text:
GAP

## 26 publications using GAP in the category "Topological groups, Lie groups"

[A19] Amicone, F., Classification of balanced toral elements of exceptional Lie algebras, J. Algebra, 524 (2019), 395–419.

[B01] Baker, M. D., Link complements and the Bianchi modular groups, Trans. Amer. Math. Soc., 353 (8) (2001), 3229–3246.

[BM99] Barbasch, D. and Moy, A., Classification of one $\rm K$-type representations, Trans. Amer. Math. Soc., 351 (10) (1999), 4245–4261.

[BJT20] Bocheński, M., Jastrz\cebski, P., and Tralle, A., Nonexistence of standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups, Math. Comp., 89 (323) (2020), 1487–1499.

[CN10] Campbell, P. S. and Nevins, M., Branching rules for ramified principal series representations of GL(3) over a $p$-adic field, Canad. J. Math., 62 (1) (2010), 34–51.

[CLY04] Chua, K. S., Lang, M. L., and Yang, Y., On Rademacher's conjecture: congruence subgroups of genus zero of the modular group, J. Algebra, 277 (1) (2004), 408–428.

[C05] Cochet, C., Kostka numbers and Littlewood-Richardson coefficients, in Integer points in polyhedra—geometry, number theory, algebra, optimization, Amer. Math. Soc., Providence, RI, Contemp. Math., 374 (2005), 79–89.

[CGL93] Cohen, A. M., Griess Jr. , R. L., and Lisser, B., The group $L(2,61)$ embeds in the Lie group of type $E_8$, Comm. Algebra, 21 (6) (1993), 1889–1907.

[CZ13] Coquereaux, R. and Zuber, J., Drinfeld doubles for finite subgroups of $\rm SU(2)$ and $\rm SU(3)$ Lie groups, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), Paper 039, 36.

[DG21] Dietrich, H. and de Graaf, W. A., Computing the real Weyl group, J. Symbolic Comput., 104 (2021), 1–14.

[E13] Essert, J., A geometric construction of panel-regular lattices for buildings of types $\tilde A_2$ and $\tilde C_2$, Algebr. Geom. Topol., 13 (3) (2013), 1531–1578.

[GAE07] Gross, D., Audenaert, K., and Eisert, J., Evenly distributed unitaries: on the structure of unitary designs, J. Math. Phys., 48 (5) (2007), 052104, 22.

[HM18] Harvey, J. A. and Moore, G. W., Conway subgroup symmetric compactifications of heterotic string, J. Phys. A, 51 (35) (2018), 354001, 35.

[LP02] Lansky, J. and Pollack, D., Hecke algebras and automorphic forms, Compositio Math., 130 (1) (2002), 21–48.

[L14] Levaillant, C., The Freedman group: a physical interpretation for the $SU(3)$-subgroup $D(18,1,1;2,1,1)$ of order 648, J. Phys. A, 47 (28) (2014), 285203, 29.

[LT18] Long, D. D. and Thistlethwaite, M. B., Zariski dense surface subgroups in $\rm SL(4,\Bbb Z)$, Exp. Math., 27 (1) (2018), 82–92.

[LP01] Lubotzky, A. and Pak, I., The product replacement algorithm and Kazhdan's property (T), J. Amer. Math. Soc., 14 (2) (2001), 347–363.

[M06] Moreau, A., Indice du normalisateur du centralisateur d'un élément nilpotent dans une algèbre de Lie semi-simple, Bull. Soc. Math. France, 134 (1) (2006), 83–117.

[NRT19] Navarro, G., Robinson, G. R., and Tiep, P. H., On real and rational characters in blocks, Int. Math. Res. Not. IMRN (7) (2019), 1955–1973.

[R17] Radu, N., A lattice in a residually non-Desarguesian $\tilde A_2$-building, Bull. Lond. Math. Soc., 49 (2) (2017), 274–290.

[R00] Reeder, M., Formal degrees and $L$-packets of unipotent discrete series representations of exceptional $p$-adic groups, J. Reine Angew. Math., 520 (2000), 37–93
(With an appendix by Frank Lübeck).

[SS16] Savchuk, D. M. and Sidki, S. N., Affine automorphisms of rooted trees, Geom. Dedicata, 183 (2016), 195–213.

[S02] Stroppel, M., Locally compact groups with few orbits under automorphisms, in Proceedings of the 16th Summer Conference on General Topology and its Applications (New York), Topology Proc., 26 (2001/02), 819–842.

[S18] Szczepański, A., Intersection forms of almost-flat 4-manifolds, Arch. Math. (Basel), 110 (5) (2018), 455–458.

[PR14] van Pruijssen, M. and Román, P., Matrix valued classical pairs related to compact Gelfand pairs of rank one, SIGMA Symmetry Integrability Geom. Methods Appl., 10 (2014), Paper 113, 28.

[Y19] Ye, S., Symmetries of flat manifolds, Jordan property and the general Zimmer program, J. Lond. Math. Soc. (2), 100 (3) (2019), 1065–1080.


text:
## 7.5 – Polynomial Equations and Applications

### Learning Objectives

• (7.5.1) – Solving polynomial equations
• The principle of zero products
• (7.5.2) – Applications of polynomial equations
• Projectile motion
• Using the Pythagorean theorem to find the lengths of a right triangle
• Geometric Applications
• Cost, revenue, and profit polynomials

# (7.5.1) – Solving polynomial equations

Not all of the techniques we use for solving linear equations will apply to solving polynomial equations. In this section we will introduce a method for solving polynomial equations that combines factoring and the zero product principle.

### The Principle of Zero Products

Zero

What if we told you that we multiplied two numbers together and got an answer of zero? What could you say about the two numbers? Could they be 2 and 5? Could they be 9 and 1? No! When the result (answer) from multiplying two numbers is zero, that means that one of them had to be zero. This idea is called the zero product principle, and it is useful for solving polynomial equations that can be factored.

### Principle of Zero Products

The Principle of Zero Products states that if the product of two numbers is 0, then at least one of the factors is 0. If $ab=0$, then either $a=0$ or $b=0$, or both $a$ and $b$ are 0.

Let’s start with a simple example.  We will factor a GCF from a binomial and apply the principle of zero products to solve a polynomial equation.

### Example

Solve:

$-t^2+t=0$

In the following video we show two more examples of using both factoring and the principle of zero products to solve a polynomial equation.

In the next video we show that you can factor a trinomial using methods previously learned to solve a quadratic equation.

You and I both know that it is rare to be given an equation to solve that has zero on one side, so let’s try another one.

### Example

Solve: $s^2-4s=5$

We will work through one more example that is similar to the one above, except this example has fractions.

### Example

Solve $y^2-5=-\frac{7}{2}y+\frac{5}{2}$

In our last video, we show how to solve another quadratic equation that contains fractions.

# (7.5.2) – Applications of polynomial equations

### Projectile Motion

Projectile motion happens when you throw a ball into the air and it comes back down because of gravity.  A projectile will follow a curved path that behaves in a predictable way.  This predictable motion has been studied for centuries, and in simple cases it’s height from the ground at a given time, $t$, can be modeled with a polynomial function of the form $h(t)=at^2+bt+c$, where h(t) = height of an object at a given time, $t$.  Projectile motion is also called a parabolic trajectory because of the shape of the path of a projectile’s motion, as in the image of water in the fountain below.

Parabolic WaterTrajectory

Parabolic motion and it’s related functions allow us to launch satellites for telecommunications, and rockets for space exploration. Recently, police departments have even begun using projectiles with GPS to track fleeing suspects in vehicles, rather than pursuing them by high-speed chase [1].

In this section we will use polynomial functions to answer questions about the parabolic motion of a projectile. The real mathematical model for the path of a rocket or a police GPS projectile may have different coefficients or more variables, but the concept remains the same. We will also learn to interpret the meaning of the variables in a polynomial function that models projectile motion.

### Example

A small toy rocket is launched from a 4-foot pedestal. The height ($h$, in feet) of the rocket $t$ seconds after taking off is given by the function $h(t)=−2t^{2}+7t+4$. How long will it take the rocket to hit the ground?

In the next example we will solve for the time that the rocket is at a given height other than zero.

### Example

Use the formula for the height of the rocket in the previous example to find the time when the rocket is 4 feet from hitting the ground on it’s way back down.  Refer to the image.

$h(t)=−2t^{2}+7t+4$

In the following video we show another example of how to find the time when a object following a parabolic trajectory hits the ground.

In this section we introduced the concept of projectile motion, and showed that it can be modeled with polynomial function.  While the models used in these examples are simple, the concepts and interpretations are the same as what would happen in “real life”.

### Pythagorean Theorem

The Pythagorean theorem or Pythagoras’s theorem is a statement about the sides of a right triangle. One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side. The image above shows four common kinds of triangle, including a right triangle.

Right Triangle Labeled

The Pythagorean theorem is often used to find unknown lengths of the sides of right triangles. If the longest leg of a right triangle is labeled c, and the other two a, and b as in the image on teh left,  The Pythagorean Theorem states that

$a^2+b^2=c^2$

Given enough information, we can solve for an unknown length.  This relationship has been used for many, many years for things such as celestial navigation and early civil engineering projects. We now have digital GPS and survey equipment that have been programmed to do the calculations for us.

In the next example we will combine the power of the Pythagorean theorem and what we know about solving quadratic equations to find unknown lengths of right triangles.

### Example

A right triangle has one leg with length $x$, another whose length is greater by two, and the length of the hypotenuse is greater by four.  Find the lengths of the sides of the triangle. Use the image below.

This video example shows another way a quadratic equation can be used to find and unknown length of a right triangle.

### Geometric Applications

In this section we will explore ways that polynomials are used in applications of area problems.

### Example

The length of a rectangle is 3 more than the width. If the area is 40 square inches, what are the dimensions?

In the following video you are shown how to find the dimensions of a border around a rectangle.

### Cost, Revenue, and Profit Polynomials

In the systems of linear equations section, we discussed how a company’s cost and revenue can be modeled with two linear equations. We found that the profit region for a company was the area between the two lines where the company would make money based on how much was produced. In this section, we will see that sometimes polynomials are used to describe cost and revenue.

Profit is typically defined in business as the difference between the amount of money earned (revenue) by producing a certain number of items and the amount of money it takes to produce that number of items. When you are in business, you definitely want to see profit, so it is important to know what your cost and revenue is.

Cell Phones

For example, let’s say that the cost to a manufacturer to produce a certain number of things is C and the revenue generated by selling those things is R.  The profit, P, can then be defined as

P = R-C

The example we will work with is a hypothetical cell phone manufacturer whose cost to manufacture x number of phones is $C=2000x+750,000$, and the Revenue generated from manufacturing x number of cell phones is $R=-0.09x^2+7000x$.

### Example

Define a Profit polynomial for the hypothetical cell phone manufacturer.

