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University Physics Volume 2

9.1Electrical Current

University Physics Volume 29.1 Electrical Current

Learning Objectives

By the end of this section, you will be able to:

• Describe an electrical current
• Define the unit of electrical current
• Explain the direction of current flow

Up to now, we have considered primarily static charges. When charges did move, they were accelerated in response to an electrical field created by a voltage difference. The charges lost potential energy and gained kinetic energy as they traveled through a potential difference where the electrical field did work on the charge.

Although charges do not require a material to flow through, the majority of this chapter deals with understanding the movement of charges through a material. The rate at which the charges flow past a location—that is, the amount of charge per unit time—is known as the electrical current. When charges flow through a medium, the current depends on the voltage applied, the material through which the charges flow, and the state of the material. Of particular interest is the motion of charges in a conducting wire. In previous chapters, charges were accelerated due to the force provided by an electrical field, losing potential energy and gaining kinetic energy. In this chapter, we discuss the situation of the force provided by an electrical field in a conductor, where charges lose kinetic energy to the material reaching a constant velocity, known as the “drift velocity.” This is analogous to an object falling through the atmosphere and losing kinetic energy to the air, reaching a constant terminal velocity.

If you have ever taken a course in first aid or safety, you may have heard that in the event of electric shock, it is the current, not the voltage, which is the important factor on the severity of the shock and the amount of damage to the human body. Current is measured in units called amperes; you may have noticed that circuit breakers in your home and fuses in your car are rated in amps (or amperes). But what is the ampere and what does it measure?

Defining Current and the Ampere

Electrical current is defined to be the rate at which charge flows. When there is a large current present, such as that used to run a refrigerator, a large amount of charge moves through the wire in a small amount of time. If the current is small, such as that used to operate a handheld calculator, a small amount of charge moves through the circuit over a long period of time.

Electrical Current

The average electrical current I is the rate at which charge flows,

$Iave=ΔQΔt,Iave=ΔQΔt,$
9.1

where $ΔQΔQ$ is the amount of net charge passing through a given cross-sectional area in time $ΔtΔt$ (Figure 9.2). The SI unit for current is the ampere (A), named for the French physicist André-Marie Ampère (1775–1836). Since $I=ΔQΔtI=ΔQΔt$, we see that an ampere is defined as one coulomb of charge passing through a given area per second:

$1A≡1Cs.1A≡1Cs.$
9.2

The instantaneous electrical current, or simply the electrical current, is the time derivative of the charge that flows and is found by taking the limit of the average electrical current as $Δt→0Δt→0$:

$I=limΔt→0ΔQΔt=dQdt.I=limΔt→0ΔQΔt=dQdt.$
9.3

Most electrical appliances are rated in amperes (or amps) required for proper operation, as are fuses and circuit breakers.

Figure 9.2 The rate of flow of charge is current. An ampere is the flow of one coulomb of charge through an area in one second. A current of one amp would result from $6.25×10186.25×1018$ electrons flowing through the area A each second.

Example 9.1

Calculating the Average Current

The main purpose of a battery in a car or truck is to run the electric starter motor, which starts the engine. The operation of starting the vehicle requires a large current to be supplied by the battery. Once the engine starts, a device called an alternator takes over supplying the electric power required for running the vehicle and for charging the battery.

(a) What is the average current involved when a truck battery sets in motion 720 C of charge in 4.00 s while starting an engine? (b) How long does it take 1.00 C of charge to flow from the battery?

Strategy

We can use the definition of the average current in the equation $I=ΔQΔtI=ΔQΔt$ to find the average current in part (a), since charge and time are given. For part (b), once we know the average current, we can its definition $I=ΔQΔtI=ΔQΔt$ to find the time required for 1.00 C of charge to flow from the battery.

