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# All Questions

10,142 questions
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23 views

### Maneuverability calculation of a catamaran ship with azimuth thruster?

In a competition about ship prototype performance, my team decided to make a small catamaran ship with azimuth thrusters. This competition was originally about speed contest but due to COVID-19 it was ...
118 views

### What is the maximum force needed to move the roller block from point A to point B and where does that maximum force occur?

Hello everyone. This project is for a very specific application and I am trying to determine the force necessary to get the block from point A to B. Specifically the maximum force necessary and at ...
68 views

### Rotation around an axis outside of the mechanism

I want to design a system of linkages that will allow me to rotate the green element around the blue point/axis without entering the red circle. The green triangle to go fully around the blue point. I ...
20 views

### Pump Terminology: Suction Piping vs Suction Waterway

Karassik and McGuire state in their 2012 book, [1] p. 415, The available energy that can be utilized to get the liquid through the suction piping and suction waterway of the pump into the impeller is ...
40 views

### How do I connect multiple water-filled syringes?

I'm trying to connect 4 separate syringes containing water and we need water-tight tubing to prevent water from leaking out of the system. The water would theoretically need to be pushed out of one ...
19 views

### how to obtain solution of this bessel equation using Frobenius method

Given this equation: $$\frac{1}{r}\frac{d}{dr}\bigg(r \frac{dT}{dr} \bigg) - \bigg(\frac{2h}{bk} \bigg)(T-T_a) = 0$$ I can let $x=r\sqrt{(2h/bk)}$ and $z=T-T_a$ to obtain the bessel equation using ...
41 views

### Purpose of TV Towers at Present [closed]

We know that different communication towers exist around the world(e.g. Tokyo Tower). They are not only used as a TV tower but for radios and touristy purpose too. This is the age of dish antenna, ...
15 views

### What is a suitable machinery for sample preparation?

A large scale steel analysis laboratory requires a machine to remove top layer of imperfections on steel lolly pop samples. Machine to be robust, low cost consumables to be used, safe with sample ...
42 views

### What is the use of secant modulus?

Secant modulus is defined as the slope of a line connecting the origin and a specified point (like yield point) in stress-strain diagrams. Instead of elastic modulus, which is the slope of the tangent ...
37 views

### Can I use loadcell for vibrating force measurement?

Is it possible to use a load cell to measure a force that is vibrating? The idea is to measure that force that is getting into the load cell in a sweep of frequencies and vibration is NOT a noise. ...
27 views

### Fixed end points optimal control problem

Given two points $(t_0,x(t_0)=x^{0})$ and $(t_1,x(t_1)=x^{1})$ in the $(t,x)$ plane, the objective is to find an optimal trajectory $x^{*}(t)$ such that the cost function \label{eq:1} ...
14 views

### Passively Controlled Heat Exchanger

I am currently working on designing a heat exchanger that "passively" controls temperature. Just for a little bit of context: By passive, I want to limit any use of regulating valves, ...
49 views

### What type to mechanical joint to achive Slide then Rotation motion

I have two parts (Green and red fixed) i want to connect them with a joint (Blue) that permit the green one to slide outwards then rotate. Side View (Right part) a simple pivot point doesn't work ...
52 views

### Is a MSC Adams simulation with smaller steps always more reliable?

In a given simulation in MSC Adams, if I use a 1 or 0.1 step increment, I get the (favorable) result I intuitively assumed would occur, but if I use step increments of 0.01 or 0.001 a different (and ...
58 views

### Can be used a lab on a chip device as a nuclear fusion device? [closed]

This is what a lab on a chip is https://en.wikipedia.org/wiki/Lab-on-a-chip I ask if could be used as a nuclear fusion device , that could rise the number of nuclear fusion per atom, knowing that ...
30 views

### Transfer function into block diagram and matrix form

I'm reading a book on Control System Design and running into the same issue I had in Kinematic Design. When things are presented in a format without any numerical examples it just appears as gibberish ...
41 views

### Why is there a difference in the deflection between the top and bottom of this truss?

In this truss scenario, why is the deflection of point 3 less than that of point 8? I'm assuming it has something to do with the shape of the truss. If we extend the shape we will get a circle and the ...
39 views

### How to estimate the tensile strength of inch diameter steel round?

The tensile strength for 1045 TGP Precision Steel Round is said to be 91 ksi. Although, different diameters must be a factor, right? Would a 1" diameter rod be 104.5 ksi or 1045 ksi? Say I want ...
15 views

### When 2 liquid-vapor chambers are connected at different temperatures, how do I determine what the pressure will be?

Hi all. I have 2 chambers initially filled with liquid and vapor (could be water and steam). The one side is then heated until all the fluid evaporates and becomes either fully vapor or superheated. ...
51 views

36 views

### How to plot bending moment diagram from shear force diagram

With conventions that I adopted shown below I drew roughly the shear force and bending moment diagram of a simply supported beam with concentrated load at the middle. Now while drawing I drew both ...
31 views

### Please explain engine thrust and aircraft weight

This is probably really easy for someone... But... My non-engineer head just cant seem to get an answer to this question in terms that are simple to digest. A new Boeing 777X has a max take off weight ...
37 views

### Optimal placement of two support points along a wooden beam

Assuming a beam that is supported at 2 points as per attached image, what would be Lc (length center) and Lo (length overhang) for maximum strength. (or load bearing capability) For actual material ...
49 views

### Is it practical to use tensile cables in a composite beam?

Many composite beams combine material that has a good tensile strength with material that has a good compression strength (like concrete and rebar). I was wondering if there has ever been a beam that ...
28 views

### Is molecular dynamics relevant for an undergraduate Mechanical Engineering degree? [closed]

I am completing my final year undergraduate study in Mechanical Engineering. I've had to have a scope change because my last project was not going to work with the pandemic. I've been given a project ...
21 views

### I'm looking for the simplest way to increase rotational speed from 120 to 5,000 rpm for a light duty device

I'm currently designing a device and want to increase speed from 120 to 5,000 rpm. Device is lightweight, hand operated, free spinning, low torque in/out. Picture a cheap fertilizer spreader from Home ...
34 views

### Would it be safe to use a cutting wheel in place of sand paper on a disc sander?

Assuming I had the cutting wheel centered and adhered correctly, on a bench mounted disc sander with a 14" disc. Could I then use that disc sander as a modified grinder? I don't see a problem ...
28 views

### Configuring UART Port to 115200 on Renesas RX130 using Code Generator

Does anyone have insights into configuring UART port on Renesas RX130 microcontroller for a bit rate of 115200? I have successfully configured for a 9600, 19200, and 38400. I can read from the MCU and ...
90 views

### How do AC/DC Adapters (Wall Wart) prevent AC Noise Coupling to DC?

I am wondering if AC noise (EMI) gets coupled to the DC side of the AC/DC adapter, what are the design methods used to isolate AC noise getting coupled to DC output in an AC/DC adapter. What are the ...
37 views

### Low pile capacity during pile driving

When driving piles, the resistance from the soil is constantly measured. What happens when the expected resistance values from the soil are not obtained? For a single pile or for multiple piles? What ...
31 views

### Marine engine firing orders

For a while, I was curious as to what the firing orders for inline 7 and inline 9 engines were. I was able to find them in datasheets like this one, which is a project planning guide for the MAN 35/44 ...
26 views

### Help understanding AC circuit voltage division math in frequency domain

I don't fully understand the section b. How does the math in fhat voltage division equation work out to be 200 ang(-113.13) volts?
17 views

### Arizona Calibrations Constants

I was reviewing this report on Implementing M-E in Arizona and found that the proposed local calibration constants for the fatigue damage model are extremely high. The bottom-up equation is shown ...
35 views

### Estimation of Beam Displacement from Acceleration

Hi Engineering Exchange, I have a beam with pinned-pinned boundary conditions (free to rotate but not free to translate). A legacy computer analysis provided me with the maximum mid-span acceleration ...
36 views

### I need help with my ESP8285. It completely stops working after a few days

I had gotten a few ESP8285s from Adafruit because I wanted to make my own 4 channel WiFi relay board. I had created a PCB for it, I have 4 relays, the ESP8285, a 28V power supply, a 5V regulator and 3....
55 views

### Simple clarification about EV charging

After reading some docs about Electric Vehicles (EV), I learned that there are 4 differents charging modes. For example, let's consider a slow recharge, where EV is connected to a single-phase ...
25 views

### How to constraint with objects out of sketch plane?

I would like to constraint the centers of two rectangles to be exactly across the part central axis (shown in the middle of cylindrical hole) Is it possible? Rectangles should be as wide as circular ...
23 views

### How to get rid of out of context references in Solidworks?

I got some sketches to have questionmarks. I found, that it means "Out-of-context external reference" I don't understand what it means and wish to get rid of it. Repair does nothing, rebuild ...
35 views

### Replace Hydrogen in an air cylinder

Can we use a gas cylinder made of A6061, meant to carry air up to 300 bar, with Hydrogen up to the same pressure? If not, what is the maximum pressure that one can safely use hydrogen with?
37 views

### Arduino DAC to Speaker

I'm working with an Arduino Due's two DAC pins. I want to output to a single 4W 5.7Ohm speaker to test my code, with the intent of eventually outputting to a set of speakers (woofer, mid-range, and ...
60 views

### Looking for dropping servo mechanism idea

My friend and I have an idea to make prototype of vending machine. We need suggestions for making strong dropping mechanism. The idea is: The user orders X item. Let's take a can of cola for example. ...