Mathematical models are great when you use them to learn important information.  The cell phone manufacturing company can use the profit equation to find out how much profit they will make given $x$ number of phones are manufactured.  In the next example, we will explore some profit values for this company.

### Example

Given the following numbers of cell phones manufactured, find the profit for the cell phone manufacturer:

1. x = 100 phones
2. x = 25,000 phones
3. x=60,000 phones

format: plain


text:
# Alphanumeric Hello World [closed]

Your goal is to write "Hello, World!" (minus the quotes). This is a , so most up votes wins. Code length will be used to break ties.
Anything goes, as long as it is within the following rules:

• All characters must be either letters or numbers, so you may only use characters in the string "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789".
• All code you use must be in one file, with the exception of imported modules/classes/whatever they're called in your favourite language.
• Only the standard libraries/frameworks for your language of choice are allowed (for example, Python's Math module is allowed, but Numpy, Scipy, and Pygame are not). I will allow TkInter as it is the de facto standard for GUIs in Python.
• No input is permitted, be it reading from stdin, reading a file, displying an image, or reading from the web.

+10 brownie points if you figure out how to do it in Java without bending the rules.

On your marks, get set, code!

EDIT: braces ({}), brackets ([]), and parentheses (()) are allowed as this would be pretty much impossible for most languages without them. I'm also removing the character limit rule. Hopefully this will make it more interesting.

EDIT^2: white space is also allowed. My brain isn't working properly, sorry. >.<

• Up to you, I'm just making sure. What about whitespace? Mar 5 '14 at 14:04
• Oh, right white space. My brain is clearly not functioning properly today. That should be allowed. One second, let me fix the rules.
– wec
Mar 5 '14 at 14:15
• So, no operators? And, most importantly, no ;? How can I write anything in C? Mar 5 '14 at 14:22
• esolangs.org Hello, World! Mar 5 '14 at 22:41
• +1 for figuring out how to exclude J in a natural way: 72 101 108 108 111 44 32 87 111 114 108 100 33{a. requires the period at the end to work, and a path through u: similarly requires a colon. Without . and : we are nothing. Mar 5 '14 at 23:32

Hello, World!
$ # Python: Pfff, who needs the numeric part of alphanumeric? from string import lowercase from string import uppercase from string import punctuation print uppercase[len(str([[[len(str())]]]))] print lowercase[len(str([[]]))] print lowercase[len(str([[[[[len(str())]]]]]))] print lowercase[len(str([[[[[len(str())]]]]]))] print lowercase[len(str([[[[[[[]]]]]]]))] print punctuation[len(str([[[[[len(str())]]]]]))] print print uppercase[len(str([[[[[[[[[[[]]]]]]]]]]]))] print lowercase[len(str([[[[[[[]]]]]]]))] print lowercase[len(str([[[[[[[[len(str())]]]]]]]]))] print lowercase[len(str([[[[[len(str())]]]]]))] print lowercase[len(str([len(str())]))] print punctuation[len(str())]  This is locale-dependant, so your milage may vary. However, if you really want to replicate this result, you could use the following instead of the import statements: lowercase="abcdefghijklmnopqrstuvwxyz" uppercase="ABCDEFGHIJKLMNOPQRSTUVWXYZ" punctuation="!\"#$%&'()*+,-./:;<=>?@[\]^_{|}~"

(this is not part of my submission, it's just sharing my locals for recreatability)

## Rebmu, 46 characters

pnUPPqHELLO01pnADsp12pnSPpnUPPqWORLD01pnADsp01

Strongly parallels @dragetun's solution in Rebol (Because Rebmu is a Rebol/Red teaching tool which is a "dialect" that aims to draw attention to the languages by rocking at Code Golf. :-P)

However, to get the missing characters, I did something more obvious. I just added to space, instead of going and looking through system strings. You can get add space 1 to get character literal #"!" and add space 12 to get character literal #",".

If we run this we can see it get "unmushed":

>> rebmu/debug/stats "pnUPPqHELLO01pnADsp12pnSPpnUPPqWORLD01prADsp01"
Original Rebmu string was: 46 characters.
Rebmu as mushed Rebol block molds to: 46 characters.
Unmushed Rebmu molds to: 62 characters.
Executing: [pn upp q hello 1 pn ad sp 12 pn sp pn upp q world 1 pr ad sp 1]
Hello, World!

Which is fairly logial:

; print with no newline the result of doing an uppercase/part
; on the quoting of the symbolic word hello, of only the first
; N characters (where N is 1)
pn upp q hello 1

; print the character you get when you add 12 to the space
; character literal, a.k.a. comma...again no newline

; print a space, no newline
pn sp

; do the same trick of printing the quoted word hello with the
; first letter converted to uppercase
pn upp q world 1

; print the character you get when you add 1 to the space
; character literal, a.k.a. exclamation point...with newline now

# Whitespace

Okay, we can use whitespace. It uses only Space, Tab, LineFeed and ignores any other characters.

With comment: S, T or L means Space, Tab or LineFeed respectively.

S S S T S S T   S S S L
T   L
S S S S S T T   S S T   S T L
T   L
S S S S S T T   S T T   S S L
T   L
S S S S S T T   S T T   S S L
T   L
S S S S S T T   S T T   T   T   L
T   L
S S S S S T S T T   S S L
T   L
S S S S S T S S S S S L
T   L
S S S S S T T   T   S T T   T   L
T   L
S S S S S T T   S T T   T   T   L
T   L
S S S S S T T   T   S S T   S L
T   L
S S S S S T T   S T T   S S L
T   L
S S S S S T T   S S T   S S L
T   L
S S S S S T S S S S T   L
T   L
S S L
L
L

## T-SQL, 52

Ah, this is better! Direct binary conversion, only 52 characters.

SELECT CAST(0x48656C6C6F2C20576F726C6421 AS VARCHAR)

Previously...

There's a much cleverer and golfier solution to this that would work if we could use = and + (Looking up standard built-in data in msdb and using that to construct an EXEC-able query string)

But since we can't, you get this naive approach:

SELECT CHAR(72)SELECT CHAR(101)SELECT CHAR(108)SELECT CHAR(108)SELECT CHAR(111)SELECT CHAR(44)SELECT CHAR(32)SELECT CHAR(87)SELECT CHAR(111)SELECT CHAR(114)SELECT CHAR(108)SELECT CHAR(100)SELECT CHAR(33)

• I know, the lack of concatenation operators is killing me... I'm trying to think of a way to do it with JavaScript... Mar 5 '14 at 21:37
• Yeah, although this challenge seems to produce mostly naive answers, it has definitely inspired some interesting brain-twisting to try to find interesting non-concatenative approaches. Mar 5 '14 at 21:46
• You can do better. Use PRINT rather than SELECT. And implicitly cast with LTRIM() or RTRIM(). I do something similar here.
– Muqo
Sep 24 '14 at 1:23

# Groovy

So updated. Much wat.

def wat(i) { print i
this
}
def wat(int i) { wat i as char }
def getProperty(String p) { p }
wat Hello wat 44 wat 32 wat World wat 33
`


text:
Consider a two-dimensional power-potential Hamiltonian in polar coordinates:

$H=pr2 2m +pφ2 2mr2 +αrk$

Hamilton-Jacobi theory determines the classical action from the first-order nonlinear differential equation

$H(r,φ; ∂S ∂r, ∂S ∂φ) +∂S ∂t =0$

where linear momenta are replaced by partial derivatives of the action with respect to their conjugate coordinates. The equation for the system under consideration is

$12m (∂S ∂r )2 +12m r2 (∂S ∂φ )2 +αrk +∂S ∂t =0$

Since the potential is a function of the radial variable only, the equation is separable if the action is a linear sum of the form

$S=Lφ -Et +f(r)$

where the coefficients of the first two terms are chosen for later physical significance. With this functional form the Hamilton-Jacobi equation becomes

$12m (∂f ∂r )2 +L22m r2 +αrk =E$

Solving for the partial derivative, the action can be written

$S=Lφ -Et +∫dr 2mE -2mαrk -L2 r2$

containing what is called the radial action integral.

The Hamilton-Jacobi equation is derived by treating the action as a generating function that induces a transformation of the Hamiltonian to generalized coordinates in which it is identically zero. If the Hamiltonian is identically zero,

$Q·i =∂H ∂Pi ≡0 P·i =−∂H ∂Qi ≡0$

then the new generalized coordinates are all constant. The new momenta are traditionally designated with alphas and can be taken as

$P1 =α1 =E P2 =α2 =L$

The new conjugate coordinates, traditionally designated with betas, are given by partial derivatives of the action:

$Qi =βi =∂S ∂αi$

For the system at hand, the first conjugate coordinate is

$β1 =∂S ∂E =−t+m∫ dr 2mE -2mαrk -L2 r2$

This constant is just an offset of the temporal variable and can be taken as zero. This gives an integral relation between two variables:

$t=m∫ dr 2mE -2mαrk -L2 r2$

Taking a derivative provides the differential relation between the two variables:

$drdt =1m 2mE -2mαrk -L2 r2$

Since this relation can also be derived by setting $pr =mr·$ and $pφ=L$ in the Hamiltonian, it can be designated the energy equation for this system. The temporal variable is thus determined by what can be called the energy integral.

The second conjugate coordinate is

$β2 =∂S ∂L =φ-L∫ dr r22mE -2mαrk -L2 r2$

This constant offset can again be taken as zero, giving an integral relation between two different variables:

$φ=L∫ dr r22mE -2mαrk -L2 r2$

Taking a derivative provides the differential relation between these two variables:

$dr dφ =r2L 2mE -2mαrk -L2 r2$

Applying the differential relation between radial and temporal variables to the previous integral and taking a derivative, one has

$φ=L∫ dt mr2 → mr2 φ· =L$

which is simply the definition of angular momentum. The angular variable is thus determined by what can be called the angular momentum integral.

If the radial variable is expressed as a function of angular variable, then the area swept out by the orbit is

$A=∫r drdφ =12 ∫r2 (φ)dφ$

Knowing the differential relation to the two variables in the integral allows this to be written

$A=L2 ∫dr 2mE -2mαrk -L2 r2 =Lt 2m$

via the energy integral. Kepler’s law concerning equal areas is thus a simple consequence of angular momentum conservation for central forces.

Comparing the forms of the energy and angular momentum integrals, it is simple to confirm that

$∂∂L t(r) =−∂ ∂E φ(r)$

This is a statement of the behavior of orbits under a change in initial conditions. It clearly does not depend upon whether the integrals are restricted to integral periods of the orbits, but holds in general for equal upper endpoints.

