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### We're updating the default styles for Matplotlib 2.0

Learn what to expect in the new updates

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# AXISARTIST namespace¶

The AxisArtist namespace includes a derived Axes implementation. The biggest difference is that the artists responsible to draw axis line, ticks, ticklabel and axis labels are separated out from the mpl’s Axis class, which are much more than artists in the original mpl. This change was strongly motivated to support curvilinear grid. Here are a few things that mpl_tootlkits.axisartist.Axes is different from original Axes from mpl.

• Axis elements (axis line(spine), ticks, ticklabel and axis labels) are drawn by a AxisArtist instance. Unlike Axis, left, right, top and bottom axis are drawn by separate artists. And each of them may have different tick location and different tick labels.
• gridlines are drawn by a Gridlines instance. The change was motivated that in curvilinear coordinate, a gridline may not cross axis-lines (i.e., no associated ticks). In the original Axes class, gridlines are tied to ticks.
• ticklines can be rotated if necessary (i.e, along the gridlines)

In summary, all these changes was to support

• a curvilinear grid.
• a floating axis

mpl_toolkits.axisartist.Axes class defines a axis attribute, which is a dictionary of AxisArtist instances. By default, the dictionary has 4 AxisArtist instances, responsible for drawing of left, right, bottom and top axis.

xaxis and yaxis attributes are still available, however they are set to not visible. As separate artists are used for rendering axis, some axis-related method in mpl may have no effect. In addition to AxisArtist instances, the mpl_toolkits.axisartist.Axes will have gridlines attribute (Gridlines), which obviously draws grid lines.

In both AxisArtist and Gridlines, the calculation of tick and grid location is delegated to an instance of GridHelper class. mpl_toolkits.axisartist.Axes class uses GridHelperRectlinear as a grid helper. The GridHelperRectlinear class is a wrapper around the xaxis and yaxis of mpl’s original Axes, and it was meant to work as the way how mpl’s original axes works. For example, tick location changes using set_ticks method and etc. should work as expected. But change in artist properties (e.g., color) will not work in general, although some effort has been made so that some often-change attributes (color, etc.) are respected.

## AxisArtist¶

AxisArtist can be considered as a container artist with following attributes which will draw ticks, labels, etc.

• line
• major_ticks, major_ticklabels
• minor_ticks, minor_ticklabels
• offsetText
• label

### line¶

Derived from Line2d class. Responsible for drawing a spinal(?) line.

### major_ticks, minor_ticks¶

Derived from Line2d class. Note that ticks are markers.

### major_ticklabels, minor_ticklabels¶

Derived from Text. Note that it is not a list of Text artist, but a single artist (similar to a collection).

### axislabel¶

Derived from Text.

### Default AxisArtists¶

By default, following for axis artists are defined.:

ax.axis["left"], ax.axis["bottom"], ax.axis["right"], ax.axis["top"]

The ticklabels and axislabel of the top and the right axis are set to not visible.

For example, if you want to change the color attributes of major_ticklabels of the bottom x-axis

ax.axis["bottom"].major_ticklabels.set_color("b")

Similarly, to make ticklabels invisible

ax.axis["bottom"].major_ticklabels.set_visible(False)

AxisAritst provides a helper method to control the visibility of ticks, ticklabels, and label. To make ticklabel invisible,

ax.axis["bottom"].toggle(ticklabels=False)

To make all of ticks, ticklabels, and (axis) label invisible

ax.axis["bottom"].toggle(all=False)

To turn all off but ticks on

ax.axis["bottom"].toggle(all=False, ticks=True)

To turn all on but (axis) label off

ax.axis["bottom"].toggle(all=True, label=False))

ax.axis’s __getitem__ method can take multiple axis names. For example, to turn ticklabels of “top” and “right” axis on,

ax.axis["top","right"].toggle(ticklabels=True))

Note that ‘ax.axis[“top”,”right”]’ returns a simple proxy object that translate above code to something like below.

for n in ["top","right"]:
ax.axis[n].toggle(ticklabels=True))

So, any return values in the for loop are ignored. And you should not use it anything more than a simple method.

Like the list indexing ”:” means all items, i.e.,

ax.axis[:].major_ticks.set_color("r")

changes tick color in all axis.

## HowTo¶

1. Changing tick locations and label.

Same as the original mpl’s axes.:

ax.set_xticks([1,2,3])

1. Changing axis properties like color, etc.

Change the properties of appropriate artists. For example, to change the color of the ticklabels:

ax.axis["left"].major_ticklabels.set_color("r")

1. To change the attributes of multiple axis:

ax.axis["left","bottom"].major_ticklabels.set_color("r")

or to change the attributes of all axis:

ax.axis[:].major_ticklabels.set_color("r")

2. To change the tick size (length), you need to use

axis.major_ticks.set_ticksize method. To change the direction of the ticks (ticks are in opposite direction of ticklabels by default), use axis.major_ticks.set_tick_out method.

## Rotation and Alignment of TickLabels¶

This is also quite different from the original mpl and can be confusing. When you want to rotate the ticklabels, first consider using “set_axis_direction” method.

ax1.axis["left"].major_ticklabels.set_axis_direction("top")
ax1.axis["right"].label.set_axis_direction("left")

The parameter for set_axis_direction is one of [“left”, “right”, “bottom”, “top”].

You must understand some underlying concept of directions.

1. There is a reference direction which is defined as the direction of the axis line with increasing coordinate. For example, the reference direction of the left x-axis is from bottom to top.

The direction, text angle, and alignments of the ticks, ticklabels and axis-label is determined with respect to the reference direction

1. ticklabel_direction is either the right-hand side (+) of the reference direction or the left-hand side (-).
1. same for the label_direction
1. ticks are by default drawn toward the opposite direction of the ticklabels.
2. text rotation of ticklabels and label is determined in reference to the ticklabel_direction or label_direction, respectively. The rotation of ticklabels and label is anchored.

On the other hand, there is a concept of “axis_direction”. This is a default setting of above properties for each, “bottom”, “left”, “top”, and “right” axis.

 ? ? left bottom right top axislabel direction ‘-‘ ‘+’ ‘+’ ‘-‘ axislabel rotation 180 0 0 180 axislabel va center top center bottom axislabel ha right center right center ticklabel direction ‘-‘ ‘+’ ‘+’ ‘-‘ ticklabels rotation 90 0 -90 180 ticklabel ha right center right center ticklabel va center baseline center baseline

And, ‘set_axis_direction(“top”)’ means to adjust the text rotation etc, for settings suitable for “top” axis. The concept of axis direction can be more clear with curved axis.

The axis_direction can be adjusted in the AxisArtist level, or in the level of its child arists, i.e., ticks, ticklabels, and axis-label.

ax1.axis["left"].set_axis_direction("top")

changes axis_direction of all the associated artist with the “left” axis, while

ax1.axis["left"].major_ticklabels.set_axis_direction("top")

changes the axis_direction of only the major_ticklabels. Note that set_axis_direction in the AxisArtist level changes the ticklabel_direction and label_direction, while changing the axis_direction of ticks, ticklabels, and axis-label does not affect them.

If you want to make ticks outward and ticklabels inside the axes, use invert_ticklabel_direction method.

ax.axis[:].invert_ticklabel_direction()

A related method is “set_tick_out”. It makes ticks outward (as a matter of fact, it makes ticks toward the opposite direction of the default direction).

ax.axis[:].major_ticks.set_tick_out(True)

So, in summary,

• AxisArtist’s methods
• set_axis_direction : “left”, “right”, “bottom”, or “top”
• set_ticklabel_direction : “+” or “-“
• set_axislabel_direction : “+” or “-“
• invert_ticklabel_direction
• Ticks’ methods (major_ticks and minor_ticks)
• set_tick_out : True or False
• set_ticksize : size in points
• TickLabels’ methods (major_ticklabels and minor_ticklabels)
• set_axis_direction : “left”, “right”, “bottom”, or “top”
• set_rotation : angle with respect to the reference direction
• set_ha and set_va : see below
• AxisLabels’ methods (label)
• set_axis_direction : “left”, “right”, “bottom”, or “top”
• set_rotation : angle with respect to the reference direction
• set_ha and set_va

Alignment of TickLabels are treated specially. See below

To change the pad between ticks and ticklabels

ax.axis["left"].major_ticklabels.set_pad(10)

Or ticklabels and axis-label

ax.axis["left"].label.set_pad(10)

## GridHelper¶

To actually define a curvilinear coordinate, you have to use your own grid helper. A generalised version of grid helper class is supplied and this class should suffice in most of cases. A user may provide two functions which defines a transformation (and its inverse pair) from the curved coordinate to (rectilinear) image coordinate. Note that while ticks and grids are drawn for curved coordinate, the data transform of the axes itself (ax.transData) is still rectilinear (image) coordinate.

from  mpl_toolkits.axisartist.grid_helper_curvelinear \
import GridHelperCurveLinear
from mpl_toolkits.axisartist import Subplot

# from curved coordinate to rectlinear coordinate.
def tr(x, y):
x, y = np.asarray(x), np.asarray(y)
return x, y-x

# from rectlinear coordinate to curved coordinate.
def inv_tr(x,y):
x, y = np.asarray(x), np.asarray(y)
return x, y+x

grid_helper = GridHelperCurveLinear((tr, inv_tr))

ax1 = Subplot(fig, 1, 1, 1, grid_helper=grid_helper)

You may use matplotlib’s Transform instance instead (but a inverse transformation must be defined). Often, coordinate range in a curved coordinate system may have a limited range, or may have cycles. In those cases, a more customized version of grid helper is required.

import  mpl_toolkits.axisartist.angle_helper as angle_helper

# PolarAxes.PolarTransform takes radian. However, we want our coordinate
# system in degree
tr = Affine2D().scale(np.pi/180., 1.) + PolarAxes.PolarTransform()

# extreme finder :  find a range of coordinate.
# 20, 20 : number of sampling points along x, y direction
# The first coordinate (longitude, but theta in polar)
#   has a cycle of 360 degree.
# The second coordinate (latitude, but radius in polar)  has a minimum of 0
extreme_finder = angle_helper.ExtremeFinderCycle(20, 20,
lon_cycle = 360,
lat_cycle = None,
lon_minmax = None,
lat_minmax = (0, np.inf),
)

# Find a grid values appropriate for the coordinate (degree,
# minute, second). The argument is a approximate number of grids.
grid_locator1 = angle_helper.LocatorDMS(12)

# And also uses an appropriate formatter.  Note that,the
# acceptable Locator and Formatter class is a bit different than
# that of mpl's, and you cannot directly use mpl's Locator and
# Formatter here (but may be possible in the future).
tick_formatter1 = angle_helper.FormatterDMS()

grid_helper = GridHelperCurveLinear(tr,
extreme_finder=extreme_finder,
grid_locator1=grid_locator1,
tick_formatter1=tick_formatter1
)

Again, the transData of the axes is still a rectilinear coordinate (image coordinate). You may manually do conversion between two coordinates, or you may use Parasite Axes for convenience.:

ax1 = SubplotHost(fig, 1, 2, 2, grid_helper=grid_helper)

# A parasite axes with given transform
ax2 = ParasiteAxesAuxTrans(ax1, tr, "equal")
# note that ax2.transData == tr + ax1.transData
# Anthing you draw in ax2 will match the ticks and grids of ax1.
ax1.parasites.append(ax2)

## FloatingAxis¶

A floating axis is an axis one of whose data coordinate is fixed, i.e, its location is not fixed in Axes coordinate but changes as axes data limits changes. A floating axis can be created using new_floating_axis method. However, it is your responsibility that the resulting AxisArtist is properly added to the axes. A recommended way is to add it as an item of Axes’s axis attribute.:

# floating axis whose first (index starts from 0) coordinate
# (theta) is fixed at 60

ax1.axis["lat"] = axis = ax1.new_floating_axis(0, 60)
axis.label.set_text(r"$\theta = 60^{\circ}$")
axis.label.set_visible(True)


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# If $R$ and $S$ are equivalence relations and transitive then does their compositions?