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# 3 Methods

## 3.1 Protein Isolation and Expression

### 3.1.1 Myosin V expression

Myosin V S1 is expressed (baculovirus system) with the first 792 amino acid residues which includes 1 IQ domain as detailed previously by the Yengo Lab (Gunther et al. 2020). Additionally, expressed myosin V contains the N-terminal tetracysteine motif, C-terminal Myc, and C-terminal FLAG tags (Trivedi et al. 2013; Trivedi et al. 2015, 2020; Gunther et al. 2019). The S217A mutation was introduced (serine to alanine) using QuikChange site-directed mutagenesis (Stratagene), co-expressed with calmodulin, and purified with FLAG affinity chromatography.

### 3.1.2 Skeletal muscle myosin II isolation

Fast skeletal muscle myosin II was isolated from chicken pectoralis muscle (Diemand Farm, Wendell, MA). All procedures during isolation were carried out in a cold room or performed on ice. Isolation was performed as previously described by the Debold lab (Woodward et al. 2020; Unger and Debold 2019; Longyear, Walcott, and Debold 2017) by Mike with a protocol similar to those of Margossian and Lowey (1982) with minor modifications. In short, chicken breast muscle is passed through a meat grinder and rinsed with 0.2M EDTA. 2 mL of Buffer A (Extraction buffer consisting of 0.3M KCl, 0.15M KPi, 20mM EDTA, 5mM MgCl2, 3.3mM ATP, and 5mM DTT at pH 6.7) is added per gram of tissue including 5mL of protease inhibitors. This is mixed for 12 minutes with an overhead stirrer. The reaction is stopped with a 4X dilution into water which is then mixed and filtered. After precipitate settles it is centrifuged at 10,800g for 10 minutes at 4C and the resulting pellet is resuspended with Buffer B (Suspension buffer consisting of 1M KCl, 60mM KPi, 20mM EDTA, and 5mM DTT at pH 6.7) and mixed gently before being left to dialyze overnight. Actomyosin is then precipitated and centrifuged at 41,171g for 1 hour at 4C and the resulting supernatant is diluted 10X with water. Clear supernatant is siphoned off and the rest is centrifuged again at 10,800g for 15 minutes at 4C. Supernatant is poured off and precipitate is resuspended with Buffer D (Resupsension buffer consisting of 3M KCl, 50mM KPi, and 5mM DTT at pH 6.7) before being dialized overnight for a second time against Buffer E (Dialysis buffer consisting of 0.6M KCl, 50mM KPi, 1mM NaN3, and 5mM DTT at pH 7.0). After the dialysis, myosin is clarified with an ultracentrifugation at for 2 hours at 4C, concentration determined, snap frozen with liquid nitrogen, and stored at -80C.

### 3.1.3 Actin isolation and labeling

Acetone powder was prepped from the remainder from the myosin isolation (with the leftovers from the filtered cheesecloth) and actin purification was performed from the resulting acetone powder as described by Pardee and Spudich (1982) with modifications. Briefly, acetone powder was finely ground and mixed with extraction buffer (2mM Tris Base, 0.2mM CaCl2, and 0.005% NaN3 at pH 8.0) and stirred with an effort to minimize creation of bubbles. The resulting solution is spun at 28960g for 20 minutes and supernatant filtered off and kept aside. Additional extraction buffer added to gel-like precipitate and centrifuged a second time with the same specs with the supernatant filtered off and combined with the previous. Actin is polymerized from the resulting supernatant by addition a final polymerization solution (50mM KCl, 2mM MgCl2, and 1mM ATP). Salt is added to slowly to prevent “salt shocking” the proteins before being left to stir overnight. The next day, a high salt wash (increase KCl to 600mM) removes tropomyosin from the f-actin and then then the sedimentation of f-actin performed by centrifuging at 205835g for 60 minutes. The precipitate is transferred to a homogenizer and resuspended with extraction buffer. A 4 day dialysis is performed with extraction buffer additionally containing ATP and DTT to de-polymerize actin. After dialysis the resulting G-actin is clarified with an ultracentrifugation at 200,000g for 60 minutes. After actin is polymerized by adding 10mM Imidazole (pH 7.0) and 1mM MgCl2 and dialyzed against final storage buffer (4mM Imidazole, 25mM KCl, 2mM MgCl2, 1mM NaN3, and 0.01mM ATP at pH 7.0). After calculation of final concentration actin is snap frozen in liquid nitrogen and stored at -80C. After isolation actin can then be labeled with 100% TRITC for use in vitro motility or mixed with a 50:50 TRITC/Biotin solution for use in the three-bead laser trap assay.

## 3.2 Laser trap assay

The UFFC data was collected under the same experimental conditions and buffers as described in the section 3.2 for the standard laser trap assay, but full length fast skeletal muscle myosin IIx was adhered to the nitrocellulose coated coverslips instead of the myosin V and anti-body combination. Since UFFC is an extension of the standard three-bead assay implementation was similar in regards to setting up a dumbbell. However, in order to increase the speed that the dumbbell was able to move through solution when the force clamp was engaged smaller 510nm beads (Bangs Lab) were coated with neutravidin (31000, Pierce) for use in the UFFC. Decreasing the radius of the beads reduces the Stoke’s drag coefficient since $$\beta=6 \pi \eta r$$, where $$\beta$$ represents the drag coefficient, $$\eta$$ the viscosity of solution, and r is the radius of the bead. Furthermore, the drag coefficient is inversely proportional to the velocity of the dumbbell setup as given by $$velocity = F_{total}/\beta$$. The smaller the drag the faster the dumbbell can move. Consequently, the increased velocity of the dumbbell setup in the un-attached state increases the time resolution and signal-to-noise ratio of the resulting data. Force was pre-determined for each experimental condition and set by calculating the bead’s displacement from the center of the trap and converted to a force since $$F = -kx$$, where $$x$$ is the bead’s displacement and $$k$$ is the stiffness of the trap. The trap stiffness was similar to the standard laser trap experiments at roughly 0.04 pN/nm calculated by analysis of the power spectrum. Positions of the traps were controlled through the AODs and a custom LabView program. Bead position was collected at 200kHz sampling frequency. The total feedback delay was around 8;micro:s, this is the total time it takes for the computers to detect changes in the beads position relative to the center of the trap, communicate the information, and for the AODs to respond to the changes.

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# Invariant characterization of vector bundles associated to a principal bundle?

I have two related questions.

Suppose I have a principal $G$-bundle $P\xrightarrow{\pi} M$. The usual construction of an associated vector bundle goes as follows. Fix some representation $\rho : G\rightarrow GL(V)$ of $G$ on the vector space $V$. Now define a right action of $G$ on the product manifold $P\times V$ by $(x,v)\mapsto (xg,\rho(g^{-1})\cdot v)$. Let $P\times_G V$ denote the orbits of $P\times V$ under this action. Then $P\times V\xrightarrow{q} P\times_G V$, where $q$ is the quotient map by $G$, is a principal $G$-bundle and $P\times_G V\xrightarrow{\pi_G} M$, where $\pi_G:[(x,v)]\mapsto \pi(x)$, is a vector bundle with typical fiber $V$.

Now suppose I have a principal connection $\Phi$ on $P\xrightarrow{\pi} M$, understood as an equivariant projection onto the vertical bundle of P that acts as the identity on vertical vectors. The (again) "standard" way to induce a linear connection on $P\times_G V\xrightarrow{\pi_G} M$ is to first define a connection $\Phi'=\Phi\times 1_V$ on $P\times V\xrightarrow{\pi\circ pr_1} M$. Then one shows that there's a unique linear connection $\tilde{\Phi}$ on the associated bundle $P\times_G V\xrightarrow{\pi_G} M$ such that $q_*\circ\Phi'=\tilde{\Phi}\circ q_*$.

Both of these constructions make me uneasy because they seem to depend on a choice of global trivialization of the bundle $P\times V\xrightarrow{\pi} P$, even though the results of the constructions appear independent of this choice. My first question is: is there a way to think of associated bundles that doesn't require you to mention a product structure?

What I am imagining is something like the following. Suppose I have a trivial vector bundle $E\xrightarrow{\wp} P$ with typical fiber $V$ over the total space of a principal $G$-bundle $P\xrightarrow{\pi} M$, and suppose I have a free right action of $G$ on $E$ satisfying $\wp(xg)=\wp(x)g$ for any $x\in E$. It certainly follows, with no additional conditions, that there's a unique manifold structure on $E/G$ such that $E\xrightarrow{q} E/G$ is a principal $G$-bundle. But is there a natural sense in which (1) this right action induces a representation $\rho$ of $G$ on $V$ that is unique up to automorphisms of $V$ and (2) $E/G\xrightarrow{\pi_G} M$, where $\pi_G$ is the unique map such that $\pi_G\circ q=\pi\circ \wp$, forms a vector bundle over $M$ isomorphic to the bundle $P\times_G V\xrightarrow{\pi_G} M$ defined above, for the representation $\rho$?