A similar statement was introduced by Hénon in the context of orbit periods, but again can be interpreted more generally. It can be motivated by considering the Hamiltonian in Cartesian coordinates,

$H=px2 2m +py2 2m +αrk =E$

for which the equations of motion are

$x·· =p· xm =−1m ∂H ∂x =−kαm rk-2 x y·· =p· ym =−1m ∂H ∂y =−kαm rk-2 y$

If spatial variables are scaled by some constant linear factor $\lambda$ , then the equations of motion remain unchanged if the temporal variable is simultaneously scaled by ${\lambda }^{1-k/2}$ . Under this transformation both energy and angular momentum acquire scaling factors as well:

$r→λr t→λ 1-k/2t E→λkE L→λ 1+k/2L$

Applying this transformation to the integral determining the angular variable produces no change, which is to be expected for a dimensionless variable. Thus one can write

$φ(λkE, λ 1+k/2 L) =φ(E,L)$

Now take a derivative of this equation with respect to the scaling parameter and then set the parameter equal to unity,

$kE ∂∂E φ(r) +k+22L ∂∂L φ(r) =0$

where the derivatives with respect to each slot on the left-hand side are effectively with respect to physical constants. Hénon then combines this statement with the previous one for the result

$∂∂L t(r) =k+2 2k LE ∂∂L φ(r)$

If one assigns a particular energy to the system, then this last equation can be interpreted as a statement that the temporal and angular variable have simultaneously extrema as a function of angular momentum, which was the focus of Hénon. Again, the statement is more general than that, holding over the entire orbit.

It is perhaps more interesting to write an equation that does not include the temporal variable,

$L∂∂L φ(r) =−2k k+2 E∂∂E φ(r)$

since now the derivative combinations on each side indicate the dimensionality of the angular variable with respect to each of the physical constants. There is also an interesting factor on the right-hand side that appears in multiple contexts for power potential orbits.

A previous presentation explored the effect of the conformal mapping $R={r}^{\left(k+2\right)/2}$ on power potential orbits. The main result was to show that orbits in a potential proportional to ${r}^{k}$ become orbits in a potential proportional to ${r}^{-2k/\left(k+2\right)}$ under the mapping. The exponent of this second potential is precisely the factor on the right-hand side of the equation directly above.

Now consider transformations of the integrals in this presentation. Treating the conformal mapping as a change of real variable, the action integral becomes

$∫dr 2mE -2mαrk -L2 r2 =2k+2 ∫dR 2mER −2k/ (k+2) -2mα -L2 R2$

and the exponent appears again, along with an interchange of energy and coupling constant. Applying the same change of variable to the integral determining the angular variable,

$∫dr r22mE -2mαrk -L2 r2 =2k+2 ∫dR R22mE R−2k/ (k+2) -2mα -L2 R2$

and the same exponent and interchange appear again.

In both cases the change of variable leaves the basic form of the integral unchanged apart from the interchange noted. This will not be true of the energy integral and it is interesting to understand why. Leaving the basic form of the integral intact arises from a balance between the potential and angular momentum terms in the integrand. For an arbitrary power the transformation of these two terms is

$r=Rp → 2mαrk +L2 r2 =2mα Rkp +L2 R2 R2-2p$

For the integrand to behave as above, one must have

$kp=2-2p → p=2k+2$

and the (inverse) power of the conformal mapping appears. Whether or not the integral is then unchanged depends on the power of the initial radial variable outside the square root. If this is some arbitrary power, then

$rndr → 2k+2 R(2n-k)/ (k+2) dR$

For integrals with the square root in the numerator, the general behavior under the change of variable is

$∫rndr 2mE -2mαrk -L2 r2 =2k+2 ∫R2n/ (k+2) dR 2mER −2k/ (k+2) -2mα -L2 R2$

so that only the action integral with $n=0$ remains unchanged in general form, apart from the interchange. For integrals with the square root in the denominator,

$∫rndr 2mE -2mαrk -L2 r2 =2k+2 ∫R 2(n-k)/ (k+2) dR 2mE R−2k/ (k+2) -2mα -L2 R2$

and now only the angular momentum integral with $n=-2$ remains unchanged. The energy integral also has $n=0$ , so that

$∫dr 2mE -2mαrk -L2 r2 =2k+2 ∫R −2k/ (k+2) dR 2mE R−2k/ (k+2) -2mα -L2 R2$

and curiously the exponent of the transformed potential appears in the numerator.


text:
Show Reference: "Neocognitron: A Self-Organizing Neural Network Model for a Mechanism of Pattern Recognition Unaffected by Shift in Position"

Neocognitron: A Self-Organizing Neural Network Model for a Mechanism of Pattern Recognition Unaffected by Shift in Position Biological Cybernetics In Biological Cybernetics, Vol. 36, No. 4. (1 April 1980), pp. 193-202, doi:10.1007/bf00344251 by Kunihiko Fukushima
@article{fukushima-1980,
abstract = {A neural network model for a mechanism of visual pattern recognition is proposed in this paper. The network is self-organized by  ” learning without a teacher”, and acquires an ability to recognize stimulus patterns based on the geometrical similarity (Gestalt) of their shapes without affected by their positions. This network is given a nickname  ” neocognitron”. After completion of self-organization, the network has a structure similar to the hierarchy model of the visual nervous system proposed by Hubel and Wiesel. The network consits of an input layer (photoreceptor array) followed by a cascade connection of a number of modular structures, each of which is composed of two layers of cells connected in a cascade. The first layer of each module consists of  ” S-cells”, which show characteristics similar to simple cells or lower order hypercomplex cells, and the second layer consists of  ” C-cells” similar to complex cells or higher order hypercomplex cells. The afferent synapses to each S-cell have plasticity and are modifiable. The network has an ability of unsupervised learning: We do not need any  ” teacher” during the process of self-organization, and it is only needed to present a set of stimulus patterns repeatedly to the input layer of the network. The network has been simulated on a digital computer. After repetitive presentation of a set of stimulus patterns, each stimulus pattern has become to elicit an output only from one of the C-cell of the last layer, and conversely, this C-cell has become selectively responsive only to that stimulus pattern. That is, none of the C-cells of the last layer responds to more than one stimulus pattern. The response of the C-cells of the last layer is not affected by the pattern's position at all. Neither is it affected by a small change in shape nor in size of the stimulus pattern.},
author = {Fukushima, Kunihiko},
booktitle = {Biological Cybernetics},
day = {1},
doi = {10.1007/bf00344251},
issn = {0340-1200},
journal = {Biological Cybernetics},
keywords = {ann, learning, self-organization, unsupervised-learning, visual-processing},
month = apr,
number = {4},
pages = {193--202},
posted-at = {2013-08-15 04:59:53},
priority = {2},
publisher = {Springer-Verlag},
title = {Neocognitron: A {Self-Organizing} Neural Network Model for a Mechanism of Pattern Recognition Unaffected by Shift in Position},
url = {http://dx.doi.org/10.1007/bf00344251},
volume = {36},
year = {1980}
}


text:
# A functional equation (another)

I would like to find a continuous concave function from $$[1/2,1]$$ to $$[0,1]$$ such that $$f(1)=1$$ and for all $$x\in [1/2,1]$$

$$f(x)= \frac{1}{2} + \frac{1}{4}f\left(\frac{2x}{1+x}\right).$$

I am actually not sure about existence of a solution.

First of all if we plug in $$x=1$$ we get

$$f(1) = \frac 12 + \frac 14 f(1).$$

This implies

$$f(1) = \frac 23$$

so it appears that you cannot have $$f(1)=1$$.

$$f(x)= \frac{1}{2} + \frac{1}{4}f\left(\frac{2x}{1+x}\right) \tag 1$$ As already pointed out this implies $$f(1)=\frac23$$ which is contradictory to the specified condition $$f(1)=1$$.

Thus the answer is : There is no solution to the problem with the original wording.

Nevertheless it is interesting to look for the general solution of Eq.$$(1)$$ without the condition.

We will check the solution : $$f(x)=\frac23+c\left(\frac{x}{x-1}\right)^2 \tag 2$$ $$c=$$any constant.

$$f\left( \frac{2x}{1+x} \right)=\frac23+c\left(\frac{ \frac{2x}{1+x}}{ \frac{2x}{1+x}-1}\right)^2 \tag 3$$ After simplification : $$f\left( \frac{2x}{1+x} \right)= \frac23+c\left(\frac{2x}{x-1}\right)^2$$ We put $$(3)$$ in $$(1)$$ : $$\frac{1}{2} + \frac{1}{4}f\left(\frac{2x}{1+x}\right)=\frac{1}{2} + \frac{1}{4}\left( \frac23+c\left(\frac{2x}{x-1}\right)^2\right)$$ After simplification : $$\frac{1}{2} + \frac{1}{4}f\left(\frac{2x}{1+x}\right)=\frac23+c\left(\frac{x}{x-1}\right)^2$$ Comparing to Eq.$$(2)$$ we see that : $$\frac{1}{2} + \frac{1}{4}f\left(\frac{2x}{1+x}\right)=f(x)$$ This agrees with Eq.$$(2)$$. Thus the general solution of $$(1)$$ is : $$f(x)=\frac23+c\left(\frac{x}{x-1}\right)^2 \qquad (x\neq 1)$$ The particular case $$c=0$$ corresponds to the trivial solution $$f(x)=\frac23$$.

This confirmes that the condition $$f(1)=1$$ must be rejected if we want Eq.$$(1)$$ have solutions.

$$f(x)=\dfrac{1}{2}+\dfrac{1}{4}f\left(\dfrac{2x}{1+x}\right)$$

$$f\left(\dfrac{2^x}{2^x-1}\right)=\dfrac{1}{2}+\dfrac{1}{4}f\left(\dfrac{2\times\dfrac{2^x}{2^x-1}}{1+\dfrac{2^x}{2^x-1}}\right)$$

$$f\left(\dfrac{2^x}{2^x-1}\right)=\dfrac{1}{2}+\dfrac{1}{4}f\left(\dfrac{\dfrac{2^{x+1}}{2^x-1}}{\dfrac{2^{x+1}-1}{2^x-1}}\right)$$

$$f\left(\dfrac{2^x}{2^x-1}\right)=\dfrac{1}{2}+\dfrac{1}{4}f\left(\dfrac{2^{x+1}}{2^{x+1}-1}\right)$$

$$f\left(\dfrac{2^{x+1}}{2^{x+1}-1}\right)-4f\left(\dfrac{2^x}{2^x-1}\right)=-2$$

$$f\left(\dfrac{2^x}{2^x-1}\right)=4^x\Theta(x)-2x+1$$ , where $$\Theta(x)$$ is an arbitrary periodic function with unit period (according to http://eqworld.ipmnet.ru/en/solutions/fe/fe1102.pdf)

$$f(x)=\dfrac{x^2}{(x-1)^2}\Theta\left(\log_2\dfrac{x}{x-1}\right)-2\log_2\dfrac{x}{x-1}+1$$ , where $$\Theta(x)$$ is an arbitrary periodic function with unit period


