, in square feet) of an infiltration practice is given by

$$A_s = V_w / D_o$$

Where:
Vw = the water treatment volume of the area contributing runoff to the practice; and
Do = the storage depth of ponded water in the practice.

The water treatment volume is given by

$$V_w = 0.0833 A_c$$

Where
0.0833 = the equivalen of 1 inch of volume control, in feet, as required by the permit; and
Ac = the impervious surface area contributing to the practice.

The entire water treatment volume is assumed to instantaneously pond in the bioinfiltration practice.

Schematic illustrating dimensions used for sizing an infiltration practice. Since the volume is calculated as an instantaneous volume, the depth (DO) represents a ponded depth and AM is the area at the top of the media or soil.

For an infiltration BMP with sloped sides, the surface area (As) is the average area of the BMP, given by

$$A_s = (A_o + A_M)/2$$

Where
Ao is the surface area at the overflow; and
AM is the surface area at the top of the infiltration media

The water treatment volume must drain with 48 hours (24 hours is RECOMMENDED if discharges from the practice are to a trout stream). The ponding depth can therefore be calculated knowing the infiltration rate of the soils underlying the practice and is given by

$$D = I_R t$$

where

D = maximum depth of practice (inches);
IR = infiltration rate of underlying soils (inches/hour); and
t = maximum drawdown time (48 hours).

Field-measured infiltration rates are preferred. If the infiltration rate has not been measured, use the table below to determine the infiltration rate of the underlying soils. The ponded depth must not exceed 3 feet regardless of the soil infiltration rate. Note the numbers in the table are intentionally conservative based on experience gained from Minnesota infiltration sites.

Caution: The table for design infiltration rates has been modified. Field testing is recommended for gravelly soils (HSG A; GW and GP soils; gravel and sandy gravel soils). If field-measured soil infiltration rates exceed 8.3 inches per hour, the Construction Stormwater permit requires the soils be amended. Guidance on amending these soils can be found here.

Design infiltration rates, in inches per hour, for A, B, C, and D soil groups. Corresponding USDA soil classification and Unified soil Classifications are included. Note that A and B soils have two infiltration rates that are a function of soil texture.*
The values shown in this table are for uncompacted soils. This table can be used as a guide to determine if a soil is compacted. For information on alleviating compacted soils, link here. If a soil is compacted, reduce the soil infiltration rate by one level (e.g. for a compacted B(SM) use the infiltration rate for a B(MH) soil).

Hydrologic soil group Infiltration rate (inches/hour) Infiltration rate (centimeters/hour) Soil textures Corresponding Unified Soil Classification
A
Although a value of 1.63 inches per hour (4.14 centimeters per hour) may be used, it is Highly recommended that you conduct field infiltration tests or amend soils.b See Guidance for amending soils with rapid or high infiltration rates and Determining soil infiltration rates.

gravel
sandy gravel

GW - well-graded gravels, sandy gravels
GP - gap-graded or uniform gravels, sandy gravels
1.63a 4.14

silty gravels
gravelly sands
sand

GM - silty gravels, silty sandy gravels

0.8 2.03

sand
loamy sand
sandy loam

B
0.45 1.14 SM - silty sands, silty gravelly sands
0.3 0.76 loam, silt loam MH - micaceous silts, diatomaceous silts, volcanic ash
C
0.2 0.51 Sandy clay loam ML - silts, very fine sands, silty or clayey fine sands
D
0.06 0.15

clay loam
silty clay loam
sandy clay
silty clay
clay

GC - clayey gravels, clayey sandy gravels
SC - clayey sands, clayey gravelly sands
CL - low plasticity clays, sandy or silty clays
OL - organic silts and clays of low plasticity
CH - highly plastic clays and sandy clays
OH - organic silts and clays of high plasticity

