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# Focus: Why Salt Clusters Form on Basement Walls

Published February 3, 2012  |  Physics 5, 15 (2012)  |  DOI: 10.1103/Physics.5.15

#### Discrete Salt Crystallization at the Surface of a Porous Medium

S. Veran-Tissoires, M. Marcoux, and M. Prat

Published February 3, 2012

Salt deposits on walls and other structures can lead to serious damage, but researchers don’t fully understand how they form. Now a French team reporting in Physical Review Letters has performed experiments and computer simulations of salt crystallization on porous structures and explains why the crystals form in discrete bunches, rather than uniformly coating the surface. The new understanding may help in the conservation of old buildings and art pieces.

“Efflorescence” is the name given to the “flowering” of salt often seen on walls in damp basements and old churches, as well as on new concrete and brick structures. It’s a sign that the porous material has been infiltrated by water seeping up from the ground or condensing from the surrounding air. As the water evaporates, any salt dissolved in the fluid will crystallize. This deposit—which may be sodium chloride or another salt like gypsum—can sometimes lead to structural damage, especially if the crystals grow inside the structure, causing internal stress. The problem is especially acute for art conservators trying to restore frescoes and old paintings that have become hidden under a salty glaze.

To better understand efflorescence, Marc Prat and his colleagues at the University of Toulouse in France investigated the factors that control the spatial organization of the salt crystals. “The practical motivation is to develop models aimed at predicting salt damage,” Prat says. They first designed an experiment that mimics the general conditions of efflorescence. They created a porous material by packing millimeter-sized glass beads into a $2$-centimeter-wide, $5$-centimeters-long Teflon cylinder. The cylinder was suspended vertically, with the bottom portion submerged in a sodium chloride solution. By capillary action, liquid wicked up to the top through the spaces between the beads. As the water evaporated from the top surface, it left salt crystals behind.

As is commonly observed, the salt formed in discrete patches, rather than uniformly over the surface. But the team also noticed that the salt crystallized preferentially on the periphery, near the cylinder wall. To explain this, they performed numerical calculations and found that the evaporation rate is higher near the cylinder wall than at the center. The evaporation rate, in turn, depends on the degree to which the surface is exposed to the circulating air above. The team varied this exposure by adjusting the height of the top surface relative to the top of the cylinder rim. These experiments convinced the researchers that higher evaporation in a certain region causes the fluid to flow up more rapidly from below, which leads to more crystals.

The team ran detailed computer simulations of the experiment, where they included the connection between evaporation and fluid flow. Representing the porous material as a network of randomly-sized pores connected by tubes of varying size, the team found that certain “high-speed” pathways allowed more rapid fluid flow and thus developed crystals more rapidly at their exit points on the surface. But instead of blocking evaporation at the pore, these crystals would boost the rate of evaporation by providing more surface area from which the fluid could evaporate. The increased evaporation would draw up fluid even faster along these high-speed pathways. In response, the flow through neighboring pathways would slow, and the corresponding pores would be “starved” of salt ions. The team says this process explains why salt crystals often form in isolation rather than in a uniform coat.

The team is developing the model into a quantitative theory that can predict where and how fast efflorescence will develop in a given situation. Such a theory could help in protecting artworks, Prat says. For instance, one could expose a valuable fresco to humid air to suppress evaporation, while pumping drier air on adjacent regions to induce efflorescence there, away from the artwork.

The paper “is based on a simple, but clever, analysis that accounts for the experimental observations quite well,” says George Scherer of Princeton University. “Without this theory, one might assume that the distribution of efflorescence represents scattered sources of salt, whereas this paper demonstrates that a uniform concentration of salty solution will still give patchy efflorescence.” He believes this insight, plus the demonstration of the importance of air flow, can help conservators judge how best to deal with efflorescence.

–Michael Schirber

Michael Schirber is a freelance science writer in Lyon, France.


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Browse Questions

# Which of the following statements do not convey the same meaning as, If a triangle is equiangular then it is an obtuse angled triangle.?

(A) A triangle is equiangular implies that it is an obtuse angled triangle. (B) A triangle is equiangular only if it is obtuse angled triangle. (C) If a triangle is not obtuse angled then it is equiangular. (D) For an obtuse angled triangle it is sufficient that it is equiangular.

Ans : (C)
If a triangle is not obtuse angled then it is equiangular.

format: plain


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# What is Aerodynamics?¶

Aerodynamics is the science of moving air and its impact on solid bodies placed in the flow field as an obstacle. Being a sub-field, most of the equations from fluid dynamics apply to aerodynamics as well, including all the governing equations, turbulence, boundary layer theory, and ideal gas assumption.

## History¶

Although wind had been widely used as a tool by humankind (windmills, sailboats) already, the beginning of the scientific study of aerodynamics only dates back to the seventeenth century. The dream of flight and a machine that is “lighter than air” was already present in ancient history. Very early attempts to describe flow quantities appear in the work of Aristotle and Archimedes, however the scientific field “aerodynamics” does not appear in their notes.

The very first aerodynamicist was Sir Isaac Newton, who developed and described the theory of flow resistance, also known as drag. Further famous scientists such as Bernoulli, Euler, Navier, and Stokes gave more precise and mathematically founded description about gas dynamics. The famous Navier-Stokes governing equations of fluid dynamics were born in 1800, however this is also the most difficult model to solve.

The desire for flight was always the main driving force behind the development of the engineering field of aerodynamics. In the early 1800’s, engineers and scientists studied the aerodynamic forces of flight. Terms like “drag”, “lift”, and “thrust” were born and the relationships between them were determined. In 1871, the first wind tunnel was built. Drag theories by engineers like Kirchhoff and Rayleigh were established. Otto Lilienthal was the first person to become successful with glider flights, and after all this research work, the Wright brothers flew for the first time with an engine powered airplane in 1903.

