text:
Orienting

Orienting¶

We're in New Zealand for a semester. Recently, Ash Patea was telling us some Māori lore about local mountains. We're based in Whanganui, learning about the mountains Tongariro, Taranaki, and Ruapehu, all of which are to our north. Ash starts all of these sessions off by telling us that we're learning his iwi's stories, but that they're not the only stories. That other stories are different, and none is "right." We then get to the part where we learn that the name "Tongariro" tells you it's a mountain to the south, and Ash says that's how they know the story we're learning isn't their story, and all of a sudden I have to write a little math post.

I don't want to spoil the punchline too much (yes I do), but there's a really neat connection between Ash's language and mathematics, about how you orient yourself in the world, and about how things can't always be consistent.

Math time. A "chart" is essentially your local set of coordinates, or how you tell yourself how to move around in the world. You can think of it like a street map in a city, or the idea of north, south, east, and west.

In [2]:
makeone()

You could describe different regions with different charts. You need to "glue" them together in the areas where they overlap.

In [3]:
makethree()

When you glue all of those charts together to cover an object, the result is called an "atlas"

In [4]:
makelots()

So, obviously, the connection I want point out is that each of these charts is like the stories of an individual iwi. They tell that iwi how to understand the world; how to get around in the world; how to orient themselves in the world. Understanding the local map tells you where you are; and knowing how to glue those local maps together tells you how to get around.

Sometimes, you can glue everything together into one consistent set of coordinates. Like in the above pictures, we could just have called the same direction "north" everywhere. Or, maybe every iwi has the same stories. Back to math language, maybe you can just have one chart to make up your whole atlas.

But the thing that really struck me is that sometimes you can't do that!¶

The most common example on the math side of things is coordinates on a sphere. You already know this: you know how latitude and longitude don't really work on at the North Pole or the South Pole? Well, it turns out that's because you can't cover a sphere with one chart!

Here's one example

In [5]:
make_half_sphere_chart()

where you can see a chart in red mapping the top half of the sphere ... but obviously not the bottom half. Like I said, though, it turns out that you simply can't have one chart that covers the whole sphere. Kind of like how, when we say "north" we don't mean the direction that goes through the inside of the earth straight towards the North Pole. We mean the direction we'd draw on the ground. And that direction is pointing somewhere different for someone in New Zealand vs. someone in the US.

In [6]:
make_north_example()

This turns out to be related to the Hairy Ball Theorem. That one says that, if you have a ball covered with hair (like my head, or a basketball), you can't comb it without at least one cowlick. To be specific, the hairs are pointing in some direction along the head, and you're fundamentally stuck with at least one place where your local directions don't agree!

Anyway, that's a bit of a digression. Back to what we were learning from Ash.¶

We were learning Māori legends, and they’re different in different geographical regions. They often conflict if you get far enough away. The specific example that came up was talking about stories of the histories of the mountains nearby. The story Ash told us was about a mountain called Tongariro. Skipping lots of details, but Tongariro is a mountain to our north. But if you understand the Māori meaning of "Tongariro" it tells you that the mountain is to the south!

Back to the math language, you're clearly looking at two different charts, and they don't match up! You have to be very careful about how you translate ("glue") between them, because they don't agree.

I think that's the coolest thing! That the way Māori stitch together culture is exactly the way mathematicians stitch together coordinates!¶

You just have to understand that they're different in different places, and you have to use that context to understand what's going on.

Pākehā (non-Māori folks) often try to say "this is THE one Māori legend" i.e. to impose some global coordinates. A really powerful example of this is the Moriori.

In the early 19th century, a couple of pākehā ethnologists were trying to knit together Māori stories into a global whole. They kept finding pieces that didn't match up. Often, these were stories of people who had been in Aotearoa/New Zealand before. It seems clear now that these were simply stories of other Māori groups, and stories of fighting between different iwi. The ethnologists, however, invented the "Moriori myth": they claimed that the Moriori were here when Māori came to Aotearoa/New Zealand. The truth is that the Māori came to New Zealand, and then some of them left to go to the Chatham Islands. The group that left became the Moriori.

The Moriori myth led to direct material harm: the idea that "the Māori did it to the Moriori first" was used by colonizers to justify ignoring the Treaty of Waitangi for a hundred years!

Back to the math for a second. It also turns out that, for most surfaces, you can't have one global set of coordinates. You have to understand the local charts, and then you have to understand how to glue them together into an atlas. Maybe we should take something socially from that.

And maybe I had to come all the way to the other side of the globe, to Aotearoa/New Zealand to figure out how to start piecing together local views of the world, to glue my charts into an atlas without just stretching one chart past its relevance.

Edits: I made my own versions of all of the images, made an image for the "north" example, cleaned up some of the language, added a bit more about Moriori, and moved the "orientable" punchline to the endnotes.

Endnotes:
(1) Māori talk about how these stories are how they orient themselves in the world. "Orient" has a meaning in math as well, and it essentially means that you don't just know how to define north/south/east/west, you also know how to define "up" and "down." A non-orientable surface is effectively one where you can have local definitions of "up" and "down" everywhere, but there is fundamentally no way to define a global coordinate system, to define "up" and "down."

And it turns out that most surfaces are non-orientable. Again, maybe we should take something from that.
(2) For the math folks, when I say surface above, I really mean Manifold.
(3) We learned the Moriori history from Jillian Wychel and David James, and I don't have a great website/link for them. The history of Māori and Moriori is not a simple one, and there's a lot of ugliness, especially as the Moriori were almost completely wiped out in the 1800s.
(4) If you're curious about the code that produced those plots, here it is!

In [1]:
from matplotlib.patches import Rectangle
from matplotlib.path import Path
import matplotlib.patches as patches
import numpy as np, matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
plt.xkcd()

def plt1(ax,offsetx=0,offsety=0,color='orange',text=True):
# Mostly taken from
# https://matplotlib.org/users/path_tutorial.html
verts = [
(0. + offsetx, 0. + offsety), # left, bottom
(0. + offsetx, 1. + offsety), # left, top
(1. + offsetx, 1. + offsety), # right, top
(1. + offsetx, 0. + offsety), # right, bottom
(0. + offsetx, 0. + offsety), # ignored
]

codes = [Path.MOVETO,
Path.LINETO,
Path.LINETO,
Path.LINETO,
Path.CLOSEPOLY,
]

path = Path(verts, codes)

patch = patches.PathPatch(path, facecolor=color, lw=2, alpha=0.5)

#arrow( x, y, dx, dy, **kwargs )
if text:
alpha=1
else:
alpha=0.2
plt.arrow( 0.5 + offsetx, 0.2 + offsety, 0.0, 0.4, fc="k", ec="k",head_width=0.05, head_length=0.1,alpha=alpha)
if text:
plt.text(0.6 + offsetx,0.3 + offsety,'North')
ax.set_xlim(-2,2)
ax.set_ylim(-2,2)
ax.set_yticklabels([])
ax.set_xticklabels([])

def makeone():
fig = plt.figure()

plt1(ax)

def makethree():
fig = plt.figure()

plt1(ax)
plt1(ax,-0.9,-0.1,'green')
plt1(ax,-0.5,-0.9,'blue')

def makelots():
fig = plt.figure()

for i in np.arange(-2,2,.8):
for j in np.arange(-2,2,.8):
plt1(ax,i,j,'grey',text=False)

plt1(ax)
plt1(ax,-0.9,-0.1,'green')
plt1(ax,-0.5,-0.9,'blue')

def plot_top(ax,n_theta,n_phi,r,fixz=False,alpha=0.5):
# Largely taken from
# https://stackoverflow.com/questions/41105754/heat-map-half-sphere-plot

#theta inclination angle
#phi azimuthal angle

theta, phi = np.mgrid[0.0:0.5*np.pi:n_theta*1j, 0.0:2.0*np.pi:n_phi*1j]

x = r*np.sin(theta)*np.cos(phi)
y = r*np.sin(theta)*np.sin(phi)
z = r*np.cos(theta)
if fixz:
z = np.ones_like(z)*2

ax.plot_surface(
x,y,z,  rstride=1, cstride=1,
color='red',
alpha=alpha, linewidth=1)

def plot_bottom(ax,n_theta,n_phi,r,alpha=0.5):
# Largely taken from
# https://stackoverflow.com/questions/41105754/heat-map-half-sphere-plot