*NOTE that this table has been updated from Version 2.X of the Minnesota Stormwater Manual. The higher infiltration rate for B soils was decreased from 0.6 inches per hour to 0.45 inches per hour and a value of 0.06 is used for D soils (instead of < 0.2 in/hr).
Source: Thirty guidance manuals and many other stormwater references were reviewed to compile recommended infiltration rates. All of these sources use the following studies as the basis for their recommended infiltration rates: (1) Rawls, Brakensiek and Saxton (1982); (2) Rawls, Gimenez and Grossman (1998); (3) Bouwer and Rice (1984); and (4) Urban Hydrology for Small Watersheds (NRCS). SWWD, 2005, provides field documented data that supports the proposed infiltration rates. (view reference list)
aThis rate is consistent with the infiltration rate provided for the lower end of the Hydrologic Soil Group A soils in the Stormwater post-construction technical standards, Wisconsin Department of Natural Resources Conservation Practice Standards.
bThe infiltration rates in this table are recommended values for sizing stormwater practices based on information collected from soil borings or pits. A group of technical experts developed the table for the original Minnesota Stormwater Manual in 2005. Additional technical review resulted in an update to the table in 2011. Over the past 5 to 7 years, several government agencies revised or developed guidance for designing infiltration practices. Several states now require or strongly recommend field infiltration tests. Examples include North Carolina, New York, Georgia, and the City of Philadelphia. The states of Washington and Maine strongly recommend field testing for infiltration rates, but both states allow grain size analyses in the determination of infiltration rates. The Minnesota Stormwater Manual strongly recommends field testing for infiltration rate, but allows information from soil borings or pits to be used in determining infiltration rate. A literature review suggests the values in the design infiltration rate table are not appropriate for soils with very high infiltration rates. This includes gravels, sandy gravels, and uniformly graded sands. Infiltration rates for these geologic materials are higher than indicated in the table.
References: Clapp, R. B., and George M. Hornberger. 1978. Empirical equations for some soil hydraulic properties. Water Resources Research. 14:4:601–604; Moynihan, K., and Vasconcelos, J. 2014. SWMM Modeling of a Rural Watershed in the Lower Coastal Plains of the United States. Journal of Water Management Modeling. C372; Rawls, W.J., D. Gimenez, and R. Grossman. 1998. Use of soil texture, bulk density and slope of the water retention curve to predict saturated hydraulic conductivity Transactions of the ASAE. VOL. 41(4): 983-988; Saxton, K.E., and W. J. Rawls. 2005. Soil Water Characteristic Estimates by Texture and Organic Matter for Hydrologic Solutions. Soil Science Society of America Journal. 70:5:1569-1578.

#### Step 7. Size outlet structure and/or flow diversion structure, if needed (Note: Steps 6, 7, 8 and 9 are iterative)

It is HIGHLY RECOMMENDED that the outlet for the infiltration practice shall safely convey stormwater using all of the following mechanisms (Infiltration Basin, Wisconsin Department of Natural Resources Conservation Practice Standard, 10/04).

Drawdown valve: infiltration systems may be designed with a drawdown valve for the removal of standing water for maintenance and winter diversion.
Emergency spillway:
Warning: A means to release discharge in excess of the infiltration volume safely into the downstream stormwater conveyance system is REQUIRED.
Freeboard: It is HIGHLY RECOMMENDED that two feet of freeboard be provided from the 100-year flood elevation of the infiltration practice to the lowest basement floor elevation of residential, commercial, industrial and institutional buildings located adjacent to the BMP, unless local requirements recommend otherwise.
Drop Structure: Infiltration trenches or subsurface infiltration systems may be designed with a drop structure sized to handle the overflow. This additional volume of stormwater may be directed into the existing stormwater system or it may be diverted to a downstream BMP.

#### Step 8. Perform ground-water mounding analysis (Note: Steps 6, 7, 8 and 9 are iterative)

Groundwater mounding, the process by which a mound of water forms on the water table as a result of recharge at the surface, can be a limiting factor in the design and performance of infiltraand seasonally saturated soils (or from bedrock) is required (5 feet RECOMMENDED) to maintain the hydraulic capacity of the practice and provide adequate water quality treatment. A groundwater mounding analysis is RECOMMENDED to verify this separation for infiltration practices.

The most widely known and accepted analytical methods to solve for groundwater mounding are based on the work by Hantush (1967) and Glover (1960). The maximum groundwater mounding potential should be determined through the use of available analytical and numerical methods. Detailed groundwater mounding analysis should be conducted by a trained hydrogeologist or equivalent as part of the site design procedure.