Solution

a. Entering the given values for charge and time into the definition of current gives
$I=ΔQΔt=720C4.00s=180C/s=180A.I=ΔQΔt=720C4.00s=180C/s=180A.$

b. Solving the relationship $I=ΔQΔtI=ΔQΔt$ for time $ΔtΔt$ and entering the known values for charge and current gives

$Δt=ΔQI=1.00C180C/s=5.56×10−3s=5.56ms.Δt=ΔQI=1.00C180C/s=5.56×10−3s=5.56ms.$

Significance

a. This large value for current illustrates the fact that a large charge is moved in a small amount of time. The currents in these “starter motors” are fairly large to overcome the inertia of the engine. b. A high current requires a short time to supply a large amount of charge. This large current is needed to supply the large amount of energy needed to start the engine.

Example 9.2

Calculating Instantaneous Currents

Consider a charge moving through a cross-section of a wire where the charge is modeled as $Q(t)=QM(1−e−t/τ)Q(t)=QM(1−e−t/τ)$. Here, $QMQM$ is the charge after a long period of time, as time approaches infinity, with units of coulombs, and $ττ$ is a time constant with units of seconds (see Figure 9.3). What is the current through the wire?
Figure 9.3 A graph of the charge moving through a cross-section of a wire over time.

Strategy

The current through the cross-section can be found from $I=dQdtI=dQdt$. Notice from the figure that the charge increases to $QMQM$ and the derivative decreases, approaching zero, as time increases (Figure 9.4).

Solution

The derivative can be found using $ddxeu=eududxddxeu=eududx$.
$I=dQdt=ddt[QM(1−e−t/τ)]=QMτe−t/τ.I=dQdt=ddt[QM(1−e−t/τ)]=QMτe−t/τ.$
Figure 9.4 A graph of the current flowing through the wire over time.

Significance

The current through the wire in question decreases exponentially, as shown in Figure 9.4. In later chapters, it will be shown that a time-dependent current appears when a capacitor charges or discharges through a resistor. Recall that a capacitor is a device that stores charge. You will learn about the resistor in Model of Conduction in Metals.

Handheld calculators often use small solar cells to supply the energy required to complete the calculations needed to complete your next physics exam. The current needed to run your calculator can be as small as 0.30 mA. How long would it take for 1.00 C of charge to flow from the solar cells? Can solar cells be used, instead of batteries, to start traditional internal combustion engines presently used in most cars and trucks?

Circuit breakers in a home are rated in amperes, normally in a range from 10 amps to 30 amps, and are used to protect the residents from harm and their appliances from damage due to large currents. A single 15-amp circuit breaker may be used to protect several outlets in the living room, whereas a single 20-amp circuit breaker may be used to protect the refrigerator in the kitchen. What can you deduce from this about current used by the various appliances?

Current in a Circuit

In the previous paragraphs, we defined the current as the charge that flows through a cross-sectional area per unit time. In order for charge to flow through an appliance, such as the headlight shown in Figure 9.5, there must be a complete path (or circuit) from the positive terminal to the negative terminal. Consider a simple circuit of a car battery, a switch, a headlight lamp, and wires that provide a current path between the components. In order for the lamp to light, there must be a complete path for current flow. In other words, a charge must be able to leave the positive terminal of the battery, travel through the component, and back to the negative terminal of the battery. The switch is there to control the circuit. Part (a) of the figure shows the simple circuit of a car battery, a switch, a conducting path, and a headlight lamp. Also shown is the schematic of the circuit [part (b)]. A schematic is a graphical representation of a circuit and is very useful in visualizing the main features of a circuit. Schematics use standardized symbols to represent the components in a circuits and solid lines to represent the wires connecting the components. The battery is shown as a series of long and short lines, representing the historic voltaic pile. The lamp is shown as a circle with a loop inside, representing the filament of an incandescent bulb. The switch is shown as two points with a conducting bar to connect the two points and the wires connecting the components are shown as solid lines. The schematic in part (c) shows the direction of current flow when the switch is closed.

Figure 9.5 (a) A simple electric circuit of a headlight (lamp), a battery, and a switch. When the switch is closed, an uninterrupted path for current to flow through is supplied by conducting wires connecting a load to the terminals of a battery. (b) In this schematic, the battery is represented by parallel lines, which resemble plates in the original design of a battery. The longer lines indicate the positive terminal. The conducting wires are shown as solid lines. The switch is shown, in the open position, as two terminals with a line representing a conducting bar that can make contact between the two terminals. The lamp is represented by a circle encompassing a filament, as would be seen in an incandescent light bulb. (c) When the switch is closed, the circuit is complete and current flows from the positive terminal to the negative terminal of the battery.