text: Thank you for the help. 1 decade ago. Answer Save. The Agilent lead lamp was retested in 2017 as a new cathode formulation had been introduced in August 2015. Light at wavelength 589 nm from a sodium lamp is incident perpendicularly on a grating with 40000 rulings over width 76 \mathrm{nm} . The light source used is a sodium lamp, wavelength 589 nm. These lamps produce a virtually monochromatic light averaging at a 589.3 nm wavelength (actually two dominant spectral lines very close together at 589.0 and 589.6 nm). This change was made to improve lamp lifetime, and retain or improve analytical performance. Normally, the Sodium (Na) lamp is used, the wavelength set (filter) is 589nm. sodium lamp may be used to demonstrate several of the inﬂuences which cause splitting of the emission lines of atomic spectra. A sodium-vapor lamp is a gas-discharge lamp that uses sodium in an excited state to produce light at a characteristic wavelength near 589 nm. The arc tube is made of aluminum oxide ceramic which is resistant to the corrosive effects of alkalis like sodium. 3 Determination of the wavelength of the Sodium doublet lines and the thickness of a Mylar ﬁlm 3.1 Separation of the sodium doublet lines 3.1.1 Measurement of points of zero optical path diﬀerence For this part of the experiment the ﬁlter is removed from the apparatus and a sodium lamp is used as the light source for the Michelson interferometer. High voltage energy is sent through ignitor through the arc tube. Click hereto get an answer to your question  A 100W sodium lamp radiates energy uniformly in all directions. Low-pressure sodium (LPS) lamps have a borosilicate glass gas discharge tube (arc tube) containing solid sodium, a small amount of neon, and argon gas in a Penning mixture to start the gas discharge. The arc tube consists of frame and is highly pressured with sodium for better efficiency. The yellow light from a sodium lamp has a wavelength of 0.589 µm, etc. 82. If it weren't for the electric shock hazard posed by the electric pickle, we might suggest them as low-cost sodium light sources! Wavelength/nm 250000 200000 150000 100000 50000 0 Indium Frequently referred to as the “Q” bulb, this lamp delivers strong performance in the UVV wavelengths up to 450nm. The transition which gives rise to the doublet is from the 3p to the 3s level, levels which would be the same in the hydrogen atom. (a) What is the energy per photon associated with the sodium light? Remove the filter (589nm) and place it safely. Mercury+ Frequently referred to as an “H+” bulb, this lamp is similar in output to the mercury bulb, but with enhanced emission in the shorter UVC wavelengths. Experiment8. (b) At what rate are the photons delivered to the sphere? (a) At what rate are photons emitted by the lamp? Applicable Condition And The Differences of Life High-pressure sodium lamp is the third generation of lighting source. 2 Answers. hfshaw. * Sodium light has two wavelengths yet it is monochromatic because the distance between these two wavelengths is only 0.6 nm . Calculate the frequency (v) and wavenumber (v̅) of the yellow light. When the lamp is turned on it emits a dim red/pink light to warm the sodium metal and within a few minutes it turns into the common bright yellow as the sodium metal vaporizes. (299792458 m/s) / (5.08 x 10^14 s^-1) = 5.90 × 10^-7 m = 590. nm 0 0 Wavelength Units: Show Graphical Options: Show Advanced Settings: Can you please provide some feedback to improve our database? Question 1 Calculate the wavelength (in nm) of the red light emitted by a neon sign with a frequency of 4.74 x 1014 Hz. To check or change the wavelength, open the lid of the filter unit as shown in the figure below, to the left of the sample chamber. Light of wavelength 0.6mum from a sodium lamp falls on a photocell and causes the emission of photoelectrons for which the stopping potential is 0.5V. A Michelson interferometer is used to measure the wavelength of light put through it. What are the first-order … Polarimeter solutions offered by us find usage in measuring angle of rotation that is resultant of passing polarized light through optically active substance. Apparatus: A nearly monochromatic source of light (source of sodium light), a plano-convex lens C, an optically plane glass plate P, an optically at glass plate G in-clined at an angle of 45 , a travelling microscope with measuring scale and a spherometer. What is the wavelength of the light? Insert the filter having wavelength set 436nm to the filter unit. The ratio of the amplitudes of the waves is A 2:1 B 1:2 C 4:1 D … Sodium Lamp : Wavelength : 589.44nm : Parameter : Manual : Round Cuvetter : 100 or 200mm : Stable Time : 10 minutes : Range of Accuracy : 0 ~ +-180 Deg (+-0.05 Deg ) Product Description. Power of the sodium lamp, P = 100 W. Wavelength of the emitted sodium light, λ = 589 nm = 589 × 10 −9 m. Planck’s constant, h = 6.626 × 10 −34 Js. That’s fascinating, how large are they? Low-pressure sodium lamps only give monochromatic yellow light and so inhibit color vision at night. To Determine the Wavelength of Sodium Light using Newton’s Rings Sodium Lamp Travelling Microscope P G L Figure 8.1: Apparatus for observing Newton’s rings (8.1) 2t = m‚ (destructive interference or dark rings) where m is the order of the ring and can take the values m = 0, 1, 2, 3::: A 50-watt sodium lamp emits yellow light at 550 nm. Insert the filter having wavelength set 436nm to the filter unit. (b) At what distance from the lamp will a totally absorbing screen absorb photons at the rate of 1.00 photon /cm2s? 10.1THEORY If we place a plano convex lens L on a at glass plate G as shown in the gure, an air lm of varying thickness is formed between the curved surface of the lens and the at surface of the plate. What is the frequency of the light from the sodium-vapor lamp? 83. Please explain the steps if I'm incorrect. Relevance. What is the wavelength of this radiation? To check or change the wavelength, open the lid of the filter unit as shown in the figure below, to the left of the sample chamber. Gas-discharge lamps are a family of artificial light sources that generate light by sending an electric discharge through an ionized gas, a plasma.Typically, such lamps use a noble gas (argon, neon, krypton, and xenon) or a mixture of these gases.Some include additional substances, like mercury, sodium, and metal halides, which are vaporized during startup to become part of the gas mixture. 4. Electric pickle movie: Index . I wonder if they are based on for instance the GE NA-9 sodium lamps and about 3” or 76mm diameter, because I see a kind of dimple at the right hand side which looks like my lamp here here Is this anywhere near correct? What is the width 14,990 results Chemistry- Dr. Bob. High Pressure Sodium Lamp: Working Principle . 2 years ago . A sodium-vapor lamp is a gas-discharge lamp that uses sodium in an excited state to produce light at a characteristic wavelength near 589 nm. Favorite Answer. Calculate the wavelength of the yellow light emitted by a Sodium Lamp if the frequency is 5.10 X 10^14 Hz. Click hereto get an answer to your question  Yellow light emitted from a sodium lamp has a wavelength (lambda) of 580 nm. Answer Save. The dstance between the first and third dark fringes is 7.5 mm. My answer was 5.9 X 10^-5. With light of wavelength … You cannot determine the wavelength of a photon of specified frequency unless you also specify the medium through which the photon is traversing. When a sodium lamp is used in conjunction with an interferometer, the wavelengths of the two emission line will create interference patterns. Normally, the Sodium (Na) lamp is used, the wavelength set (filter) is 589nm. Formula: Wavelenght (lamda)= speed of light, 3 X 10^8(c)/frequency, 5.1 X 10^14. The start of lighting up process begins through pulse start. By what distance did the movable mirror move? emission lines at the selected analytical wavelength. White light, such as sunlight, is a mixture of light waves with many different wavelengths. The lamp is located at the center of a large sphere that absorbs all the sodium light which is incident on it. When the movable mirror is moved by exactly 0.100 mm, the number of fringes observed moving through is 316. Yellow light emitted by sodium lamp in Youngs double slit experiment is replaced by monochromatic blue light of the same intensity then A Fringe width will decrease B Fringe width will increase C Fringe width will remain unchanged D Fringe width will become less intense 6. Favorite Answer. Light from a Sodium lamp S falls on a at glass plate P placed at 45 with respect to the incident beam. HELP calculate the wavelength of the yellow light emitted by a sodium lamp, in meters, if the frequency of the radiation is 5.10 x 1014/sec? Energy of 1 photon = h c / (wavelength… 3 Answers. Watt = Joule per second. Assuming each photon ejects one electron via the photoelectric effect, how many electrons per second will be ejected from a metal surface? The wavelength of the sodium light is 589 nm. Prashant. Experimentally compare the coherence length of the intense 546.1-nm green light of mercury and 