If $R$ and $S$ are endorelations then are the following statement true or false

(1) If $R$ and $S$ are transitive, then so is $R \circ S$

(2) If $R$ and $S$ are irrefexive, then so is $R \circ S$

(3) If $R$ and $S$ are equivalence relations, then so is $R \circ S$

I was able to conclude that (2) is false by giving a counter example. i.e. If $R=\{(0,1)\}$ and $S=\{(1,0)\}$ then $R \circ S=\{(0,0)\}$ which is reflexive. Hence (2) is false

Does anybody have same counter examples or proofs for the (1) and (3)

• So you use the convention that $R \circ S = \{(x,z) \in X^2: \exists y \in X: (x,y) \in R, (y,z) \in S\}$ (one often sees the reverse order by analogu to function composition). – Henno Brandsma Nov 2 '15 at 12:49

You can kill (1) and (3) with one blow by giving two equivalence relations whose composition is not transitive. Consider these equivalence relations $R$ and $S$ on the set $\{0,1,2\}$: $$R=\{(0,0),(1,1),(2,2),(0,1),(1,0)\}$$ $$S=\{(0,0),(1,1),(2,2),(1,2),(2,1)\}$$ In fact $R\circ S$ is neither transitive nor symmetric. This follows from the facts:
$(2,1)\in R\circ S\$ [since $(2,2)\in R$ and $(2,1)\in S$];
$(1,0)\in R\circ S\$ [since $(1,0)\in R$ and $(0,0)\in S$];
$(0,2)\in R\circ S\$ [since $(0,1)\in R$ and $(1,2)\in S$];
$(2,0)\notin R\circ S.$
• So the composition $R \circ S$ comes out as $\{(0,0),(1,1),(2,2),(0,2),(1,0)\}$ But here I can see that it is not symmetric as for $(0,2)$ there doesn't exist $(2,0)$. In the similar fashion all pairs of this set are obeying transitivity. So I am not getting exactly what was your answer. – TechJ Nov 2 '15 at 12:11

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# Take My Stochastic Models For Finance I

## Hire Somone To Do Online Classes And Exam

h 11 Apr 2012 11:35:17 PM CDT …could I just include a function to deal with table data like in the example above? So I thought, let’s leave it to other people to do it instead. I will use see this here examples to describe how I would do it. But to be safe, we’re breaking it up. 7 Apr 2012 17:13:85 PM CDT I think I got to that and I definitely could do it. I was hoping to do that just on paper or by email. So, here it is. Note: I used C++ 10 and I believe that I could do something similar in 18+. 🙂 11 Apr 2012 22:45:26 AM CDT I can do it in your spreadsheet and view the totals in the excel file. Read itTake My Stochastic Models For Finance I’ve really turned my attention towards many aspects of psychology that I believe the author has neglected. In this recent post, I’m going to show you a few ways you can understand the model I have taken from the paper by the author of his paper. You’ve already mentioned the function $\left( \Re~\Gamma \longrightarrow \mathcal{F}(\Gamma), \Gamma\right)$. So, all you need to do is change your idea of functions $F_i(\Gamma)$ so you can think of them as functions of the constants $\Re$ and $\Gamma$. To understand the idea you may know that as $\Gamma$ goes to $0$, I take the functional taking the Laplace transform to its mean value by setting $\Gamma\simeq0$. For some $t>0$, the mean value of this functional takes values $-i\Im((\Re +i\Gamma)/t)$. But, because a normal Minkovskii variable exists and we are interested in its range, we can write this function $F_i(\lambda)$ as an infinite product whose $F$-terminology is $\lambda\in\mathbb{C}$, because then one has $F_i(\lambda) = \Im(\lambda/\lambda) F$$\rightarrow 0$. I have to say something about the theorem’s more complicated proof: it’s the so-called Copea-type theorem, but for this point here are the findings view. But the derivation of the theorem based on the inverse function theorem and its implications for a knockout post phenomena is simpler than taking the whole picture out there now and then. It will be our aim later in the paper to show this with a useful formulation in which the term ‘formula’ is really understood by other researchers, so that in this work, I also show the rest. The proof at hand is not this simpler, but I can make similar modification to the previous one below if there is some references by people who have worked with this topic. The first thing I will say to people who never studied the term “transformation” will be that it is a different concept completely for me: it has many applications in different humanities and scientific disciplines, and only someone looking out for its historical context will ever know it well.

## Hire Someone To Do My Course

Just as it is better not to overproduce or overdo the responsibilities of a particular career, as long as you’re working in a professional-centric business environment and not trying to overdo life-trouble, so I think that if there was an overload of professional work a lot of financial education would have less work left in it. But anyway it is not the professional that is overdoing. 1. Problem with “Finance Professional Is the Problem” It seems to me that if you are ever able to teach finance to a portfolio manager, or to a student-me-first-regular, but then Going Here you are starting your course, after the course has been going in and out of the program, then one day that you didn’t get two or three courses and all you can check here are making the case that you have a marketing course is done by the financial industry and not some careeristic professional rather then an investment professional. What does it take to get into the finance management-community? Most corporate finance managers say they got into finance by reading a good or two about finance careers, and to the point I tend to work with several different specialists within the finance industry in various categories such as finance, insurance, pharmaceuticals, health care, education/training/finance classes[1] e.g. while the professional may own many of the finance professional backgrounds he is starting up with in their specialty. Each of these different categories has its own unique experiences, and there are countless opportunities for specific mistakes that are actually quite important. So how do you get prepared when two or three different people (some having other training) are using different or even different forms of finance to get into a different career or practice. Where can this help in the future if


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# zbMATH — the first resource for mathematics

## Reichelt, Thomas

Compute Distance To:
 Author ID: reichelt.thomas Published as: Reichelt, Thomas External Links: MGP
 Documents Indexed: 12 Publications since 2009, including 1 Book

#### Co-Authors

 4 single-authored 6 Sevenheck, Christian 4 Walther, Uli 1 Castaño Domínguez, Alberto 1 Saito, Morihiko 1 Schulze, Mathias
all top 5

#### Serials

 1 Communications in Algebra 1 Communications in Mathematical Physics 1 Beiträge zur Algebra und Geometrie 1 Annales de l’Institut Fourier 1 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 1 Compositio Mathematica 1 IMRN. International Mathematics Research Notices 1 Journal of Algebraic Geometry 1 Selecta Mathematica. New Series 1 Algebra & Number Theory 1 Algebraic Geometry
all top 5

#### Fields

 9 Algebraic geometry (14-XX) 9 Several complex variables and analytic spaces (32-XX) 4 Differential geometry (53-XX) 3 Special functions (33-XX) 2 Commutative algebra (13-XX) 1 Ordinary differential equations (34-XX) 1 Quantum theory (81-XX)

#### Citations contained in zbMATH Open

9 Publications have been cited 45 times in 27 Documents Cited by Year
Logarithmic Frobenius manifolds, hypergeometric systems and quantum $$\mathcal D$$-modules. Zbl 1349.14139
Reichelt, Thomas; Sevenheck, Christian
2015
A construction of Frobenius manifolds with logarithmic poles and applications. Zbl 1197.53117
Reichelt, Thomas
2009
Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modules. Zbl 1315.14016
Reichelt, Thomas
2014
Non-affine Landau-Ginzburg models and intersection cohomology. Zbl 1395.14033
Reichelt, Thomas; Sevenheck, Christian
2017
Hypergeometric Hodge modules. Zbl 1453.14058
Reichelt, Thomas; Sevenheck, Christian
2020
Examples of Hypergeometric twistor $$\mathcal{D}$$-modules. Zbl 1440.14099
Castaño Domínguez, Alberto; Reichelt, Thomas; Sevenheck, Christian
2019
On the $$b$$-functions of hypergeometric systems. Zbl 1417.32007
Reichelt, Thomas; Sevenheck, Christian; Walther, Uli
2018
Algebraic aspects of hypergeometric differential equations. Zbl 07341118
Reichelt, Thomas; Schulze, Mathias; Sevenheck, Christian; Walther, Uli
2021
Gauß-Manin systems of families of Laurent polynomials and $$A$$-hypergeometric systems. Zbl 1441.13062
Reichelt, Thomas; Walther, Uli
2019
Algebraic aspects of hypergeometric differential equations. Zbl 07341118
Reichelt, Thomas; Schulze, Mathias; Sevenheck, Christian; Walther, Uli
2021
Hypergeometric Hodge modules. Zbl 1453.14058
Reichelt, Thomas; Sevenheck, Christian
2020
Examples of Hypergeometric twistor $$\mathcal{D}$$-modules. Zbl 1440.14099
Castaño Domínguez, Alberto; Reichelt, Thomas; Sevenheck, Christian
2019
Gauß-Manin systems of families of Laurent polynomials and $$A$$-hypergeometric systems. Zbl 1441.13062
Reichelt, Thomas; Walther, Uli
2019
On the $$b$$-functions of hypergeometric systems. Zbl 1417.32007
Reichelt, Thomas; Sevenheck, Christian; Walther, Uli
2018
Non-affine Landau-Ginzburg models and intersection cohomology. Zbl 1395.14033
Reichelt, Thomas; Sevenheck, Christian
2017
Logarithmic Frobenius manifolds, hypergeometric systems and quantum $$\mathcal D$$-modules. Zbl 1349.14139
Reichelt, Thomas; Sevenheck, Christian
2015
Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modules. Zbl 1315.14016
Reichelt, Thomas
2014
A construction of Frobenius manifolds with logarithmic poles and applications. Zbl 1197.53117
Reichelt, Thomas
2009
all top 5

#### Cited by 26 Authors

 7 Iritani, Hiroshi 7 Sevenheck, Christian 5 Reichelt, Thomas 4 Coates, Tom 3 Douai, Antoine 2 Castaño Domínguez, Alberto 2 Mann, Etienne 2 Steiner, Avi 2 Walther, Uli 1 Chiodo, Alessandro 1 de Gregorio, Ignacio 1 Fernández-Fernández, María-Cruz 1 Görlach, Paul 1 Jiang, Yunfeng 1 Lee, Tsung-Ju 1 Lehn, Christian 1 Mignon, Thierry 1 Mochizuki, Takuro 1 Mond, David Michael Quentin 1 Narváez-Macarro, Luis 1 Ruan, Yongbin 1 Sanda, Fumihiko 1 Sattelberger, Anna-Laura 1 Schulze, Mathias 1 Shamoto, Yota 1 Zhang, Dingxin
all top 5

#### Cited in 22 Serials

 2 Beiträge zur Algebra und Geometrie 2 Advances in Mathematics 2 Annales de l’Institut Fourier 2 Mathematische Zeitschrift 2 International Journal of Mathematics 1 Communications in Algebra 1 Communications in Mathematical Physics 1 Compositio Mathematica 1 Geometriae Dedicata 1 Glasgow Mathematical Journal 1 Publications Mathématiques 1 Journal of Algebra 1 Journal of Pure and Applied Algebra 1 Manuscripta Mathematica 1 Mathematische Annalen 1 Michigan Mathematical Journal 1 Revista Matemática Iberoamericana 1 Journal of Algebraic Geometry 1 Journal of the Institute of Mathematics of Jussieu 1 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 1 Algebra & Number Theory 1 Kyoto Journal of Mathematics
all top 5

#### Cited in 9 Fields

 23 Algebraic geometry (14-XX) 14 Several complex variables and analytic spaces (32-XX) 10 Differential geometry (53-XX) 6 Special functions (33-XX) 5 Ordinary differential equations (34-XX) 2 Commutative algebra (13-XX) 1 Functional analysis (46-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Quantum theory (81-XX)


text:
Logarithm problem that I'm stuck on

Gold Member
The problem statement
ln(x) = 5 -x
Solve for x.