This leads to the second question, which is (largely) independent of the first. Suppose I am given (1) a principal bundle $P\xrightarrow{\pi} M$ with structure group $G$, (2) a trivial vector bundle $E\xrightarrow{\wp} P$ with typical fiber $V$, and (3) a right action of $G$ on $E$, all such that $E\xrightarrow{q} E/G$ is a principal G-bundle, $E/G\xrightarrow{\pi_G} M$ is a vector bundle, and $(\wp,\pi_G)$ is a principal bundle homomorphism from $E\xrightarrow{q} E/G$ to $P\xrightarrow{\pi} M$. (I get all of this from the standard associated bundle construction.) Does a principal connection on $P\xrightarrow{\pi} M$ induce a canonical linear connection on $E/G\xrightarrow{\pi_G} M$ that can be characterized without mentioning a trivialization?

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# Geometry and Topology Seminar

Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.

The Seminar usually takes place on Fridays at 10:40 AM via Zoom (please contact the seminar organizers for the Zoom link), not in Room 617 on the sixth floor of Wachman Hall.

• Friday November 20, 2020 at 10:40, over Zoom
Random hyperbolic geometry

Bram Petri, Sorbonne Université

Abstract: In graph theory, random constructions are a classical subject of study. Besides giving one an idea of what a "typical" graph on a large number of vertices looks like, these constructions have also been used to solve certain extremal problems. That is, sometimes it is easier to prove that a random graph has extremal properties than to explicitly construct graphs with these properties. This last idea is often called the probabilistic method. In this talk, I will speak about how these methods can be applied in (mostly low-dimensional) hyperbolic geometry.

• Friday November 6, 2020 at 10:40, over Zoom
Infinitely many virtual geometric triangulations

David Futer, Temple University

Abstract: Every cusped hyperbolic 3-manifold should admit a decomposition into a union of positively oriented ideal tetrahedra. Somewhat shockingly, the question of whether such a geometric triangulation exists is still open. Luo, Schleimer, and Tillmann proved that geometric ideal triangulations of this sort exist in some cover of every cusped 3-manifold. We extend their result by showing that (almost) every cusped hyperbolic 3-manifold has a single cover admitting an infinite sequence of geometric ideal triangulations. The proof involves double coset separability of peripheral subgroups. This is joint work with Neil Hoffman.

• Friday October 23, 2020 at 10:00, over Zoom
Covers from simple closed curves

Tarik Aougab, Haverford College

Abstract: In joint work with Max Lahn, Marissa Loving, and Sunny Yang Xiao, we showed recently that a regular cover of surfaces is determined by which closed curves on the base surface admit simple elevations to the cover. This theorem was motivated by a question related to pairs of isospectral hyperbolic surfaces arising from Sunada’s construction, but the assumption of regularity prevented the result from being applicable in that context. In joint work with Max, Marissa, and Nick Miller, we are now able to drop this assumption. We’ll discuss the motivation, some of the ideas used in the arguments, and where there is still work to do.

• Friday October 16, 2020 at 10:40, over Zoom
Extensions of word hyperbolic groups that satisfy a quantitative law alternative

Thomas Ng, Technion

Abstract: Given two groups we can look for a new group containing the first group as a normal subgroup so that the quotient is the second group. Such groups are called group extensions, and arise naturally when studying fiber bundles. I will describe work in progress with Robert Kropholler and Rylee Lyman on how hyperbolicity of the normal subgroup can be leveraged to show that the extension inherits a strengthened quantitative Tits alternative from the quotient group. Along the way we will show that any nonelementary subgroup of the fundamental group of a fibered 3-manifold whose fiber has negative Euler characteristic contains a 6-short free subgroup.

• Friday October 9, 2020 at 10:40, over Zoom
Geometric and group theoretic perspectives on curve intersection

Edgar Bering, Technion

Abstract: Given two simple closed curves a and b on a surface, their geometric intersection number is the minimum number of intersection points over all pairs of curves homotopic to (a, b). This number can be characterized combinatorially, by describing the complementary regions of the two curves, or with the help of hyperbolic geometry. In this talk I will introduce a different point of view, characterizing both simple curves and their intersections group-theoretically. I will then discuss the generalization of the group theoretic point of view to settings far removed from surfaces.

• Friday October 2, 2020 at 10:40, over Zoom
Totally geodesic surfaces in twist knot complements II

Rebekah Palmer, Temple University

Abstract: The study of surfaces has been essential in studying the geometry and topology of the 3-manifolds that contain them. In particular, there has been considerable work in understanding the existence of totally geodesic surfaces in hyperbolic 3-manifolds. Most recently, Bader, Fisher, Miller, and Stover showed that having infinitely many maximal totally geodesic surfaces implies that the 3-manifold is arithmetic. In this talk, we will discuss an infinite family of twist knot complements containing exactly 1 totally geodesic surface, and how to use covering space theory to extend this result to present examples of infinitely many non-commensurable (non-arithmetic) hyperbolic 3-manifolds that contain exactly $k$ totally geodesic surfaces for every positive integer $k$. This is joint work with Khanh Le.

• Friday September 25, 2020 at 10:40, over Zoom
Totally geodesic surfaces in twist knot complements I

Khánh Lê, Temple University

Abstract: The study of surfaces has been essential in studying the geometry and topology of the 3-manifolds that contain them. In particular, there has been considerable work in understanding the existence of totally geodesic surfaces in hyperbolic 3-manifolds. Most recently, Bader, Fisher, Miller, and Stover showed that having infinitely many maximal totally geodesic surfaces implies that the 3-manifold is arithmetic. In a joint work with Rebekah Palmer, we present examples of infinitely many non-commensurable (non-arithmetic) hyperbolic 3-manifolds that contain exactly $k$ totally geodesic surfaces for every positive integer $k$. In this talk, I will start with some background in hyperbolic geometry and discuss some ideas behind the proof of the main results. I will also talk about applications of these ideas in showing that a family of twist knot complements is not right-angled and examples of maximal Fuchsian subgroups of infinite covolume in hyperbolic 3-manifolds.

• Friday September 18, 2020 at 10:40, over Zoom
Graphs of curves and arcs quasi-isometric to big mapping class groups

Anschel Schaffer-Cohen, University of Pennsylvania

Abstract: When is a big mapping class group quasi-isometric to a graph whose vertices are curves or arcs on the underlying surface? I will describe a classification of those surfaces whose mapping class groups admit a quasi-isometry to a graph of curves, and provide a specific example of a big mapping class group quasi-isometric a graph of arcs. As a bonus, I will show that the mapping class group of a plane minus a Cantor set is Gromov-hyperbolic. This talk should be accessible to anyone comfortable with geometric group theory and surface topology.

• Friday September 11, 2020 at 10:40, over Zoom
Non-uniquely ergodic trees in the boundary of Outer space

There exist non-uniquely ergodic arational laminations on a surface of genus $g \geq 2$. That is, there exists an arational lamination which supports two transverse measures that are not scalar multiples of each other. In analogy, one asks if 'arational' trees in the boundary of Outer space support metrics that are not scalar multiples of each other. In this talk, I will first talk about laminations on surfaces. Then we will see some examples of trees in Outer space and understand what it means for a tree to be non-uniquely ergodic. Time permitting, I will describe the construction of a "non-geometric", arational, non-uniquely ergodic $F_n$-tree. This is joint work with Mladen Bestvina and Jing Tao.

• Wednesday March 11, 2020 at 14:30, Wachman 617
Hyperbolic volume and bounded cohomology

James Farre, Yale University

Abstract: A natural notion of complexity for a closed manifold $M$ is the smallest number of top dimensional simplices it takes to triangulate $M$.  Gromov showed that a variant of this notion called simplicial volume gives a lower bound for the volume of $M$ with respect to any (normalized) Riemannian metric.  The heart of his proof factors through the dual notion of bounded cohomology.  I will define bounded cohomology of discrete groups illustrated by some examples coming from computing the volumes of geodesic simplices in hyperbolic space.  Although bounded cohomology is often an unwieldy object evading computation, we give some conditions for volume classes to be non-vanishing in low dimensions.  We then ask, When do higher dimensional volume classes vanish?’’

• Friday February 28, 2020 at 16:00, Haverford College, room Sharpless S113
Right-veering open books and the Upsilon invariant

Diana Hubbard, CUNY

PATCH Seminar, at Haverford College

Abstract: Fibered knots in a three-manifold $Y$ can be thought of as the binding of an open book decomposition for $Y$. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon. I will discuss some applications of this work, including an application to the Slice-Ribbon conjecture.

There will also be a background talk on this topic at 11:00am.