Figure 1: A double decker airplane in the early days of aircrafts

But the study of aerodynamics around airfoils did not stop here. Lanchester, Kutta, Zhukovsky, and Prandtl developed theories to better describe flow circulation around wings. While the First and the Second World Wars boosted the development of aircraft and the science of aerodynamics, the next breakthrough happened in 1947, when the sound barrier was broken for the first time with a Bell X-1 rocket-powered aircraft. At this point, the understanding of subsonic and supersonic flows had matured, however the evolution was still to continue. With the appearance of computers, the numerical calculation of the governing equations became possible. Computational Fluid Dynamics models and software started to appear on the horizon. Nowadays, CFD is utilized in almost every development process where airflow is involved and plays an important role.

## Flow classification and equations¶

An aerodynamic flow field can be classified based on the following physical parameters: velocity, pressure, and temperature. Additionally, density and viscosity can be used as well for classification. The most common flow classification practices are velocity and pressure-based.

### Velocity-based classification¶

Subsonic flows are flow fields in which the air speed does not exceed the local speed of sound, that is $$M<1$$. Typical examples are common ground vehicles like cars, trains, F1 cars, and sport airplanes. $$M$$ represents the Mach number which is the flow velocity past a boundary. For more information about the mach number, please read our article about incompressible flow.

Transonic flows include both subsonic and supersonic flow regimes $$0.8<M<1.2$$. Typically, commercial airplanes fly in this speed region. A supersonic flow zone appears when the otherwise subsonic fluid accelerates above sound speed due to surface curvatures (e.g. suction side of the wing).

Supersonic flows are defined as having velocities greater than the speed of sound everywhere in the domain ($$M>1$$). Military jets and gun bullets are good examples for this velocity.

We could also include hypersonic flows, where the speed is much higher than the speed of sound. There is no accurate definition of this value ($$M>>1$$). Some hypersonic rocket-powered airplanes and spacecrafts can be mentioned as examples.

### Density / Pressure-based classification¶

Incompressible flow refers to a flow where the material (air) density remains constant. Under Mach 0.3 this is a good assumption for industrial flows and helps engineers make the calculation / simulation simpler. Please note that an incompressible flow does not mean that the fluid has to be incompressible itself.

A flow can be considered as compressible if the density changes along a streamline. Generally, this is the case for subsonic flows above Mach 0.3. Transonic, supersonic, and hypersonic flows are all compressible.

Figure 2: Velocity and Compressibility of flows

## Aerodynamic quantities¶

As mentioned before, aerodynamics is the study of forces on a solid body placed in an airflow. In order to better describe and understand these effects as well as to be able to compare cases easier, one can use mathematical quantities. These parameters include drag, lift, moment, center of pressure and pressure coefficient, to name a few important ones.

### Drag¶

One of the most important quantity in aerodynamics is the resistance of the fluid, also known as drag or drag force. This force is acting opposite to the motion of the solid body relative to the air thus causing losses. This drag force depends on the relative velocity as well as on the shape of the solid body. The drag force can be computed using the following equation:

$F_{d}={\tfrac {1}{2}}\rho \,u^{2}\,c_{d}\,A \tag{1}$

where

• $${\displaystyle F_{d}}$$ is the drag force,
• $${\displaystyle \rho }$$ is the density of the fluid,
• $${\displaystyle v}$$ is the speed of the object relative to the fluid,
• $${\displaystyle A}$$ is the cross-sectional area, and
• $${\displaystyle c_{d}}$$ is the drag coefficient which is dependent on the shape of the object and the Reynolds number.

### Lift¶

Another component of the resulting force on a solid body is called lift or lift force. This component is perpendicular to the oncoming flow direction. The term “lift” is coming from the aeronautics field, where it is an important quantity that makes flying actually possible for objects heavier than air. But lift force is generated in case of propellers, helicopter rotors and even on car bodies and wind turbines. Usually, an asymmetric body or a symmetric body with an angle of attack produces lift force. Lift can be determined by the following equation:

${\displaystyle L={\tfrac {1}{2}}\rho v^{2}SC_{L}} \tag{2}$

where

• $$L$$ is lift force,
• $$ρ$$ is air density,
• $$v$$ is speed,
• $$S$$ is the area, and
• $${\displaystyle C_{L}}$$ is the lift coefficient

### Moment¶

Aerodynamic moment or torque is produced by the aerodynamic force on a body (wing, car on any solid) making it rotate in the flow field. The rotation happens when this turning force is acting outside the center of pressure or the aerodynamic center.

Figure 3: Aerodynamic quantities on a car

If you imagine the air stream going from the front of the car over the engine hood, you can imagine that the air wants to lift the car at about the point where the headlights are. This is where the moment illustrated in figure 3 is generated.

### Center of Pressure¶

The Center of Pressure (CP) is a point where the force generated by the total sum of the surface pressure acts. This force can be calculated as the surface integral of the vectorial pressure field, and can be used to calculate the stability of an aerodynamic body (bullets, wings, cars). For example, when analyzing a weapon’s bullet aerodynamics, the distance between the CP and the center of gravity can create a rotating moment thus making the projectile less accurate. Easily speaking you can imagine the Center of Pressure as something like the center of gravity where the average weight of an object is. The center of gravity of a hammer for example is far away from the middle because the handle is usually a lot lighter than the head. The same goes for aerodynamics. The Center of Pressure is the point where the average aerodynamic forces drag and lift/downforce exert. The center of Pressure allows for example engineers to balance the lift of an aircraft.