When the switch is closed in Figure 9.5(c), there is a complete path for charges to flow, from the positive terminal of the battery, through the switch, then through the headlight and back to the negative terminal of the battery. Note that the direction of current flow is from positive to negative. The direction of conventional current is always represented in the direction that positive charge would flow, from the positive terminal to the negative terminal.

The conventional current flows from the positive terminal to the negative terminal, but depending on the actual situation, positive charges, negative charges, or both may move. In metal wires, for example, current is carried by electrons—that is, negative charges move. In ionic solutions, such as salt water, both positive and negative charges move. This is also true in nerve cells. A Van de Graaff generator, used for nuclear research, can produce a current of pure positive charges, such as protons. In the Tevatron Accelerator at Fermilab, before it was shut down in 2011, beams of protons and antiprotons traveling in opposite directions were collided. The protons are positive and therefore their current is in the same direction as they travel. The antiprotons are negativity charged and thus their current is in the opposite direction that the actual particles travel.

A closer look at the current flowing through a wire is shown in Figure 9.6. The figure illustrates the movement of charged particles that compose a current. The fact that conventional current is taken to be in the direction that positive charge would flow can be traced back to American scientist and statesman Benjamin Franklin in the 1700s. Having no knowledge of the particles that make up the atom (namely the proton, electron, and neutron), Franklin believed that electrical current flowed from a material that had more of an “electrical fluid” and to a material that had less of this “electrical fluid.” He coined the term positive for the material that had more of this electrical fluid and negative for the material that lacked the electrical fluid. He surmised that current would flow from the material with more electrical fluid—the positive material—to the negative material, which has less electrical fluid. Franklin called this direction of current a positive current flow. This was pretty advanced thinking for a man who knew nothing about the atom.

Figure 9.6 Current I is the rate at which charge moves through an area A, such as the cross-section of a wire. Conventional current is defined to move in the direction of the electrical field. (a) Positive charges move in the direction of the electrical field, which is the same direction as conventional current. (b) Negative charges move in the direction opposite to the electrical field. Conventional current is in the direction opposite to the movement of negative charge. The flow of electrons is sometimes referred to as electronic flow.

We now know that a material is positive if it has a greater number of protons than electrons, and it is negative if it has a greater number of electrons than protons. In a conducting metal, the current flow is due primarily to electrons flowing from the negative material to the positive material, but for historical reasons, we consider the positive current flow and the current is shown to flow from the positive terminal of the battery to the negative terminal.

It is important to realize that an electrical field is present in conductors and is responsible for producing the current (Figure 9.6). In previous chapters, we considered the static electrical case, where charges in a conductor quickly redistribute themselves on the surface of the conductor in order to cancel out the external electrical field and restore equilibrium. In the case of an electrical circuit, the charges are prevented from ever reaching equilibrium by an external source of electric potential, such as a battery. The energy needed to move the charge is supplied by the electric potential from the battery.

Although the electrical field is responsible for the motion of the charges in the conductor, the work done on the charges by the electrical field does not increase the kinetic energy of the charges. We will show that the electrical field is responsible for keeping the electric charges moving at a “drift velocity.”


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# Prove of refractive index

1. Jul 18, 2005

### primarygun

Can anyone show me a proof of "refractive index= (real depth)\(apparent depth) "?
I found the proof in my book has a mistakes and I found some contradicts to this equation.

Last edited: Jul 18, 2005
2. Jul 18, 2005

### Pengwuino

Why do you say there are mistakes and contradictions?

3. Jul 18, 2005

### primarygun

The book said used a pair of similar triangles to infer it but the triangles are not similar.
Do you want a picture? If you want, I can upload it now.

4. Jul 18, 2005

### primarygun

Hope it helps you bring me out of the troubles.