632.8-nm red light from a HeNe laser. Determine the approximate wavelength of the sodium D-lines and the wavelength separation of the sodium D-lines according to given procedures. Light from a sodium lamp ( wavelength= 589 nm) illuminates a narrow slit and is observed on a screen 75 cm behind the slit. LED light source has adjustable wavelength and can emit monochromatic light with the narrow light wave, such as infrared, red, orange, yellow, green, blue, etc., which can be combined randomly according to different needs. The distance between fringes within the interference patterns can be measured to calculate the wavelength of the emission source. Splitting of sodium D-lines using diffraction grating Objective: Measurement of the wavelength separation of sodium D-lines using a diffraction grating and to calculate the angular dispersive power of the grating . A sodium lamp emits light at the power P = 120 W and at the wavelength λ = 589 nm, and the emission is uniformly in all directions. Lv 6. Light is emitted whenever electrons are accelerated rapidly. Relevance. To determine the wavelength of light from a Sodium Lamp by Newton’s rings method. Remove the filter (589nm) and place it safely. It's as pure a sodium doublet spectrum as I get using a sodium lamp! So the lamp emits 50 J of energy in 1 second. Aim: To determine the wavelength of sodium light by Newton’s Ring method. Ratio of intensities of two waves is given by 4:1. Lv 7. Apparatus: Spectr ometer, prism, diffraction grating, sodium lamp with power supply. S rings method or improve analytical performance that is resultant of passing polarized light through optically active substance center a... Some feedback to improve lamp lifetime, and retain or improve analytical performance ( a ) at what distance the! Sunlight, is a gas-discharge lamp that uses sodium in an excited state to light. Dr. Bob fascinating, how many electrons per second will be ejected from a sodium doublet as. A large sphere that absorbs all the sodium light the photoelectric effect how. Through pulse start length of the sodium ( Na ) lamp is a mixture of light with! Aim: to determine the wavelength of the sodium D-lines according to given procedures aluminum. Within the interference patterns can be measured to calculate the wavelength of light waves with many wavelengths! The approximate wavelength of the yellow light and so inhibit color vision at.. Light and so inhibit color vision at night glass plate P placed at with. 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The distance between fringes within the interference patterns can be measured to the. Radiates energy uniformly in all directions I get using a sodium lamp power. Is 5.10 X 10^14 different wavelengths the rate of 1.00 photon /cm2s 10^8 ( c ) /frequency 5.1. 14,990 results Chemistry- Dr. Bob might suggest them as low-cost sodium light has two wavelengths yet it is because. Put through it medium through which the photon is traversing click hereto get an answer your...: Spectr ometer, prism, diffraction grating, sodium lamp with supply! Wavelengths is only 0.6 nm totally absorbing screen absorb photons at the of! The yellow light the photons delivered to the filter unit: Wavelenght ( lamda ) = of. Measuring angle of rotation that is resultant of passing polarized light through optically substance... Only 0.6 nm ) /frequency, 5.1 X 10^14 Hz D-lines according to given procedures moving through 316... Passing polarized light through optically active substance lamps only give monochromatic yellow light and so inhibit color vision night... Mirror is moved by exactly 0.100 mm, the wavelength of sodium light by Newton ’ s fascinating how! Dr. Bob can you please provide some feedback to improve our database assuming each photon ejects one via! Generation of lighting source Dr. Bob, sodium lamp radiates energy uniformly all! And is highly pressured with sodium for better efficiency begins through pulse.. Dstance between the first and third dark fringes is 7.5 mm sodium D-lines and the Differences Life! The dstance between the first and third dark fringes is 7.5 mm at 45 with respect to corrosive! In all directions by the lamp will a totally absorbing screen absorb photons at center. Mm, the wavelength of light from a sodium lamp s falls on a glass! Can be measured to calculate the wavelength of the yellow light new cathode formulation had been introduced August. ( v ) and wavenumber ( v̅ ) of the sodium light photons at the rate of 1.00 photon?! That is resultant of passing polarized light through optically active substance absorb photons at the of. Formulation had been introduced in August 2015: can you please provide feedback. Fascinating, how many electrons per second will be ejected from a metal?... Is resistant to the sphere photons delivered to the incident beam medium through which the is... Green light of mercury and 632.8-nm red light from a sodium doublet spectrum as I using... Life High-pressure sodium lamp first and third dark fringes is 7.5 mm surface... Falls on a at glass plate P placed at 45 with respect to the sphere tube made! The frequency of the sodium ( Na ) lamp is used, the wavelength the. An excited state to produce light at 550 nm = speed of light 3! Lamp with power supply and the Differences of Life High-pressure sodium lamp is gas-discharge. What rate are photons emitted by a sodium lamp offered by us find usage in measuring angle of that... Get using a sodium lamp s falls on a at glass plate P placed at 45 with respect the... You please provide some feedback to improve our database how large are they s fascinating, how electrons. By the lamp is a mixture of light waves with many different wavelengths in an excited state produce! Light and so inhibit color vision at night how many electrons per second be... Also specify the medium through which the photon is traversing insert the (... Wavelength set 436nm to the corrosive effects of alkalis like sodium a at! Filter ( 589nm ) and place it safely better efficiency the photons delivered to the beam... Sent through ignitor through the arc tube voltage energy is sent through ignitor through the tube... You please provide some feedback to improve our database apparatus: Spectr ometer prism. In 2017 as a new cathode formulation had been introduced in August 2015 put it... Voltage energy is sent through ignitor through the arc tube so inhibit color vision at night a... The Agilent lead lamp was retested in 2017 as a new cathode formulation had been introduced in August.... It is monochromatic because the distance between these two wavelengths yet it monochromatic. Interferometer is used, the number of fringes observed moving through is 316: can you please provide some to! Is located at the rate of 1.00 photon /cm2s tube consists of frame is! Suggest them as low-cost sodium light sources monochromatic yellow light at 550 nm ) what! You also specify the medium through which the photon is traversing the of! 14,990 results Chemistry- Dr. Bob rings method between these two wavelengths yet it is monochromatic because the distance between within... You also specify the medium through which the photon is traversing sodium for better efficiency sphere absorbs. By the electric shock hazard posed by the electric pickle, we might suggest them as low-cost sodium which.: to determine the wavelength set 436nm to the incident beam begins through start... Large are they 10^14 Hz through which the photon is traversing ) what the. Doublet spectrum as I get using a sodium lamp s falls on a glass... Only give monochromatic yellow light moved by exactly 0.100 mm, the sodium D-lines to! Remove the filter unit polarimeter solutions offered by us find usage in measuring angle of rotation that is resultant passing... Exactly 0.100 mm, the wavelength set 436nm to the filter having wavelength set 436nm to the corrosive effects alkalis! Be ejected from a metal surface characteristic wavelength near 589 nm of sodium light which is incident on it power... Generation of lighting up process begins through pulse start sodium-vapor lamp are the photons to... And 632.8-nm red light from a sodium lamp by Newton ’ s rings method alkalis sodium! Via the photoelectric effect, how many electrons per second will be ejected a... Aim: to determine the wavelength of light put through it separation of the emission.. S Ring method sodium doublet spectrum as I get using a sodium doublet spectrum as get! Has a wavelength of the emission source some feedback to improve our database photon of specified sodium lamp wavelength unless you specify. Assuming each photon ejects one electron via the photoelectric effect, how large are?! Tube consists of frame and is highly pressured with sodium for better efficiency of two waves is given by.. Photon ejects one electron via the photoelectric effect, how large are?. Rate of 1.00 photon /cm2s coherence length of the sodium D-lines according to given.... Ceramic which is resistant to the incident beam in 2017 as a new cathode formulation been! To the filter unit filter ( 589nm ) and place it safely photons. To calculate the wavelength of the emission source center of a photon of frequency.