### Pressure Coefficient¶

One of the most widely used non-dimensional numbers to visualize pressure on the surface of the aerodynamic body is the pressure coefficient $$C_p$$. This quantity describes the relative pressures in an incompressible flow, and can be calculated as

$C_p={p-p_{\infty}\over {\frac{1}{2}}\rho_{\infty}V_{\infty}^{2}}={p-p_{\infty}\over p_{0}-p_{\infty}} \tag{3}$

where

• $$p$$ is the static pressure at the location where pressure coefficient is calculated,
• $$p_{\infty }$$ is the static pressure in the free stream,
• $$p_{0}$$ is the total pressure in the free stream,
• $$\rho _{\infty }$$ is the free stream fluid density,
• $$V_{\infty }$$ is the free stream velocity of the fluid, or the velocity of the body through the fluid.

It can be seen that using $$C_p$$, different scenarios of flow / body velocities, pressures and densities can be compared for the same geometry. $$C_p$$ visualization is broadly used in the automotive industry where different study concepts are compared to each other in a virtual wind tunnel.

## Boundary Layer¶

In aerodynamics, the description, measurement, and simulation of near wall velocity and other quantities are always a challenge. As the flow decelerates rapidly near a solid wall, the viscosity effects become significant in a thin region also known as boundary layer. Boundary layers are classified into two main groups: laminar boundary layers at low Re numbers and turbulent boundary layers at higher flow speeds.

## Turbulence¶

Turbulence plays a very significant role in aerodynamics. Turbulence is by nature a chaotic, irregular phenomena observed in lower viscosity fluids. In general, unsteady vortices appear in many sizes in the flow field and interact with each other and the solid body which usually generates them. Turbulence can be “good” when, for example, one wants to generate thinner boundary layers and “bad”, when it leads to large separations, therefore increased drag and losses. Describing and simulating turbulence is still an ongoing project, however many modeling approaches exist. In CFD, to achieve accurate aerodynamic results, turbulence needs to be taken into account properly. This can be done by either modeling it or resolving the eddies (LES, DES). Eddies are swirls in a fluid flow regime that create reverse current.

Figure 4: Eddies in a simulation of a cylinder standing in a fluid flow and a real life example of an island creating eddies in the clouds$$^4$$.

## Convergence¶

As mentioned before, the aerodynamic quantities such as drag and lift are the key outcomes of a simulation. Therefore, their convergence to a steady or quasi-steady state is to be monitored and evaluated carefully before the residual targets are reached. When the simulation is stopped too early, the lift and drag values may not be fully converged, resulting in an inaccurate result. Faster convergence and shorter simulation time can be reached by using higher relaxation factors as well as by proper initialization.

Further information about convergence in numerical simulations can be found in this article.

## Applications¶

Aerodynamics is present and plays an important role in almost all the transportation devices and big structures we have around.

### Aerospace¶

As described in the History section, the first studies about aerodynamics appeared with the desire of flight. Later on, the aerospace industry and its continuous research and development on aircraft required many mathematical models, measurement devices, wind tunnels and so on, all designed to understand aerodynamics better. Nowadays it is a key field of studies to reduce emissions, reduce environmental noise and improve human comfort. The terms lift, drag, and moment commonly appear in the design process of an aircraft.

Figure 5: Aerodynamics in Aerospace

### Automotive¶

The automotive industry is very competitive these days. With the desire to build environmentally friendly or even zero emission vehicles, the reduction of drag is a key aspect of the development process. Brake cooling and HVAC airflow can also be mentioned here as some of the main topics of aerodynamics. The importance of lift force appears mostly when studying stability and when designing race vehicles.

### Sports¶

With the continuous improvement of performance in sports, the aerodynamics of race bikes, swimmers, motorbikes, or Formula 1 cars is becoming more and more important. In a competition, where even milliseconds count, the reduction of the drag by 0.1% can distinguish the winner from the others. Modern CFD tools allow the engineers to simulate tiny to extremely large scale bodies in many different scenarios. As shown in figure 6, the aerodynamic profile of a biker can have significant influence on the power that needs to be provided by the biker to achieve higher speeds. Especially the shape of the rims and the seatpost have a huge influence on the aerodynamic performance since they can create a lot of turbulence. Turbulence should be avoided for the best possible performance.

Figure 6: Road biker and aerodynamic analysis of a road biker

### Internal flows: HVAC and ductwork¶

Aerodynamics in HVAC (Heating, Ventilation and Air Conditioning) and pipe flow applications are also important. Here usually pressure losses are calculated as a result of viscous aerodynamic forces, separation at corners and bendings and sudden expansions.

Figure 7: CFD analysis of piping work at the manifold of an engine

### Wind load on buildings and urban aerodynamics¶

In case of tall buildings and skyscrapers, due to their large surface area, forces due to wind loads can be quite significant. This can be handled as “drag” force of the solid obstacle placed in the flow. Besides the wind load force on the structure, pedestrian comfort can be also assessed at lower levels. Separations on the structure as well as high speed regions can affect humans, making it sometimes impossible to even walk. Building and urban aerodynamicists simulate those scenarios with CFD tools these days.

Figure 8: Simulation of Burj Kalifa to determine wind loads

### Wind turbine¶

A very important technology towards sustainable energy resources is the wind turbine. As it usually consists of 3 aerofoils placed in an airflow, all the aspects of aerodynamics are present. Drag is used as a measure of the efficiency of the turbine, and lift as a measure of the power extracted from the wind. Moments are used to calculate the loads on the blades. SimScale provides a very interesting Project Spotlight about the optimization of wind turbine plates. Go and check it out if you are interested in finding out more about this interesting topic.