5. Jul 18, 2005

### Pengwuino

Ugh, basic geometry :(

I think they are considered similar triangles.

6. Jul 18, 2005

### primarygun

is it correct?

7. Jul 18, 2005

### Pengwuino

yah im pretty sure it is... hopefuly someone will verify though... its 3am here :D

8. Jul 18, 2005

### primarygun

But, I think the equation should be correct as I saw it in many books.
We need the someone's help;

9. Jul 18, 2005

### HallsofIvy

Staff Emeritus
What DEFINITION of 'refractive index' does your book give? (It is possible to use "refractive index= (real depth)/(apparent depth)" as the definition.)

10. Jul 18, 2005

### primarygun

index of refraction.

11. Jul 30, 2005

if the two triangles were similar, wouldn't r = i, or 90 degrees - r = i?

not sure if giving the ratio of the sines will show this. :grumpy:

12. Jul 30, 2005

### sniffer

i derived this equation before. the formula is just an approximation for small angle case, where $sin\theta=tan\theta$ for small angle. i.e. it is for the case where you almost look vertically down to the object from above. you can derive it easily by drawing slender triangles. very easy.

if you get stuck i can post a proper solution with the drawing.

13. Jul 30, 2005

### Gokul43201

Staff Emeritus
The triangles are NOT similar. That derivation is just extremely poorly worded (or was written by someone who knew the result and "made up" the proof). What it is actually using is the small angle approximation that sniffer mentions.

Anyway, I believe Primary's doubts have long been resolved.

14. Jul 30, 2005

### Nerro

According to the definition these triangles áre similar because when you shrink one side the become congruent which is apparently enough to qualify them as "similar" (I have looked this up.).

Frankly I thought the explanation was pretty straightforward...

Last edited: Jul 30, 2005

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# Closed embedding into a normal Hausdorff space and left lifting property

I am trying to understand the characterization of the class of closed embeddings into a normal Hausdorff space as the class of continuous maps satisfying the left lifting property with respect to a unique map, presented in https://ncatlab.org/nlab/show/colimits+of+normal+spaces. The last change made in this page is anonymous but I have found a paper not giving more details in https://mishap.sdf.org/.

First of all, if the rule is $$a if and only if $$b$$ is in the closure of $$a$$, the poset given in the nLab page is the wrong one: it should be $$a>bd because the closed points are $$a,c,e$$ (there is also a typo in the description of the closed points) and the points $$b,d$$ are open.

For the author of this note (I guess M. Gavrilovich), a continuous map is a closed embedding into a normal Hausdorff space if and only if it satisfies the LLP with respect to the map $$g:\{a>bdb=c=d

Could someone explain what seems to be evident for the author of this note please ?

What I can understand is that a map $$\varnothing\to X$$ satisfies the LLP with respect to $$g$$ if and only if $$X$$ is normal (but not necessarily Hausdorff). Indeed, assume the LLP. Take two disjoint closed subsets $$A$$ and $$E$$ of $$X$$ taken to the closed points $$a$$ and $$e$$ respectively of $$\{a>b=c=d; the other points of $$X$$ are taken to the open point $$b=c=d$$. Then the existence of the lift $$\ell:X\to \{a>bd provides two closed subsets $$F_1=\ell^{-1}(\{a,b,c\})$$ and $$F_2=\ell^{-1}(\{c,d,e\})$$. Then $$F_1^c \cap F_2^c=\varnothing$$. And $$E\subset F_1^c$$ and $$A\subset F_2^c$$. Hence $$X$$ is normal. Conversely, if $$X$$ is normal, then the LLP is satisfied.

I do not understand either why the Tietze extension theorem is mentioned which is a characterization of normal Hausdorff spaces.

• Could you be more specific about which of Gavrilovich's papers you are referencing (and where in the paper the statement is found). May 19, 2021 at 11:34
• @Tyrone I managed to get in touch with him (it turns out that his Google email does not work, the other one works). It seems that the statement is still unproved when the source is not the empty set. You will find the statement (without proof) that $X$ is normal iff $\varnothing\to X$ satisfies the LLP with respect to $g$ in his paper "A diagram chasing formalisation of elementary topological properties" page 7. May 19, 2021 at 13:11

The answer by KP Hart to extending disjoint open subsets of a normal Hausdorff space shows that the characterisation fails as stated for normal Hausdorff spaces but might work for hereditarily normal Hausdorff spaces.