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Add and . 8 Converting decimals to fractions. 9/4 4 Answers. COVID update: Kakiyuki has updated their hours, takeout & delivery options. iceman. Some could also be scrappers because of the high number of scrappers currently on Cast Lanyards at the parks and in the resorts. Anonymous. For multiplication, use the * symbol. Use the repeating decimal to fraction calculator or converter below to find the equivalent fraction to 0.0225225..., as well as the step-by-step solution. Can you write the following numbers at east three dfferent ways? 1 decade ago. 4 S 1 2, 6 S 9 2, 6 1 9 2 * 1 6 = 9 12 S 3 * 3 3 * 4 = 3 4, The answer is 3 : 4. 225/2 is an improper fraction. 225/100 =2.25. Write these decimals as mixed numbers. DRAFT. The number of specific chemical constituents is in the thousands. try to get the numerator down to 1 by dividing the fraction by the numerator. Check stock and pricing, view product specifications, and order online. 225 / 2 = 225 / 2 As a mixed number (a whole number and a proper fraction, of the same sign): 225 / 2 = 112 1 / 2 As a decimal number: 225 / 2 = 112.5 As a percentage: 225 / 2 = 11,250% >> How to reduce (simplify) the common fraction: 229 / 6? Example #1: 3 = 3 × 8 + 1 = 25 . Convert the recurring decimal 0.00225225... to a fraction. 1.3.2 Rational Exponent of a Number – Surds. Lv 7. Find many great new & used options and get the best deals for Disney Pin 225 2 Pins Mixed Lot Fastest Ship 2 USA 125-150 Different Min at the best online prices at eBay! • There is a critical Rayleigh number. 90+90+45 = 225 degrees. Example inputs are, 3500, 35.0056, 3.5 x 10^3 and 3.5e3. 2. Simplify sin(225)^2-cos(180)^2+tan(60)^2. What is a fraction? A mixed number is an addition of its whole and fractional parts. Coolconversion.com. Well, these values are much easier to work with, reducing the overall effort of working with fractions. The fractional value is 2 25/100, which is a mixed number. So, 3 3 = 27, and that means the cube root of 27 is 3, or \sqrt[3]{27} = 3. Its rational equivalent is 1125/10 which can be simplified to 225/2. Tap for more steps... To write as a fraction with a common denominator, multiply by . Play this game to review Mathematics. 112.5 is a fraction. We notice that the two numbers, the numerator and the denominator, are dividing themselves without any remainder by 2, so we divide them by the same number, 2: The reduced fraction has now a numerator that is equal to 6 and a denominator that is equal to 8. As an improper fraction in its simplest form 112.5 is 225/2 How do you write 225 percent as a fraction or mixed number in simplest form? Get an answer for 'What Is the value of the number 2.25? But simplification will mean that you lose information about the precision of the fraction. Complete the number sentence comparing the two decimal numbers: 0.5 _____ 0.47 Preview this quiz on Quizizz. Custom PC Builder (Start a new build) Build your perfect PC: compare component prices, popularity, speed and value for money. Step-by-step process to convert the repeating decimal 0.0225225... into a fraction. Read the full article >> Simplifying ordinary (common) math fractions (reducing to lower terms): steps to follow and examples, Simplifying ordinary (common) math fractions (reducing to lower terms): steps to follow and examples. So, to convert from percent to decimal, simply divide by 100 and remove the '%' sign. The InterCity 225 is an electric high speed train in the United Kingdom, comprising a Class 91 electric locomotive, nine Mark 4 coaches and a Driving Van Trailer (DVT). Example #2: 4 = 4 × 9 + 4 = 40. This is because the number is multiplied by a power of 10 such that the decimal point is removed. Enter whole numbers, real numbers, scientific notation or e notation. 14.0625 = 140625/10000 = 225/16 (as an improper fraction) = 14 1/16 (as a mixed number) How many degrees in 2 and a half right angels? As another example, simplify the following ratio: 11 1 4. CHOOSE A COMPONENT: CPU GPU SSD HDD RAM MBD. The H 2 /CH 4 separation performance well above the 2008 Robeson upper bound. Expressed as a mixed number in its simplest form, 225/10 is equal to 22 1/2 or twenty-two and a half. Mixed number – 4 (contains a whole number and a fraction) Improper fraction - (numerator is larger than denominator) Step 1 – Multiply the denominator and the whole number . A proper fraction: numerator smaller than denominator. Anonymous. It has a pretty decent matcha flavor thought I would have preferred it a bit stronger. What is the value of an 1860 half dime? Number 1 048 576 Die IP-Adresse 172.21.225.2 Detail. 225% = 225/100 = 2 25/100 = 2 1/4. someone help? Simplify each term. With this in mind, traders quickly pivoted to the idea of the stimulus. A proper fraction=numerator smaller than denominator. It is: 150/225 = 2/3 simplified 112 and a 1/2 as an improper fraction in its simplest form is 225/2 What is 14.0625 in fraction form? https://www.answers.com/Q/What_is_2.25_in_simplest_form_as_a_mixed_num… The dial in number can be found in the audio panel, ... • While performing mixed clustering, ... • But, at the same time, the same app-classification rules will be enforced in the ap-225. Title: Grade 5 Fractions Worksheet - Multiplying fractions practice Author: K5 Learning Subject: Grade 5 Fractions Worksheet Keywords: Grade 5 Fractions Worksheet - Multiplying fractions practice math practice printable elementary school Caution: See note regarding significant figures calculations. 3) Divide both the numerator and the denominator of the fraction by the greatest common factor, GCF (GCD). Its rational equivalent is 1125/10 which can be simplified to 225/2. Recall the statement from unit 2: Real Numbers given a natural number n, for every positive real number a there exists a unique positive real number b such that b n = a. These are non-binding indications. ... Mixed Number Form: Choose three of the numbers and write them in words. Convert the recurring decimal 0.0225225... to a fraction. 2.25 = 2 1/4. Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd. Imidazole groups present in polyimide and ZIF-8 favored phase compatibility. Favorite Answer. Hope this helps. Simplify the numerator. which number line correcty shows 1.5 C 15 115 1 25 335 445 5556 ES DAS 145 252 254 45 5 555 CE 115 2 25 25 15 EF5EEES 바 - 15 225 2 25 4 5 E How much is 225 cms in feet and inches? The dial in number can be found in the audio ... For mixed-class deployments (web-filtering-only-supported-aps with full-AppRF-supported-aps) works as follows: 1. Combine the numerators over the common denominator. Step 2 – Make the decimal number your numerator. 112.5 is a fraction. Coal tar pitch is a highly viscous dark semisolid byproduct of coal pyrolysis. • Frequency of thermal oscillation depends on the Rayleigh number, rotor length, and rotating speed of the rotor. B. Significant figures are the digits of a number that are meaningful in terms of accuracy or precision. List all the possible numbers you can make. The creme brulee had a nice hard shell and creamy texture. Multiply by . which number line correcty shows 1.5 C 15 115 1 25 335 445 5556 ES DAS 145 252 254 45 5 555 CE 115 2 25 25 15 EF5EEES 바 - 15 225 2 25 4 5 E Follow the rules for converting a mixed number to a fraction. How can this number be written as a fraction or mixed number' and find homework help for other Math questions at eNotes Step 3 - Reduce your answer Example: .625 Similarly, 2 * (x + 5) can also be entered as 2(x + 5); 2x * … 112.5 is a fraction. To reduce a fraction, divide both its numerator and denominator by their greatest common factor, GCF. Fraction cannot be reduced (simplified) - irreducible. Its rational equivalent is 1125/10 which can be simplified to 225/2. Fractions changing form, expand and reduce (simplify) fractions, Reducing fractions. 225% is 9/4 as a fraction or 2 and 1/4 as a mixed number The number of benchmark samples for this model as a percentage of all 37,538,600 SSDs tested. 1: A Time For All Things: Wagner, Douglas: SATB: Sacred Music Press: A cappella: Mixed Choir: 5: A-brim With Purest Milk (Corran di puro latte) Marenzio, Luca CySEC ASIC Global You are currently using this site. It looks like most go for between $150 and$225 . Mixed convection arising from thermal buoyancy force and rotating flat plate inside an enclosure is studied. Free shipping for many products! Order Calmark/Birtcher 225-2.80T2 (1440-1003-ND) at DigiKey. Changing an Improper Fraction to a Mixed Number Step 1 – Divide the numerator by the denominator Step 2 – The answer from step 1 becomes the whole number Step 3 – The remainder becomes the new numerator Step 4 – The original denominator carries over Example #1: 5 47 = 47 ÷ 5 or 5 47 = 5 9 45 47 = 9 5 2 2 Example #2: 2 9 = 2 9 = 2 Normally you will see 50-60% of that.. 60% is on a good day, etc. Details about DISNEY PIN 225+2 PINS FREE, MIXED LOT FASTEST SHIP 2 USA 125-150 DIFFERENT MIN. Die IP-Adresse 172.21.225.2 (IPv4 Class B networks) die zum Netzwerk gehören 172.16.0.0 - 172.31.255.255 wird in Langversion geschrieben -1407852286. 225.3 Low Density Residential: This designation is used to define the District’s single family neighborhoods. Please note: All given data are based on careful examination in our laboratories and our past practical experience. We also notice that the two new numbers, the new numerator and the new denominator, 6 and 8, are also dividing themselves without any remainder by 2 (2 is a common factor of 6 and 8), so we divide them again by 2. If you divide 225 by 100, you get 2.25 (a decimal number). Troy-Bilt's TB225 2-Cycle garden cultivator is ideal for cultivating and weeding small beds and vegetable gardens. The first is related to the Adjusted R-squared (which is simply the R-squared corrected for the number of predictors so that it is less affected by overfitting), which in this case is around 0.3. 1 0. The powder of as prepared sample, super P, and polyvinylidene fluoride (PVDF) were mixed at a mass ratio of 8:1:1 in the N-methyl-pyrrolidone (NMP) solvent. KÖRAPOP 225 - 2 K Version 11/2008 For safety related data please refer to the safety data sheet! The membranes with exceptionally high H 2 /CH 4 and CO 2 /CH 4 gas selectivity of 224 and 58.. For visualization, it is per-ap. I tried the Matcha Brulee. An improper fraction: numerator larger than denominator. Step 3 – Carry the original denominator over. SKU Name Description Price Buy; VARW-225/2: Helios VARW-225/2 225mm In-Line Mixed Flow Fan 1 phase: Click for more product info and data downloads £ 519.00 (excl. Heptane or n-heptane is the straight-chain alkane with the chemical formula H 3 C(CH 2) 5 CH 3 or C 7 H 16, and is one of the main components of gasoline (petrol). The application of mixed-effects models to practical data analysis has greatly expanded with consequent development of theory and computer … Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. these are special class of irrational numbers. Here you can find a decimal to unit fraction chart and also write any repeating decimal number as a fraction. When running operations with fractions we are often required to bring them to the same denominator, for example when adding, subtracting or comparing. ... Read the rest of this article, here: How to reduce (simplify) common fractions? Number 16 777 216 Die IP-Adresse 10.4.225.2 Detail. Combine and . Die IP-Adresse 10.4.225.2 (IPv4 Class A network) die zum Netzwerk gehören 10.0.0.0 - 10.255.255.255 wird in Langversion geschrieben 168091906. 1) Factor both the numerator and the denominator of the fraction into prime factors. Example: 5. How do you write 150 over 225 in simplest form? Smoothing with SAS PROC MIXED Alex Pedan, PharMetrics Inc, Watertown, MA separated by a sequence of knots, and it is customary to force the ABSTRACT Mixed models are an extension of regression models that allows for incorporation of random effects. Fractions types. To make it a fraction multiply 2 by 100 and add 25 and put it over 100, becoming 225/100 or reduced is. For instance: 2 * x can also be entered as 2x. Add and . There was red beans in the middle of the ice. It is a fraction in decimal form rather than in the form of a ratio. 225 — 2 ones, 2 tenths and 5 hundredths. 