#theta inclination angle
#phi azimuthal angle

theta, phi = np.mgrid[0.0:0.5*np.pi:n_theta*1j, 0.0:2.0*np.pi:n_phi*1j]

x = r*np.sin(theta)*np.cos(phi)
y = r*np.sin(theta)*np.sin(phi)
z = -r*np.cos(theta)

ax.plot_surface(
x,y,z,  rstride=1, cstride=1,
color='blue',
alpha=alpha, linewidth=1)

def plot_circles(ax,includetop=True,color='k'):
# Largely taken from
# https://stackoverflow.com/questions/32424670/python-matplotlib-drawing-3d-sphere-with-circumferences
u = np.linspace(0, 2 * np.pi, 100)
v = np.linspace(0, np.pi, 100)

elev = 10.0
rot = 80.0 / 180 * np.pi

#calculate vectors for "vertical" circle
a = np.array([-np.sin(elev / 180 * np.pi), 0, np.cos(elev / 180 * np.pi)])
b = np.array([0, 1, 0])
b = b * np.cos(rot) + np.cross(a, b) * np.sin(rot) + a * np.dot(a, b) * (1 - np.cos(rot))
horiz_front = np.linspace(0, np.pi, 100)
ax.plot(np.sin(u),np.cos(u),0,color=color, linestyle = 'dashed')
ax.plot(np.sin(horiz_front),np.cos(horiz_front),0,color=color)

if includetop:
ax.plot(np.sin(u),np.cos(u),2.0,color=color, linestyle = 'dashed')
ax.plot(np.sin(horiz_front),np.cos(horiz_front),2.0,color=color)

vert_front = np.linspace(np.pi / 2, 3 * np.pi / 2, 100)
ax.plot(a[0] * np.sin(u) + b[0] * np.cos(u), b[1] * np.cos(u), a[2] * np.sin(u) + b[2] * np.cos(u),
color=color, linestyle = 'dashed')
ax.plot(a[0] * np.sin(vert_front) + b[0] * np.cos(vert_front),
b[1] * np.cos(vert_front), a[2] * np.sin(vert_front) + b[2] * np.cos(vert_front),
color=color)

def plot_lines(ax):
ax.plot(xs=[0.75,.75],ys=[-0.75,-0.75],zs=[0,2],color='k',alpha=1.0)
ax.plot(xs=[0.75,.75],ys=[0.75,0.75],zs=[0,2],color='k',alpha=1.0)

ax.plot(xs=[0,0],ys=[0,0],zs=[1,1.65],color='k')
ax.plot(xs=[0,0],ys=[0,0],zs=[1.65,2.0],color='k',linestyle='dashed',alpha=0.75)

ax.plot(xs=[-0.75,-.75],ys=[-0.5,-0.5],zs=[0.5,1.5],color='k',alpha=1.0)
ax.plot(xs=[-0.75,-.75],ys=[-0.5,-0.5],zs=[1.5,2.0],color='k',linestyle='dashed',alpha=0.75)
ax.plot(xs=[-0.75,-.75],ys=[-0.5,-0.5],zs=[0,0.5],color='k',linestyle='dashed',alpha=0.75)

ax.plot(xs=[-0.75,-.75],ys=[0.5,0.5],zs=[0.5,1.5],color='k',alpha=1.0)
ax.plot(xs=[-0.75,-.75],ys=[0.5,0.5],zs=[1.5,2.0],color='k',linestyle='dashed',alpha=0.75)
ax.plot(xs=[-0.75,-.75],ys=[0.5,0.5],zs=[0,0.5],color='k',linestyle='dashed',alpha=0.75)

def make_half_sphere_chart():
fig = plt.figure(figsize=(10, 8))

plot_top(ax,30,30,1,alpha=0.25)
plot_bottom(ax,30,30,1,alpha=0.25)
plot_circles(ax,color='darkgrey')
plot_top(ax,30,30,1,fixz=True,alpha=0.7)
plot_lines(ax)

ax.set_xlim([-2.2,2.2])
ax.set_ylim([-2.2,2.2])
ax.set_zlim([-2.2,2.2])

ax.set_aspect('equal')
ax.view_init(elev = 20, azim = 0)
ax.set_yticklabels([])
ax.set_xticklabels([])
ax.set_zticklabels([])

def make_north_example():
fig = plt.figure(figsize=(10, 8))

plot_top(ax,30,30,1,alpha=0.25)
plot_bottom(ax,30,30,1,alpha=0.25)
plot_circles(ax,includetop=False,color='darkgrey')

offsetx = 0
offsety = 0
alpha=1.0
# These arrows aren't mathematically correct ... but they look reasonable!
ax.quiver([0],[-.13],[-0.9],[ 0],[-.1],[.75],color='k')
ax.text(0,0,-.75,'North',fontsize=18)

ax.quiver([0],[-.16],[0.5], [0],[.1],[.75],color='k')
ax.text(0,0,.75,'North',fontsize=18)

ax.set_xlim([-1.5,1.5])
ax.set_ylim([-1.5,1.5])
ax.set_zlim([-1.5,1.5])

ax.set_aspect('equal')
ax.view_init(elev = 20, azim = 0)
ax.set_yticklabels([])
ax.set_xticklabels([])
ax.set_zticklabels([])


text:
# Integral domain with two elements that do not have a gcd

I have the following example of an integral domain with two elements that do not have a gcd from wikipedia:

$R = \mathbb{Z}\left[\sqrt{-3}\,\,\right],\quad a = 4 = 2\cdot 2 = \left(1+\sqrt{-3}\,\,\right)\left(1-\sqrt{-3}\,\,\right),\quad b = \left(1+\sqrt{-3}\,\,\right)\cdot 2.$

The elements $2$ and $1 + \sqrt{−3}$ are two "maximal common divisors" (i.e. any common divisor which is a multiple of $2$ is associated to $2$, the same holds for $1 + \sqrt{−3}$), but they are not associated, so there is no greatest common divisor of a and b.

I can understand it, but how can I prove or strictly explain that $2$ and $1 + \sqrt{−3}$ are two "maximal common divisors" and they are not associated?

-
What can the norm of a common divisor be? What elements have those norms? –  Qiaochu Yuan May 5 '11 at 2:21
@Yuan I found the concept of multiplicative norms, defined by $N(a+b\sqrt{-3})=a^2+3b^2$ in here. Do you mean this? –  Gobi May 8 '11 at 5:11

A simple but general way to deduce that this gcd fails to exist is by failure of Euclid's Lemma.

LEMMA $\rm\ \ (a,b)\ =\ (ac,bc)/c\quad$ if $\rm\ (ac,bc)\$ exists $\rm\quad$ (GCD distributive law )

Proof $\rm\quad d\ |\ a,b\ \iff\ dc\ |\ ac,bc\ \iff\ dc\ |\ (ac,bc)\ \iff\ d|(ac,bc)/c$

But generally $\rm\ (ac,bc)\$ need not exist, as is most insightfully viewed as failure of

EUCLID'S LEMMA $\rm\quad a\ |\ bc\$ and $\rm\ (a,b)=1\ \Rightarrow\ a\ |\ c\quad$ if $\rm\ (ac,bc)\$ exists.

Proof $\ \$ If $\rm\ (ac,bc)\$ exists then $\rm\ a\ |\ ac,bc\ \Rightarrow\ a\ |\ (ac,bc) = (a,b)\:c = c\$ by the Lemma.

Therefore if $\rm\: a,b,c\:$ fail to satisfy the Euclid Lemma $\Rightarrow\:$, namely if $\rm\ a\ |\ bc\$ and $\rm\ (a,b) = 1\$ but $\rm\ a\nmid c\:$, then one immediately deduces that the gcd $\rm\ (ac,bc)\$ fails to exist.$\:$ For the special case that $\rm\:a\:$ is an atom (i.e. irreducible), the implication reduces to: atom $\Rightarrow$ prime. So it suffices to find a nonprime atom in order to exhibit a pair of elements whose gcd fails to exist. This task is a bit simpler, e.g. for $\rm\ \omega = 1 + \sqrt{-3}\ \in\ \mathbb Z[\sqrt{-3}]\$ we have that the atom $\rm\: 2\ |\ \omega'\: \omega = 4\:,\:$ but $\rm\ 2\nmid \omega',\:\omega\:,\:$ so $\rm\:2\:$ is not prime. Therefore your gcd $\rm\: (2\:\omega,\ \omega'\:\omega)\ =\ (2+2\sqrt{-3},\:4)\$ fails to exist in $\rm\ \mathbb Z[\sqrt{-3}]\:$.