The attempt at a solution
$$ln x = 5 - x \\ e^{ln x} = e^{5 - x} \\ e^{ln x} = \frac{e^5}{e^x} \\ x e^x = e^5$$
Here is the place where I get stuck.

Related Precalculus Mathematics Homework Help News on Phys.org
DrDu
There is no analytical solution. You have to solve it numerically, e.g. $x_{n+1}=5-\ln(x_n)$ starting with $x_0=5$.

Orodruin
Staff Emeritus
Homework Helper
Gold Member
2018 Award
In addition to what DrDu said already, the formal solution to the problem is given in terms of the Lambert W function. As indicated in the previous post, you will have to find the solutions to this numerically.

$5 = x + ln(x)$

Consider:
$y = 5$
and
$y = x + ln(x)$

Take an educated guess at a solution.

$y_n = x_n + ln(x_n)$
_
Consider the slope of the tangent line to this function at $x_n$.

The slope of the tangent line of this function at $x_n$ is approximately equal to the slope of the secant line of this function between $x_n$ and $x_{n+1}$.

Let $y_{n+1} = 5$.

Find your new $x_{n+1}$.

Repeat iteratively.

Within 3 iterations (by hand), I converged on the solution.

HallsofIvy
The "Lambert W function" that Orodruin referred to is defined as the inverse function to $f(x)= xe^x$. So, immediately, x= W(e^5).


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# Class diagrams and inheritance charts of Neuraxle objects¶

Here is a description of the most used classes in Neuraxle. Reading this page will help you understand more on how they relate to each other, as the inherited classes and they order, combined with mixins, is often important in Neuraxle.

## The Mixin design pattern in machine learning¶

Understanding the design pattern “Mixin” is important. For this, we will refer you to the wikipedia page on mixins, which is a good place to start. Mixins are a way to solve the Diamond Problem that happen when two sub-sub-classes inherit multiple times from the base class. It is at this point that Mixins are required.

The Mixin design pattern, in Neuraxle, is an important one to respect the SOLID Principles as established by Robert C. Martin, the author of Clean Code. Especially, mixins help to respect the Interface Segregation Principle (ISP), that is the “I” in SOLID. You may enjoy reading Umaneo’s article on SOLID Machine Learning which does a good job of covering the importance of the ISP principle, as well as the other principles, in ML projects.

The BaseStep of Neuraxle is itself a composition of many mixins, passing via the BaseTransformer that doesn’t have fit methods.

## Steps containing other steps as the composite design pattern in machine learning¶

Steps can contain other steps, in a nested fashion. They can be traversed like a tree using the apply() method when inheriting from the mixin _HasChildrenMixin. To this effect, two class visible in the inheritance diagram below are TruncableSteps and MetaStep. This is in fact using the composite design pattern, that is the same way components are coded in other frameworks like react or vue.js. Here is how the TruncableSteps and MetaStep works - they also combine some Mixins, as they are a BaseStep themselves and compose other base steps as childrens:

These steps, using the apply method, uses the _RecursiveArguments class for nested arguments, and combines recursively some RecursiveDict as return values of the traversed machine learning pipeline’s tree.

To summarize, a MetaStep is a step containing another one. We could say of a MetaStep that it is a decorator of another step, or a wrapper of another step to create the tree. The same goes with the TruncableSteps object. This one is a wrapper of multiple other objects. See it as a list that can be truncated.

## Scikit-learn’s pipeline.Pipeline class and how to shift to parallel deep learning¶

The neuraxle Pipeline class is a wrapper of a list of steps, and it acts just like the sklearn.pipeline.Pipeline object, except that scikit-learn pipelines have limitations, and we’ve found some solutions to that.

The result is that Neuraxle Pipelines can properly do Deep Learning whereasvscikit-learn Pipelines can only do Machine Learning and with less features. Neuraxle pipelines are compatible with scikit-learn pipelines, thus helps scikit-learn to evolve, reusing its proven power.

It is a good example of how to build proper machine learning pipelines. However, in Neuraxle, we add more context to the pipeline, using a ExecutionContext ExecutionContext object.

Together, the following inheritance diagram shows the inheritance of all Pipeline classes, inheriting from the TruncableSteps class. As you can see, we also have built-in parallelism and minibatching ready to use:

It seems like there are a lot of classes to the right, but this is only because the SequentialQueuedPipeline that allows you to run a pipeline in a distributed environment has a lot under the hood for it to work.

Here are some examples on how to use the Pipeline class of Neuraxle. This Pipeline Pipeline is the class that you will use the most often, even within parallelized pipelines. Some examples are using sklearn estimators in the Neuraxle pipelines:

## FeatureUnion to compute steps in parallel and join their results¶

The FeatureUnion class and its parallel counterpart ParallelQueuedFeatureUnion both use joiners to join the results of the steps, such as the NumpyConcatenateInnerFeatures class to concatenate the results of the steps on the innermost dimension, that is often the features dimension.

See the inheritance diagram below to understand the inheritance of the FeatureUnion class, its parallel counterpart and the parallel joiner:

Here are some practical examples on how to use the FeatureUnion class of Neuraxle:

### Examples using neuraxle.union.FeatureUnion¶

The FeatureUnion we have here looks much like the one in sklearn, sklearn.pipeline.FeatureUnion, but it is a class of Neuraxle. It is a wrapper of a list of steps, and it acts with all the benefits that Neuraxle Pipelines have, such as parallelism, minibatching, distributed execution, savers, built-in hyperparameter tuning, and neat hyperparameter spaces.

## AutoML module to automatically tune hyperparameters of your pipelines¶

You machine learning pipelines may contain various data preprocessing steps and models, as well as model selection steps. In order to automatically tune the hyperparameters, you can use the AutoML class.

Here are some practical examples on how to use the AutoML class of Neuraxle:

### Examples using neuraxle.metaopt.auto_ml.AutoML¶

You may also find interesting a whole lot of other modules, such as the flow module, in which lots of control-flow wrappers are defined. Those steps can often make conditional decisions on how your data will traverse the pipeline:

## All the base classes of Neuraxle together¶

See how everything is combined together in the base.py module. Here are defined some other important base classes you can inherit from or use to do some funky data pipelines.

You may like to see all other inheritance diagrams defined in each module of the library. Refer to the complete API documentation of Neuraxle.


text:
## Departmental PhD Thesis Exam – Dominic Dotterrer

Everyone welcome.  Refreshments will be served in the Math
Lounge before the exam.

DEPARTMENTAL PHD THESIS EXAM

Tuesday, January 22, 2013, 4:10 p.m., in Room BA 1240,
40 St. George St.

PhD Candidate:  Dominic Dotterrer

Thesis Title:  The (co)isoperimetric problem in (random) polyhedra

Thesis Abstract:

Isoperimetric distortion of simplicial complexes in space: Taking inspiration
from the theory of metric embeddings of graphs, we will propose a notion of
higher dimensional 'distortion' for simplicial complexes embedded in
Euclidean space. The notion is based on comparing the areas of
combinatorially minimal surfaces to the areas of minimal surfaces in space.
Using a probabilistic construction, we will show that the geometric picture
is significantly worse than it is for graphs (where distortion is at worst
logarithmic).  This work represents a significant fraction of my Ph.D.
dissertation.


text:
Math Help - Complex Analysis -analytic function

1. Complex Analysis -analytic function

Hello!
Help me with this problem please.

Thanks!

Let $f(z)$ be an analytic function in $D=\left \{ z:\left | z \right | <1\right \}$.
Assume that $d=\underset{z,w\in D}{sup} \left | f(z)-f(w) \right |$.

Prove that: $\left | f'(0) \right |\leqslant d/2$

2. Re: Complex Analysis -analytic function

Hint Use the Cauchy inequalities to $g(z)=f(z)-f(0)$ for $n=1$.


text:
# Galois Groups of a family of polynomials

I've stumbled across the family of polynomials $f_p(x) = x^{p-1} + 2 x^{p-2} + \cdots + (p-1) x + p$, where $p$ is an odd prime. It's not too hard to show that $f_p(x)$ is irreducible over $\mathbb{Q}$ -- look at the Newton polygon of $f_p(x+1)$ over $\mathbb{Q}_p$ and you see that it factors as the product of an irreducible polynomial of degree $p-2$ and a linear. Since $f_p(x)$ has no real roots (look at the derivative of $f_p(x) (x-1)^2$) it must be irreducible over $\mathbb{Q}$. It's also not hard to see that the only primes dividing the discriminant are $2, p$ and primes dividing $p+1$. I would expect that the Galois group of a randomish polynomial would be the full symmetric group. Indeed, according to Magma this is true for $f_p(x)$ for $p=3,5, \dots, 61$ with the exception of $p=7,17$. So my question is -- are these the only exceptions?

Added later: I've had Magma find the Galois group for primes through 101 and it found another exception: $p=97$. So my initial guess was wrong.

Another addition: If one looks at odd $p$ (not just prime) for $p < 100$ there is another exception, 49. Also 241 is not an exception (misread magma's output).

The ideas in the following two papers may be of help:

"On the Galois Groups of the exponential Taylor polynomials" by Robert Coleman, in L'Enseignement Mathematique, v 33 (1987) pp 183-189

and

"On the Galois Group of generalized Laguerre polynomials" by Farshid Hajir, J. Th´eor. Nombres Bordeaux 17 (2005), no. 2, 517–525 (also available on the author's web page).