Let $$C$$ be a closed subset of a normal Hausdorff space $$X$$. Any two open disjoint subsets $$U$$ and $$V$$ of a closed subset $$C$$ of $$X$$ (i.e. $$U$$ and $$V$$ are open in $$C$$) can be "extended" to disjoint open subsets $$U'\supset U'$$ and $$V'\supset V$$ of $$X$$ >such that $$U=U'\cap C$$ and $$V=V'\cap C$$, and $$U'\cap V'=\emptyset$$.

$$C \to X \rightthreetimes \{ U < x > V \}\to \{U=x=V\}$$

Let $$C$$ be a closed subset of a normal Hausdorff $$X$$. For any two closed subsets $$A'$$ and $$B'$$ of $$X$$, any two open subsets $$U\supset A$$ and $$V\supset B$$ of $$C$$ separating $$A=A'\cap C$$ and $$B=B'\cap C$$, i.e. $$U\cap V=\emptyset$$, there exist open subsets $$U'\supset A'$$ and $$V'\supset B'$$ of $$X$$ "extending" $$U$$ and $$V$$, and separating $$A'$$, and $$B'$$, i.e. $$A'\subset U'$$, and $$B'\subset V'$$, and $$U=U'\cap C$$, and $$V=V'\cap C$$, and $$U'\cap V'=\emptyset$$.

$$C \to X \rightthreetimes \{ a > u < x > v < b \}\to \{a > u=x=v < b \}$$

In fact, being hereditarily normal is also a lifting property :

Recall hereditarily normal means that any two separated subsets $$A$$ and $$B$$, i.e. such that there are open neighbourhoods $$U\supset A$$ and $$V\supset B$$ such that $$U \cap B= A \cap V= \emptyset$$, are separated by neighbourhoods $$U'\supset A$$ and $$V'\subset B$$, i.e. $$U'\cap V'=\emptyset$$.

Let us write the LLP in two notations, < and $$\to$$. Recall that our convention is that $$\{o is the same as $$\{o\rightarrow c\}$$, and here o is open and c is closed.

$$\emptyset \to X \rightthreetimes \{ x > au \approx u' > u > uv < v < v'\approx bv < x \} \to \{ x > au \approx u' = u > uv < v = v'\approx bv < x \}$$

$$\emptyset \to X \rightthreetimes \{ x \leftarrow au \leftrightarrow u' \leftarrow u \leftarrow uv \rightarrow v \rightarrow v'\leftrightarrow bv \rightarrow x \} \to \{ x \leftarrow au \leftrightarrow u' = u \leftarrow uv \rightarrow v = v'\leftrightarrow bv \rightarrow x \}$$

(In this notation, $$U$$ corresponds to the preimage of $$\{au,u,uv\}$$ and $$\{au,u',u,uv\}$$, i.e. the subsets of points whose notation contains letter $$u$$, and similarly for $$v$$. Letter $$x$$ stands for points of $$X$$ "in general position", i.e. outside of $$U$$ and $$V$$.)

This LLP also holds for a closed inclusion into a hereditarily normal space. Hence, this should be enough to conclude that a colimit of closed inclusions into a hereditarily normal space is also a closed inclusion into a hereditarily normal space.

• I would vote this up, but I just cannot stand the notation. Jul 8 at 19:14


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# A Lisp version of the Schwartzian transform

Randal Schwartz stole Lisp's decorate-sort-undecorate idiom when he came up with what we'd later call the Schwartzian transform. All of my Lisp knowledge disappeared through nine half lives, but I gave it a shot in SBCL. I'm aiming for an idiomatic example I can use in a presentation about the history of the Schwartzian transform.