### Hydrodynamics¶

As a very similar field to aerodynamics, hydrodynamics needs to be mentioned here as well. Hydrodynamics deals with liquid flows, mostly water. While usually incompressible and low speed, water flows experience the same effects as air flow scenarios: boundary layers, turbulence, and forces acting on a solid body are also present.

## Aerodynamics & SimScale¶

Most of the transient or steady, compressible or incompressible aerodynamic flows can be accurately simulated with CFD software. SimScale offers a browser-based platform for engineers, where a CAD design can be meshed and simulated, and the results post-processed easily. Additionally, you can have a look at the public projects and choose a case which fits your problem best, so you can get started even faster. In the simulation below you can see the airpressure behind a biker. As you can see, it would be very beneficial for a different biker to bike just behind him because the pressure is lower and he would nede less power to keep the same speed.

Animation 1: Aerodynamic simulation of a road biker with SimScale.

## Resources¶

$$^1$$: Von Karman, Theodore (2004). Aerodynamics: Selected Topics in the Light of Their Historical Development. Dover Publications. ISBN 0-486-43485-0. OCLC 53900531

$$^2$$: Anderson, John D. (1997). A History of Aerodynamics: And Its Impact on Flying Machines. Cambridge University Press. ISBN 0-521-45435-2. OCLC 228667184.

$$^3$$: Anderson, John D. (2007). Fundamentals of Aerodynamics (4th ed.). McGraw-Hill. ISBN 0-07-125408-0. OCLC 60589123.


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# 10.5 Graphing quadratic equations  (Page 6/15)

 Page 6 / 15

The quadratic equation $h=-16{t}^{2}+128t+32$ is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height? Round answers to the nearest tenth.

It will take 4 seconds to reach the maximum height of 288 feet.

A toy rocket shot upward from the ground at a rate of 208 ft/sec has the quadratic equation of $h=-16{t}^{2}+208t$ . When will the rocket reach its maximum height? What will be the maximum height? Round answers to the nearest tenth.

It will take 6.5 seconds to reach the maximum height of 676 feet.

Access these online resources for additional instruction and practice graphing quadratic equations:

## Key concepts

• The graph of every quadratic equation is a parabola.
• Parabola Orientation For the quadratic equation $y=a{x}^{2}+bx+c$ , if
• $a>0$ , the parabola opens upward.
• $a<0$ , the parabola opens downward.
• Axis of Symmetry and Vertex of a Parabola For a parabola with equation $y=a{x}^{2}+bx+c$ :
• The axis of symmetry of a parabola is the line $x=-\frac{b}{2a}$ .
• The vertex is on the axis of symmetry, so its x -coordinate is $-\frac{b}{2a}$ .
• To find the y -coordinate of the vertex we substitute $x=-\frac{b}{2a}$ into the quadratic equation.
• Find the Intercepts of a Parabola To find the intercepts of a parabola with equation $y=a{x}^{2}+bx+c$ :
$\begin{array}{cccc}\hfill {\text{y}}\mathbf{\text{-intercept}}& & & \hfill {\text{x}}\mathbf{\text{-intercepts}}\\ \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}x=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}y.& & & \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}y=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}x.\end{array}$
• To Graph a Quadratic Equation in Two Variables
1. Write the quadratic equation with $y$ on one side.
2. Determine whether the parabola opens upward or downward.
3. Find the axis of symmetry.
4. Find the vertex.
5. Find the y -intercept. Find the point symmetric to the y -intercept across the axis of symmetry.
6. Find the x -intercepts.
7. Graph the parabola.
• Minimum or Maximum Values of a Quadratic Equation
• The y - coordinate of the vertex of the graph of a quadratic equation is the
• minimum value of the quadratic equation if the parabola opens upward.
• maximum value of the quadratic equation if the parabola opens downward.

## Practice makes perfect

Recognize the Graph of a Quadratic Equation in Two Variables

In the following exercises, graph:

$y={x}^{2}+3$

$y=\text{−}{x}^{2}+1$

In the following exercises, determine if the parabola opens up or down.

$y=-2{x}^{2}-6x-7$

down

$y=6{x}^{2}+2x+3$

$y=4{x}^{2}+x-4$

up

$y=-9{x}^{2}-24x-16$

Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find the axis of symmetry and the vertex.

$y={x}^{2}+8x-1$

$x=-4$ $\left(-4,-17\right)$

$y={x}^{2}+10x+25$

$y=\text{−}{x}^{2}+2x+5$

$x=1$ $\left(1,6\right)$

$y=-2{x}^{2}-8x-3$

Find the Intercepts of a Parabola

In the following exercises, find the x - and y -intercepts.

$y={x}^{2}+7x+6$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,6\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-1,0\right),\left(-6,0\right)$

$y={x}^{2}+10x-11$

$y=\text{−}{x}^{2}+8x-19$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,19\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none}$

$y={x}^{2}+6x+13$

$y=4{x}^{2}-20x+25$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,25\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(\frac{5}{2},0\right)$

$y=\text{−}{x}^{2}-14x-49$

Graph Quadratic Equations in Two Variables

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.