Note that if the gcd $\rm\: (ac,bc)\$ fails to exist then this implies that the ideal $\rm\ (ac,bc)\$ is not principal. Therefore we've constructively deduced that the failure of Euclid's lemma immediately yields both a nonexistent gcd and a nonprincipal ideal.

That the $\Rightarrow$ in Euclid's lemma implies that Atoms are Prime $\rm(:= AP)$ is denoted $\rm\ D\ \Rightarrow AP\$ in the list of domains closely related to GCD domains in my post here. There you will find links to further literature on domains closely related to GCD domains. See especially the referenced comprehensive survey by D.D. Anderson: GCD domains, Gauss' lemma, and contents of polynomials, 2000.

See also my post here for the general universal definitions of $\rm GCD,\: LCM$ and for further remarks on how such $\iff$ definitions enable slick proofs, and see here for another simple example of such.

-
Thanks for your great answer! I understood this and knew some other interesting concepts. Later I want to study them. –  Gobi May 8 '11 at 5:05

format: plain


text:
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# Re: How can I check for the existence of a glyph in TeX?

```At 11:52 AM 9/7/98 +0100, Rebecca and Rowland wrote:

>I'm writing a package to automate the use of Euro symbols (the idea is that
>you type \euro and get a Euro symbol no matter which fount you're using or
>whether or not it's actually got a Euro).

>One thing I'd like to do is try and ensure that if the fount really *does*
>have a Euro, you get to use it.  To do this, I need to be able to test for
>the existence of a real Euro glyph in a fount.

>The immediately obvious test:

>\settowidth{\@tempdima}{\texteuro}

>\ifdim\@tempdima=0pt...

>fails because \texteuro might well have been defined to be a `Missing
>glyph' rule and therefore have non-zero width (and height and depth).

>So...  Is there any way of seeing if a glyph *really* exists *from TeX*?

This is a tough one, since Knuth worked with fonts that had no missing glyphs.
Hence there is little support for this sort of thing.  For example, missing
glyph
messages do not appear on screen in most TeX implementations, only in the
log file (which means they are mostly never seen), and then do not contain
any context information (so it can be very hard to find out where they
com from).  In any case just testing the character causes an error message
to be emitted if it is not there. I suppose you could read in the log file
and
check afterwards :-)?

Regards, Berthold.

```


text:
anonymous one year ago What do I do, I am unsure how to interpret it as a geometric series. http://i.imgur.com/pyZd2ip.png Thanks

1. Empty

You can make a substitution to make it easier perhaps? $x=(1+c)^{-1}$ $\sum_{n=1}^\infty x^n =8$

2. jtvatsim

I agree with @empty. You can then use the infinite series formula for geometric series to get a value for x. Then solve for c after. That is, 1/(1-x) = 8 solve for x. You will get x = something. Then, x = 1/(1+c) so, something = 1/(1+c) solve for c.

3. anonymous

Ah yes thanks guys!

4. anonymous

Let $$x = 1+c$$ Partial sum is$s_n = x^{-1} + x^{-2} + x^{=3} + ... + x^{-n}$Multiply both sides by $$x$$$xs_n = 1 + x^{-1} + x^{-2} + ... + x^{-n+1}$Subtract 1st equation from 2nd equation$(x-1)s^n = 1 + x^{-n}$$s_n = \frac{ 1 + x^{-n} }{ x-1 }$Sum of infinite series is$\lim_{n \rightarrow \infty} s_n = \lim_{n \rightarrow \infty} \frac{ 1 + x^{-n} }{ x-1 } = 8$$\frac{ 1 }{ x-1 } = 8$Solve for $$x$$ and then for $$c$$.


text:
Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# If the Roots of X 2 − B X + C = 0 Are Two Consecutive Integers, Then B2 − 4 C is - Mathematics

MCQ

If the roots of $x^2 - bx + c = 0$ are two consecutive integers, then b2 − 4 c is

#### Options

• 0

• 1

• 2

• none of these.

#### Solution

1

Given equation:

$x^2 - bx + c = 0$

Let $\alpha \text { and } \alpha + 1$ be the two consecutive roots of the equation.

Sum of the roots = $\alpha + \alpha + 1 = 2\alpha + 1$

Product of the roots = $\alpha (\alpha + 1) = \alpha^2 + \alpha$

$\text { So, sum of the roots }= 2\alpha + 1 = \frac{- \text { Coeffecient of } x}{\text { Coeffecient of } x^2} = \frac{b}{1} = b$

$\text { Product of the roots } = \alpha^2 + \alpha = \frac{\text { Constant term }}{\text { Coeffecient of }x^2} = \frac{c}{1} = c$

$\text { Now, } b^2 - 4c = \left( 2\alpha + 1 \right)^2 - 4\left( \alpha^2 + \alpha \right) = 4 \alpha^2 + 4\alpha + 1 - 4 \alpha^2 - 4\alpha = 1$

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook


text:
Open Access

# Functions on surfaces and incompressible subsurfaces

### Abstract

Let $M$ be a smooth connected compact surface, $P$ be either a real line $\mathbb R$ or a circle $S^1$. Then we have a natural right action of the group $D(M)$ of diffeomorphisms of $M$ on $C^\infty(M,P)$. For $f\in C^\infty(M,P)$ denote respectively by $S(f)$ and $O(f)$ its stabilizer and orbit with respect to this action. Recently, for a large class of smooth maps $f:M\to P$ the author calculated the homotopy types of the connected components of $S(f)$ and $O(f)$. It turned out that except for few cases the identity component of $S(f)$ is contractible, $\pi_i O(f)=\pi_i M$ for $i\geq3$, and $\pi_2 O(f)=0$, while $\pi_1 O(f)$ it only proved to be a finite extension of $\pi_1D_{Id}M\oplus\mathbb Z^{l}$ for some $l\geq0$. In this note it is shown that if $\chi(M)<0$, then $\pi_1O(f)=G_1\times\cdots\times G_n$, where each $G_i$ is a fundamental group of the restriction of $f$ to a subsurface $B_i\subset M$ being either a $2$-disk or a cylinder or a Mobius band. For the proof of main result incompressible subsurfaces and cellular automorphisms of surfaces are studied.

### Article Information

 Title Functions on surfaces and incompressible subsurfaces Source Methods Funct. Anal. Topology, Vol. 16 (2010), no. 2, 167-182 MathSciNet MR2667811 Copyright The Author(s) 2010 (CC BY-SA)

### Authors Information

Sergiy Maksymenko
Institute of Mathematics of NAS of Ukraine, Tereshchenkivska st. 3, Kyiv, 01601 Ukraine

### Citation Example

Sergiy Maksymenko, Functions on surfaces and incompressible subsurfaces, Methods Funct. Anal. Topology 16 (2010), no. 2, 167-182.

### BibTex

@article {MFAT553,
AUTHOR = {Maksymenko, Sergiy},
TITLE = {Functions on surfaces and incompressible subsurfaces},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {16},
YEAR = {2010},
NUMBER = {2},
PAGES = {167-182},
ISSN = {1029-3531},
MRNUMBER = {MR2667811},
URL = {http://mfat.imath.kiev.ua/article/?id=553},
}


text:
# FINA Congress Votes ‘NO’ on Underwater Cameras

July 26th, 2013

In what was a wild, multi-round vote, sources tell us that the FINA Technical Congress has voted to NOT allow underwater cameras for the judgement of disqualifications at major meets.

Those who we spoke to weren’t sure even they could exactly pinpoint what happened in the vote, but after two rounds, a few switches back-and-forth between which side of the issue put one ‘yay’ and which side of the issue put one ‘nay,’ ultimately the measure failed by about a dozen votes.

Who voted which direction was unclear; Denmark were the formal proposers of the rule, so they voted yes, as did the Americans, but after that things became murky.