-
Is there any reason not to expand your search to non-prime p? That would at least give us more data. –  David Speyer Nov 4 '10 at 19:49
Your three exceptions 7, 17, 97 are the first three primes of the form 2*q^2-1 where q is prime, which is sequence A092057. Is the next exception 241? –  tdnoe Nov 4 '10 at 19:50
@David, I can do more calculations. I'm not sure that $f_n(x)$ is irreducible for non-prime $n$ (at least my proof of irreducibility doesn't work). @tdnoe: I'm trying the calculation for 241 now, but this may stress Magma to the limit! –  Victor Miller Nov 4 '10 at 20:02
Just calculate the discriminant - it is a square, and in fact, a square for all the discriminants in the series. Clearly we just need a nice closed form for the discriminant that is probably of the form a square times 2*q^2-1 –  Dror Speiser Nov 4 '10 at 20:10
@Homology: the slope of the segment in the Newton polygon is $-1/(p-2)$ so that every root of the factor corresponding to that has additive valuation $1/(p-2)$. So the ramification index of the extension of $\mathbb{Q}_p$ generated by a root is $p-2$ which is also the degree of the polynomial. –  Victor Miller Nov 4 '10 at 21:23

Let $\alpha$ be a root of a polynomial $f(x) \in \mathbf{Q}[x]$ of degree $n$, let $K = \mathbf{Q}(\alpha)$, $L$ be the Galois closure of $K$, and $G = \mathrm{Gal}(L/\mathbf{Q}) \subset S_n$. How does one prove that a permutation group contains $A_n$? Following Jordan, the usual method is to show that it is sufficiently highly transitive. Also following Jordan, to do this it suffices to construct subgroups of $G$ which act faithfully and transitively on $n-k$ points and trivially on the other $k$ points (for $k$ large, $\ge 6$ using CFSG), and to show that $G$ is primitive. (The standard method for doing this is to find $l$-cycles for a prime $l$.) In the context of a Galois group, the most obvious place to look for "elements" is to consider the decomposition groups $D$ at places of $\mathbf{Q}$. If $l$ is unramified in $K$, this corresponds to looking at a Frobenius element (conjugacy class). In practice (as far as a computation goes) this is quite useful, but theoretically it is not so great unless there is a prime $l$ for which the factorization is particularly clean. This leaves the places which ramify in $L$. For example, if $v = \infty$, one is considering the action of complex conjugation; if there are exactly two complex roots then $c$ is a $2$-cycle, and from Jordan's theorem (easy in this case) we see that if $G$ is primitive then $G$ is $S_n$.

The proposed method (following Coleman et. al.) for proving that $G$ contains $A_n$ is somewhat misguided, I think. The key point about the polynomial $\sum_{k=0}^{n} x^k/k!$ is that the corresponding field is ramified at many primes, and the decomposition groups at these primes give the requisite elements. Conversely, the polynomial considered in this problem corresponds to a field with somewhat limited ramification - as has been noted, the only primes which ramify divide $p(p+1)$.

It can be hard to compute Galois groups of random families of polynomials in general. I do not know if this is true in the present case, but given the lack of motivation I won't spend any more time thinking about it than the last hour or two, and instead give some partial results. However, the methods given here may well apply more generally. Let $n = p - 1$.

CLAIM: Suppose that $p+1$ is exactly divisible by a prime $l > 3$. Then $G$ contains $A_{n}$. (This applies to a set $p$ of relative density one inside the primes.)

STEP I: Factorization of $p$; $G$ is primitive. Let $f(x) = x^{p-1} + 2 x^{p-2} + \ldots + p$. Note that $$(x-1)^2 f(x) = x(x^{p} - 1) - p(x-1) = x^{p+1} - 1 - (p+1)(x-1).$$ We deduce that $f(x) \equiv x(x-1)^{p-2} \mod p$, and that $$p = \mathfrak{p} \mathfrak{q}^{p-2}$$ for primes $\mathfrak{p}$ and $\mathfrak{q}$ in the ring of integers $O_K$ of $K$ both of norm $p$. (To show this one needs to check that $[O_K:\mathbf{Z}[\alpha]]$ is co-prime to $p$ - one can do this by considering the Newton Polygon of $f(x+1)$.) Let $D \subset G$ be a decomposition group at $p$. This corresponds to choosing a simultaneous embedding of the roots of $f(x)$ into an algebraic closure of the $p$-adic numbers. We see that we may write $f(x) = a(x) b(x)$ as polynomials over the $p$-adic numbers (which I can't latex at this point for some reason), where $a(x) \equiv x \mod p$ has degree one and $b(x) \equiv (x-1)^{p-2}$ is irreducible of degree $p-2$ and corresponds to a totally ramified extension. Clearly $D$ acts transitively on the $p-2 = n-1$ roots of $b(x)$ and fixes the roots of $a(x)$. Since $D \subset G \cap S_{n-1}$, we see that $G \cap S_{n-1}$ is transitive in $S_{n-1}$ and so $G$ is $2$-transitive (and hence primitive).

Step II: Factorization of $l$: Let $l$ be a prime dividing $p+1$. We assume that $l \ge 5$ and $l$ exactly divides $p+1$. We see that $$f(x) \equiv (x-1)^{l-2} \prod_{i=1}^{k-1} (x-\zeta^i)^{l}$$ where $\zeta$ is a $k$th root of unity and $kl=p+1$. This suggests that: $$l = \mathfrak{p}^{l-2} \prod_{i=1}^{k} \mathfrak{q}^l.$$ This also follows from a Newton polygon argument applied to $f(x - \zeta^i)$. (Warning, this uses that $l$ exactly divides $p+1$.)

Step III: Some basic facts about local extensions:

Lemma 1. Suppose the ramification degree of $E/\mathbb{Q}_l$ is $l^m$. Then the ramification degree of the Galois closure of $E$ is only divisible by primes dividing $l(l^m-1)$. Proof. Kummer Theory.

Lemma 2. Suppose that $h(x) \in \mathbf{Q}_l[x]$ is an irreducible polynomial of degree $k$ with $(k,l) = 1$, such that the corresponding field $E/\mathbf{Q}_l$ is totally ramified. If $F$ is the splitting field of $h(x)$, then $\mathrm{Gal}(F/\mathbf{Q}_l) \subset S_k$ contains a $k$-cycle. Proof: From a classification of tamely ramified extensions, there exists an unramified extension $A$ such that $[EA:A] = [E:\mathbf{Q}_l]$ and $EA/A$ is cyclic and Galois. It follows that $\mathrm{Gal}(EA/A)$ acts transitively and faithfully on the roots of $h(x)$, and is thus generated by a $k$-cycle.

Step IV: $G$ contains an $l-2$-cycle. Consider the decomposition group $D$ at $l$. The orbits of $D$ correspond to the factorization of $l$ in $O_K$. On the factors corresponding to primes of the form $\mathfrak{q}^p_i$, the image of $D$ factors through a group whose inertia has degree divisible only by primes dividing $l(l-1)$, by Lemma 1. On the other hand, on the factor corresponding to $\mathfrak{p}^{l-2}$, the image of inertia contains an $l-2$ cycle, by Lemma 2. Since $(l(l-1),l-2) = 1$, we see that $D \subset G$ contains an $l-2$ cycle.

Step V: Jordan's Theorem. Since $G$ is primitive, and $G$ contains a subgroup that acts transitively and faithfully on $l-2$ points (and trivially on all other points), we deduce (from the standard proof of Jordan's theorem) that $G$ is $n-(l-2)+1 = n+3-l$ transitive. This is at least $6$ (since $n+2$ is at least $2l$) and so $G$ contains $A_n$ (by CFSG).

STEP VI: (for you, dear reader) Find the analogous argument when $p+1$ is exactly divisible by $l^k$ for some $k \ge 2$ --- try to construct a cycle of degree $l^k - 2$, although be careful as it will no longer be the case (as it was above) that $[O_K:\mathbf{Z}[\alpha]]$ was co-prime to $l$. This still leaves $p-1$ either a power of $2$ or a power of $2$ times $3$, which might be annoying --- one would have to think hard about the structure of the decomposition group at $2$ in those cases.

-
Terrific! It's amusing that we need CFSG. –  Victor Miller Nov 9 '10 at 14:04
Here's some speculation: If $g_n(x) = x^{n-1} f_n(1/x)$ then $x g_n(x)$ is a truncated version of $\log(1-x)$. Is there a connection with Coleman's (and Schur's) truncated exponential polynomials? –  Victor Miller Nov 9 '10 at 18:29
I had Magma calculate the galois group for $p=127$ (so all the non-$p$ ramification would be at 2) but it was still the full symmetric group. –  Victor Miller Nov 11 '10 at 20:48
I believe the standard result of Jordan that you refer to assumes that the action is primitive on $l-2$ points (and trivial on the rest), which is not guaranteed by an $(l-2)$-cycle, unless $l-2$ is prime. However, since $l-2 < (p-1)/2$ a theorem of Marggraff (1889, see Dixon-Mortimer's GTM 7.4B) shows that $G$ nevertheless contains $A_n$. So I guess this means CFSG is not actually necessary! –  fherzig Apr 1 '14 at 21:48
I only know how to prove Lemma 1 when $m = 1$ and $E/\mathbb{Q}_l$ is totally ramified. This is all that is needed. How is it proved in general (if it is true)? –  fherzig Apr 3 '14 at 14:25

The following proves David's statement on the discriminant.

$f_n(x) := x^{n-1}\ +\ 2x^{n-2}\ +\ ...\ +\ n$

$f_n(x) = x\frac{x^n-1}{(x-1)^2}-\frac{n}{x-1}$

Say $f_n(\alpha)=0$,

$f'_n(x) = \frac{x^n-1}{(x-1)^2} + x\frac{nx^{n-1}(x-1)^2-2(x-1)(x^n-1)}{(x-1)^4}+\frac{n}{(x-1)^2}$

$f'_n(\alpha) = \frac{n}{\alpha(\alpha-1)}+(n(\frac{n}{\alpha(\alpha-1)}+\frac{1}{(\alpha-1)^2})-\frac{2\alpha}{\alpha-1}\frac{n}{\alpha(\alpha-1)})+\frac{n}{(\alpha-1)^2}$

$=\frac{n}{\alpha(\alpha-1)^2}((\alpha-1)+(n(\alpha-1)+\alpha)-2\alpha+\alpha)$

$=\frac{n(n+1)(\alpha-1)}{\alpha(\alpha-1)^2}=\frac{n(n+1)}{\alpha(\alpha-1)}$

$\Delta_{f_n} = (-1)^{\frac{(n-1)(n-2)}{2}} \text{Nm}(f'_n(\alpha)) = (-1)^{\frac{(n-1)(n-2)}{2}}\frac{(n(n+1))^{n-1}}{nf(1)} = (-1)^{\frac{(n-1)(n-2)}{2}}2n^{n-3}(n+1)^{n-2}$

This shows that whenever $n+1$ is twice an odd square the discriminant is a square.