$y={x}^{2}+6x+5$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,5\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-1,0\right),\left(-5,0\right);$
axis: $x=-3;\phantom{\rule{0.2em}{0ex}}\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(-3,-4\right)$

$y={x}^{2}+4x-12$

$y={x}^{2}+4x+3$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,3\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-1,0\right),\left(-3,0\right);$
axis: $x=-2;\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(-2,-1\right)$

$y={x}^{2}-6x+8$

$y=9{x}^{2}+12x+4$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,4\right)\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{2}{3},0\right);$
axis: $x=-\frac{2}{3};\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{2}{3},0\right)$

$y=\text{−}{x}^{2}+8x-16$

$y=\text{−}{x}^{2}+2x-7$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,-7\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none};$
axis: $x=1;\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}\left(1,-6\right)$


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# American Institute of Mathematical Sciences

May  2013, 18(3): 601-641. doi: 10.3934/dcdsb.2013.18.601

## Mathematics of traveling waves in chemotaxis --Review paper--

 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  November 2012 Revised  November 2012 Published  December 2012

This article surveys the mathematical aspects of traveling waves of a class of chemotaxis models with logarithmic sensitivity, which describe a variety of biological or medical phenomena including bacterial chemotactic motion, initiation of angiogenesis and reinforced random walks. The survey is focused on the existence, wave speed, asymptotic decay rates, stability and chemical diffusion limits of traveling wave solutions. The main approaches are reviewed and related analytical results are given with sketchy proofs. We also develop some new results with detailed proofs to fill the gap existing in the literature. The numerical simulations of steadily propagating waves will be presented along the study. Open problems are proposed for interested readers to pursue.
Citation: Zhi-An Wang. Mathematics of traveling waves in chemotaxis --Review paper--. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 601-641. doi: 10.3934/dcdsb.2013.18.601
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text:
# Mosfet Sparkler

So I decided to experiment with power MOSFETS (IRF510). The wife is out of town and it's easier to catch stuff on fire when she isn't here. The circuit is shown below. Eventually I would like to supply the gate with a 5V pulse from an Arduino. However, to get a feel for the chip I decided to supply the gate with a constant 5Vdc. Here is the circuit:

The max Vth for this FET is 4 volts. The Vd to Id curve shows about 1 amp for the setup above. I also opened up a text book and put the Id formula to use. Id = Kn(Vgs-Vth)^2. I assumed the forward transconductance would work for Kn, so I plugged it in and ended up with 1A (the datasheet shows a higher forward trans value of 2. The package reads 1).

I had a multimeter setup between vd and the drain pin. For some reason the current reported just above 2 amps before the gate was even turned on. Then it would slowly decrease to about 700mA then it would increase to about an amp and POOF! Smoke and tiny silicon ambers. I tried this with two different chips. For the second chip I tried it with a 1ms pulse followed by a 500ms delay, I didn't let this chip catch flames, but the temp was reading well over 190F.

Everything I have read points to the gate voltage not being high enough, causing the FET to not fully turn on. I guess I can kind of see how that is possible. All of the papers and books I have read discuss how the Rds decreases as the gate voltage increases. A lower voltage would in turn cause a greater resistance, generating lots of heat and wasted power. If the FET wasn't meant to operate at these voltages, why are they included in the characteristic graphs? I'm certainly no expert, but I don't think the constant gate voltage is the problem.

If this is indeed the problem would a heatsink help? Or are there any suggestions for supplying a higher voltage to the gate? I would really like to figure out how to resolve this issue before running away to another product. Eventually I am going to make two sweet ass motor controllers with these guys :) Please help me!!! MOSFETS make for expensive fire crackers.

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When the MOSFET turns on it will die if driven fully on. When it does it will in some cases short from Drain to Gate wityh drain to source open circuit. In such a case it will connect +12V to the arduino drive pin. A series drive resistor and a voltage clamp on the processor side which both are designed to allow the Arduino to survive +12V to gate are required. || With no load apart from the MOSFET if you drive it fully on then you must get 10 - 20W or more dissipation in the MOSFET. | What are you trying to achieve. A hammer is often an easier way of achieving the main effect. – Russell McMahon Aug 18 '12 at 12:47
@RussellMcMahon Thanks for your help. In the long rung I hope to put together a stepper motor controller. I am on the kick of trying to build everything instead of buying it. I think every ee student should do this! – atomSmasher Aug 19 '12 at 5:33

Even if the graphs in the datasheet say 1 A at 5 V you should at a load to limit the current. FET characteristics can vary widely, just look at $V_{GS(th)}$: minimum 2V, maximum 4 V. For some FETs even a 3:1 range is given. Your 5 V may be 1 V above $V_{GS(th)}$, but it might as well be 3 V over it. The 1 A is typical, but $R_{DS}$ may already be low. Note that at 10 V this will be as low as 0.5 Ω. Now you have the full 12 V across the FET, and if $R_{DS}$ would go to even 2 Ω this would cause a dissipation of 72 W, which without any cooling is way too much for the poor thing.

BTW, even the 12 W of a 1 A current would be too much without cooling, though I wouldn't expect it to explode then.

So make sure you add a load so that the FET won't see the full 12 V across it.

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Thanks for your help! It's funny that you mention the 12v Vds being too high. Eventually I am going to build a stepper motor controller using these fets. The motors I am going to use are rated for 3.6 volts and 3A per phase. Do you suggest I change the Vds to 4 volts and experiment? This might be a stupid question, and I apologize for my ignorance, but how do motor controllers get away with tiny resistors? I am only familiar with using 1/4W resistors. If I cut my voltage to 4 volts that still puts about 4w on the resistor, I can see the .25W resistor bursting in flames. – atomSmasher Aug 19 '12 at 5:45
@atomsmasher - If you have the motors already and have a 3.6 V source then use the motors as load, they're made for it. I don't know why you would want 4 V instead of 3.6 V. I checked the IRF510 datasheet again, and I think you need a better one. This one has an $R_{DS(ON)}$ of more than 0.5 $\Omega$, but that's at 10 V gate voltage, at 5 V will be higher. You may lose 20 % of your 3.6 V in there, and the motor 30 % of its power. The NTD4960N has an $R_{DS(ON)}$ of only 12.5 $\Omega$ at 5 V, so it won't even get warm at 1 A, and voltage drop will be negligible. – stevenvh Aug 19 '12 at 9:41
Awesome! Thanks for your help. – atomSmasher Aug 19 '12 at 18:02
@atomsmasher - You're welcome. That 12.5 $\Omega$ should be 12.5 m$\Omega$, obviously, but I guess you already knew that :-) – stevenvh Aug 19 '12 at 18:06