There have been a number of reasons bandied about as to why various parties were against the cameras. Among them were:

• interference with broadcasters’ underwater cameras
• costs associated with installing
• parity across the world (can every national/international level meet afford to install such cameras?)
• How exactly the rules should be worded/administered
• Deciding what exactly can be reviewed

So still, we emerge from yet another FINA Technical Congress with no solution to one of the most controversial issues in swimming over the last decade: how to monitor what happens underwater, specifically with regard to the breaststroke underwater pullout. Our readers will recall that at least half of the Olympic final could be observed on camera doing more than the one allowed underwater dolphin kick off of the start.

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8 years ago

So, people are going to keep cheating? I agree with no extra dolphin kicks in breaststroke but no underwater cameras either? Come on…

Breaststroke BS
8 years ago

Total BS. You let them cheat and admit to cheating and show the cheating on live TV, yet they totally get away with it. I feel bad for the poor breaststrokers that work their behinds off and lose by a hair because the cheaters are allowed to cheat. FINA should be ashamed of themselves. I have lost all respect.

8 years ago

Agreed completely

Lane Four
8 years ago

So who has been dropping  into the pockets of FINA to bring about such a decision? This smells of something we don’t know about. Look the other way and hey, here’s another dime to go into your pocket. But let’s keep this a secret amongst ourselves.

K-man
8 years ago

Im thinking some of the same people that allowed Cielo off with a warning for his masking agent a few years back.

Mr Krabbs
8 years ago

I agree with you completely. Unlimited dolphin kicks off the wall ruins the nature of breaststroke, but not installing underwater cameras ruins the entire competition and FINA’s reputation.

boknows34
8 years ago

At the very least there should be underwater cameras at the World Championships, Pan Pacs, Europeans and Olympics.

aswimfan
8 years ago

This.

I agree 100%

newswim
8 years ago

I wonder why they couldn’t agree on a “compromise” where video “could be used as an option” if it met certain technical specifications?.That way, not every meet need have video but we could be sure that the “big” meets would use it (Olympics, Worlds, USA Nationals, etc).

SwimFanFinland
8 years ago

How about world records achieved at meetings without underwater cameras?

newswim
8 years ago

You need to start somewhere…..better to have them at major international meets (and big USA/NCAA meets) than no where.

weirdo
8 years ago

I agree there should be cameras. I think those people voting aren’t officials, coaches or athletes! They don’t have a clue!

C Martin
8 years ago

I liken this rule to the automatic review for home runs in baseball. The decision was pretty much split for all parties, but once they passed it you can see what it has done for the sport.

Honestly, to have the excuse that you wouldn’t stop cheating dead in its tracks because it costs too much is complete bull. If you have a multi-million dollar facility you can afford to install a few cameras. Good job FINA.

C Martin
8 years ago

Braden, I can see where you’re coming from, but I wasn’t comparing the budgets of the MLB with that of FINA and other governing bodies. The baseball reference was used for the actual rule/proposal and the voting surrounding it.

You can’t put a price on cheating (technically, you could, but if you want a clean sport, you have to take the appropriate measures to keep it clean). If a third-world country can’t afford cameras, the right thing to do would be to loan or give them cameras, as you said. But for FINA to make the excuse that there isn’t enough money for them to buy cameras doesn’t seem to be right.

If a tree falls in the… Read more »

commonsense
8 years ago

It should be mandatory for at the very least at the World Championships, Pan Pacs, Europeans and Olympics!! There is no excuse for that and saying there isn’t funding. if other countries and meets can’t afford it thats fine, but these meets should have it!

Philip Johnson
8 years ago

Alright van der Burgh and others, you just got the green light.

Coach Bambam
8 years ago

This line says a lot – interference with broadcasters’ underwater cameras -> I interpret that to mean its about the revenues instead of compliance.

Its unfortunate that this will not be instituted but it might make its way in due process.

Phil
8 years ago

And that’s a BS excuse. Look at the underwater cameras in Indy or Omaha. There were broadcasting cameras as well as DQ-judging cameras underwater. Neither interfered with the other.

coach Bambam
8 years ago

I agree – even at the Grand Prix in Santa Clara someone with a go pro did some great shots on turns (even caught a DQ).

No need for FINA to be stuck in the 80’s – technology is around to improve things and it shows not just in sports – medicine, engineering, automotive…cameras help.

SWIMGUY12345
8 years ago

This is probably the worst ruling in awhile. Could have at LEAST put them at the Olympics, world championship, or whatever international championship meet there is for the year. Agreed, cost is an issue and not every country can afford the cameras, but at the biggest meet of the year they are completely necessary. World records have been broken as a result of not having cameras (Van De Cheater). Really perplexed by this.


text:
max_cardinality {igraph} R Documentation

Maximum cardinality search

Description

Maximum cardinality search is a simple ordering a vertices that is useful in determining the chordality of a graph.

Usage

max_cardinality(graph)

Arguments

 graph The input graph. It may be directed, but edge directions are ignored, as the algorithm is defined for undirected graphs.

Details

Maximum cardinality search visits the vertices in such an order that every time the vertex with the most already visited neighbors is visited. Ties are broken randomly.

The algorithm provides a simple basis for deciding whether a graph is chordal, see References below, and also is_chordal.

Value

A list with two components:

 alpha Numeric vector. The 1-based rank of each vertex in the graph such that the vertex with rank 1 is visited first, the vertex with rank 2 is visited second and so on. alpham1 Numeric vector. The inverse of alpha. In other words, the elements of this vector are the vertices in reverse maximum cardinality search order.

Author(s)

Gabor Csardi csardi.gabor@gmail.com

References

Robert E Tarjan and Mihalis Yannakakis. (1984). Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal of Computation 13, 566–579.

is_chordal

Examples

## The examples from the Tarjan-Yannakakis paper
g1 <- graph_from_literal(A-B:C:I, B-A:C:D, C-A:B:E:H, D-B:E:F,
E-C:D:F:H, F-D:E:G, G-F:H, H-C:E:G:I,
I-A:H)
max_cardinality(g1)
is_chordal(g1, fillin=TRUE)

g2 <- graph_from_literal(A-B:E, B-A:E:F:D, C-E:D:G, D-B:F:E:C:G,
E-A:B:C:D:F, F-B:D:E, G-C:D:H:I, H-G:I:J,
I-G:H:J, J-H:I)
max_cardinality(g2)
is_chordal(g2, fillin=TRUE)

[Package igraph version 1.3.5 Index]


text:
# Change which default python install runs .py scripts a the command prompt

I have 3 installations of python on my machine currently, the ArcGIS install, the QGIS install, and an Anaconda install. The Anaconda is the on that I use predominantly. If I run a .py script (checkSource.py) with the following lines at the command prompt I get different results depending how I call it :

import sys
sys.prefix

C:\Users\RDebbout> checkSource.py
'C:\Python27\ArcGISx6410.3'

C:\Users\RDebbout> python checkSource.py
'C:\Users\RDebbout\AppData\Local\Continuum\Anaconda'

Will running .py scripts at the command line always search for the C:\Python27 install first? Is there a way to change this? I never do anything with the ArcGIS install other than import arcpy into the Anaconda install through the .pth file and would like to keep it sidelined from being used at the command prompt, so if I run a .py file there I want it to default to the Anaconda install, not the Arc install

• C:\Python27\ArcGISx6410.3\python.exe C:\Users\RDebbout\checkSource.py etc. – Midavalo May 16 '16 at 18:33
• this is a solution to explicitly call which python to use when running a script, however I am looking for a way to set up my environment variables so that it doesn't find the ArcGIS python when running the .py file. – rickD May 16 '16 at 18:56

## 1 Answer

You can do this with a batch file by explicitly pointing to your ArcGIS install:

start C:\Python27\ArcGIS10.3\python.exe C:\Path\to_the\script\to_run.py

You can apply the same logic to the cmd and point it to your desired python interpreter...If you are using a more robust Python IDE, you may even be able to point that to which version to use when running your script.