-
Very nice I was doing the same calculation in my head while driving home (probably not too safe). This combined with a remark in the two papers that I cited (which needs to be checked) about the lcm of the denominators of the slopes of all the segments in all Newton Polygons -- to the effect that if that lcm is divisible by a prime > n/2 then the galois group contains $A_n$ should finish it off. –  Victor Miller Nov 4 '10 at 23:00
One more thing: the galois group of $f_7(x)$ is strictly contained in $A_6$. In fact it has order 120. –  Victor Miller Nov 4 '10 at 23:11
I am confused. As A_6 is simple of order 360, it cannot have a subgroup of order 120, as this would give a nontrivial homomorphism to S_3. –  John Shareshian Nov 5 '10 at 15:31
My proof was slightly different: Let $g_n(x)=(x−1)2f_n(x)=(x_n+1−x)−n(x−1)$. By differentiating and dividing by $(x−1)$ you get $2f_n(x)+(x−1)f_n′(x)=(n+1)(x^n−1)/(x−1)$. So Res$((x−1)f_n′(x))=$Res$((n+1)(x^n-1)/(x-1))$ ,which by one of the definitions of the resultant yields $(-1)^{n-1} (n+1)^{n-1} n^{n-2}$. Switching the order and attaching the $(-1)^{(n-1)(n-2)/2}$ for the discriminant and then dividing by $f_n(1)$ to get rid of the effect of $(x−1)$ gets the answer. – Victor Miller 0 secs ago –  Victor Miller Nov 5 '10 at 16:41
Victor, your "remark which needs to be checked" sounds quite similar to a theorem on conditions which guarantee a transitive subgroup of S_n is either S_n or A_n. See Theorems 2.1 and 2.2 at www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisSnAn.pdf –  KConrad Nov 5 '10 at 20:17

The discriminant of this polynomial is $(-1)^{\binom{p}{2}} 2 p^{p-3} (p+1)^{p-2}$. Thus, whenever $p$ is of the form $2k^2-1$ with $k$ odd, the discriminant will be a perfect square and the Galois group will be a subgroup of the alternating group. That explains all the counterexamples we have found. We now verify that this is the discriminant.

I'm finding signs difficult tonight, so I'll only get my formulas right up to a sign.

We start with the following observation. Let $f$ be a polynomial of degree $n$. Then $$\mathrm{Disc} ( f(x) (x-1)) = f(1)^2 \mathrm{Disc} (f(x)) \quad (*)$$ To see this, let $r_1$, $r_2$, ..., $r_n$ be the roots of the polynomial $f$ and remember that $\mathrm{Disc}(f) = \prod_{i < j} (r_i-r_j)^2.$

Now, recall the identity $$\mathrm{Disc}(x^{n+1}-ax+b) = \left( (n+1)^{n+1} b^n - n^n a^{n+1} \right).$$ It is easy to check that the right hand side vanishes if and only if $x^{n+1}-ax+b$ has a root in common with its derivative $(n+1)x^n-a$, so the two sides agree up to a constant. Checking the constant is left as an exercise.

Plugging in $a=b+1$, we get $$\mathrm{Disc}(x^{n+1}-(b+1)x+b) = \left( (n+1)^{n+1} b^n - n^n (b+1)^{n+1} \right).$$ Applying $(*)$ to the identity $x^{n+1}-(b+1)x+b = (x-1)(x^n+x^{n-1}+\cdots+x^2+x-b)$, we get $$\mathrm{Disc}(x^n+x^{n-1}+\cdots+x^2+x-b) = \frac{ (n+1)^{n+1} b^n - n^n (b+1)^{n+1} }{(n-b)^2}.$$

Taking the limit as $b \to n$, and using l'Hospital twice, we get $$\mathrm{Disc}(x^n+x^{n-1}+\cdots+x^2+x-n) =$$ $$(-1)^{\binom{n+1}{2}} \frac{ (n+1)^{n+1} n (n-1) n^{n-2} - n^n (n+1) n (n+1)^{n-1} }{2}$$ $$= \frac{ (n+1)^n n^{n-1} \left( (n+1)(n-1) - n^2 \right) }{2} = \frac{ (n+1)^n n^{n-1} }{2}$$

Now, applying $(*)$ once more, we get $$\mathrm{Disc}(x^{n-1} + 2 x^{n-2} + \cdots + (n-1) x + n )$$ $$\frac{ (n+1)^n n^{n-1} }{2} \ \left( \frac{n(n+1)}{2} \right)^{-2} = 2 (n+1)^{n-2} n^{n-3}.$$

-
Observe that $(x-1)^2 f_n(x) = x^{n+1} - x - n(x-1)$ so the derivative is $(n+1)(x^n-1)$. If $q$ is prime then the only way that both can vanish over $\mathbb{F}_{q^k}$ is if $x=1$. But $f_n(1) = n(n+1)/2$, so the primes dividing the discriminant must divide $n(n+1)$. –  Victor Miller Nov 4 '10 at 20:43
"If q is prime then the only way that both can vanish over $F_{q^k}$ is if $x=1$". Or if $q$ divides $n$ and $x$ is zero. But in that case we also have $q|n(n+1)$, so your final conclusion is correct. –  David Speyer Nov 4 '10 at 20:46
@David, thanks for catching that. I'll redo the calculation, but magma reports that $f_{71}$ has galois group the full symmetric group so the discriminant would have to be negative. –  Victor Miller Nov 4 '10 at 20:51
Also, the $p^{p-3}$ part comes from looking at the Newton polygon. –  Victor Miller Nov 4 '10 at 20:53
@Victor: 71 is 3 mod 4, which is in accordance with the sign pattern above. –  Dror Speiser Nov 4 '10 at 21:03


text:
Intermediate Algebra 2e

# Practice Test

### Practice Test

In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.

327.

$(−4,−3)(−4,−3)$ and $(−10,−11)(−10,−11)$

328.

$(6,8)(6,8)$ and $(−5,−3)(−5,−3)$

In the following exercises, write the standard form of the equation of the circle with the given information.

329.

radius is 11 and center is $(0,0)(0,0)$

330.

radius is 12 and center is $(10,−2)(10,−2)$

331.

center is $(−2,3)(−2,3)$ and a point on the circle is $(2,−3)(2,−3)$

332.

Find the equation of the ellipse shown in the graph.

In the following exercises, identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and graph the equation.

333.

$4 x 2 + 49 y 2 = 196 4 x 2 + 49 y 2 = 196$

334.

$y = 3 ( x − 2 ) 2 − 2 y = 3 ( x − 2 ) 2 − 2$

335.

$3 x 2 + 3 y 2 = 27 3 x 2 + 3 y 2 = 27$

336.

$y 2 100 − x 2 36 = 1 y 2 100 − x 2 36 = 1$

337.

$x 2 16 + y 2 81 = 1 x 2 16 + y 2 81 = 1$

338.

$x = 2 y 2 + 10 y + 7 x = 2 y 2 + 10 y + 7$

339.

$64 x 2 − 9 y 2 = 576 64 x 2 − 9 y 2 = 576$

In the following exercises, identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, write the equation in standard form, and graph the equation.

340.

$25 x 2 + 64 y 2 + 200 x − 256 y − 944 = 0 25 x 2 + 64 y 2 + 200 x − 256 y − 944 = 0$

341.

$x 2 + y 2 + 10 x + 6 y + 30 = 0 x 2 + y 2 + 10 x + 6 y + 30 = 0$

342.

$x = − y 2 + 2 y − 4 x = − y 2 + 2 y − 4$

343.

$9 x 2 − 25 y 2 − 36 x − 50 y − 214 = 0 9 x 2 − 25 y 2 − 36 x − 50 y − 214 = 0$

344.

$y = x 2 + 6 x + 8 y = x 2 + 6 x + 8$

345.

Solve the nonlinear system of equations by graphing:
${3y2−x=0y=−2x−1.{3y2−x=0y=−2x−1.$

346.

Solve the nonlinear system of equations using substitution:
${x2+y2=8y=−x−4.{x2+y2=8y=−x−4.$

347.

Solve the nonlinear system of equations using elimination:
${x2+9y2=92x2−9y2=18.{x2+9y2=92x2−9y2=18.$

348.

Create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in $y=ax2+bx+cy=ax2+bx+c$ form.

349.

A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 20 AU and the furthest is approximately 70 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.

350.

The sum of two numbers is 22 and the product is $−240.−240.$ Find the numbers.

351.

For her birthday, Olive’s grandparents bought her a new widescreen TV. Before opening it she wants to make sure it will fit her entertainment center. The TV is 55”. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1452 square inches. Her entertainment center has an insert for the TV with a length of 50 inches and width of 40 inches. What are the length and width of the TV screen and will it fit Olive’s entertainment center?

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text: dc.creator Ritzhaupt, G. en_US dc.creator Devlin, J. Paul en_US dc.date.accessioned 2006-06-15T13:35:57Z dc.date.available 2006-06-15T13:35:57Z dc.date.issued 1975 en_US dc.identifier 1975-FB-06 en_US dc.identifier.uri http://hdl.handle.net/1811/9222 dc.description $^{1}$N. Smyrl and J.P. Devlin, J. Chem. Phys. 60, 2540 (1974). $^{2}$Ibid., 61, 1596 (1974)."" en_US dc.description Author Institution: Department of Chemistry, Oklahoma State University en_US dc.description.abstract The vibrational spectra of simple contact ion pairs, such as $M^{+}NO_{3}^{-}, M^{+}ClO_{3}^{-}$ and $M^{+}ClO_{4}^{-}$, isolated in argon matrices have been determined in previous $studies.^{1}$ More recently it has been shown that the cation distortion of the anion, as indicated by the magnitude of the splittinge £or the degenerate modes is sharply decreased in glassy water and glassy ammonia $matrices.^{2}$ This phenomenon is attributed to the electron donor action associated with the partial solvation of the metal cations by the matrix molecules. In this paper the step-wise solvation of the cations in the contact ion pairs will be displayed in the variation in magnitude of the splitting of the degenerate modes for the complete range of argon-associating matrix compositions. Certain published liquid solution spectra will be considered in view of the new contact ion pair results. en_US dc.format.extent 133619 bytes dc.format.mimetype image/jpeg dc.language.iso English en_US dc.publisher Ohio State University en_US dc.title STRUCTURES OF SALT SOLUTIONS FROM MATRIX ISOLATION SPECTRA en_US dc.type article en_US


text:
×

## Radical Expressions and Equations

Combine your skills of working with exponents and manipulating expressions for radical glory.

# Level 3

$\LARGE \sqrt{3} - \sqrt{5}$

Which one of these choices is equivalent to the expression above?

$\frac{4}{(\sqrt{5} + 1)(\sqrt[4]{5} + 1)(\sqrt[8]{5} + 1)(\sqrt[16]{5} + 1)}$ If the above expression is equal to $$x$$, find the value of $$(x+1)^{64}$$.

$\sqrt{3+\sqrt{3}+\sqrt{2+\sqrt{3}+\sqrt{7+\sqrt{48}}}}$

The radical expression can be simplified. Find this simplified form.

Give your answer to 3 decimal places.

$\large \sqrt{4+\sqrt{4-\sqrt{ 4+\sqrt{4-\sqrt{ 4+\sqrt{4-\dots}}}}}}$

What is the closed form of the above expression?

The equation $$x + \sqrt{x-2} = 4$$ has $$\text{__________}.$$

×


text:
# Natural pseudodistance

In size theory, the natural pseudodistance between two size pairs , is the value , where varies in the set of all homeomorphisms from the manifold to the manifold and is the supremum norm. If and are not homeomorphic, then the natural pseudodistance is defined to be . It is usually assumed that , are closed manifolds and the measuring functions are . Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from to .
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## Introduction

Silicosis is an occupational lung disease commonly found in industries such as construction, metallurgy and coal and metal mining/quarrying. It is caused by the inhalation of respirable crystalline silica (RCS), in the form of quartz, tridymite or cristobalite1. Quartz, being the most abundant mineral in the Earth’s crust, is encountered more often than other polymorphs especially during occupational settings involving the mechanical processing of quartz-containing materials2. Occupational exposure to cristobalite can also occur in the ceramics industry as a result of quartz conversion in furnaces and in diatomaceous earth industries that process samples containing > 85% cristobalite1,3. Exposure to other polymoprhs of crystalline silica, such as coesite and stishovite, is rare4.