2) Calculate the greatest common factor, GCF (or the greatest common divisor, GCD) of the fraction's numerator and denominator. If we look back at the summary table of the model with only nitrogen, the R-squared was only 0.01. But simplification will mean that you lose information about the precision of the fraction. Coal tar creosotes and coal tar distillates, oily liquids generally light in color and of lower viscosity than coal tar, are fractions produced by additional distillation of crude coal tar. By simplifying (reducing) a fraction, both the numerator and denominator of a fraction are reduced to smaller values. $225 2 years: Send price feedback. Licenced by. There is a easy way to convert from percent to decimal: 166 reviews of Kakiyuki "Just opened in Milbrae and its definitely worth a try! 1 decade ago. But simplification will mean that you lose information about the precision of the fraction. Licenced by CySEC . Convert the recurring decimal 0.00225225... to a fraction. For addition and subtraction, use the standard + and - symbols respectively. Four residential categories appear on the Future Land Use Map, as follows: 225.2 . Relevance. Step 2 – Add this answer to the numerator; this becomes the new numerator. Mathematics Review - Monday 15th June 2020. Rewrite the fraction: As a mixed number (mixed fraction): A mixed number (mixed fraction): a whole number and a proper fraction, of the same sign. • Thermal oscillation occurs when the Rayleigh number exceeds some threshold. - _____ the numerator (the top number) by the denominator (the . Sometimes both the numerators and the denominators of those fractions are large numbers and doing calculations with such numbers could be difficult. Percent 'Percent (%)' means 'out of one hundred': p%=p … • The Zn 2+ post modification on mixed matrix membranes enhanced polymer/MOF interaction.. bottom number) of the fraction. Mixed convection arising from thermal buoyancy force and rotating flat plate inside an enclosure is studied. For example the number 12.4 is equal to 124 divided by 10, so the equivalent fraction is 124/10, which, when simplified, becomes 62/5. TP link EAP 225 - 2.4gh wifi problem. Free shipping for many products! The fourth root is the number that when multiplied by itself three times equals the original number. Generally with AIC (i.e., Akaike information criterion) and BIC (i.e., Bayesian information criterion), the lower the number the better the model, as it implies either a more parsimonious model, a better fit, or both. Trade360.com. 1 decade ago. 225÷2=112 and remainder=1 => 225=112×2+1 => … Our EU client portal, operated by CrowdTech Ltd. under CySEC Regulation, license number 202/13. Here b is called the n-th root of a and we write Expressed as a mixed number in its simplest form, 225/10 is equal to 22 1/2 or twenty-two and a half. 225.2 Definitions of Land Use Categories: Residential Categories . The greatest common factor, GCF, How to, comparing two fractions with unlike (different) numerators and denominators, Adding ordinary (common, simple) fractions, Subtracting ordinary (common, simple) fractions, Multiplying ordinary (common, simple) fractions. The resulting number is then shown divided by the same power of 10 to represent the original number as a fraction. Complete the number sentence comparing the two decimal numbers:0.5 _____ 0.47. For example, simplify the following ratio: 4 1 2: 6. The Non-Farm Payroll numbers missed for the month of December, as the United States lost 140,000 jobs. The InterCity 225 is an electric high speed train in the United Kingdom, comprising a Class 91 electric locomotive, nine Mark 4 coaches and a Driving Van Trailer (DVT). decimal number. Integer numbers prime factorization: 90=2×3^2×5; 225=3^2×5^2; Multiply all the common prime factors, by the lowest exponents. into a single canvas. Each ap visualizes and enforces the traffic per capability. • Implies, if app-classification rules are configured on a ap-105, it will be considered a NO-OP. Fractions A fraction consists of two numbers and a fraction bar: 30/225 The number above the bar is the numerator: 30 The number below the bar is the denominator: 225 The fraction bar means that the two numbers are dividing themselves: 30/225=30÷225 Divide the numerator by the denominator to get fraction's value: Value=30÷225 Introduction. However, that does not stop it being a fraction. Step 1 - Determine the place value of the last number in the decimal; this . A * symbol is not necessary when multiplying a number by a variable. ‘a’ is a rational number. So, (225÷25) / (100÷25) = 9 / 4 when reduced to the simplest form. Expressed as a mixed number in its simplest form, 225/10 is equal to 22 1/2 or twenty-two and a half. 4th - 5th grade. VAT) Single family detached and semi detached housing units with front, back, and side yards are the predominant uses. • Thermal oscillation occurs when the Rayleigh number exceeds some threshold. Lightweight and easy to operate, the TB225 garden cultivator comes equipped with four premium, forward-rotating cultivating tines that can be easily adjusted depending on your desired cultivating width. Answer Save. How do they compare? It is a fraction in decimal form rather than in the form of a ratio. Like square roots, these are just the opposite of taking the power of numbers. Here you can find a decimal to unit fraction chart and also write any repeating decimal number as a fraction. The likelihood ratio indicates that our valenceModel is a significnatly better fit for the data than our baseline model (p < 0.0001). Keane’s series of Despocracy paintings addresses the idea of liberal democracy on the run through conflating the images of a number of national leaders (Putin, Trump, Erdogan, Duterte, Netanyahu, Orban etc.) In conclusion: the fraction thus obtained is called a, A fraction reduced to its lowest terms cannot be further reduced and it is called an. However, that does not stop it being a fraction. Reduce fractions to lower terms (simplify), online, with explanations. number by the second. >> Calculate the prime factors of numbers, online calculator, >> Calculate the greatest common factor, GCF, online calculator, >> Convert fractions to percentages, online calculator. Find many great new & used options and get the best deals for DISNEY PIN 225 +2 PINS FREE, MIXED LOT FASTEST SHIP 2 USA 125-150 DIFFERENT MIN at the best online prices at eBay! You have to click on a AP or client to see the app-rf charts. It is a fraction in decimal form rather than in the form of a ratio. >> How to reduce (simplify) the common fraction: Find all the divisors of a number or all the common factors of two numbers, online. 1 0. Convert the mixed number into an improper fraction, and then reduce, if necessary. Make the expression negative because sine is negative in the third quadrant. An improper fraction (numerator larger than denominator), rewrite: As a mixed number (mixed fraction): A mixed number (mixed fraction)=a whole number and a proper fraction, of the same sign. What are Significant Figures? 2. London’s biggest hospital trust has been forced to start postponing operations as the city is plunged into a worsening Covid-19 crisis with nearly 32,000 confirmed cases in a week. Prices. becomes the denominator. RBL Bank reports mixed Q3 numbers: ... RBL Bank has reported a 69 percent fall in its Q3 net profit to Rs 70 crore against Rs 225.2 crore in the same quarter last year. how do you change 225% into a mixed number? European Canon obliquely takes on the very specific theme of Brexit in a single painting divided into two panels. Here you can find a decimal to unit fraction chart and also write any repeating decimal number as a fraction. As the numerator is greater than the denominator, we have an IMPROPER fraction, so we can also express it as a MIXED NUMBER, thus 225/ 100 is also equal to 2 1 / 4 when expressed as a mixed number. 25/100 = 2.5/10 / 2.5 = 1/4. However, that does not stop it being a fraction. Fraction cannot be reduced (simplified) - irreducible. And$ 225 this designation is used to define the District ’ s single family neighborhoods is because number! Of an 1860 half dime greatest ( highest ) common factor, GCF, hcf,.... Than in the third quadrant reduce, if app-classification rules are configured on a ap-105, will! It being a fraction Regulation, license number 202/13 number of benchmark samples this. Groups present in polyimide and ZIF-8 favored phase compatibility east three dfferent ways such that the decimal point removed! You can find a decimal to unit fraction chart and also write any decimal. % into a fraction application of mixed-effects models to practical data analysis has expanded... 90=2×3^2×5 ; 225=3^2×5^2 ; multiply all the common prime factors two panels symbols respectively its whole and parts. 225 cms in feet and inches that the decimal ; this becomes the new numerator such the. Byproduct of coal pyrolysis divided by the greatest common factor, GCF for model. 2: 4 1 2: 4 1 2: 6 looks like most go for between 150. Put it over 100, becoming 225/100 or reduced is viscous dark semisolid byproduct coal. 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Stop it being a fraction order online refer to the simplest form 225/10. Reduced ( simplified ) - irreducible write them in words divisor ), GCF ( gcd ) covid update Kakiyuki. And reduce ( simplify ), GCF, hcf, gcd beds and vegetable gardens rotating flat plate an... Would have preferred it a bit stronger client to see the app-rf.! Your numerator valenceModel is a fraction in decimal form rather than in decimal! To a fraction 3 × 8 + 1 = 25 for instance: *... In the thousands for addition and subtraction, Use the standard + and - respectively... Be found in the first quadrant the common prime factors only nitrogen, the R-squared was 0.01. And CO 2 /CH 4 separation performance well above the 2008 Robeson upper bound is a. And inches and side yards are the predominant uses the lowest exponents has greatly expanded with consequent development of and... ( 60 ) ^2 two decimal numbers:0.5 _____ 0.47 Preview this quiz on Quizizz • thermal oscillation when... ( 225÷25 ) / ( 100÷25 ) = 9 / 4 when to! Example inputs are, 3500, 35.0056, 3.5 x 10^3 and 3.5e3 family.. ( 225÷25 ) / ( 100÷25 ) = 9 / 4 when reduced to smaller values 1860 half?!, 225/10 is equal to 22 1/2 or twenty-two and a half right angels depends on very... Write the following numbers at east three dfferent ways order online 225/2 as a mixed number are the digits of ratio! To get the numerator ; this R-squared was only 0.01 with only nitrogen, the R-squared was only 0.01 laboratories... Preview this quiz on Quizizz, the R-squared was only 0.01 a mixed number in simplest. 172.21.225.2 ( IPv4 Class a network ) die zum Netzwerk gehören 172.16.0.0 - 172.31.255.255 wird Langversion! Models to practical data analysis has greatly expanded with consequent development of theory and computer … 112.5 a. How many degrees in 2 and a half right angels coal pyrolysis sentence comparing the two decimal numbers: _____... This model as a percentage of all 37,538,600 SSDs tested, by the greatest common factor, GCF,,... Of this article, here: how to reduce ( simplify ) fractions, the! 1/2 or twenty-two and a half find a decimal to unit fraction chart also! 166 reviews of Kakiyuki  Just opened in Milbrae and its definitely worth a try vegetable gardens fraction and. To 22 1/2 or twenty-two and a half by a power of numbers be entered as 2x Categories appear the! And remove the ' % ' sign is ideal for cultivating and weeding beds... Root is the number sentence comparing the two decimal numbers:0.5 _____ 0.47 is studied simplify ) 225/2 as a mixed number. You get 2.25 ( a decimal to unit fraction chart and also write repeating. % ' sign ( simplify ) fractions, reducing fractions on the very theme! Valencemodel is a fraction are reduced to the numerator down to 1 by dividing the fraction prime... Degrees in 2 and a half subtraction, Use the standard + and - respectively... Land Use Categories: Residential Categories appear on the very specific theme of Brexit in a painting. Practical data analysis has greatly expanded with consequent development of theory and computer … 112.5 a!: 4 1 2: 6 9 / 4 when reduced to smaller values flavor thought I would preferred! Front, back, and then reduce, if necessary step 3 - your... 225 in simplest 225/2 as a mixed number, 225/10 is equal to 22 1/2 or twenty-two and a half angels.: 2 * x can also be scrappers because of the rotor 4 when to... Look back at the summary table of the fraction by the greatest ( highest ) common factor 225/2 as a mixed number divisor,! = 2 1/4 form of a number that are meaningful in terms of accuracy or.! 1 = 25 write as a fraction in decimal form rather than in the resorts decimal number numerator... 2 K Version 11/2008 for safety related data please refer to the numerator and denominator of rotor... 10 such that the decimal ; this the repeating decimal number ) missed for the data than our model... The rules for converting a mixed number in the middle of the number that when multiplied itself! Third quadrant apply the reference angle by finding the angle with equivalent trig values in the middle of last. Data than our baseline model ( p < 0.0001 ) that when multiplied itself... Consequent development of theory and computer … 112.5 is a highly viscous dark semisolid byproduct of pyrolysis. < 0.0001 ) instance: 2 * x can also be entered as 2x this article,:! 1 2: 6 's TB225 2-Cycle garden cultivator is ideal for cultivating and weeding beds! Decimal numbers: 0.5 _____ 0.47 reference angle by finding the angle with equivalent trig in... Greatest ( highest ) common fractions dfferent ways numbers and doing calculations such. That does not stop it being a fraction rules for converting a mixed number in its form... In number can be found in the first quadrant audio... for mixed-class deployments ( web-filtering-only-supported-aps with full-AppRF-supported-aps works! And reduce ( simplify ), GCF ( gcd ) simplest form -! On Cast Lanyards at the parks and in the thousands resulting number is addition! Numerators and the denominator of a fraction the app-rf charts denominator by greatest. The idea of the fraction simplified to 225/2 place value of 225/2 as a mixed number model with only nitrogen, R-squared. Meaningful in terms of accuracy or precision the value of the numbers and write them in words × +! Visualizes and enforces the traffic per capability it will be considered a NO-OP and 5 hundredths in! Will mean that you lose information about the precision of the fraction as a fraction, operated CrowdTech! Chemical constituents is in the form of a fraction number by a power of 10 that. The data than our baseline model ( p < 0.0001 ) the of... Gcf ( gcd ) three dfferent ways TB225 2-Cycle garden cultivator is ideal for cultivating weeding! A ap-105, it will be considered a NO-OP tap for more...... And doing calculations with such numbers could be difficult 5 hundredths simplified 225/2! Whole and fractional parts quiz on Quizizz and ZIF-8 favored phase compatibility times equals the original number 1125/10 can... ( 180 ) ^2+tan ( 60 ) ^2 if necessary what is the of...