#### Step 9. Determine pre-treatment volume and design pre-treatment measures (Note: Steps 6, 7, 8 and 9 are iterative)

See the section on pre-treatment for specific pre-treatment design guidance.

#### Step 10. Check volume, peak discharge rates and period of inundation against State, local and watershed organization requirements (Note: Steps 6, 7, 8 and 9 are iterative)

Follow the design procedures identified in the unified sizing criteria section of the Manual to determine the volume control and peak discharge requirements for water quality, recharge, channel protection, overbank flood and extreme storm.

Perform hand calculations or model the proposed development scenario using a surface water model appropriate for the hydrologic and hydraulic design considerations specific to the site (see also the section on stormwater modeling). This includes defining the parameters of the infiltration practice defined above: elevation and area (defines the storage volume), infiltration rate and method of application (effective infiltration area), and outlet structure and/or flow diversion information. The results of this analysis can be used to determine whether or not the proposed design meets the applicable requirements. If not, the design will have to be re-evaluated (back to Step 5).

The following items are specifically REQUIRED by the MPCA Permit

Warning:
• Volume - Infiltration or filtration systems shall be sufficient to infiltrate or filter a water quality volume of 1 inch of runoff from the new impervious surfaces created by the project. If this criterion is not met, increase the storage volume of the practice or treat excess water quality volume (Vwq) in an upstream or downstream BMP (see Step 5). Retrofit and supplemental systems do not need to meet this requirement, provided new impervious surfaces are not created.
• Peak Discharge Rates - Since most infiltration systems are not designed for quantity control they generally do not have peak discharge limits. However outflow must be limited such that erosion does not occur down gradient.
• Period of Inundation - Infiltration practices shall discharge through the soil or filter media in 48 hours or less. Additional flows that cannot be infiltrated or filtered in 48 hours should be routed to bypass the system through a stabilized discharge point. The period of inundation is defined as the time from the high water level in the practice to 1 to 2 inches above the bottom of the facility. This criterion was established to provide the following: wet-dry cycling between rainfall events; unsuitable mosquito breeding habitat; suitable habitat for vegetation; aerobic conditions; and storage for back-to-back precipitation events.

Other design requirements may apply to a particular site. The applicant should confirm local design criteria and applicability (see Step 3).

#### Step 11. Prepare Vegetation and Landscaping Plan

A landscaping plan for an infiltration basin should be prepared to indicate how the enhanced swale system will be stabilized and established with vegetation. Landscape design should specify proper grass species and wetland plants based on specific site, soils and hydric conditions present along the channel. Further information on plant selection and use occurs in the Minnesota plant lists section.

#### Step 12. Prepare Operation and Maintenance (O&M) Plan

See Operation and Maintenance section for guidance on preparing an O&M plan.

#### Step 13. Prepare Cost Estimate

See Cost Considerations section for guidance on preparing a cost estimate that includes both construction and maintenance costs.


text:
# Boolean Alegebra

1. Feb 20, 2008

### needhelp83

if $$A \oplus B=C$$, then $$C \oplus B=A$$, and $$A \oplus C=B$$ (use substitution)

$$C \oplus B=A$$

$$C \overline{B} + \overline{C} B = A$$

$$(A \oplus B) \overline{B} + (\overline {A \oplus B}) B = A$$

$$(A \overline{B} + \overline {A} B ) \overline {B} + (AB+ \overline{A} \overline{B})B=A$$

$$A \overline{B} \overline {B} + \overline {A} B \overline{B} + ABB + \overline{A} \overline{B}B=A$$

$$AB + 0 + AB + 0 = A$$

AB = A

I don't know how to get A = A

Any help?

Last edited: Feb 20, 2008
2. Feb 20, 2008

### needhelp83

Am I doing the steps correctly?

3. Feb 21, 2008

### needhelp83

Anybody know how to solve

4. Feb 21, 2008

### CEL

You made a mistake in line 6. You should have:
$$A\overline{B} + 0 + AB + 0 = A$$

$$A(\overline{B} + B) = A$$

$$A.1 = A$$

$$A = A$$