I suspect the reason you read the current as 2A before the gate was "turned on" is because the gate was floating (i.e. not pulled to ground. This is due to the gate being very high impedance (just touching it will make a difference) You need a resistor between gate and ground (e.g. 10k) to prevent this happening.

As Steven says, it would be wise to use something to limit the current. If you don't limit the current, as long as the supply is capable of much more than 2A, the full 12V will be across the FET, so 1A * 12V = 12W. At 12W, for a TO-220 package you certainly need a heatsink.
Without a heatsink, the Tja (junction to ambient) is 62°C/W, and the max operating temperature is 175°C, so assuming an ambient temperature of 25°C the FET can only handle around (175 - 25) / 62 = 2.33W without a heatsink. This is very forgiving, you would typically design for an ambient of e.g. up to 70°C and a max temp of 125°C

Note that if you want control the current accurately, you need to use some feedback rather than trying to rely on the variable Vth and transconductance parameters. A current sense resistor and opamp combined with the FET can make a simple constant current circuit (components can obviously be changed to suit, the LM741 is obsolete but people still insist on using it):

Another thing to note is the SOA (Safe Operating Area) curves are given for different pulse widths, and often DC is omitted if the FET is designed for switching, so make sure you take all this into account when designing your circuit. Notice the SOA curves are specified for a case temperature of 25°C.

Few useful inks:
Thermal Resistance
Selecting a heatsink
MOSFET SOA guide

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Thank you for the excellent response! I wanted to give everyone a correct answer for this question, but the laws of physics won't allow it. I am defaulting to the first response, but you still get +1 and a pat on the back!! In response to the ground pin floating, is it OK to tie the Arduino gnd pin to my makeshift hacked computer power supply gnd? I keep meaning to pick up a fire extinguisher, but I always forget :\$ – atomSmasher Aug 19 '12 at 5:37
No probs, glad to be of help. I was referring to the MOSFET gate floating, which just means it's not connected to anything. It only needs to be tied to local ground to keep the gate-source voltage at 0V and the MOSFET turning on (by accumulation of charge on the extremely high impedance gate - you can make a good static detector with a MOSFET - LED in series with resistor on drain and gate connected to an "antenna") – Oli Glaser Aug 19 '12 at 6:20
Oli, what's that "1/10" spec next to the potmeter? – stevenvh Aug 20 '12 at 18:29
@Steven - Yeh, I noticed that too. According to the blog I got it from, it means it's set to a ratio of 1:10 (i.e. the pot resistance doesn't really matter) I couldn't find anything better on Google images, and didn't have time to draw it in LTSpice at that point. Think I should change it? – Oli Glaser Aug 20 '12 at 18:47
@Oli - probably not. It's been there for 2 days and I seem to be the only one asking about it. Though I would have use concrete values if it were my schematic, like a 10k pot in series with a 1k resistor. That would have been clear. BTW, I see nor reason why you wouldn't turn the input all the way to zero. Wait, I'll fix it for you ....... There, done! :-) – stevenvh Aug 20 '12 at 18:54
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text:
# Is there a name for a differences versus differences plot?

I'm aware of the Bland–Altman plot where the differences between two measures of a given parameter (obtained with two different methods) are plotted against their averages.

What I need to plot, are the differences between two measures of parameter $A$, against the difference between two measures of parameter $B$.

Both parameters are properties associated to the same physical phenomena (star clusters to be more precise, for example age and distance).

Is there a name for such a plot?

To make it more clear, this is the plot I'm producing:

where $\Delta A=A_1-A_2$. Each of those is the value of parameter $A$, obtained with a different method, i.e.: "Method 1" and "Method 2" (same for $B$).

The curves are iso-density lines, for a fitted 2D Gaussian Kernel on the data.

• I think you just named it perfectly well. I've not come across a name in literature. In addition to names in your references, the Bland-Altman plot was used earlier by P.D. Oldham. – Nick Cox Apr 2 '16 at 0:36
• Thanks Nick. I actually thought of naming it a "Delta plot", since I use the $\Delta$ symbol for the differences. – Gabriel Apr 2 '16 at 2:07
• Speckman, P. L., Rouder, J. N., Morey, R. D., & Pratte, M. S. 2008. Delta plots and coherent distribution ordering. American Statistician 62(3): 262-266 used that term. They don't have exclusive rights to the name, but that implies caution. – Nick Cox Apr 2 '16 at 8:32
• Incidentally that paper doesn't make clear that the main ideas are all in the Wilk and Gnanadesikan paper there cited. See also stata-journal.com/sjpdf.html?articlenum=gr0027 if interested. I am not saying that they have your idea, just that they use the term you like. – Nick Cox Apr 2 '16 at 8:41
• @Glen_b Yes, that and Oldham, P. D. 1962. A note on the analysis of repeated measurements of the same subjects. Journal of Chronic Diseases 15: 969–977. Tukey was plotting difference versus mean in the early 1960s, I think; I don't recall if there are examples in his Exploratory data analysis. In one sense it's an easy variation on residual versus fitted, explicit in Anscombe and Tukey Technometrics 1963 and also used by Neyman and friends in the 1950s. . – Nick Cox Apr 3 '16 at 7:48


text:
# T1 Measurement

Now that we have measured the basic qubit properties — readout resonator frequency, qubit resonance frequency and qubit drive pulse amplitudes — and calibrated our system accordingly, we can use these to obtain information about the properties of our qubit. First, we will do a simple measurement of the lifetime T1 of the qubit.