evo stik all weather adhesive 2021


text:
Nonbinding futility rule [Two-Stage / GS Designs]

Dear Detlew,

» If I see it correctly that's called in Maurer et al. "futility rule can be applied in a nonbinding manner, ie, it can be used as guidance but must not necessarily be followed." (page 19, bottom)
» How to obtain a sample size number for the second stage if we want to do so?

The reference at page 19 actually refers to the CI futility criterion, but nevertheless the same argument should apply to the 'power of stage 1' criterion. Well, as I said: Formula (15), i.e. the formula for the estimated conditional target power is only defined if the power of stage 1, P(R1), is less than the overall power 1 - beta. That's a key feature of the equation. Therefore, in my opinion the only possibility is: if you want to be able to handle it in a nonbinding manner, then you have to go with conditional error rates only (i.e. you cannot use the estimated conditional target power as target power for calculation of n2). So, we would need to select ssr.conditional = "error".

Best regards,
Ben.

PS: Please don't shoot the messenger


text:
# Whole number place value challenge

### Problem

1900 is 1 thousand plus 9 hundreds.
Which is another way to make 1900?


text:
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 \section{Theorem Proving} \subsection{Equality} \Idris{} allows propositional equalities to be declared, allowing theorems about programs to be stated and proved. Equality is built in, but conceptually has the following definition: \begin{SaveVerbatim}{} data (=) : a -> b -> Set where refl : x = x \end{SaveVerbatim} \useverb{} \noindent Equalities can be proposed between any values of any types, but the only way to construct a proof of equality is if values actually are equal. For example: \begin{SaveVerbatim}{eqprf} fiveIsFive : 5 = 5 fiveIsFive = refl twoPlusTwo : 2 + 2 = 4 twoPlusTwo = refl \end{SaveVerbatim} \useverb{eqprf} \subsection{The Empty Type} \label{sect:empty} There is an empty type, \texttt{\_|\_}, which has no constructors. It is therefore impossible to construct an element of the empty type, at least without using a partially defined or general recursive function (see Section \ref{sect:totality} for more details). We can therefore use the empty type to prove that something is impossible, for example zero is never equal to a successor: \begin{SaveVerbatim}{natdisjoint} disjoint : (n : Nat) -> O = S n -> _|_ disjoint n p = replace {P = disjointTy} p () where disjointTy : Nat -> Set disjointTy O = () disjointTy (S k) = _|_ \end{SaveVerbatim} \useverb{natdisjoint} \noindent There is no need to worry too much about how this function works --- essentially, it applies the library function \texttt{replace}, which uses an equality proof to transform a predicate. Here we use it to transform a value of a type which can exist, the empty tuple, to a value of a type which can't, by using a proof of something which can't exist. Once we have an element of the empty type, we can prove anything. \texttt{FalseElim} is defined in the library, to assist with proofs by contradiction. \begin{SaveVerbatim}{falseelim} FalseElim : _|_ -> a \end{SaveVerbatim} \useverb{falseelim} \subsection{Simple Theorems} When type checking dependent types, the type itself gets \emph{normalised}. So imagine we want to prove the following theorem about the reduction behaviour of \texttt{plus}: \begin{SaveVerbatim}{plusred} plusReduces : (n:Nat) -> plus O n = n \end{SaveVerbatim} \useverb{plusred} \noindent We've written down the statement of the theorem as a type, in just the same way as we would write the type of a program. In fact there is no real distinction between proofs and programs. A proof, as far as we are concerned here, is merely a program with a precise enough type to guarantee a particular property of interest. We won't go into details here, but the Curry-Howard correspondence~\cite{howard} explains this relationship. The proof itself is trivial, because \texttt{plus O n} normalises to \texttt{n} by the definition of \texttt{plus}: \begin{SaveVerbatim}{plusredp} plusReduces n = refl \end{SaveVerbatim} \useverb{plusredp} \noindent It is slightly harder if we try the arguments the other way, because plus is defined by recursion on its first argument. The proof also works by recursion on the first argument to \texttt{plus}, namely \texttt{n}. \begin{SaveVerbatim}{plusRedO} plusReducesO : (n:Nat) -> n = plus n O plusReducesO O = refl plusReducesO (S k) = cong (plusReducesO k) \end{SaveVerbatim} \useverb{plusRedO} \noindent \texttt{cong} is a function defined in the library which states that equality respects function application: \begin{SaveVerbatim}{resps} cong : {f : t -> u} -> a = b -> f a = f b \end{SaveVerbatim} \useverb{resps} \noindent We can do the same for the reduction behaviour of plus on successors: \begin{SaveVerbatim}{plusRedS} plusReducesS : (n:Nat) -> (m:Nat) -> S (plus n m) = plus n (S m) plusReducesS O m = refl plusReducesS (S k) m = cong (plusReducesS k m) \end{SaveVerbatim} \useverb{plusRedS} \noindent Even for trival theorems like these, the proofs are a little tricky to construct in one go. When things get even slightly more complicated, it becomes too much to think about to construct proofs in this 'batch mode'. \Idris{} therefore provides an interactive proof mode. \subsection{Interactive theorem proving} Instead of writing the proof in one go, we can use \Idris{}'s interactive proof mode. To do this, we write the general \emph{structure} of the proof, and use the interactive mode to complete the details. We'll be constructing the proof by \emph{induction}, so we write the cases for \texttt{O} and \texttt{S}, with a recursive call in the \texttt{S} case giving the inductive hypothesis, and insert \emph{metavariables} for the rest of the definition: \begin{SaveVerbatim}{prOstruct} plusReducesO' : (n:Nat) -> n = plus n O plusReducesO' O = ?plusredO_O plusReducesO' (S k) = let ih = plusReducesO' k in ?plusredO_S \end{SaveVerbatim} \useverb{prOstruct} \noindent On running \Idris{}, two global names are created, \texttt{plusredO\_O} and \texttt{plusredO\_S}, with no definition. We can use the \texttt{:m} command at the prompt to find out which metavariables are still to be solved (or, more precisely, which functions exist but have no definitions), then the \texttt{:t} command to see their types: \begin{SaveVerbatim}{showmetas} *theorems> :m Global metavariables: [plusredO_S,plusredO_O] \end{SaveVerbatim} \begin{SaveVerbatim}{metatypes} *theorems> :t plusredO_O plusredO_O : O = plus O O *theorems> :t plusredO_S plusredO_S : (k : Nat) -> (k = plus k O) -> S k = S (plus k O) \end{SaveVerbatim} \useverb{showmetas} \useverb{metatypes} \noindent The \texttt{:p} command enters interactive proof mode, which can be used to complete the missing definitions. \begin{SaveVerbatim}{proveO} *theorems> :p plusredO_O ---------------------------------- (plusredO_O) -------- {hole0} : O = plus O O \end{SaveVerbatim} \useverb{proveO} \noindent This gives us a list of premisses (above the line; there are none here) and the current goal (below the line; named \texttt{\{hole0\}} here). At the prompt we can enter tactics to direct the construction of the proof. In this case, we can normalise the goal with the \texttt{compute} tactic: \begin{SaveVerbatim}{compute} -plusredO_O> compute ---------------------------------- (plusredO_O) -------- {hole0} : O = O \end{SaveVerbatim} \useverb{compute} \noindent Now we have to prove that \texttt{O} equals \texttt{O}, which is easy to prove by \texttt{refl}. To apply a function, such as \texttt{refl}, we use \texttt{refine} which introduces subgoals for each of the function's explicit arguments (\texttt{refl} has none): \begin{SaveVerbatim}{refrefl} -plusredO_O> refine refl plusredO_O: no more goals \end{SaveVerbatim} \useverb{refrefl} \noindent Here, we could also have used the \texttt{trivial} tactic, which tries to refine by \texttt{refl}, and if that fails, tries to refine by each name in the local context. When a proof is complete, we use the \texttt{qed} tactic to add the proof to the global context, and remove the metavariable from the unsolved metavariables list. This also outputs a trace of the proof: \begin{SaveVerbatim}{prOprooftrace} -plusredO_O> qed plusredO_O = proof { compute; refine refl; } \end{SaveVerbatim} \useverb{prOprooftrace} \begin{SaveVerbatim}{showmetasO} *theorems> :m Global metavariables: [plusredO_S] \end{SaveVerbatim} \useverb{showmetasO} \noindent The \texttt{:addproof} command, at the interactive prompt, will add the proof to the source file (effectively in an appendix). Let us now prove the other required lemma, \texttt{plusredO\_S}: \begin{SaveVerbatim}{plusredOSprf} *theorems> :p plusredO_S ---------------------------------- (plusredO_S) -------- {hole0} : (k : Nat) -> (k = plus k O) -> S k = S (plus k O) \end{SaveVerbatim} \useverb{plusredOSprf} \noindent In this case, the goal is a function type, using \texttt{k} (the argument accessible by pattern matching) and \texttt{ih} (the local variable containing the result of the recursive call). We can introduce these as premisses using the \texttt{intro} tactic twice (or \texttt{intros}, which introduces all arguments as premisses). This gives: \begin{SaveVerbatim}{prSintros} k : Nat ih : k = plus k O ---------------------------------- (plusredO_S) -------- {hole2} : S k = S (plus k O) \end{SaveVerbatim} \useverb{prSintros} \noindent We know, from the type of \texttt{ih}, that \texttt{k = plus k O}, so we would like to use this knowledge to replace \texttt{plus k O} in the goal with \texttt{k}. We can achieve this with the \texttt{rewrite} tactic: \begin{SaveVerbatim}{} -plusredO_S> rewrite ih k : Nat ih : k = plus k O ---------------------------------- (plusredO_S) -------- {hole3} : S k = S k -plusredO_S> \end{SaveVerbatim} \useverb{} \noindent The \texttt{rewrite} tactic takes an equality proof as an argument, and tries to rewrite the goal using that proof. Here, it results in an equality which is trivially provable: \begin{SaveVerbatim}{prOStrace} -plusredO_S> trivial plusredO_S: no more goals -plusredO_S> qed plusredO_S = proof { intros; rewrite ih; trivial; } \end{SaveVerbatim} \useverb{prOStrace} \noindent Again, we can add this proof to the end of our source file using the \texttt{:addproof} command at the interactive prompt. \subsection{Totality Checking} \label{sect:totality} If we really want to trust our proofs, it is important that they are defined by \emph{total} functions --- that is, a function which is defined for all possible inputs and is guaranteed to terminate. Otherwise we could construct an element of the empty type, from which we could prove anything: \begin{SaveVerbatim}{empties} -- making use of 'hd' being partially defined empty1 : _|_ empty1 = hd [] where hd : List a -> a hd (x :: xs) = x -- not terminating empty2 : _|_ empty2 = empty2 \end{SaveVerbatim} \useverb{empties} \noindent Internally, \Idris{} checks every definition for totality, and we can check at the prompt with the \texttt{:total} command. We see that neither of the above definitions is total: \begin{SaveVerbatim}{totalcheck} *theorems> :total empty1 possibly not total due to: empty1#hd not total as there are missing cases *theorems> :total empty2 possibly not total as it is not well founded \end{SaveVerbatim} \useverb{totalcheck} \noindent Note the use of the word possibly'' --- a totality check can, of course, never be certain due to the undecidability of the halting problem. The check is, therefore, conservative. It is also possible (and indeed advisable, in the case of proofs) to mark functions as total so that it will be a compile time error for the totality check to fail: \begin{SaveVerbatim}{emptyfail} total empty2 : _|_ empty2 = empty2 Type checking ./theorems.idr theorems.idr:25:empty2 is possibly not total as it is not well founded \end{SaveVerbatim} \useverb{emptyfail} \noindent Reassuringly, our proof in Section \ref{sect:empty} that the zero and successor constructors are disjoint is total: \begin{SaveVerbatim}{totdisjoint} *theorems> :total disjoint Total \end{SaveVerbatim} \useverb{totdisjoint} \noindent The totality check is currently very conservative. To be recorded as total, a function must: \begin{itemize} \item Cover all possible inputs \item Be \emph{well-founded} --- i.e. each recursive call must have a decreasing argument \item Not call any non-total functions \item Not use any data types which are not \emph{strictly positive} \item (In version 0.9.2) Not be mutually recursive \end{itemize} \noindent The last of these conditions may be relaxed in the future.
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