• +1 When setting up scheduled tasks, I ALWAYS explicitly point to the full path of python.exe that I am wanting to use. And yes, in some IDEs (PyCharm, for example), you can set the interpreter on a per-project basis for use in dev and testing. – Chad Cooper May 18 '16 at 15:48
• Yes, I do this too just to be safe...explicit is better than implicit :) – crmackey May 18 '16 at 15:51


text:
# Resolving vectors into components

#### Gringo123

I have just looked at a problem in an A level physics book which asks you to work out the forward force acting on a boat. There are 7 N of force going straight ahead and a further 5 N of force going 30 degrees to the left. In order to work out the forward force of the boat the book says this:

First you need to find the amount of the 5 N force that acts in the forward direction, using trigonometry:

Part of 5 N force in forward direction = 5 cos 30° = 4.3 N

Then this can be added to the 7 N force:

4.3 + 7 = 11.3 N force in the forward direction.

How do I work out that 5 cos 30° = 4.3 N?

Related Other Physics Topics News on Phys.org

#### phyzmatix

It stems from the trig identity

$$\cos \theta = \frac{x}{r}$$

In this case, x represents your forward direction (draw yourself a sketch to see) and r represents the force vector. The rest is then simply algebraic manipulation of this identity and a final application of the trigonometric rules to find the value of x.

Makes sense?

### Physics Forums Values

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text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Dynamical robust ${H}_{\infty }$ filtering for nonlinear uncertain systems: an LMI approach. (English) Zbl 1202.93157
Summary: A new approach to robust ${H}_{\infty }$ filtering for a class of nonlinear systems with time-varying uncertainties is proposed in the LMI framework based on a general dynamical observer structure. The nonlinearities under consideration are assumed to satisfy local Lipschitz conditions and appear in both state and measured output equations. The admissible Lipschitz constants of the nonlinear functions are maximized through LMI optimization. The resulting ${H}_{\infty }$ observer guarantees asymptotic stability of the estimation error dynamics with prespecified disturbance attenuation level and is robust against time-varying parametric uncertainties as well as Lipschitz nonlinear additive uncertainty.
##### MSC:
 93E11 Filtering in stochastic control 93C10 Nonlinear control systems 93B36 ${H}^{\infty }$-control 93D20 Asymptotic stability of control systems


text:
# 1.3: Concepts Quiz

In a Word document, number your paper 1 – 12 for the terms. Opposite each number, match the statement that best describes the term. Place your name at the bottom of the page following the number 12. (THIS ASSIGNMENT IS WORTH 12 POINTS)

Terms:

1. Backstage view

2. Columns

3. Workbook

4. Groups

5. Dialog Box Launcher

7. Document Window

8. Formula Bar

9. Worksheet window

10. Range

11. Rows

12. Zoom slider

Statement:

a. Selected data. For example, selecting B1 to G8

b. When clicking this feature a pull down menu appears

c. To better view a portion of your worksheet, this increases the font size without actually changing the font.

d. This object can contain up to 255 worksheets.

e. This contains a grid of a possible 1,049,576 rows.

f. These are vertical parts of a spreadsheet

g. Always the number in the cell reference.

h. When a specific tab key is chosen, the resulting ribbon is divided into what?

i. The intersection of a row and a column is called this.

j. When formulas are inserted, they display in the cell and also in what other place?

k. The result of pressing the file key and seeing a display of tasks such as Open, Save, Print

l. This item is located in the upper right corner of the screen

CC licensed content, Original

1.3: Concepts Quiz is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.


text:
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# In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree

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In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

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21 Jun 2017, 00:44
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In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degrees greater than ∠C. The value of angle B is

(A) 30
(B) 40
(C) 50
(D) 60
(E) 70
[Reveal] Spoiler: OA

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In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

### Show Tags

21 Jun 2017, 00:49
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Bunuel wrote:
In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degrees greater than ∠C. The value of angle B is

(A) 30
(B) 40
(C) 50
(D) 60
(E) 70

Let $$∠C = x$$

$$∠B = x + 10$$

$$∠A = x + 10 + 10 = x + 20$$

$$∠A + ∠B + ∠C = 180$$

$$x + 20 + x + 10 + x = 180$$

$$3x + 30 = 180$$

$$3x = 180 - 30 = 150$$

$$x = 150/3 = 50$$

$$∠C = 50$$

$$∠B =x + 10 = 50 + 10 = 60$$ . Answer (D) ...
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Joined: 22 May 2016
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In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

### Show Tags

21 Jun 2017, 11:16
Bunuel wrote:
In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degrees greater than ∠C. The value of angle B is

(A) 30
(B) 40
(C) 50
(D) 60
(E) 70

Define all angles in terms of ∠B (= b, ∠A = a, ∠C = c).

Quote:
∠A is 10 degrees greater than ∠B

a = b + 10
Quote:
∠B is 10 degrees greater than ∠C

b = c + 10, or

c = b - 10

a + b + c = 180, substitute values in bold type

(b + 10) + b + (b - 10) = 180

3b = 180

b = 60
[Reveal] Spoiler:
D

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In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

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21 Jun 2017, 13:35
1
KUDOS
The sum of all the angles in a triangle is 180 Degree.
If the three angles have to differ by 10 and add to 180,
the only possibility(with a difference of 10) is when the three angles are 50,60 and 70.

Since ∠B is 10 degree greater than ∠C and 10 degree smaller than ∠A, it must be 60(Option D)
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Re: In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

### Show Tags

21 Jun 2017, 18:36
In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degrees greater than ∠C. The value of angle B is

(A) 30
(B) 40
(C) 50
(D) 60
(E) 70

Solution:
Given:
a=b+10
b=c+10
Also, we know
a+b+c=180.----eq (1)

Therfore subsitute the value of a and c in the eq(1)
we ger
b+10+b+b-10=180.
3b=180.
b=60.

So Option D
Math Expert
Joined: 02 Aug 2009
Posts: 5776
Re: In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

### Show Tags

21 Jun 2017, 19:51
Bunuel wrote:
In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degrees greater than ∠C. The value of angle B is

(A) 30
(B) 40
(C) 50
(D) 60
(E) 70

Hi,

Here actually difference of 10 doesn't matter. Any difference but similar would give us SAME answer as 60.. WHY?

The three angles differ by certain amount so they are in AP and central angle B would be the MEAN
So 3*B =180..... B=60
D
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Re: In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

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21 Jun 2017, 21:19
If all three angles are equal: 60, 60, 60

If the three angles are in Arithmetic Progression with a difference of 10: 50, 60, 70

So the central angle is 60 only. Hence D answer
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Re: In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

### Show Tags

22 Jun 2017, 04:50
Bunuel wrote:
In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degrees greater than ∠C. The value of angle B is

(A) 30
(B) 40
(C) 50
(D) 60
(E) 70

Let the /_C = x
So, /_ B = x +10
/_ A = x +20

A + B + C = x+20 + x +10 +x = 180
-> 3x +30 = 180
-> x= 150/3 = 50

So, /_B = 50 +10 = 60

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In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

### Show Tags

22 Jun 2017, 08:09
Bunuel wrote:
In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degrees greater than ∠C. The value of angle B is

(A) 30
(B) 40
(C) 50
(D) 60
(E) 70

Let the value of $$∠A = a°$$ , $$∠B = b°$$ and $$∠C = c°$$

We know, $$∠A + ∠B + ∠C = 180°$$ Or, $$a° + b° + c° = 180°$$
Quote:
∠A is 10 degrees greater than ∠B, and ∠B is 10 degrees greater than ∠C

Value of $$∠B$$ = $$( c° + 10° )$$
Value of $$∠A$$ = $$( b° + 10° )$$ => $$( c° + 10° ) + 10°$$ = $$c° + 20°$$
Value of $$∠C$$ = c°

Now, $$(c° + 20°)$$ + $$( c° + 10° )$$ + $$c°$$ = $$180°$$

Or, $$3c° + 30° = 180°$$

Or, $$3c° = 150°$$

So, c° = 50°

Now, as Value of $$∠B$$ = $$( c° + 10° )$$ = $$60°$$

Thus, answer will be (D) 60°
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Re: In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

### Show Tags

24 Jun 2017, 08:34
Bunuel wrote:
In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degrees greater than ∠C. The value of angle B is

(A) 30
(B) 40
(C) 50
(D) 60
(E) 70

We can let angle C = x, angle B = 10 + x, and angle A = 20 + x. Since there are 180 degrees in a triangle, we have:

x + x + 10 + x + 20 = 180

3x + 30 = 180

3x = 150

x = 50

Thus, angle B is 50 + 10 = 60 degrees.