This works (and I know I can generalize it further), but I wonder how Lisp-y it is:

(require :sb-posix)
(defun schwartzian-files-mtime ( glob-pattern )
(map
'list
#'cdr
(stable-sort
(map
'list (
lambda (x) (
cons
(sb-posix:stat-mtime (sb-posix:stat x))
x
)
)
(directory glob-pattern)
)
#'<
:key
#'car
)
)
)

(schwartzian-files-mtime "/etc/*")

• Welcome to codereview! I hope you get some nice answers ! More, feel free to visit our Perl section and answer whatever question might look interesting for you – Grajdeanu Alex. Aug 11 '16 at 11:37
• It's also not spelled LISP the same way that it's not spelled PERL. – ferada Aug 11 '16 at 13:32
• @ferada LISP is LISt Processing. Perl is a name which is not an acronym. Often, LISP is capitalised as Lisp but I would posit that as wrong. – cat Aug 11 '16 at 14:42
• Which LISP? The original LISP? This looks kinda like Scheme but more like CL. Is it CL? Which interpreter do i use to run it, and why isn't this confusing to anyone else? there are like, 10 different kinds of "real" LISP. – cat Aug 11 '16 at 14:44
• @cat (map 'list ...) is certainly CL. Have you missed the I gave it a shot in sbcl in the quesiton? – Will Ness Aug 11 '16 at 16:39

## 2 Answers

What is not enough LISP-y in your function is the style used for the parentheses (they should not be left alone as last element of a line and they should be on the last line of the enclosed form). Another minor point is that you can use the shorter mapcar instead of map 'list.

Here is a version of your function (generalized for any function applied to the data and any predicate) which follows the more conventional style used in Common Lisp.

(defun schwartzian-transform (list costly-function predicate)
"sort a list of objects over the value of a function applied to them,
by applying the Schwartzian Transform (https://en.wikipedia.org/wiki/Schwartzian_transform)
the parameters are the list, the function, and the predicate for the sort."
(mapcar #'cdr
(stable-sort (mapcar (lambda (x)
(cons (funcall costly-function x) x))
list)
predicate
:key #'car)))

(require :sb-posix)
(schwartzian-transform
(directory "/etc/*")
(lambda (x) (sb-posix:stat-mtime (sb-posix:stat x)))
#'<=)

• No documentation string? – Gareth Rees Aug 11 '16 at 13:06
• Nice transformation! The only thing I would change is to add space after lambda, although that's a small one. – MatthewRock Aug 11 '16 at 13:25
• @MatthewRock, thanks! Sometimes I miss that space. – Renzo Aug 11 '16 at 13:27
• @WillNess, in ANSI Common Lisp both forms are equivalent. In the Google Common Lisp Style Guide both forms are allowed. I prefer the form without the #' since it is shorter but still very clear. – Renzo Aug 11 '16 at 20:45
• Yes I know they're equivalent, I had that impression about the style... Thanks for the link. – Will Ness Aug 11 '16 at 23:22

What's certainly extremely very non-LISPy is your opening parens ending the line of code like that. With that, mapcar and explicit #' in front of lambda, it could look like

(defun schwartzian-files-mtime ( glob-pattern )
(mapcar #'cdr
(stable-sort
(mapcar #'(lambda (x)
(cons (sb-posix:stat-mtime (sb-posix:stat x))
x))
(directory glob-pattern))
#'<
:key #'car)))

This looks natural to me, but still may be off w.r.t. official style guidelines. It has a "quick-n-dirty" feel to it, like something you'd scribble on the spot (not unlike Randal's original piece of code), not a proper production code with documentation string etc..