• Friday February 28, 2020 at 14:30, Haverford College, room Sharpless S113
Convergence of normalized Betti numbers in nonpositive curvature

Ian Biringer, Boston College

PATCH Seminar, at Haverford College

Abstract: We’ll show that if $X$ is any symmetric space other than 3-dimensional hyperbolic space and $M$ is any finite volume manifold that is a quotient of $X$, then the normalized Betti numbers of M are “testable", i.e. one can guess their values by sampling the complex at random points. This is joint with Abert-Bergeron-Gelander, and extends some of our older work with Nikolov, Raimbault and Samet. The content of the recent paper involves a random discretization process that converts the "thick part" of $M$ into a simplicial complex, together with an analysis of the "thin parts" of $M$. As a corollary, we can prove that whenever $X$ is a higher rank irreducible symmetric space and $M_i$ is any sequence of finite volume quotients of $X$, the normalized Betti numbers of the $M_i$ converge to the "$L^2$-Betti numbers" of $X$.

There will also be a background talk on this topic at 9:30am.

• Wednesday February 12, 2020 at 14:30, Wachman 617
Smooth Fibrations of the 3-Sphere by Simple Closed Curves

Leandro Lichtenfelz, University of Pennsylvania We show that the moduli space of all smooth fibrations of a 3-sphere by oriented simple closed curves has the homotopy type of a disjoint union of a pair of 2-spheres, which coincides with the homotopy type of the finite-dimensional subspace of Hopf fibrations. In the course of the proof, we present a pair of entangled fiber bundles in which the diffeomorphism group of the 3-sphere is the total space of the first bundle, whose fiber is the total space of the second bundle, whose base space is the diffeomorphism group of the 2-sphere. This is joint work with D. DeTurck, H. Gluck, M. Merling and J. Yang.

• Wednesday February 5, 2020 at 14:30, Wachman 617
An Analog to the Curve Complex for FC Type Artin groups

Rose Morris-Wright Brandeis University

Artin groups are a generalization of braid groups that provide a rich field of examples and counter-examples for many algebraic, geometric, and topological properties. Any given Artin group contains many subgroups isomorphic to other Artin groups, creating a hierarchical structure similar to that of mapping class groups. I generalize and unify the work of Kim and Koberda on right angled Artin groups and the work of Cumplido, Gonzales-Meneses, Gebhardt, and Wiest on finite type Artin groups, to construct a simplicial complex in analogy to the curve complex. I will define this complex, and discuss some properties that this complex shares with the curve complex of a mapping class group.

• Wednesday January 29, 2020 at 14:30, Wachman 617
From Hierarchical to Relative Hyperbolicity


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# Enhancing screenshots with PIL

When I wrote a post last week about using Flickr as a CDN, there was a paragraph that bothered me:

One thing did change when I started using Flickr to host screenshots: I had to change my screenshot format from PNG to JPEG. Flickr accepts PNGs, but it saves the resized images as JPEGs, and any transparent areas in the original PNG become black in the resized JPEGs. Since many of my screenshots are of windows with the semi-transparent shadow, the resized versions on Flickr look like shit. With JPEG screenshots, the window shadow backgrounds are white, which is just what I want.

While a white background to the window shadow is generally OK when I include a screenshot here, it isn’t always “just what I want.” For example, when I want to put a window screenshot in an update to a post, the white background of the screenshot looks stupid against the light green of the update <div>. And when I need to mix screenshots of multiple windows with those of single windows, there’s a clash of styles because my multiple window screenshots usually have a bit of my Desktop color (Solid Aqua Dark Blue) showing.1

Unfortunately, the screencapture command, which my snapflickr program calls, doesn’t have an option to layer in a background color. So I needed to do the layering myself. Since snapflickr is written in Python, the natural way to add the background is through the Python Imaging Library (PIL).

I was a little apprehensive. I’ve used PIL several times in the past and never had any real trouble installing it, but I hadn’t installed it under Lion, and in particular I hadn’t installed it under Lion after Xcode moved from /Developer to /Applications. Gabe Weatherhead (Macdrifter) has tweeted recently about the problems he had getting PIL installed, and I was not inclined to follow in his footsteps (he eventually got it installed via Macports, a system I never want to see again). The comments on this post suggest that Apple’s movement of Xcode, combined with the fact that it no longer installs the command line tools with Xcode by default, has turned the installation of PIL into an almost superhuman effort.

But I really wanted that background color in my screenshots, and I didn’t want to do it through AppleScripting Acorn if I could help it. So I started by installing the Command Line Tools through Xcode’s Preferences:

Then I crossed my fingers and typed

sudo easy_install pil

It worked on the first try! Oh, there were compilation warnings aplenty, most of which had to do with casting 32-bit values as 64-bit (or something like that, I didn’t save them), but when I was done, I had a PIL that was usable. Apparently, PIL just likes me better than it likes Gabe.

The right library makes all the difference in the world. Adding a background to my screenshots was very easy with PIL. Here are the lines I had to add to snapflickr:

 95: # Add a desktop background if it's a shadowed window screenshot.
96: snap = Image.open(fn)
97: if snap.mode == 'RGBA':
98:   # Crop it to a more reasonable size,
99:   # and paste it over a solid-colored background.
100:   cropbox = (margin[0],
101:              margin[1],
102:              snap.size[0]-margin[2],
103:              snap.size[1]-margin[3])
104:   snap = snap.crop(cropbox)
105:   bg = Image.new('RGB', snap.size, bgcolor)
106:   bg.paste(snap, None, snap)
107:   bg.save(fn)

We start on Line 96 by opening the screenshot, which has already been saved to the Desktop as a PNG file and has been given the name stored in the fn variable. Since I want to add the background color only to window screenshots, and window screenshots in PNG format have an alpha channel to handle the shadow transparency, the test on Line 97 will return true only for window screenshots. The remaining lines will be skipped for rectangular area screenshots.

I like the shadow Apple adds to windows, but when you take a screenshot with a shadow, there’s a lot of extra margin beyond the shadow itself. Lines 100-104 crop out the extra using a margin tuple defined earlier in the program as

margin = (25, 5, 25, 35)

These are how many pixels get cropped off the left, top, right, and bottom sides. I chose them after a bit of experimentation because I thought they gave a decent look to the result.

Line 105 creates a solid background image of the same size as the newly-cropped screenshot and filled with the color defined near the top of the program as

bgcolor = (61, 101, 156)

which are the decimal RGB values for Solid Aqua Dark Blue.

Line 106 pastes the screenshot on top of the background. The third argument to paste is the mask. According to the PIL documentation, calling paste with three arguments

updates only the regions indicated by the mask. You can use either “1”, “L” or “RGBA” images (in the latter case, the alpha band is used as mask).

So using snap as both the source and the mask is like using two layers in a program like Photoshop or Acorn. The screenshot is set in a layer in front of the background layer, which is visible through the transparent parts. The image is then flattened into a single layer.

The new image with the background color is then saved, overwriting the original screenshot. Because the bg image is in RGB mode (no alpha channel), so is the saved image. When the image is uploaded to Flickr, the original size will be a PNG and all the other sizes will be JPEGs, but there’ll be no funny black background areas in the resized images because Flickr won’t have to deal with transparency in the original.

It’s entirely possible that I’ll fiddle around some more with the margins, trying different crop amounts to see if I like something better. But I think the basic structure of the code is set.

1. Actually, what’s usually showing in a multi-window screenshot is Backdrop. Although I like to keep a tidy Desktop, there are usually at least a few icons on it. Backdrop is an app that covers up the Desktop—and any app windows it’s stacked in front of—with a nice solid color. Great for screenshots and screencasts.


text:
# HugeCTR Core Features

## Summary of Core Features

In addition to single-node and full-precision training, HugeCTR supports a variety of features that are described in the following topics.

NOTE: Multi-node training and mixed precision training can be used simultaneously.

## Model Parallel Training

HugeCTR natively supports both model parallel and data parallel training, making it possible to train very large models on GPUs. Features and categories of embeddings can be distributed across multiple GPUs and nodes. For example, if you have two nodes with 8xA100 80GB GPUs, you can train models that are as large as 1TB fully on GPU. By using the embedding training cache, you can train even larger models on the same nodes.

To achieve the best performance on different embeddings, use various embedding layer implementations. Each of these implementations target different practical training cases such as:

• LocalizedSlotEmbeddingHash: The features in the same slot (feature field) will be stored in one GPU, which is why it’s referred to as a “localized slot”, and different slots may be stored in different GPUs according to the index number of the slot. LocalizedSlotEmbedding is optimized for instances where each embedding is smaller than the memory size of the GPU. As local reduction for each slot is used in the LocalizedSlotEmbedding with no global reduction between GPUs, the overall data transaction in the LocalizedSlotEmbedding is much less than the DistributedSlotEmbedding.

Note: Make sure that there aren’t any duplicated keys in the input dataset.