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Re: In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

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24 Jun 2017, 09:25
IMO D
A+B+C=180
B+10+B+B-10=180
3B=180
B=60
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Re: In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree [#permalink]

### Show Tags

16 Dec 2017, 03:21
Here,<A+<B+<C=180 and B=C+10, A=B+10=>A=C+10+10 =>A=C+20
Hence, C+20+C+10+C=180
C=150/3=50

Re: In Triangle ABC, ∠A is 10 degrees greater than ∠B, and ∠B is 10 degree   [#permalink] 16 Dec 2017, 03:21
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text:
# To find the average rate of change of f between x = 0 and x = 2 and x = 15 and x = 50

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2, Problem 61RE

a.

To determine

## To find the average rate of change of f between x=0 and x = 2 and x=15 and x = 50

Expert Solution

The average rate of change of f between x=0 and x = 2 is 12 and of x=15 and x = 50 is 12 .

### Explanation of Solution

Given: The given function is f(x)=12x6

Calculation:

f(x)=12x6

=f(50)f(15)5015=193235=383235=352(135)=12

We get,

and f(2)=12(2)6           =16            =5

So we get ,

f(2)=5

Hence the average rate of change

=f(2)f(0)20=5(6)2=5+62=12

Hence the average of change of f between.

x=0x=2 is 12

Now we find f(15)and f(50) as follows;

f(15)=12(15)6        =15122        f(15) =32

So, we get

f(15) =32

And

f(50)=12(50)6        =256         =19

We get

f(50) =19

Hence the average rate of change is,

=f(50)f(15)5015=193235=383235=352(135)=12

Hence the average of change of f between.

x=15x=50 is 12

b.

To determine

Expert Solution


text:
# Accurate Polynomial Evaluation in Floating Point

What are the most accurate algorithms for evaluating a polynomial using floating point arithmetic? The internet seems to suggest that Horner's method is commonly used. In particular I have a cubic polynomial of the form f(x) = a x^3 + b x^2 + c x + d, which I am evaluating in the naive way with floating point arithmetic, and was curious whether some recursive approach or some refactoring would give a more accurate result. That is, a result closer to what I would get if I were to evaluate the polynomial exactly or with a metric ton of precision in Mathematica.

• A key question, also, is: what is your input? Sometimes you get coefficients in different "bases", for instance you may want to evaluate the polynomial such that $p(0)=0$, $p(1)=5$, $p(2)=3$, $p(3)=5$ in $x=10$ (interpolation). – Federico Poloni Sep 1 '20 at 14:52
• Are you dealing with piecewise cubic splines? – Brian Borchers Sep 2 '20 at 15:48

## 3 Answers

Horner is indeed the most stable way to evaluate a polynomial (and you get the bonus of evaluating its derivatives with not too much extra cost). Higham presents a nice error analysis of the algorithm (Accuracy and stability of numerical algorithms, 2nd edition, p.94). He also presents an algorithm that includes a running error bound so you have an idea on the difference between the real value and what Horner calculated (Algorithm 5.1 on p.95).

If your case is limited to cubic polynomials and you're worried about loop overhead, you could unroll the loops by hand. But I doubt you will gain much. Stick to the tried and tested.

• The accuracy of the Horner scheme can be improved by using compensated arithmetic. See: Philippe Langlois and Nicolas Louvet. "How to ensure a faithful polynomial evaluation with the compensated Horner algorithm." In 18th IEEE Symposium on Computer Arithmetic (ARITH'07), pp. 141-149. IEEE, 2007 (draft online); Stef Graillat, Philippe Langlois, and Nicolas Louvet. "Algorithms for Accurate, Validated and Fast Polynomial Evaluation", Japan J. Indust. Appl. Math., 26 (2009), 191–214 (online) – njuffa Jun 19 '16 at 18:46

The stability of Horner can be improved by subtracting a constant from each $$x$$ in the Horner form.

As an alternative you may use the barycentric Lagrange basis which has the same evaluation complexity as the Newton basis. The barycentric formulation is also used in the MATLAB Chebfun implementation. For details see papers of L. N. Trefethen.


text:
# Groups. I am stumped again.

1. May 17, 2004

### Nexus[Free-DC]

Okay, I'm really scratching my head here.

If an Abelian group A has three generators x,y,z and they are subject to three defining relations, say something like

x+y+z=0
x-y-z=0
2x-2y+3z=0

then I can solve for x,y,z and find A as a direct sum of cyclic groups, Z_x + Z_y + Z_z.

But what do I do if the three equations are not linearly independent? I get left with everything in terms of x and I can't just plug in the numbers.

Thanks,
N.

2. May 17, 2004

### NateTG

I don't understand

There's some context missing:

Let's say you've got generators $$\{x,y,z\}$$ and the relations:
$$xy=yx$$
$$yz=zy$$
and
$$zx=xz$$
Which are equivalent to the claim that the group is abelian.

Now we add the following relations:
$$0=0$$

So at this point, this group is equivalent to $$\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$$ and the system of defining relations is clearly not linearly independant.

3. May 17, 2004

### matt grime

you probably oughtn't to confuse additive and multiplicative notation in the same thread

4. Sep 17, 2004

### mathwonk

nexus, you are talking about a representation of your abelian group as a cokernel of a map which is not necessarily injective.

by the way your example is non sense as your groupo as described is zero.

i.e. you gave so many relations that everything was trivial.

In general if you have n generators, that emans you map a direct sum of n copies of Z onto the group. then telling what say r relations are, is giving a generating set for the kernel of that map, hence it lets you map another direct sum of r copies of Z onto the kernel.

So now you have a linear map from r copies of Z, to n copies of Z, given by a matrix of integers. and your group is the cokernel of this map.

To get the explicit structure of the group, you just diagonalize the matrix using row and column operations. Then you can easily see the group structure, as the quotient of the direct sums then becoems the direct sum of the quotients of the diagonal maps, i.e. one dimensional maps.

this is the usual proof of the structure theorem for finitely generated abelian groups.


text:
# The Tank's leaking!

A tank with a small hole at the bottom has been filled with water and kerosene (specific gravity=0.8). The height of water is 3 m and that of kerosene is 2 m. When the hole is opened the velocity of fluid coming out from it is nearly: (take g=10 m$$s^{-2}$$ and density of water= 1000 kg $$m^{-3}$$)

×


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# If function f(x) satisfies f (1-x) = f (2+x) for all

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If function f(x) satisfies f (1-x) = f (2+x) for all x, which of the following can be true?

1. f(x) = $$3-x$$
2. f(x) = $$1-x^2$$
3. f(x) = $$2 - x - x^2$$
4. f(x) = $$2x - (1-x)^2$$
5. f(x) = $$(1-x) / (2 + x)$$

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Re: If function f(x) satisfies f (1-x) = f (2+x) for all  [#permalink]

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18 Sep 2018, 18:32
The answer is C. The question is asking for a function symmetric about x=-0.5. A, D & E are not even functions, so they can't be symmetric. B is symmetric about 0. And C simplifies to

2.25-(x+0.5)^2.

Thus it is symmetric about x= -0.5.
Re: If function f(x) satisfies f (1-x) = f (2+x) for all &nbs [#permalink] 18 Sep 2018, 18:32
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# If function f(x) satisfies f (1-x) = f (2+x) for all

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# On the automorphisms of quantum Weyl algebras

Kitchin, Andrew, Launois, Stephane (2019) On the automorphisms of quantum Weyl algebras. Journal of Pure and Applied Algebra, 223 (4). pp. 1514-1530. ISSN 0022-4049. (doi:10.1016/j.jpaa.2018.06.016) (KAR id:53002)

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https://doi.org/10.1016/j.jpaa.2018.06.016

## Abstract

Motivated by Weyl algebra analogues of the Jacobian conjecture and the Tame Generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl-Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of U+q(so5).