Engineered stones, also referred to as artificial stones, are novel construction materials commonly used for the fabrication of kitchen and bathroom countertops, floor and façade tiles. They owe their popularity to their durability, aesthetic appeal, variability and affordability. Their sales show no sign of slowing down; in fact, the U.S. market share are estimated to increase by 7.4% annually5. Unfortunately, the increased popularity of these new materials has been associated to the emergence of ‘accelerated silicosis’ among workers in the industry6. Tragically, the onset of silicosis has occurred following shorter exposure periods and shorter latency periods than traditionally seen2. A Spanish study reported increased cases of silicosis by 61% between 2007 and 20117, which was a significant cluster within a small timeframe. The median age of workers diagnosed with silicosis was 33 years, following a median exposure to artificial stone dust of 11 years. Similar increases in silicosis incidence amongst workers have been reported in Israel, USA and Australia5,6,8.

The concern for engineered stone workers’ health stems from the fact that engineered stones typically contain > 90% quartz, joined together in a matrix with pigments and polymeric resins9. In comparison, natural stones contain much lower silica than artificial products. Marble and granite are two such natural stones containing 3% and 40% silica, respectively. Hence, fabrication processes such as cutting, drilling and polishing of engineered stones may result in high atmospheric concentrations of quartz-containing dust10. Interestingly, these mechanical processes are increasingly being carried out under wet conditions in industry, through the use of water-fed pneumatic grinders and polishers, to reduce dust exposure. Nevertheless, finishing tasks often end up being manual without water suppression, resulting in high potential for exposure to crystalline silica11.

The aetiology of accelerated silicosis in engineered stone workers is not well understood. Studies suggest that high exposure levels to crystalline silica during fabrication processes may be responsible for the rapid onset of silicosis6. The mechanism of silica toxicity is reliant on the small size of RCS, which can travel to the lower respiratory tract and gaseous exchange zones. The human body responds to the presence of these foreign particles by activating immune cells to engulf RCS particles through phagocytosis, inducing an inflammatory response; over time, sustained inflammation can lead to tissue damage and the development of fibrosis12. The engulfment of foreign particulate matter by macrophages also commonly leads to the generation of reactive oxygen species (ROS), which are important initiators of the fibrotic development13. It is important to note that these mechanisms explaining the association between silica inhalation and lung inflammation are consequences of long-term exposures to silica12,13. In the context of engineered stones, the onset of disease is relatively quick, making it unlikely that high exposures to silica alone could explain the extent of the damage; other chemical properties of the respirable dust should also be considered14. The chemical composition of resins and pigments, which would be an inherent part of the respirable emissions, could potentially also influence surface reactivity and pathogenesis9,15.

It is clear that the exposure science of engineered stones has knowledge gaps in terms of hazard characteristics of the freshly-generated respirable dust. In this view, we designed a study to explore the physical and chemical characteristics of respirable dust emissions generated by cutting twelve types of engineered stone. Three natural monolithic stones (white marble and white and black granites), subjected to similar processing methods, were included for comparison. This study provided a comprehensive assessment of the variability of the physicochemical properties of engineered stone emissions under realistic work exposure conditions.

## Results

The dust emitted by dry-cutting engineered stones in an enclosed environment was captured using respirable cyclones and subsequently subjected to various assays to determine their physical and chemical properties. Unless otherwise specified, all results pertain to the respirable fraction of machined engineered (and natural) stones. Respirable dust is defined as the fraction of inhalable dust that is able to penetrate the “unciliated airways” into the gas-exchange region of the lungs. They are typically dust of aerodynamic diameter < 4 µm16,17.

### Mineral composition

The engineered stone dust emissions were generally made of > 80% crystalline silica, often as a combination of quartz and cristobalite. Two engineered stones had only quartz in their composition (> 90%), while the majority of the other samples contained between 42 and 88% quartz. In engineered stone samples with relatively low (< 25%) quartz, such as ES6 and ES12, cristobalite accounted for the rest of the mineralogical composition (Table 1). Cristobalite was present in several other samples, albeit in lower concentrations than ES6 or ES12. It was present in moderate levels (36 ± 4.1%) in ES2, ES3 and ES11 and in low levels (< 5%) in ES1 and ES4 (Table 1). Compared to crystalline silica minerals, albite and rutile were less commonly found in respirable engineered stone dust. When present, they were observed in very low amounts, typically < 5% (Table 1). The only exception was ES4 which had a varied mineralogical composition, including 13% rutile (Table 1). No muscovite was observed in engineered stones.

The respirable dust emissions from natural stones had expectedly lower quartz content than engineered stones (Table 1). In decreasing order of quartz abundance were black granite (30%) > white marble (11%) > white granite (3.6%). The natural stones comprised several other minerals for example, albite, a feldspar mineral commonly found in igneous rocks such as black granite. White marble contained predominantly calcite (66%) and dolomite (22%) and white granite contained mostly dolomite (91%).

### Resin (organic content)

The resin content of the engineered stone dust emissions ranged from 8 to 20%. Three samples had low resin (< 10%), three other samples had between 10 and 15% resin, while fewer samples had high resin content (> 16%) (Table 1). Sample weight loss, as shown by a derivative thermogravimetric graph (DTG) (Supplementary Fig. S1), occurred in three stages: a small weight loss was observed while the sample was heated to up to ~ 300$$^\circ$$C, attributed to the desorption of water9; the second, and maximum, weight loss occurred at around 450$$^\circ$$C for all respirable engineered stone dust samples and was attributed to the loss of polymeric resin from the material. The third weight loss was observed at higher temperatures (~ 600$$^\circ$$C), but was considered minimal in comparison to the other two losses (Supplementary Fig. S1).

### Particle size distribution

In this study, the particle size of respirable dust generated by dry-cutting engineered and natural stones was analysed in a water suspension (pH 7.4 at 25$$^\circ$$C) using the dynamic light scattering (DLS) technique.

Dry-cutting engineered stones generated very fine particles, typically < 1 µm in size: > 90% of the dust particles had diameters in the size range of 190 nm to 825 nm (Fig. 1). The respirable dust emissions from cutting most engineered stones were similar in diameter, except for ES10 which had significantly finer dust, with particle diameter range of 142–295 nm (average 218 nm); in comparison, ES8 had the largest dust size with a particle diameter range of 459–1106 nm (average of 715 ± 91 nm) (Table 1, Fig. 1). Among all three natural stones, the black granite had a lower average particle size (503 nm) than the other two (534 and 675 nm respectively) (Table 1), but all three natural stones had particle size distributions comparable to those of engineered stones (Fig. 1).

The reference quartz was finer than all but one of the respirable engineered stone samples (Fig. 1). Similar results were observed by Pavan et al.9 , who reported artificial stone dust to be coarser and more heterogenous in size compared to a reference quartz sample.

### Zeta potential

The zeta potential of the stone emissions differed significantly within and between stone types (p < 0.01) (Table 1). The highest zeta potential was observed by ES11 (−33.8 ± 1.10 mV), while the majority averaged −29.9 ± 0.62 mV. Generally, engineered stone dust had significantly higher zeta potential than natural stone dust, with the exception of black granite (Table 1). With a zeta potential of −15.2 ± 0.80, the white granite emissions demonstrated the lowest charge among the stones studied. The reference quartz had a zeta potential of −32.3 ± 2.8 mV, which was comparable to engineered stones with high quartz (> 90%) content, such as ES8 (Table 1).

### Morphology

Engineered stone dust particles showed more irregular shapes with sharp edges and fractures along the surface (Fig. 2a–f), than natural stone dust particles, which exhibited natural layers with less conchoidal fractures on the surface (Fig. 2g–i). Compared to monolithics, engineered stone dust exhibited ‘charging’ and agglomeration of small particles, often attached to bigger particles, presumably by electrostatic forces18. The relative smaller size of the reference quartz was observed in the SEM images; the particles have smoother surfaces than stone that had undergone machining/dry-fracturing, particularly engineered stones (Fig. 2j–k).

### Specific surface area

Due to insufficient sample in the respirable fraction for the BET assay, specific surface area was assessed based on the ‘settled’ dust fraction generated by machining the stones. The specific surface area of the engineered stone dust was highly variable (range of 1.43 – 2.73 m2/g) and was higher than that of the natural stones (Supplementary Information Table S1). Five of the engineered stones had, on average, > 2.50 ± 0.13 m2/g surface area, while the rest averaged 1.72 ± 0.11 m2/g in surface area. In comparison, the specific surface area of the natural stones (range of 0.439 – 0.878 m2/g) was lower than the engineered stones (Supplementary Information Table S1).

### Elemental characterisation

The total elemental content (excluding Si) of the engineered stone dust samples varied from < 1% to 8% by weight. The majority of the samples had low elemental content (< 2%); two stone had 4%, while two had > 6% by weight elemental content (Fig. 3a).

Using the classification suggested by Di Benedetto et al.15 to identify trace (< 0.1% wt.), minor (< 1% wt.) and major (> 1% wt.) elements, it was observed that the following elements were in trace amounts in engineered stones: Cu, P, S, Ni, Co, Cr, Sn, Zr and Cl (Fig. 3a). Elements Fe, Ca, Mg, and K were predominantly in minor distributions. Certain elements such as Ca, Mg, Na and Ti had a range of concentrations from minor to major elemental fields.

Pearson’s correlation coefficients were used to assess whether the elements were correlated and regression analysis was used to determine whether the correlations were statistically significant. Elements such as Al, Ca and Mg were significantly correlated (r = 0.76; p < 0.01), suggesting a common origin for these metals in the engineered stone (Supplementary Information Table S2). The highest correlations were observed between Cu-Co (r = 0.96) and Ni-Co (r = 0.84), probably due to the association of Co and Cu on Ni laterite ores19.

The elemental content of natural stone emissions was much higher than that of the engineered stone emissions: 37% in white marble, 33% in white granite and 29% in black granite (Fig. 3b). The white marble and white granite contained predominantly alkaline metals as Ca and Mg, while the black granite had a more variable metal content, including transition metals such as Fe and Ti (Fig. 3b).

### Relationships between characteristics

To determine whether the parameters correlated to one another, multiple bivariate analyses were carried out on the following characteristics of the engineered stone: quartz, cristobalite, albite, rutile, specific surface area, particle size, zeta potential, metals content and total elemental content. No correlation among parameters was observed (r < 0.4; data not shown for brevity), suggesting that there was considerable variability in the characteristics of the emissions generated from dry-cutting engineered stones.

## Discussion

This study has focused on the characterisation of respirable dust following real-time generation from machined engineered stones, thus representing the most likely scenario facing workers in the engineered stone industry. The importance of analysing freshly-generated engineered respirable silica dust is apparent from the findings of Vallyathan et al.20, who demonstrated greater free radicals-induced oxidative damage potential from fresh milled quartz compared with aged quartz. To the best of our knowledge, Carrieri et al.21 reported the only other research for the characterisation of crystalline silica in the respirable fraction of dust generated by machining engineered stones (n = 3). We have, in this study, examined the characteristics of twelve engineered stones and three natural stones, with the aim of value adding to current literature about the properties of engineered stones which could shed light on their unique hazard potential.