• DistributedSlotEmbeddingHash: All the features, which are located in different feature fields / slots, are distributed to different GPUs according to the index number of the feature regardless of the slot index number. That means the features in the same slot may be stored in different GPUs, which is why it’s referred to as a “distributed slot”. Since global reduction is required, the DistributedSlotEmbedding was developed for cases where the embeddings are larger than the memory size of the GPU. DistributedSlotEmbedding has much more memory transactions between GPUs.

Note: Make sure that there aren’t any duplicated keys in the input dataset.

• LocalizedSlotEmbeddingOneHot: A specialized LocalizedSlotEmbedding that requires a one-hot data input. Each feature field must also be indexed from zero. For example, gender: 0,1; 1,2 wouldn’t be considered correctly indexed.

## Multi-Node Training

Multi-node training makes it easy to train an embedding table of arbitrary size. In a multi-node solution, the sparse model, which is referred to as the embedding layer, is distributed across the nodes. Meanwhile, the dense model, such as DNN, is data parallel and contains a copy of the dense model in each GPU (see Fig. 2). With our implementation, HugeCTR leverages NCCL for high speed and scalable inter-node and intra-node communication.

To run with multiple nodes, HugeCTR should be built with OpenMPI. GPUDirect RDMA support is recommended for high performance. For more information, refer to our dcn_2node_8gpu.py file in the samples/dcn directory on GitHub.

## Mixed Precision Training

Mixed precision training is supported to help improve and reduce the memory throughput footprint. In this mode, TensorCores are used to boost performance for matrix multiplication-based layers, such as FullyConnectedLayer and InteractionLayer, on Volta, Turing, and Ampere architectures. For the other layers, including embeddings, the data type is changed to FP16 so that both memory bandwidth and capacity are saved. To enable mixed precision mode, specify the mixed_precision option in the configuration file. When mixed_precision is set, the full FP16 pipeline will be triggered. Loss scaling will be applied to avoid the arithmetic underflow (see Fig. 5). Mixed precision training can be enabled using the configuration file.

Fig. 1: Arithmetic Underflow

## SGD Optimizer and Learning Rate Scheduling

Learning rate scheduling allows users to configure its hyperparameters, which include the following:

• learning_rate: Base learning rate.

• warmup_steps: Number of initial steps used for warm-up.

• decay_start: Specifies when the learning rate decay starts.

• decay_steps: Decay period (in steps).

Fig. 6 illustrates how these hyperparameters interact with the actual learning rate.

Fig. 2: Learning Rate Scheduling

## Embedding Training Cache

Embedding Training Cache (ETC) gives you the ability to train a large model up to terabytes. It’s implemented by loading a subset of an embedding table, which exceeds the aggregated capacity of GPU memory, into the GPU in a coarse-grained, on-demand manner during the training stage.

This feature currently supports both single-node and multi-node training. It supports all embedding types and can be used with Norm, Raw, and Parquet dataset formats. We revised our criteo2hugectr tool to support the key set extraction for the Criteo dataset.

You can view the HugeCTR Continuous Training example notebook to learn how to use this feature with the Criteo dataset. The Criteo dataset is a common use case, but embedding training cache is not limited to this dataset.

## HugeCTR to ONNX Converter

The HugeCTR to Open Neural Network Exchange (ONNX) converter (hugectr2onnx) is a Python package that can convert HugeCTR models to ONNX. It can improve the compatibility of HugeCTR with other deep learning frameworks since ONNX serves as an open-source format for AI models.

After training with our HugeCTR Python APIs, you can get the files for dense models, sparse models, and graph configurations, which are required as inputs when using the hugectr2onnx.converter.convert API. Each HugeCTR layer will correspond to one or several ONNX operators, and the trained model weights will be loaded as initializers in the ONNX graph. You can convert both dense and sparse models or only dense models. For more information, refer to the onnx_converter directory of the HugeCTR repository on GitHub and the hugectr2onnx_demo.ipynb sample notebook.

## Hierarchical Parameter Server

HugeCTR Inference Hierarchical Parameter Server implemented a hierarchical storage mechanism between local SSDs and CPU memory, which breaks the convention that the embedding table must be stored in local CPU memory. Distributed Database layer allows utilizing Redis cluster deployments, to store and retrieve embeddings in/from the RAM memory available in your cluster. The Persistent Database layer links HugeCTR with a persistent database. Each node that has such a persistent storage layer configured retains a separate copy of all embeddings in its locally available non-volatile memory.

Try out our hugectr_wdl_prediction.ipynb Notebook. For more information, refer to Distributed Deployment.

## Sparse Operation Kit

The Sparse Operation Kit (SOK) is a Python package that wraps GPU-accelerated operations that are dedicated for sparse training or inference cases. The package is designed to be compatible with common deep learning frameworks such as TensorFlow.

SOK provides a model-parallelism GPU embedding layer. In sparse training or inference scenarios, such as click-through-rates, there are large number of parameters that do not fit into memory on a single GPU. Common deep learning frameworks do not support model-parallelism (MP). As a result, it is difficult to fully utilize all available GPUs in a cluster to accelerate the whole training process.


text:
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Get Full Access to Atkins' Physical Chemistry - 11 Edition - Chapter 4a - Problem E4a.2(a)
Get Full Access to Atkins' Physical Chemistry - 11 Edition - Chapter 4a - Problem E4a.2(a)

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ISBN: 9780198769866 2042

## Solution for problem E4A.2(a) Chapter 4A

Atkins' Physical Chemistry | 11th Edition

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1
Problem E4A.2(a)

The difference in chemical potential of a particular substance between two regions of a system is $$+7.1 \ \mathrm{kJ} \ \mathrm{mol}^{-1}$$. By how much does the Gibbs energy change when 0.10 mmol of that substance is transferred from one region to the other?

Text Transcription:

+7.1 kJ mol^-1

Step-by-Step Solution:
Step 1 of 3

Intro to Organic Chemistry 1- Week 1 notes 8/21/18 Key Terms Lecture 1 Review from General Chemistry ● Atom ○ Nucleus (center) no volume but mass=1 amu (protons and neutrons) ○ Electrons- electronic cloud=volume; mass=.0005 amu ○ Neutral atoms- #of proton=# of electrons ● Symbol ○ Top number- neutrons ○ Bottom number-protons ○ Different forms of the same element can have different masses. These are called isotopes ● The Four Quantum Numbers ○ 1. Principal Quantum Number (n)- main energy level. n=1, 2, 3… ○ 2. Secondary Quantum Number (l) - type of orbital. Value depends on value of n. l= 0, 1… n-1 ○ 3. Magnetic Quantum Number (m l - # of possible orientationml1...0…+1 In 3D space for each type of orbital ○ 4. Spin Quantum Number (m ) 2sossible values. +½ or -½ ● Carbon is the most important element. Going to be focusing mostly on the nonmetals of the second row ● To make it organic it has to have carbon and hydrogen ● Potential energy is inverse of stability ● Uncertainty Principle- it is impossible to know the electron’s velocity and it’s position simultaneously (the atom is fuzzy) ● Potential Energy increases as the distance from the nucleus increases (less stable) ● Orbital- a region in 3D space where there is a high probability of finding the electron ○ Only 2 electrons can occupy an orbital (they hav

Step 2 of 3

Step 3 of 3

##### ISBN: 9780198769866

Atkins' Physical Chemistry was written by Aimee Notetaker and is associated to the ISBN: 9780198769866. The answer to “?The difference in chemical potential of a particular substance between two regions of a system is $$+7.1 \ \mathrm{kJ} \ \mathrm{mol}^{-1}$$. By how much does the Gibbs energy change when 0.10 mmol of that substance is transferred from one region to the other?Text Transcription:+7.1 kJ mol^-1” is broken down into a number of easy to follow steps, and 46 words. The full step-by-step solution to problem: E4A.2(a) from chapter: 4A was answered by Aimee Notetaker, our top Chemistry solution expert on 04/25/22, 03:45PM. This textbook survival guide was created for the textbook: Atkins' Physical Chemistry, edition: 11. This full solution covers the following key subjects: . This expansive textbook survival guide covers 327 chapters, and 1120 solutions. Since the solution to E4A.2(a) from 4A chapter was answered, more than 209 students have viewed the full step-by-step answer.

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text:
# How can I slash through a formula?

What is the MathJax syntax to produce a big slash through a formula? I've seen it used but can't find an example now.

• It is probably better to ask this in Meta. It isn't about math after all. Jan 29, 2014 at 8:24
• Jan 29, 2014 at 8:24
• $\require{cancel} \cancel{\sum \! \! \! \! \! \! \large \int}$... Jan 29, 2014 at 11:41
• @draks... Here is a better version: $$\require{cancel}\cancel{\color{#01020b}{\rlap{\displaystyle\,\int}{\sum}}}$$
– user93957
Jan 29, 2014 at 12:38
• @Aðøbe touché... Jan 29, 2014 at 12:40
• Haha, very good, clever both of you ;)
– MPW
Jan 29, 2014 at 12:53

It's probably the \cancel command.