Item Type: Article 10.1016/j.jpaa.2018.06.016 Weyl Algebras, The Tame Generators Problem, Endomorphisms Q Science > QA Mathematics (inc Computing science)Q Science > QA Mathematics (inc Computing science) > QA150 Algebra Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Stephane Launois 10 Dec 2015 11:37 UTC 06 May 2020 03:13 UTC https://kar.kent.ac.uk/id/eprint/53002 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-7252-8515


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## Where communities thrive

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##### Activity
• 09:00
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Mohit Gupta
@mohitacecode
Sorry I meant that I want to apply as mentor.
2 replies
Kalevi Suominen
@jksuom
I think that it would suffice to add one's name to this mentor list: https://github.com/sympy/sympy/wiki/GSoC-Ideas#potential-mentors
Mohit Gupta
@mohitacecode
Okay, thanks @jksuom
aplund
@aplund:matrix.org
[m]
Can anyone help me with this NotAlgebraic exception: https://gist.github.com/aplund/888da79a96fb295e7d614f43780119f9
Miguel Salazar de Troya
@salazardetroya
Hello, can sympy be used to nondimensionalize equations? Like it is done here: https://en.wikipedia.org/wiki/Non-dimensionalization_and_scaling_of_the_Navier%E2%80%93Stokes_equations . Basically, replace the variables with their nondimensional counterparts and lump the parameters into nondimensional numbers. Thanks.
Ivan
@Tookser
Hello!
What is the best way to interpret something like 'a+b+c' as syntaxic tree of depth 3, not 2? In other words, how to make expression tree become binary-unary?
3 replies
AndoBando
@AndoBando
Should cse work for tensors? I have some big Christoffel symbol I want to lambdify with cse, but i don't think it has any effect. If I flatten it it works fine.
faze-geek
@faze-geek
What is the best way to simplify a rational expression in SymPy . simplify(),cancel() ....etc etc there are many methods but I want to one which is the least costly.
orbisvicis
@orbisvicis:matrix.org
[m]

According to the docs it depends on the expression to be simplified. You might need to provide some assumptions for your symbols and in general avoid simplify() if cost is a concern.

https://docs.sympy.org/latest/tutorial/simplification.html

Gaurav Dhingra
@gxyd

Hi all,

I'm trying to use SymPy's parse_latex function to parse mixed-fractions. But the issue that's happening is, the tree representation of mixed-fraction is same as if the integer part of mixed fraction were getting multiplied with fractional part of the mixed fraction.

Minimal code example:

>>> from sympy.parsing.latex import parse_latex
>>> from sympy import srepr
>>> e1 = "1 \\frac{1}{3}"
>>> e2 = "1 \\times \\frac{1}{3}"
>>> expr1 = parse_latex(e1)
>>> expr2 = parse_latex(e2)
>>> print(srepr(expr1))
Mul(Integer(1), Pow(Integer(3), Integer(-1)))
>>> print(srepr(expr2))
Mul(Integer(1), Pow(Integer(3), Integer(-1)))

>>> srepr(expr1) == srepr(expr2)
True

(possible solution): Is writing a transformation rule the best possible solution to tackle this? https://docs.sympy.org/latest/modules/parsing.html#parsing-transformations-reference

Anutosh Bhat
@anutosh491
Hello All,
My Gsoc proposal (Improving series Expansions and Limit Computations) is almost about complete (just 3- 4 code snippets need to be added and I'm just finalizing them and some reframing if any ). I've received a basic review from the potential Mentor and also a decent to good review from the potential co-mentor . I would like to request for a second iteration of reviews from the org members and members related to the series/limits module in particular .
The proposal is exhaustive enough for a 175 hour project, to leave some room for bug fixes and implementations if I finish early . Hence I would also like to request if the members could validate the timeline proposed as that was a bit confusing for me to frame . Also now Gsoc projects can be extended to 22 weeks (midway) than the standard 12 week project , so like how do I consider or integrate this if required with the proposed timeline in my proposal?
All suggestions are welcome
Christopher Smith
@smichr
Does anyone know what sort of matrix analysis involves checking to find a plausible subset of variables for which a solution might be found for an underdetermined system (more variables than equations)? A trivial case would involve a diagonal system: a solution for variables on the diagonal is possible. But if it's not a diagonal system what method without actually solving equation -- just working with position of non-zero elements in the matrix -- is used to tell if a solution is plausible or not?
Christopher Smith
@smichr
I see unanswered question here except for a numerical system. But I am considering plausibility of the sysmbol, not numeric, case.
faze-geek
@faze-geek
Can some one help me on auto simplification of trigo identities in SymPy .For example I want sin(x)2 + cos(x)2 -1 to simplify and give me 0 in a expression ( to prevent self from returning as ouput )
Kalevi Suominen
@jksuom
@faze-geek This is for simplification of trigonometric expressions: https://docs.sympy.org/latest/modules/simplify/fu.html#sympy.simplify.fu.fu
Christopher Smith
@smichr
Is there a canonical way to tell if two matrices are the same except for permutation of rows and columns?
Christopher Smith
@smichr
Hmm...I hadn't asked exactly that way yesterday but today I find this.
Varenyam Bhardwaj
@airpods69
I was wondering if anything is supposed to be done for sympy/sympy#23336
It does seem do-able but i might need a little guidance with it
Chen Tao
@jagt

Folks I have a expression I don't know how to simplify:

from sympy import *
from sympy.abc import w, x, y, z

expr = (w - x) * (x - y) + (y - z) * (z - w)
expanded = expand(expr)
back = factor(expr)

I want back to be something similar to expr. I tried factor/simplify/cancel and stuff and seems non of these works.
Anyone know what should I try?

Kuldeep Borkar Jr
@KuldeepBorkar

Hi @jagt factor() takes a polynomial and factors it into irreducible factors over the rational numbers. Note: factors may not be irreducible if the input is no longer a polynomial over the rationals.

To confirm what will be the factors you can do the following:
factor_list() instead of factor which will return the expected factors or output of factor() in a more structured way.

Chen Tao
@jagt
@KuldeepBorkar I tried factor_list which doesn't work in my case. I assume factor_list and factor have similar functionality but rather different output?
SamG-01
@SamG-01
Why are the DifferentialOperator and qapply functions specific to the sympy.physics.quantum.operator package and to Wavefunction objects? I feel like differential operators would be useful to have as a more general tool for all types of functions, not just Wavefunctions.
Micah
@micahscopes
anyone else noticing that the documentation is down?
Liumeo
@eagleoflqj
SymPy Beta supports dark theme now!
Talos878
@Talos878
Wooo!!
HeroCC
@HeroCC

Hi all, I had a quick question on covariance calculations. I'm no expert, but for some reason Sympy is different than what I'd expect. I'm probably using the wrong function somewhere. Here is a quick snippet:

covariance(FiniteRV('Jimmy', density={-.20: .40, .30: .60}), FiniteRV('Walter', density={.40: .40, -.20: .60}))
# expect -0.072

And the equation I'm using: https://wikimedia.org/api/rest_v1/media/math/render/svg/18143cb5b668ff2cc739385a5916d9317bb74df0

HeroCC
@HeroCC
I'm no stats genius so please excuse me if there is a better RV to be using here
HeroCC
@HeroCC
Another example:
q2rush = FiniteRV('Rush', density={-0.25: .30, 0.35: .70})
q2oberman = FiniteRV('Oberman', density={0.45: .30, -0.17: .70})

covariance(q2rush, q2oberman)
(.30 * (-0.25 - E(q2rush))*(0.45 - E(q2oberman))) + (.70 * (0.35 - E(q2rush))*(-0.17 - E(q2oberman)))
Constantin Mateescu
I'm totally disappointed with how slow sympy is at finding the angle between lines that are given in terms of points with symbolic coordinates. Is there a flag to disable symbolic computation across all sympy objects? Having to remember to call .evalf() is far from ideal... I had to spend a great deal of time understanding why my code was running incredibly slow.
Liumeo
@eagleoflqj
Initial LaTex input support added to SymPy Beta
Ivan Petukhov
@satels
@eagleoflqj Get url of this site, pls?
Liumeo
@eagleoflqj
sympy-beta.vercel.app Not stable on iOS, tested on Win/Linux/macOS/Android
Richard Fuhr
@rdfuhr