To measure the lifetime of the qubit state, the qubit is brought to its excited state. We will then wait a variable time before measuring the state of the qubit. Averaging over many measurements will yield the probability of still finding the qubit in its excited state after this time.

The experimental parameters for the T1 experiment are the range of delay times and the number of steps we will iterate the delay over. We also define the pulse used to drive the qubit from the ground to its excited state, using the Gaussian waveform already used in the Rabi experiment, together with the newly determined π-pulse amplitude.

# delay range for ramsey pulses
t1_min = 0.
t1_max = 5 * qubit_parameters['T1']
# how many delay points to sweep
t1_num = 60

# how many averages per point: 2^n_average
n_average = 10

# set up sweep parameter
t1_sweep = LinearSweepParameter(uid="t1_delay", start=t1_min, stop=t1_max, count=t1_num)

# T1 excitation pulse - qubit pi pulse
x180 = pulse_library.gaussian(
uid="x180", length=qubit_parameters['qb_len'], amplitude=qubit_parameters['pi_amp']
)

The definition of the experiment follows the same lines as before, with the sweep parameter now being the delay time between the qubit excitation pulse and the readout of the qubit state.

# Create Experiment
exp_t1 = Experiment(
uid="Qubit T1",
signals=[
ExperimentSignal("drive"),
ExperimentSignal("measure"),
ExperimentSignal("acquire"),
],
)

## define experiment sequence
# outer loop - real-time averaging
with exp_t1.acquire_loop_rt(uid="t1_shots", count=pow(2, n_average), averaging_mode=AveragingMode.SEQUENTIAL, acquisition_type=AcquisitionType.INTEGRATION):
# inner loop - real-time sweep delay between pi pulse and readout
with exp_t1.sweep(uid="t1_sweep", parameter=t1_sweep):
# qubit pi pulse and following delay
with exp_t1.section(uid="qubit_excitation", length=t1_max+2*x180.length, alignment=SectionAlignment.RIGHT):
exp_t1.play(signal="drive", pulse=x180)
exp_t1.delay(signal="drive", time=t1_sweep)
# readout pulse and data acquisition
)


text:
# Math Help - I need help with functions.

1. ## I need help with functions.

1) Let f be the function defined by f(x) = 2x^4 + 5. Find f(a+h), f(2a-h).

2) Let f be the function defined by f(x) = -3x^2 – 3. Find f(1/2), f(-3/4).

3) Let f(x) = 3x – 4 and g(x) = (-1/3)x^2 – 4x. Find f+g, f-g, f/g, f∙g, f°g, g°f.

Thanks!!

2. 1) As $f(x)=2x^4+5$,

\begin{aligned}
f(a+h)&=2(a+h)^4+5\\
&=2(a^4+4a^3h+6a^2h^2+4ah^3+h^4)+5.
\end{aligned}

$f(a-h)$ can be found similarly.

2) If $f(x)=-3x^2-3$, then $f\left(\frac{1}{2}\right)$ and $f\left(-\frac{3}{4}\right)$ can be found by evaluating

$-3\left(\frac{1}{2}\right)^2-3$

and

$-3\left(-\frac{3}{4}\right)^2-3.$

3) If $f(x)=3x-4$ and $g(x)=-\frac{1}{3}x^2-4x$, then

\begin{aligned}
(f+g)(x)&=(3x-4)+\left(-\frac{1}{3}x^2-4x\right)\\
(f-g)(x)&=(3x-4)-\left(-\frac{1}{3}x^2-4x\right)\\
(f/g)(x)&=\frac{3x-4}{\left(-\frac{1}{3}x^2-4x\right)}\\
(f\cdot g)(x)&=(3x-4)\cdot\left(-\frac{1}{3}x^2-4x\right)\\
\end{aligned}

\begin{aligned}
(f\circ g)(x)&=\left(3\left(-\frac{1}{3}x^2-4x\right)-4\right).\\
\end{aligned}

Hope this helps.


text:
# Challenge Data

### Real-time train crowding forecasting by SNCF-Transilien

#### Description

##### Dates

Started on Jan. 6, 2023

##### Challenge context

SNCF-Transilien is the SNCF commuting network operator covering the Paris region. It runs more than 6,200 trains per day and moves 3.4 million of passengers. SNCF-Transilien gathers an increasing amount of data on the number of passengers on-board through infra-red door sensors. This increasing amount of data allows SNCF-Transilien to provide new services to passengers and improve its operational performance.

##### Challenge goals

The aim of this challenge is to give SNCF-Transilien the tools to provide an accurate train occupancy rate forecasting. Thus, deliver precise real time crowding information (RTCI) to its passengers through digital services.

When a traveler is waiting for a k-train at station s, the train may be several stations ahead, but we want to give the travelers the most consistent information possible about how busy the train will be when they board it. To do this, we need to predict for each train the actual occupancy rates at the next stations. Here, we propose to predict, more simply, the occupancy rate at the next station only.

### Dataset structure

k is used to specify the train, s for the station and d for the day. One stop is given by the triplet k, s, d.