\require{cancel}
\dfrac{\cancel{9}}{2t}\cdot\dfrac{t}{\cancel{9}}

will produce: $$\require{cancel} \dfrac{\cancel{9}}{2t}\cdot\dfrac{t}{\cancel{9}}$$ For a slightly better appearance you can use the following code:

\require{cancel}
\dfrac{\color{red}{\cancel{\color{black}{9}}}}{2t}
\cdot
\dfrac{t}{\color{red}{\cancel{\color{black}{9}}}}

which produces: $$\require{cancel} \dfrac{\color{red}{\cancel{\color{black}{9}}}}{2t} \cdot \dfrac{t}{\color{red}{\cancel{\color{black}{9}}}}$$

There is also the \cancelto command:

\require{cancel}
\cancelto{\dfrac{\sqrt{x^2+x+2}}{\sqrt{\beta+\alpha}}}{1}

$$\require{cancel} \cancelto{1}{\dfrac{\sqrt{x^2+x+2}}{\sqrt{\beta+\alpha}}}$$

• I tried \cancel but the parser didn't know it. Now I see from your example that I needed \require{cancel} first! didn't know about that. Thanks! FOLLOWUP: Is this just how you use packages?
– MPW
Jan 29, 2014 at 12:21
• @MPW You're welcome! But I don't understand what you mean by your last question.
– user93957
Jan 29, 2014 at 12:37
• The equivalent of LaTeX \usepackage? Or maybe LaTeX already has a \require command? I'm a little rusty...
– MPW
Jan 29, 2014 at 12:55
• I learn something new everyday on math SE. Jan 29, 2014 at 14:41
• @MPW As far as I know (my LaTeX knowledge is very poor) the \usepackage and \require are equivalent. (I'm maybe wrong).
– user93957
Jan 30, 2014 at 12:33
• @AveMaleficum We all learn something new everyday on Math.SE :-)
– user93957
Jan 30, 2014 at 12:34
• I believe \require is a MathJax command. It fills roughly the same role that \usepackage does in LaTeX. Feb 2, 2014 at 0:50


text:
# American Institute of Mathematical Sciences

• Previous Article
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June 2018, 23(4): 1581-1600. doi: 10.3934/dcdsb.2018062

## Two codimension-two bifurcations of a second-order difference equation from macroeconomics

 School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China

* Corresponding author: sf_deng@sohu.com, sfdeng@vt.edu

Received  May 2017 Revised  August 2017 Published  February 2018

Fund Project: The paper was supported by the National Natural Science Foundation of China (No. 11371314 and No. 11771197), the Guangdong Natural Science Foundation of China (No. 2017A030313030), the High-Level Talent Project of Colleges and Universities in Guangdong Province (No. QBS201501), the Startup Foundation for Doctors of Lingnan Normal University (No. ZL1605), and the Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (Grant No. 2015QYJ06)

In this paper we mainly investigate two codimension-two bifurcations of a second-order difference equation from macroeconomics. Applying the center manifold theorem and the normal form analysis, we firstly give the parameter conditions for the generalized flip bifurcation, and prove that the system does not produce a strong resonance. Then, we compute the normal forms to obtain the parameter conditions for the Neimark-Sacker bifurcation, from which we present the conditions for the Chenciner bifurcation. In order to verify the correctness of our results, we also numerically simulate a half stable invariant circle and two invariant circles, one stable and one unstable, arising from the Chenciner bifurcation.

Citation: Jiyu Zhong, Shengfu Deng. Two codimension-two bifurcations of a second-order difference equation from macroeconomics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1581-1600. doi: 10.3934/dcdsb.2018062
##### References:
 [1] D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University, Cambridge, 1990. [2] J. Carr, Application of Center Manifold Theory, Springer, New York, 1981. doi: 10.1007/978-1-4612-5929-9. [3] S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University, Cambridge, 1994. [4] S. Elaydi, An Introduction to Difference Equations, 3rd edition, Springer, New York, 2005. doi: 10.1007/978-1-4757-9168-6. [5] H. A. El-Morshedy, On the global attractivity and oscillations in a class of second-order difference equations from macroeconomics, J. Differ. Equ. Appl., 17 (2011), 1643-1650. doi: 10.1080/10236191003730548. [6] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors, Springer, New York, 1983. [7] G. Iooss, Bifurcation of Maps and Applications, Mathematical Studies, 36, North Holland, Amsterdam, 1979. [8] C. M. Kent and H. Sedaghat, Global stability and boundedness in $x_{n+1} = cx_n+f(x_n-x_{n-1})$, J. Differ. Equ. Appl., 10 (2004), 1215-1227. doi: 10.1080/10236190410001652829. [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer, New York, 1998. doi: 10.1007/978-1-4757-2421-9. [10] S. Li and W. Zhang, Bifurcations in a second-order difference equation from macroeconomics, J. Differ. Equ. Appl., 14 (2008), 91-104. doi: 10.1080/10236190701483145. [11] J. Liu, Z. Yu and W. Zhang, Invariant curves for a second-order difference equation modelled from macroeconomics, J. Differ. Equ. Appl., 21 (2015), 757-773. doi: 10.1080/10236198.2015.1040008. [12] P. A. Samuelson, Interaction between themultiplier analysis and the principle of acceleration, Rev. Econ. Stat., 21 (1939), 75-78. doi: 10.2307/1927758. [13] H. Sedaghat, A class of nonlinear second-order difference equations from macroeconomics, Nonlinear Anal., 29 (1997), 593-603. doi: 10.1016/S0362-546X(96)00054-5. [14] H. Sedaghat, Regarding the equation $x_{n+1} = cx_n+f(x_n-x_{n-1})$, J. Differ. Equ. Appl., 8 (2002), 667-671. doi: 10.1080/10236190290032525. [15] H. Sedaghat, Global attractivity, oscillations and chaos in a class of nonlinear, second order difference equations, Cubo, 7 (2005), 89-110. [16] I. Sushko, T. Puu and L. Gardini, A Goodwin-type model with cubic investment function, in Business cycle dynamics: models and tools (eds. T. Puu and I. Suchko), Springer, (2006), 299-316. doi: 10.1007/3-540-32168-3_12. [17] W. Wang, Analytic invariant curves of nonlinear second order equation, Acta Mathematica Scientia, 29 (2009), 415-426. doi: 10.1016/S0252-9602(09)60041-2. [18] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, New York, 2003. doi: 10.1007/b97481.

show all references

##### References:
 [1] D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University, Cambridge, 1990. [2] J. Carr, Application of Center Manifold Theory, Springer, New York, 1981. doi: 10.1007/978-1-4612-5929-9. [3] S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University, Cambridge, 1994. [4] S. Elaydi, An Introduction to Difference Equations, 3rd edition, Springer, New York, 2005. doi: 10.1007/978-1-4757-9168-6. [5] H. A. El-Morshedy, On the global attractivity and oscillations in a class of second-order difference equations from macroeconomics, J. Differ. Equ. Appl., 17 (2011), 1643-1650. doi: 10.1080/10236191003730548. [6] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors, Springer, New York, 1983. [7] G. Iooss, Bifurcation of Maps and Applications, Mathematical Studies, 36, North Holland, Amsterdam, 1979. [8] C. M. Kent and H. Sedaghat, Global stability and boundedness in $x_{n+1} = cx_n+f(x_n-x_{n-1})$, J. Differ. Equ. Appl., 10 (2004), 1215-1227. doi: 10.1080/10236190410001652829. [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer, New York, 1998. doi: 10.1007/978-1-4757-2421-9. [10] S. Li and W. Zhang, Bifurcations in a second-order difference equation from macroeconomics, J. Differ. Equ. Appl., 14 (2008), 91-104. doi: 10.1080/10236190701483145. [11] J. Liu, Z. Yu and W. Zhang, Invariant curves for a second-order difference equation modelled from macroeconomics, J. Differ. Equ. Appl., 21 (2015), 757-773. doi: 10.1080/10236198.2015.1040008. [12] P. A. Samuelson, Interaction between themultiplier analysis and the principle of acceleration, Rev. Econ. Stat., 21 (1939), 75-78. doi: 10.2307/1927758. [13] H. Sedaghat, A class of nonlinear second-order difference equations from macroeconomics, Nonlinear Anal., 29 (1997), 593-603. doi: 10.1016/S0362-546X(96)00054-5. [14] H. Sedaghat, Regarding the equation $x_{n+1} = cx_n+f(x_n-x_{n-1})$, J. Differ. Equ. Appl., 8 (2002), 667-671. doi: 10.1080/10236190290032525. [15] H. Sedaghat, Global attractivity, oscillations and chaos in a class of nonlinear, second order difference equations, Cubo, 7 (2005), 89-110. [16] I. Sushko, T. Puu and L. Gardini, A Goodwin-type model with cubic investment function, in Business cycle dynamics: models and tools (eds. T. Puu and I. Suchko), Springer, (2006), 299-316. doi: 10.1007/3-540-32168-3_12. [17] W. Wang, Analytic invariant curves of nonlinear second order equation, Acta Mathematica Scientia, 29 (2009), 415-426. doi: 10.1016/S0252-9602(09)60041-2. [18] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, New York, 2003. doi: 10.1007/b97481.
Bifurcation diagram for $q_5>0$
Bifurcation diagram of system (20)
Bifurcation diagram for $q_5<0$
Bifurcation diagram for $\mathcal{L}>0$
Chenciner bifurcation of system (44) in the case $c_5(0)<0$
Chenciner bifurcation of system (3) in the case $\mathcal{L}<0$
Invariant circles arising from Chenciner bifurcation
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# When should you rest?