The following code plots the graph of a simple piecewise-constant function f
such that f(x) = 1 for x between 0 and 1
and f(x) = -1 for x between 1 and 2
and f(x) = 0 otherwise.
In this plot, the graph is plotted for x between -1 and 3.
However, vertical lines are drawn at the values of x where there are discontinuities.
Is there a way to plot the graph of this function such that the vertical lines (which should not be part of the graph) are not drawn?

from sympy import symbols
from sympy.plotting import plot
from sympy import Piecewise
x = symbols('x')
s = Piecewise((1, (x >= 0) & (x <= 1)), (-1, (x >= 1) & (x <= 2)), (0, True))
plot(s,(x,-1,3))

Above is the graph that is generated by the above code.
Lin Hsu
@chaosink
Seems I cannot integrate (x**2+1)**-1.5
Liumeo
@eagleoflqj
Whenever possible, use rational instead of float. integrate((x**2+1)**(-S(3)/2))
Liumeo
@eagleoflqj
3D plot support for binary function added to SymPy Beta
Lin Hsu
@chaosink
Thanks! @eagleoflqj
Saksham ALok
@Saksham-13
Hello,
My name is Saksham(19), I am a first-year computer science student who has just ventured into open source. Sympy intrigued me as it is a python based alternative to MATLAB which has been a little irritating to use in my college coursework.
I have some experience in python and have built bots and projects using different libraries but have never worked on something of this size. I am hoping to find some lead here to see what to work on and where to learn the required prerequisites for that. Please guide me regarding this matter. Also, I would be giving this work first priority after my internals ends in 5 days.
Thanks
PS... My first PR got merged
ETHEM UZER
@ethemuzer777_gitlab
With the help of this modified version, you can configure the privacy of your chats in much more detail and completeness. Hide activity status, double check, forward reminders, and more. Also change the theme of the application and translate the messages you receive in the chat window. Everything is possible with WhatsApp Plus Red : https://modpree.com/whatsapp-plus-rojo/
Christopher Smith
@smichr
@costica1234_twitter what problem are you trying to solve that you need to find the angles? I wonder if this would be of interest: sympy/sympy#22644
Liumeo
@eagleoflqj
SymPy Beta now plots for unary complex function
Gaurav Dhingra
@gxyd
Something that might be helpful for people here:
https://github.blog/2022-05-19-math-support-in-markdown/
fmwaapps


text:
## Introduction to Programming using Python 1st Edition

$[ ' welc', 'me']$
We find that the proper steps are: s1="Welcome" $s1.split('o')=$ $[ ' welc', 'me']$


text:
Quote by Saladsamurai I don't see ant difference between 20 & 29 and 19 & 29.
That's right : although 20 is not prime, the linear combinations of (20,29) also make up all possible integers, because 20 is still prime with 29.
 Recognitions: Gold Member This thread is getting out of hand. Disgusted pending moderation?
 Blog Entries: 1 Recognitions: Homework Help You're only allowed to date someone 2/3rds of your age and 3/2ths your age. 19*3/2=28.5 so a 19 year old can date up to a 28 year old without being creepy. I think that's the rule

 Quote by Office_Shredder You're only allowed to date someone 2/3rds of your age and 3/2ths your age. 19*3/2=28.5 so a 19 year old can date up to a 28 year old without being creepy. I think that's the rule
Fine, that means the 19 year old must wait for one more year.
 2/3rds??? So a 21-year-old can date a 14-year-old??? A 24-year-old can date a 16-year-old??? No, the formula is b=.5a + 7. So, he can date someone as young as half his age plus 7 14 14 16 15 18 16 20 17 22 18 29 21 ... 80 47 Seraphin, when's your birthday?? Ask him for a coffee on your birthday.

Recognitions:
Gold Member
Staff Emeritus
 Quote by DaveC426913 No, the formula is b=.5a + 7.
Really? Woo hoo! That means I can date my TA! Just wait until I tell my boyfriend. Um, oh, wait...I think there might be a flaw there somewhere.

Blog Entries: 1
Recognitions:
 Quote by Office_Shredder You're only allowed to date someone 2/3rds of your age and 3/2ths your age. 19*3/2=28.5 so a 19 year old can date up to a 28 year old without being creepy. I think that's the rule
I thought the rule was:
$$age_{min}=\frac{age_{you}}{2}+7$$

To the OP... Let's see...
• He's a male graduate student, in his upper 20s, in mathematical sciences.

Based purely upon stereotypes, the presumption that he's single (and attracted to females) and that you have no severely hideous deformities, just bat your eyelashes, and he should be all over that.

Unless he's still TAing you, in which case, you should probably wait until next semester.

As with many academic life issues, Jorge Cham of PhdComics has some commentary:
http://www.phdcomics.com/comics/archive.php?comicid=35
http://www.phdcomics.com/comics/archive.php?comicid=49
http://www.phdcomics.com/comics/archive.php?comicid=50

(Seriously though, even though I've never dated any of my TAs--and later, students--just do your standard thing when you're interested in a guy, although you might want to crank it up a notch since there's a possibility that he's entirely clueless about such things)

 Quote by MATLABdude I thought the rule was: $$age_{min}=\frac{age_{you}}{2}+7$$
Dude. Come to the meetings! You're ten minutes late.

Blog Entries: 1
Recognitions:
 Quote by DaveC426913 Dude. Come to the meetings! You're ten minutes late.

Say, y'know what Seraphin, he's probably into coffee...
(Where's the tongue-in-cheek smilie when you need it? Yes, it was on the first page, which is why this comment is topical!)

Awe, thanks guys for all the advice! I was a bit afraid people would go "Ewr, what? Don't bother!" but it's reassuring that this doesn't sound wildly impossible. The fear of rejection is still huge, but at least I know what guys must go through all the time now.

Matlabdude: Awe, really? No fears of cradle-robbing? I suppose you're right. I'm afraid some lucky lady might already have him. He has an adorable Irish accent, after all...

But quite honestly, thank you a million times everyone, I appreciate the advice quite a lot!

My birthday's not till June, unfortunately! I think I might just crack and jump his bones if I wait that long though. Decisions, decisions...

Blog Entries: 1
Recognitions:
 Quote by Seraphin ... Matlabdude: Awe, really? No fears of cradle-robbing? I suppose you're right. I'm afraid some lucky lady might already have him. He has an adorable Irish accent, after all...
Uh oh, you didn't mention handsome charming foreigner... This throws everything out the window!

(Just kidding; make like Nike and Just Do It™. DISCLAIMER: No promotional considerations paid by Nike for this advertisement.)
 I'd like to know the results of this. So many possible endings... Will he be married and not tell her till after the...? Will he go to have the coffee and after having been flung upon still not get it and just leave to never be seen again? Could he turn out to be prince charming?
 Recognitions: Homework Help Too much analysis, Too little action! Don't think!, Act! Good luck!

Blog Entries: 1
Recognitions:
 Quote by Cyclovenom Too much analysis, Too little action! Don't think!, Act! Good luck!
Truth. The OP, Seraphin, is an angel (or at least her username is almost one). What could possibly go wrong?
http://en.wikipedia.org/wiki/Seraphim

No liability is assumed if something goes terribly, terribly wrong.

 Quote by Seraphin Anyways, the major thing stopping me is the age difference (he's 29 and I'm 20) since I'm worried he wouldn't want to date someone so much younger than him and try as I might, I can't think of any way to get closer to him so I can get to know him better.
Just go for it. Generally males dont have any issues having relationships with younger partners. If anything, your age is working for you, not against you. Go get him tiger :P

Don't dwell on rejections , should they happen.

A "cliche" quote for you:

"Twenty years from now you will be more disappointed by the things you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover."

Mark Twain

 Quote by rootX but that would make him a pedophile.
Anything legal is fair game.

Quote by DanP
Quote by rootX
 Quote by Saladsamurai Yeah. Oh! if only you were still 19 and he was 29....."We're both in our prime..."
but that would make him a pedophile.
Anything legal is fair game.
Right. So, instead of a pedophile, he'd merely be a creepy, dirty old man.


text:
# Trigonometry Examples

Convert to Polar Coordinates (-1,1)
Convert from rectangular coordinates to polar coordinates using the conversion formulas.
Replace and with the actual values.
Find the magnitude of the polar coordinate.
Raise to the power of .
One to any power is one.