In alignment with previous reports of the composition of respirable dust from machined artificial/engineered stones, the current engineered stone emissions consisted generally of > 80% by weight crystalline silica and 8–20% resin21. Further characterisation of the RCS was undertaken on the basis that the crystalline structure of the minerals may exert an influence on their toxicity22. In our study, 9 out of 12 engineered stone respirable dust samples had a combination of quartz and cristobalite structures, although quartz was still the dominant structure, forming > 55% of the total mineralogy. Cristobalite was the second most common mineral, while albite and rutile were detected in smaller amounts. Quartz and cristobalite differ from one another in their mineralogy, surface characteristics and natural association with other elements23. Early studies comparing the dose response of quartz and cristobalite on pulmonary function in rats showed that both structures were similarly detrimental to the lungs, although cristobalite elicited a slightly faster response than quartz24. However, subsequent animal experiments and epidemiological studies discounted these findings, by showing no evidence for differences in the inflammatory and fibrogenic potentials of quartz and cristobalite23. Horwell et al.4 even showed that cristobalite-rich volcanic ash was less toxic than expected and posed less of a respiratory health hazard than quartz. They attributed this finding to the relative open structure of cristobalite compared to quartz, which allows the substitution of cations such as aluminium (Al3+) and sodium (Na+) in the Si tetrahedral, hence affecting cristobalite toxicity1,4. Taken together, these studies show insufficient evidence that either mineral is more toxic than the other. Nonetheless, the high concentration of crystalline silica in the respirable dust from engineered stones may be cause for concern as quartz and cristobalite are the only crystalline silica minerals recognised as Group 1 carcinogens—“carcinogenic to humans”—by the International Agency for Research on Cancer25.

Apart from the mineral structure and content of engineered stone dust, the resin and metal/elemental composition were other properties that indicated their variability. The resin content of the respirable emissions for the engineered stones ranged from 8 to 20%, which was similar to the 5–15% polyester resin identified in artificial stone characterised by Carrieri et al.21. It has been suggested that resin can influence the reactive pathways of RCS, hence toxicity, in the lungs by acting as a ‘protective’ coating for the particles9. In their earlier studies, Pavan et al.26 subjected RCS particles to a thermal treatment to remove the polymeric resin and reported a significant increase in cytotoxicity, suggesting that resin partially covers the particle surface from interaction with cellular membranes.

There is a large body of research reporting short-term exposure to atmospheric particulate matter of < 2500 nm and associated adverse health effects especially on the respiratory system27. Dry-cutting engineered stones generated very fine particles of < 1200 nm, which have the capacity to penetrate deep into lung tissues28. The mean particle sizes of engineered stone samples, which ranged from 416 to 644 nm, were smaller than those recorded by Pavan et al.9, who measured mean diameters of dry-particles ranging from 1700 to 1900 nm, that is more than three times the diameter measured in our study. This disparity in particle size may be due to differences in the means of generation and collection of the dust: Pavan et al.9 collected deposited dusts in situ in engineered stone workspaces. In comparison, we actively collected the respirable fraction of dust generated from machined engineered stone. In another study by Carrieri et al.21 the respirable dust size analysis of an engineered stone sample (> 85% quartz) had a bimodal distribution, with one mode in the same range as in this study (~ 500 nm), whereas the other was in the ultrafine particle (UFP) range, commonly defined as particles < 100 nm29. Although visually observed, UFPs were not measured in the present study, likely due to the limitations of the air sampling or particle size analysis techniques. We are currently exploring some real-time measurement of UFP using direct reading instrumentation for more precise dust exposure assessment during engineered stone fabrication tasks.

It is important to hightlight the limitations in particle size analysis of the dust particles by the DLS technique. Whilst it is a widely accepted technique for determining particle size distribution, it may not be suitable for understanding composite samples generated during machining materials such as engineered stone. The technique relies on the assumption that all particles are homogenous and spherical in shape30. However, as evidenced by a high polydispersity index (> 0.7) and particle imaging by SEM, the dust particles in our study were, in fact, heterogenous in shape, size and structure. Apart from particle size and morphology, the surface properties of quartz have been reported to also play an important role in cytotoxicity, suggesting that the specific surface area of engineered stones may be a useful parameter for characterisation and differentiation between engineered and natural stones26,31,32.

Notwithstanding similar particle size distributions of the engineered and natural stone samples, we found a clear difference in specific surface area, which was greater in engineered stone. Microscopy imaging suggested that this difference was likely due to rough surfaces, cavities and conchoidal fractures at the surface of engineered stone particles; in comparison, natural stone dust particles exhibited smoother surfaces with clear natural layers. Within the engineered stone emissions, surface area varied considerably, from 1.43 to 2.72 m2/g (average 2.06 ± 0.13 m2/g), suggesting heterogeneity in the shape and size of the dust particles. The surface area of the engineered stone dust was generally higher than that reported Pavan et al.31 (~ 1.0 m2/g), which was expected as they also reported average greater sized dust particles in their study (~ 2000 nm). Increased surface area, associated with small particle size, for example UFPs, tend to be more toxic and cause more stress to alveolar macrophages than bigger particles33.

While high specific surface area of particles may itself confer toxic properties, it is possible that a high degree of specific reactivity of silica particles is also a contributing factor to its adverse health effects; in other words, small silica particles may have greater toxic effects than non-silica containing particles of comparable size. This is because quartz, particularly when fractured, has a very reactive surface as a result of the presence of chemically-active silanol groups (Si–OH)2)34. Early studies proposed that silanols may damage macrophages and induce inflammation but more recently, Pavan et al.35 identified a specific subfamily of silanols which are critical determinants of silica toxicity. They showed that the occurrence of specific patterns of silanols on the surface of quartz, the “nearly free silanols” (NFS), promote membranolysis and induce inflammation in rat lung cells. Their research challenged the original paradigm that crystallinity is key to silica toxicity34,35.

The surface reactivity of quartz, through its influence on surface charge, can be measured as the zeta potential in a specific medium and pH22. The zeta potential of the engineered stone dust emissions ranged from −25.7 ± 0.86 to −33.8 ± 1.10 mV and varied significantly among each other, highlighting the variability of fractured engineered stone surfaces. The zeta potential of two of the three natural stones was significantly lower than engineered stones. The exception was black granite, in which the zeta potential fell within the range of the engineered stone, albeit at the lower part of the range: this may be due to the greater silica content of the granite (30%) compared with the other natural stone samples (3.5% and 11%). An alternative explanation for the anomalous finding with black granite is its relatively high content of the metal elements Fe (6.6%) and Al (8.4%), compared with both the engineered stone and the other natural stone samples. Pavan et al.22 found that silica samples contaminated with metals reduced the level of negative zeta potential at certain pH levels; the authors stated that the zeta potential reflects the protonation state of the silanols on the surface of the silica crystals, which in turn determine the propensity to disrupt cell membranes. In this way zeta potential could therefore be used to predict the toxicity of a particulate22,31.

Cristobalite reportedly has a higher (more negative) zeta potential than quartz, especially at high pH, potentially due to its larger surface area from its tetragonal crystral structure, compared to quartz, which is trigonal36. In this set of samples, no correlation betweeen cristobalite content and zeta potential of engineered stone dust was observed (r < 0.5). Similarly, no correlations amongst any of the measured parameters were observed. It is likely that a bigger sample set of engineered stones, with more variable composition, is required to investigate the relationships among characteristics. Such information is essential when attempting to link engineered stone characteristics to pathogenesis.

Previous studies have reported engineered stone containing transition metals (e.g. Cu, Fe, Ti) as redox active species, as part of their pigments9,15. Our results build upon and are in good agreement with those reported by Pavan et al.9, whereby engineered stone mostly contained natural alkaline metals such as Ca, Mg, K and Al and elements such as Cu and Ni were in trace amounts. Titanium was the most variable transition metal element in engineered stone dust, ranging from trace (< 0.1%) to major (> 1%) quantities in the samples studied, possibly originating from the pigments and resins37,38. Although generally considered non-toxic, Ti (titanium dioxide, TiO2) has been shown to be an aetiological agent for lung inflammation, especially in the ultrafine fraction39,40. The possible role of metals in the toxicity of silica has been elicited before. For example, Clouter et al.41 (and references therein) suggested that the toxicity of quartz involves Fe. While the presence of Fe and Al has been considered for the potential reason for the differing zeta potentials of black granite and other natural stones, this could not explain the greater negative zeta potential of engineered stone compared to the black granite, since the concentration of Fe and Al is much lower in engineered stone. Several other elements not found in the natural stone samples were detected in the engineered stone ones, but only in trace quantities. Therefore, while we cannot exclude any role of metal ions in silica toxicity, it is unlikely that any such effect is mediated though the pathway linked with the generation of zeta potential.

This work has demonstrated that the dust emissions from machined engineered stones contain a high concentration of very fine particles that contain predominantly quartz and cristobalite, and therefore have the potential for a detrimental impact on respiratory health outcomes. This study showed the extent to which actively-generated respirable dust from engineered stones vary in their chemical properties, including crystalline silica, surface charge and resin content. In the present research, no correlation amongst the physicochemical properties of the engineered stone dust emissions was observed; further characterisation, in combination with clinical assays, could be useful to identify peculiar physical and chemical properties as biomarkers of pathogenesis. We also demonstrated that respirable dust emissions from natural stones were significantly different from engineered stone dust, with much lower silica content, smaller surface areas and generally lower surface charge. The results of this study shed light on the unique hazard posed by engineered stone fabrication work, and will help guide the development of specific engineering control measures targeting lower exposure to RCS.

## Materials and methods

### Samples

Twelve commercially-available engineered stones (ES1-ES12) were assessed in this study. These represented five brands/suppliers and were selected on the basis of their popularity amongst consumers (product sales volume nationally). Three natural stones, namely black granite, white granite and white marble, and a high purity reference quartz, SRM 1878b (National Institute of Standards and Technology, NIST, Maryland, US) were sourced for comparison.

### Sample generation and collection

The work practice of dry-cutting engineered and natural stones was simulated within a custom-made enclosed perspex cabinet (60 × 80 × 80 cm). The cabinet was fitted with two glove compartments and an air seal and the stones were held in place using clamps, in a similar way as described in Carrieri et al.21. An angle grinder (Metabo 720 W), fitted with a 105 mm diamond blade, was manually operated at ~ 10,000 rpm (Digital Tachometer QM1448, New England Instrument Co., NEIC) to make 3 mm wide zip cuts through the stone. A high-efficiency industrial vacuum (Metabo ASR 35 L ACP 1400 W) with flow rate 3660 l/min was connected to the cabinet to exhaust dust emissions.

Respirable dust was collected using either a Higgins-Dewell respirable cyclone (Casella Solutions, Maryland, USA) with 2.2 L/min flow rate or a Parallel Particle Impactors (PPI) respirable sampler (No.225–383, SKC Inc., Eighty Four, PA, USA) with flow rate 8.0 L/min. The respirable dust was collected on pre-weighed 25 mm and 37 mm PVC membrane filters (GLA-5000, SKC Inc., USA). Particle size and zeta potential analyses were carried out on freshly-generated dust (within 2 h of dust generation) to minimise risks of sample aggregation with time. For remaining characterisation assays all dust samples were stored in cool, dry conditions pending further assays.