• You are in an airport and are worried about missing your flight.

• So you start to run the long distance to the flight gate at a steady speed of 10km/h (you have bags and are not very fit).

• Luckily some stretches of the journey are covered by moving walkways on which you also run at 10km/h.

• However, you don't have the energy to run all the way and must rest for 1 minute in total at some point before you get to the gate.

• When you rest you walk at 5km/h.

To get there fastest, should you rest on a moving walkway or on some part in between them?

(Please give a full explanation in any answer rather than just a simple guess.)

• Please feel free to suggest better tags if possible. – Lembik Nov 20 '14 at 21:22
• “Luckily some stretches of the journey are covered by moving walkways on which you also run at 10km/h.” 10 km/h in addition to however fast the walkways move? – Ry- Nov 20 '14 at 21:26
• For the record, you shouldn't run for the gate. The best thing to do is to notify the staff on your arrival and they can arrange a transport; infinitelegroom.com/gallery/… – Richard Nov 20 '14 at 22:09
• I freaked out for a minute here; thought you said I run at 10 km/s – Justin Nov 21 '14 at 2:37
• Lateral thinking answer: rest just before the gate, where there is probably a line and you are on sight of the personnel :) – clabacchio Nov 21 '14 at 13:20

$D=$distance to gate

$W=$distance on walkways

$S=$Speed of walkways

$X=$distance travelled while standing

You time to get there assuming you stop off walkway is $$T_r=\frac{D-W-X}{10}+\frac{W}{10+S}+\frac{X}{5}$$ $$X=1/12$$

If you stop on walkway: $$T_w=\frac{D-W}{10}+\frac{W-X}{10+S}+\frac{X}{S+5}$$ $$\frac{X}{S+5}=1/60$$ $$X=S/60+1/12$$ $$T_w=\frac{D-W}{10}+\frac{W-S/60-1/12}{10+S}+1/60$$ $$T_w=\frac{D-W}{10}+\frac{W}{10+S}-\frac{S+5}{600+60S}+1/60$$

$$Tr-Tw=\frac{-1}{120}+\frac{S+5}{600+60S}>0$$

Please note that if $S=0$ then the result is zero.

You should stop on the walkway because you will let the walkway do work to move you during that time.

• Why it works, you spend more time on the walkway where you always move faster. – kaine Nov 20 '14 at 21:54
• You killed the other answer :) – Lembik Nov 20 '14 at 21:57
• Both of them apparently. I wasn't sure I should answer as I wasn't sure it was tricky enough to could as a puzzle... apparently it is... – kaine Nov 20 '14 at 21:59
• +1 for laying out your variables first. I wish more mathematicians would do that instead of expecting people to guess what the letters stand for or wade through text to find out. – Pharap Nov 21 '14 at 1:25
• I think you have the inequality the wrong way round at the end. – Lembik Nov 21 '14 at 21:49

If the distance to the destination is $D$ km and the moving walkway moves at speed $v$ km/h and carries you distance $D_w$ km then:

• If you run all the time on the moving walkway it will take you $\frac{D_w}{v+10}$ hours and if you rest off the moving moving walkway you will do $\frac{1}{60}$ hours resting and $\frac{D - D_w - \frac{5}{60}}{10}$ hours running; and
• If you run all the time off the moving walkway then it will take you $\frac{D-D_w}{10}$ hours and $\frac{D_w - \frac{v+5}{60}}{v+10}$ hours running on the walkway and $\frac{1}{60}$ hours resting on the walkway.

Both options have $\frac{1}{60}$, $\frac{D-D_w}{10}$ and $\frac{D_w}{v+10}$ terms which can be cancelled out so the comparison is between:

• $-\frac{\frac{v+5}{60}}{v+10}=-\frac{1}{60}+\frac{1}{12(v+10)}$ hours if you rest on the walkway; and
• $-\frac{5}{600} = -\frac{1}{120}$ hours if you rest off the walkway.

These two times can only be equal if $v=0$ and if $v>0$ then $-\frac{1}{60}+\frac{1}{12(v+10)} < -\frac{1}{120}$.

Therefore, less time is spent overall if you rest on the moving walkway.

You have to travel distance $d$. At some point, you will stop and rest for one minute, spending time $t$ minutes on a walkway. (If you stopped off of a walkway, $t = 0$. Otherwise, $0 < t \leq 1$.)

During your rest time, you will move $(w + r)t + r(1 - t) = wt + rt + r - rt = wt + r$, where $w$ is the speed of the walkway and $r$ is your running speed.

If you have $W$ metres of walkway out of the distance $d$, without resting, you would cover the distance in $\dfrac{W}{w + r} + \dfrac{d - W}{r}$ minutes. With your rest time, you have $(w + r)t$ less moving walkway, so your non-rest time becomes $\dfrac{W - (w + r)t}{w + r} + \dfrac{d - W}{r} = \dfrac{W}{w + r} + \dfrac{d - W}{r} - t$.

$$\dfrac{W}{w + r} + \dfrac{d - W}{r} - t + 1$$

. Minimizing this means making $t$ as high as possible – 1 – meaning you should spend all your rest time on a moving walkway.

Probably.

• This is the answer. Probably. – Nit Nov 21 '14 at 12:07

No need for any complex calculation!

Let $r$ be your run speed, $w$ your walking speed, and $m$ be the moving walkway speed. If you rest for some time $T$, then you cover the distance $(w+m) \, T$ if you're on the walkway and $w\,T$ otherwise, whereas the distances covered would be $(r+m)T$ and $r\,T$ if running. Let's compare the distance covered by walking for a time $T$ on the walkway and spending the same amount of time running on the unmoving ground, and the distance covered if running on the unmoving ground and walking on the walkway. (I assume that the rest time isn't long enough to cover the whole unmoving ground or the whole walkway.)

• Run on the walkway, walk on unmoving ground: $(r+m) T + w T$
• Walk on the walkway, run on unmoving ground: $(w+m) T + r T$

So whether you choose to rest on the walkway or off it, there is a time period $2T$ (composed of two intervals which may not be consecutive) during which you'll cover the distance $(r+w+m) T$.

Now, how long will it take to cover the rest of the distance? If you run on the walkway, then you'll have less walkway to cover during the time not already accounted for, therefore it will take longer to cover the distance. Hence it's better to run on the walkway.

Illustrated graphically — === is the walkway, --- is unmoving ground, +++ means running, ... means walking, and ←————→ is one of two intervals of time $T$:

• Walk on the walkway:

=============================---------------------------------
++++++++++++++++++++.........+++++++++++++++++++++++++++++++++
←———————→←————→

• Walk on the unmoving ground:

=============================---------------------------------
+++++++++++++++++++++++++++++...++++++++++++++++++++++++++++++
←——————————→←—→

In the second case, more unmoving ground is covered outside the period of time of duration $2T$, therefore the second case requires more time.

Just observe the behavior of crowds, and you get to the right answer: people mostly stand still and rest on escalators, they tend not to do so on staircases. There is a good reason for this.

Key is to realize you have to cover a given distance in a minimum amount of time (rather than maximize the distance covered in a given amount of time). So you have to minimize your slowness (time spend per unit distance), rather than maximize your speed (distance covered per unit time).

For the specific problem posted here: resting in-between moving walkways increases your slowness from $1/10$ to $1/5$ h/km, a net increase of $1/10$ h/km. Denoting the speed of the moving walkways in km/h by $v$, it follows that resting on the walkways increases your slowness from $1/(v+10)$ to $1/(v+5)$ h/km. For any positive $v$ this increase is less than $1/10$ h/km.

Here is another very simple argument. Suppose you are currently 1 meter away from the next moving walkway. Consider two possibilities:

• You may move the 1 meter before resting and move along with the walkway.

• You rest right away, and afterwards move the 1 meter to the walkway.

Both possibilities cost the same time (the time of resting plus the time of walking one meter). Clearly, the first possibility brings you farther ahead.

Hence without any calculation: You should rest on the moving walkway.

• note that the problem statement implies "resting" to mean "walking a full minute at reduced speed". – Johannes Jan 26 '15 at 11:49

Not sure if this is right after seeing the other answers, but I think it doesn't matter if you rest on the moving walkway or not (maybe someone can correct me?)

EDIT: Apparently it does matter when you choose to walk. It seems walking on the Moving Walkway is the better option.

In my previous example I was only calculating with time and not distances. Using my previous reasoning and adding distance this time, my calculations are as followed:

I'll assume the moving walkway moves at 10km/h (same as run speed). This leaves us with 4 speeds:

• Run without MW: 166.7 M/Min
• Walk without MW: 83.3 M/Min
• Run with MW: 333.3 M/Min
• Walk with MW: 250 M/Min

(MW = moving walkway)(numbers have been rounded)

Let's also assume we have to walk 1km, of which the last half is a Moving Walkway (so 500m normal walkway, 500m moving walkway).