• A Xtrain.csv dataset with 31 119 rows (i.e. 31 119 stops k, s, d) and 12 columns
• A Xtest.csv dataset with 13 752 rows (i.e. 13 752 stops k, s, d) and 12 columns
• A Ytrain.csv dataset with 31 119 rows and one column, giving the target variable to be predicted for the training set.
• A Ysample.csv dataset with 13 752 rows and one column with random values between [0, 1], which is a submission example that you can use to prepare your own submission

### Data origin

The data comes from infra-red sensors located above each door of the rolling stocks (NAT, R2N) in Île-de-France, measuring the number of alighting and boarding passengers per door. This data is captured in real time and is accessible only at the train scale for this challenge. The crowding factor or occupancy rate is the number of passenger on-board divided by the total capacity of the train.

We want you to predict the occupancy rate in real time using the near past. We introduce p and q notations to describe the past: * p is used for the past values of the previous trains at the same station * q is used to describe previous stops for the same train.

The actual occupancy rate is: p0q0 ∈ [0,1] which is crowding factor/occupancy rate, noted Y(k,s,d), for a stop which is a train k at a station s for a day d.

The columns, i.e., the features, are split into 6 contextual variables such as day, train id, etc. and 6 lag variables:

##### Contextual variables
• date d: YYYYMMDD is the date of the train id run
• train k: 1, 2, …, 55 train id (unique by day d)
• station s: AA, AB, AC,… is the station id
• hour: HH:00:00 is the time slot
• way: 0, 1 is the train way with 0 (toward Paris) or 1 (toward suburb)
• composition: 1, 2 is the number of train units (one or two)
##### Lags variables
• p1q0 ∈ [0,1] is the occupancy rate of the previous train k-1 at the same station s
• p2q0 ∈ [0,1] is the occupancy rate of the second previous train k-2 at the same station s
• p3q0 ∈ [0,1] is the occupancy rate of the third previous train k-3 at the same station s
• p0q1 ∈ [0,1] is the occupancy rate of the same train k at the previous station s-1
• p0q2 ∈ [0,1] is the occupancy rate of the same train k at the second previous station s-2
• p0q3 ∈ [0,1] is the occupancy rate of the same train k at the third previous station s-3

The lag variables are NA if the lag does not exist for the stop because of timetable structure. All other missing values have been imputed by a business algorithm.

For more details on the data set or on the benchmark model, see our paper One-Station-Ahead Forecasting of Dwell Time, Arrival Delay and Passenger Flows on Trains Equipped with Automatic Passenger Counting (APC) Device.

##### Benchmark description

The benchmark used in this challenge is based on a multiple linear model. Each node is defined by the intersection of a train id and a station id. We use a simple linear model with at most one neighbor in p and q.

$Y_{k, s} = \beta^0_{k, s} + \beta^{1, 0}_{k, s} y_{k-1, s} + \beta^{0, 1}_{k, s} y_{k, s-1} + \varepsilon_{k, s} \, \, ,$

when a neighborhood exists. In R syntax, it is:

lm(p0q0 ~ p1q0 + p0q1, data = dataByNode)

#### Files

Files are accessible when logged in and registered to the challenge

Data scientist


text:
# Splitting infinite sets into disjoint sets of equal cardinality

When given an infinite set $X$, it seems to me very reasonable that one can 'split' it up into two disjoint subsets $A$ and $B$ such that all three have the same cardinality. For countable sets, this is rather easy, for then we can index the elements in $X$ by $(x_n)_{n \in \mathbb{N}}$, and we can take $A$ as the elements indexed by even $n$, and $B$ those by odd ones. Of course, countability isn't needed per se; with a set indexed by $\mathbb{R}$, it can be split up rather easily as well. Where I run into difficulty is a general infinite set, where this indexing trick doesn't seem to work, because the indexing set isn't 'known' well enough (in the sense that I can't make an easy choice, as with $\mathbb{N}$ or $\mathbb{R}$).

My question is if this is possible for any infinite set $X$.

• You should first decide/specify what you mean by "same cardinality". Cardinality for finite sets is pretty obvious, for infinite sets they all end up being "equally infinite". To me it feels like you want them to be of equal measure. – N.Bach May 15 '17 at 12:30
• @N.Bach "equal cardinality" is a strictly defined mathematical term. Two sets have the same cardinality if and only if there exists a bijection between them. – 5xum May 15 '17 at 12:31
• Not an expert: assuming Choice, you can well-order your set, i.e. it has the same cardinality as some ordinal $\alpha$. I guess the ordinal $\alpha+\alpha$ does have the same cardinality too. A bijection between your set $X$ and $\alpha+\alpha$ will give you such a partition. But this is just a sketch. Maybe the difficulty is now to construct the bijection $f:\alpha\leftrightarrow \alpha+\alpha$. – M. Winter May 15 '17 at 12:33

In general, if $X$ is infinite, $\vert X \vert = \vert X \vert+\vert X \vert$, ie. there exists a bijection between $X$ and $X\sqcup X$, where $X\sqcup X$. denotes the disjoint union. There's an obvious injection from $X$ to $X\sqcup X$, and there's an injection from $X\sqcup X$ to $X\times X$. Since $X$ is infinite, we have $\vert X \vert = \vert X\times X\vert$ (as a well-known consequence of AC), and so $\vert X \vert = \vert X\sqcup X \vert$. Thus, there exists a bijection $f: X \to X\sqcup X$. Now let $A = f^{-1}[\{0\}\times X], B = f^{-1}[\{1\}\times X]$. Then $A,B$ are disjoint and have the same cardinality as $X$, and $X=A\cup B$.
• In fact, every infinite set $X$ being in bijection with $X\times X$ is equivelant to the axiom of choice. – Jason May 15 '17 at 15:48