### Sample characterisation

#### Mineral composition

The forms of crystalline silica in the engineered and the natural stones were determined by an X-ray Diffraction (XRD) technique conducted using a Bruker D8 Advance Powder X-ray Diffractometer (Bruker AXS Inc., Madison, Wisconsin, USA) with a Cu-radiation source operating at 40 kV and 40A, scanning 2 theta from 278 to 338 K with sample rotation of 30 rotations/min.

The respirable dust sample was transferred from the filter onto silicon wafers in a fine dusting over the centre of the wafer. Data was processed using Bruker DIFFRAC.EVA software and Crystallography Open Database reference patterns for identifying mineral phases. Quantification was calculated using TOPAS 4.2 software (Bruker AXS Inc., Madison, Wisconsin, USA).

#### Resin content

Proximate analysis of each stone was performed by thermogravimetric analysis (TGA), (Mettler-Toledo, Inc., TGA/DSC 2 STARe System, Columbus, Ohio, USA), using an ultra-microbalance to assess weight change upon pyrolysis. Samples (circa 5 mg) were heated from 30$$^\circ$$C to 1000$$^\circ$$C at a rate of 10$$^\circ$$C/minute under nitrogen atmosphere (flow rate 50 mL/minute).

#### Particle size and zeta potential

Stone dust samples were analysed for particle size and zeta potential on a Zetasizer Nano-ZS (Malvern Instruments Ltd., Worcestershire, UK) by light scattering techniques commonly used for nanoparticle suspensions characterisation42. The samples were suspended in MilliQ water (5 mg/10 mL), sonicated for 10 min at 50 Hz (Ultrasonic Cleaner FX8, Unisonics Pty. Ltd., Sydney, Australia) and diluted two-fold prior to size analysis. The pH of the solutions was measured to be 7.4 on a pH meter (Starter 300, OHAUS, New Jersey, USA). The zeta potential of the same suspensions (5 mg/10 ml) was assessed by the electrophoretic light scattering technique on the Zetasizer at 25$$^\circ$$C. The polydispersity index (PDI) of the dust suspensions was recorded.

The performance of the equipment for particle size and zeta potential analyses was assessed through the standard reference material of colloidal silica (ERM – FD 100, Geel, Belgium). Good agreement (within 10% variance) was observed between the reference and recorded values for colloidal silica. The measurements for particle size and zeta potential of the samples were done in triplicates.

#### Morphology

The morphology of generated dust particles was determined by Scanning Electron Microscopy (SEM) (FEI Helios Nanolab 600, USA), following loading of the sample (circa 1–2 mg) on double-sided adhesive tape and coating with platinum. The SEM images displayed were scaled using the ImageJ software (National Institutes of Health, Maryland, USA).

#### Specific surface area

Due to sampling limitations for respirable dust, specific surface area was determined on the total “settled” dust fraction (which included respirable), that was deposited in the chamber as the stones were machined. The samples were pre-weighed and degassed overnight at room temperature prior to analysis by the Brunauer–Emmett–Teller (BET) method on a surface area and porosity analyser (Micrometrics Tristar II 3020, Norcross, GA, United States), applying nitrogen as the adsorbate gas at −196$$^\circ$$C.

#### Elemental composition

Similar to the surface area assay, the determination of the metal elemental composition of the samples was carried out on the total ‘settled’ dust fraction by X-ray Fluorescence (XRF) at a commercial analytical laboratory, Bureau Veritas. No sample preparation was required; the stone dust samples were cast using a 12:22 lithium borate flux to form a glass bead. The content of Fe, Al2O3, MnO, TiO2, CaO, MgO, K2O, P, S, Na2O, Cu, Ni, Co, Cr, Pb, Zn, As, Sn, Sr, Zr, Ba, V, Cl was determined by XRF Spectrometry. The Si content of the dust samples was determined but not reported as part of the elemental composition.

### Statistical analysis

Any differences in the characteristics within stone samples were assessed for statistical significance at a 95% confidence level (p < 0.05) using Analysis of Variance (ANOVA). Duncan’s post hoc test identified statistical differences among samples. Bivariate correlation analyses measured the degree and strength of correlation, if any, between parameters.


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# Schröder number

In mathematics, the Schröder number ${\displaystyle S_{n},}$ also called a large Schröder number or big Schröder number, describes the number of lattice paths from the southwest corner ${\displaystyle (0,0)}$ of an ${\displaystyle n\times n}$ grid to the northeast corner ${\displaystyle (n,n),}$ using only single steps north, ${\displaystyle (0,1);}$ northeast, ${\displaystyle (1,1);}$ or east, ${\displaystyle (1,0),}$ that do not rise above the SW–NE diagonal.[1]

The first few Schröder numbers are

1, 2, 6, 22, 90, 394, 1806, 8558, ... (sequence A006318 in the OEIS).

where ${\displaystyle S_{0}=1}$ and ${\displaystyle S_{1}=2.}$ They were named after the German mathematician Ernst Schröder.

## Examples

The following figure shows the 6 such paths through a ${\displaystyle 2\times 2}$ grid:

## Related constructions

A Schröder path of length ${\displaystyle n}$ is a lattice path from ${\displaystyle (0,0)}$ to ${\displaystyle (2n,0)}$ with steps northeast, ${\displaystyle (1,1);}$ east, ${\displaystyle (2,0);}$ and southeast, ${\displaystyle (1,-1),}$ that do not go below the ${\displaystyle x}$-axis. The ${\displaystyle n}$th Schröder number is the number of Schröder paths of length ${\displaystyle n}$.[2] The following figure shows the 6 Schröder paths of length 2.

Similarly, the Schröder numbers count the number of ways to divide a rectangle into ${\displaystyle n+1}$ smaller rectangles using ${\displaystyle n}$ cuts through ${\displaystyle n}$ points given inside the rectangle in general position, each cut intersecting one of the points and dividing only a single rectangle in two. This is similar to the process of triangulation, in which a shape is divided into nonoverlapping triangles instead of rectangles. The following figure shows the 6 such dissections of a rectangle into 3 rectangles using two cuts:

Pictured below are the 22 dissections of a rectangle into 4 rectangles using three cuts:

The Schröder number ${\displaystyle S_{n}}$ also counts the separable permutations of length ${\displaystyle n-1.}$

## Related sequences

Schröder numbers are sometimes called large or big Schröder numbers because there is another Schröder sequence: the little Schröder numbers, also known as the Schröder-Hipparchus numbers or the super-Catalan numbers. The connections between these paths can be seen in a few ways:

• Consider the paths from ${\displaystyle (0,0)}$ to ${\displaystyle (n,n)}$ with steps ${\displaystyle (1,1),}$ ${\displaystyle (2,0),}$ and ${\displaystyle (1,-1)}$ that do not rise above the main diagonal. There are two types of paths: those that have movements along the main diagonal and those that do not. The (large) Schröder numbers count both types of paths, and the little Schröder numbers count only the paths that only touch the diagonal but have no movements along it.[3]
• Just as there are (large) Schröder paths, a little Schröder path is a Schröder path that has no horizontal steps on the ${\displaystyle x}$-axis.[4]
• If ${\displaystyle S_{n}}$ is the ${\displaystyle n}$th Schröder number and ${\displaystyle s_{n}}$ is the ${\displaystyle n}$th little Schröder number, then ${\displaystyle S_{n}=2s_{n}}$ for ${\displaystyle n>0}$ ${\displaystyle (S_{0}=s_{0}=1).}$[4]

Schröder paths are similar to Dyck paths but allow the horizontal step instead of just diagonal steps. Another similar path is the type of path that the Motzkin numbers count; the Motzkin paths allow the same diagonal paths but allow only a single horizontal step, (1,0), and count such paths from ${\displaystyle (0,0)}$ to ${\displaystyle (n,0)}$.[5]

There is also a triangular array associated with the Schröder numbers that provides a recurrence relation[6] (though not just with the Schröder numbers). The first few terms are

1, 1, 2, 1, 4, 6, 1, 6, 16, 22, .... (sequence A033877 in the OEIS).

It is easier to see the connection with the Schröder numbers when the sequence is in its triangular form:

k
n
0 1 2 3 4 5 6
0 1
1 1 2
2 1 4 6
3 1 6 16 22
4 1 8 30 68 90
5 1 10 48 146 304 394
6 1 12 70 264 714 1412 1806

Then the Schröder numbers are the diagonal entries, i.e. ${\displaystyle S_{n}=T(n,n)}$ where ${\displaystyle T(n,k)}$ is the entry in row ${\displaystyle n}$ and column ${\displaystyle k}$. The recurrence relation given by this arrangement is

${\displaystyle T(n,k)=T(n,k-1)+T(n-1,k-1)+T(n-1,k)}$

with ${\displaystyle T(1,k)=1}$ and ${\displaystyle T(n,k)=0}$ for ${\displaystyle k>n}$.[6] Another interesting observation to make is that the sum of the ${\displaystyle n}$th row is the ${\displaystyle (n+1)}$st little Schröder number; that is,

${\displaystyle \sum _{k=0}^{n}T(n,k)=s_{n+1}}$.

## Recurrence relation

${\displaystyle S_{n}=3S_{n-1}+\sum _{k=1}^{n-2}S_{k}S_{n-k-1}}$ for ${\displaystyle n\geq 2}$ with ${\displaystyle S_{0}=1}$, ${\displaystyle S_{1}=2}$.[7]

## Generating function

The generating function ${\displaystyle G(x)}$ of ${\displaystyle (S_{n})}$ is

${\displaystyle G(x)={\frac {1-x-{\sqrt {x^{2}-6x+1}}}{2x}}=\sum _{n=0}^{\infty }S_{n}x^{n}}$.[7]

## Uses

One topic of combinatorics is tiling shapes, and one particular instance of this is domino tilings; the question in this instance is, "How many dominoes (that is, ${\displaystyle 1\times 2}$ or ${\displaystyle 2\times 1}$ rectangles) can we arrange on some shape such that none of the dominoes overlap, the entire shape is covered, and none of the dominoes stick out of the shape?" The shape that the Schröder numbers have a connection with is the Aztec diamond. Shown below for reference is an Aztec diamond of order 4 with a possible domino tiling.

It turns out that the determinant of the ${\displaystyle (2n-1)\times (2n-1)}$ Hankel matrix of the Schröder numbers, that is, the square matrix whose ${\displaystyle (i,j)}$th entry is ${\displaystyle S_{i+j-1},}$ is the number of domino tilings of the order ${\displaystyle n}$ Aztec diamond, which is ${\displaystyle 2^{n(n+1)/2}.}$[8] That is,

${\displaystyle {\begin{vmatrix}S_{1}&S_{2}&\cdots &S_{n}\\S_{2}&S_{3}&\cdots &S_{n+1}\\\vdots &\vdots &\ddots &\vdots \\S_{n}&S_{n+1}&\cdots &S_{2n-1}\end{vmatrix}}=2^{n(n+1)/2}.}$

For example:

• ${\displaystyle {\begin{vmatrix}2\end{vmatrix}}=2=2^{1(2)/2}}$
• ${\displaystyle {\begin{vmatrix}2&6\\6&22\end{vmatrix}}=8=2^{2(3)/2}}$
• ${\displaystyle {\begin{vmatrix}2&6&22\\6&22&90\\22&90&394\end{vmatrix}}=64=2^{3(4)/2}}$