If we choose to walk on the normal walkway the whole trip will take us:

• 1 minute walking
• 2.5 minutes running
• 1.5 minutes running (on the moving walkway)

Total: 5 minutes

If we choose to walk on the moving walkway the whole trip will take us:

• 3 minutes running
• 1 minute walking (on the moving walkway)
• 0.75 minutes running (on the moving walkway)

Total: 4.75 minutes

This difference can be explained by looking at the difference in speeds in ratio. The difference in walking/running on the normal walkway is 1:2, which means you are 50% slower when walking. On the moving walkway the ratio is 3:4, so you're only 25% slower.

The amount of time you save depends on the speed of the moving walkway. The greater the speed, the more time you save if you rest on it.

I could try to add a calculation here but I'd probably end up copying the other answers :)

My previous (faulty) reasoning:

We have 4 different speeds (I changed to meters/minute for easy calculation):

• Run without MW: 166.6667 M/Min
• Walk without MW: 83.3333 M/Min
• Run with MW: 333.3333 M/Min
• Walk with MW: 250 M/Min

(MW = moving walkway)

Only two minutes of time are relevant (all other time will be spent running):

• One minute without the moving walkway
• One minute with the moving walkway

This gives us two options

A. Run the first minute, walk the second (on the MW)
B. Walk the first minute, run the second (on the MW)

Option A gives us 166.6667 + 250 = 416.6667 meters
Option B gives us 83.3333 + 333.3333 = 416.6666 meters
(I think we can neglect the minor difference, which is probably caused by rounding)

A different way to look at it, is that the moving walkway will always move you 166.6667 meters per minute extra, regardless if you walk or not. If you split this value from the rest in the calculation, it becomes more obvious:
Option A gives us 166.6667 + (166.6667 + 83.3333)
Option B gives us (166.6667 + 166.6667) + 83.3333

• your flaw is in "Only two minutes of time are relevant (all other time will be spent running)" since if it were to make a difference, one of the two approaches would have less overall minutes. The distance is the same in both cases but you can't assume the time is the same and then use that to prove the time is the same. Also, you have not been told the speed of the walkway. – Kate Gregory Nov 21 '14 at 12:52
• Ah you're right. I misread the part of the Moving Walkway speed (it's run speed apparently). You're also right about the distance I didn't take into account. I'll extend my answer in a few minutes. Thanks for pointing it out :) – Deruijter Nov 21 '14 at 12:58

Lets say we have to rest 1 hour instead of 1 min for easier calculation. Speed increase is linear if you run\walk on moving walkways(MV). Distance you will cover on MV: 1)run (10 + 10) * 1 = 20 km 2) rest (10 + 5) * 1 = 15 km Distance you will cover on road: 1) run 10 * 1 = 10 km 2) rest 5 * 1 = 5 km What we see here is that difference covered distance is constant to 5km, so the answer is doesn't matter, take a rest whenever you want.

UPD: Another way of thinking. We can translate to new coordinate system, where all speed is decreased by 5km\h. This means we run 5km\h and MV is also 5km\h and now rest means stop moving. Obviously we get the same answer - doesn't matter.

• I'm afraid not. You're correct that there is a running time that compensates the walking time either way, and the distance covered during the walking time plus compensating time is the same. But while the remaining distance is therefore the same either way, it isn't split in the same way between the walkway and the steady ground. It's better to leave more walkway for the rest of the time. – Gilles 'SO- stop being evil' Nov 22 '14 at 1:38


text:
# Abstract

We introduce a new language representation model called BERT, which stands for Bidirectional Encoder Representations from Transformers. Unlike recent language representation models, BERT is designed to pre-train deep bidirectional representations by jointly conditioning on both left and right context in all layers. As a result, the pre-trained BERT representations can be fine-tuned with just one additional output layer to create state-of-the-art models for a wide range of tasks, such as question answering and language inference, without substantial task-specific architecture modifications.

BERT is conceptually simple and empirically powerful. It obtains new state-of-the-art results on eleven natural language processing tasks, including pushing the GLUE bench-mark to 80.4% (7.6% absolute improvement), MultiNLI accuracy to 86.7% (5.6% absolute improvement) and the SQuAD v1.1 question answering Test F1 to 93.2 (1.5 absolute improvement), outperforming human performance by 2.0.

# BERT

### Input Representation

There are two downsides to such an approach. The first is that we are creating a mismatch between pre-training and fine- tuning, since the [MASK] token is never seen during fine-tuning. To mitigate this, we do not always replace “masked” words with the actual [MASK] token. Instead, the training data generator chooses 15% of tokens at random, e.g., in the sentence my dog is hairy it chooses hairy. It then performs the following procedure:

• Task #2: Next Sentence Prediction
Specifically, when choosing the sentences A and B for each pre-training example, 50% of the time B is the actual next sentence that follows A, and 50% of the time it is a random sentence from the corpus. For example:

### Pre-training Procedure

emmm…However…

Training of $BERT_{BASE}$ was performed on 4 Cloud TPUs in Pod configuration (16 TPU chips total).

Training of $BERT_{LARGE}$ was performed on 16 Cloud TPUs (64 TPU chips total). Each pre-training took 4 days to complete.

### Comparison of BERT and OpenAI GPT

• GPT is trained on the BooksCorpus (800M words); BERT is trained on the BooksCorpus (800M words) and Wikipedia (2,500M words).
• GPT uses a sentence separator ([SEP]) and classifier token ([CLS]) which are only introduced at fine-tuning time; BERT learns [SEP], [CLS] and sentence A/B embeddings during pre-training.
• GPT was trained for 1M steps with a batch size of 32,000 words; BERT was trained for 1M steps with a batch size of 128,000 words.
• GPT used the same learning rate of 5e-5 for all fine-tuning experiments; BERT chooses a task-specific fine-tuning learning rate which performs the best on the development set.

# Experiments

In this section, we present BERT fine-tuning results on 11 NLP tasks.

### GLUE Datasets

• MNLI Multi-Genre Natural Language Inference is a large-scale, crowdsourced entailment classification task. Given a pair of sentences, the goal is to predict whether the second sentence is an entailment, contradiction, or neutral with respect to the first one.
• QQP Quora Question Pairs is a binary classification task where the goal is to determine if two questions asked on Quora are semantically equivalent.
• QNLI Question Natural Language Inference is a version of the Stanford Question Answering Dataset which has been converted to a binary classification task. The positive examples are (question, sentence) pairs which do contain the correct answer, and the negative examples are (question, sentence) from the same paragraph which do not contain the answer.
• SST-2 The Stanford Sentiment Treebank is a binary single-sentence classification task consisting of sentences extracted from movie reviews with human annotations of their sentiment.
• CoLA The Corpus of Linguistic Acceptability is a binary single-sentence classification task, where the goal is to predict whether an English sentence is linguistically “acceptable” or not.
• STS-B The Semantic Textual Similarity Bench-mark is a collection of sentence pairs drawn from news headlines and other sources. They were annotated with a score from 1 to 5 denoting how similar the two sentences are in terms of semantic meaning.
• MRPC Microsoft Research Paraphrase Corpus consists of sentence pairs automatically extracted from online news sources, with human annotations for whether the sentences in the pair are semantically equivalent.
• RTE Recognizing Textual Entailment is a binary entailment task similar to MNLI, but with much less training data.
• WNLI Winograd NLI is a small natural language inference dataset deriving from. The GLUE webpage notes that there are issues with the construction of this dataset, and every trained system that’s been submitted to GLUE has has performed worse than the 65.1 baseline accuracy of predicting the majority class. We therefore exclude this set out of fairness to OpenAI GPT. For our GLUE submission, we always predicted the majority class.

The Standford Question Answering Dataset (SQuAD) is a collection of 100k crowdsourced question/answer pairs. Given a question and a paragraph from Wikipedia containing the answer, the task is to predict the answer text span in the paragraph. For example:

### Named Entity Recognition

To evaluate performance on a token tagging task, we fine-tune BERT on the CoNLL 2003 Named Entity Recognition (NER) dataset. This dataset consists of 200k training words which have been annotated as Person, Organization, Location, Miscellaneous, or Other (non-named entity).

### SWAG

The Situations With Adversarial Generations (SWAG) dataset contains 113k sentence-pair completion examples that evaluate grounded commonsense inference.

# Conclusion

Recent empirical improvements due to transfer learning with language models have demonstrated that rich, unsupervised pre-training is an integral part of many language understanding systems. In particular, these results enable even low-resource tasks to benefit from very deep unidirectional architectures. Our major contribution is further generalizing these findings to deep bidirectional architectures, allowing the same pre-trained model to successfully tackle a broad set of NLP tasks.

While the empirical results are strong, in some cases surpassing human performance, important future work is to investigate the linguistic phenomena that may or may not be captured by BERT.