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At this page I want to present the Thomson Ring Experiment using a capacitor as energy source (Fig. 278 556 556 556 556 556 556 556 556 556 556 278 278 584 584 584 556 1015 667 667 Newton's ring experiment mainly contains following questions… Q.What are Newton’s Rings? endobj Date of experiment: 16/03/2012 Date of report: 30/03/2012 Investigation of Interference Patterns and Newton’s Rings Experiment conducted by Sultan LeMarc. Objective – To determine the wavelength of sodium light using Newton’s ring method. In this diffraction grating experiment you will observe a pattern of diffracted light. Study Resources. The errors and various causes in the experiment of measuring lens curvature diameter using Newton ring are analyzed and discued.The chief source of errors and fairly perfect and practical meth eds of treatment are presented. 278 278 556 556 556 556 556 556 556 584 611 556 556 556 556 500 556 500] /Name/F6 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 /Type/Encoding Physclips provides multimedia education in introductory physics (mechanics) at different levels. If the radius of curvature of the lens is 100 cm, find the wave length of the light. <> /Name/F1 stream /Subtype/Type1 /FontDescriptor 21 0 R 7 0 obj In case if they are not able to complete, they should get attestation from the staff in the record note book for the readings taken 3 0 obj For air . A central dark fringe can be located between the glass plate and a lens. 444 1000 500 500 333 1000 556 333 889 0 0 0 0 0 0 444 444 350 500 1000 333 980 389 At the point of contact the thickness of air film (i) The Newton’s ring experiment can be also used to find the wavelength of a monochromatic light. Diffraction Pattern Measurements using a Laser From Eq. After refraction and reflection two rays 1 and 2 are obtained. What the . /FontDescriptor 9 0 R Performing the simulation: The … The rotational inertia of a ring with consistent density is dependent of its mass and the inner and outer radius. In my experiments I mostly used a 150uF capacitor bank charged up to several kVs. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 /Filter[/FlateDecode] The main piece of apparatus for this experiment is a force table, that is, a horizontal disk with degree markings on it. 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 Feel free to Subscribe and please do hit the thumbs up button ... Newton's Ring Experiment - Duration: 3:28. These rays interfere each other producing alternate bright and dark rings. >> As we know that in the newton's rings the central fringe is always dark in the reflected system, is there any method by which we can obtain the central fringe as bright in the reflected system? 16 0 obj The relationship between the mass and the radii is described as . Consider light of wave length 'l' falls on the lens. Figure 2: Two microscope slides are stacked to form an 'air wedge', which produces dark and light fringes (top). Then the specific rotation of the sugar solution, from the formula is; S = 10θ / LC. �:t{D��09v���3����O��8��=�T���q�p�[&�t,���-�/Z:Q�=Vʞ�O~(}K���D���uY��hw�g��:�h�ҺC�{��Y�Fei 722 1000 722 667 667 667 667 389 389 389 389 722 722 778 778 778 778 778 570 778 556 333 1000 556 556 333 1000 667 333 1000 0 0 0 0 0 0 333 333 350 556 1000 333 1000 /Encoding 7 0 R 889 667 611 611 611 611 333 333 333 333 722 722 722 722 722 722 722 564 722 722 722 Light, interference, thin films. 500 500 500 500 500 500 500 564 500 500 500 500 500 500 500 500] So, rotation produced by the sugar solution = X-Y = 15 0 [15 degrees is the difference, its interpretation is important.]. 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 The experiment is performed when there is an air film between the plano-convex lens and the optically plane glass plate.The diametre of the m th and the (m+p) th dark rings are determined with the help of a travelling microscope. Physics with animations and video film clips. =R2¡2tR+t2+r. <> Then wavelength is calulated using equation. 4 0 obj The infinitesimal area of each ring $$dA$$ is therefore given by the length of each ring ($$2 \pi r$$) times the infinitesimmal width of each ring … EXPERIMENT No. Let the radius of curvature of the convex lens is R and the radius of ring is 'r'. �����mR�l�\$U�+b��~wX��h�\�!�8b-�;q@9V��Z�ٷ��&~�qw�q�0�J��@�#��9��1�=\�����l��>���'�p��8 ˬ[K�^#s���y��E]���v=Vh�*��R��KqX-�2�op�J�,G5c�!SN8�g8�P:vx�����|(tٻ�o�s��V���o��� Calculations. Figure 8.3:Measure the diameter of the central ring (Dc) and the positions of the rings on the left hand (L1to LM) and right hand sides (R1to RM). 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 After charging up the capacitor C to a certain voltage U 0, the capacitor will be discharged through a flat/spiral coil. and Ring Purpose: The purpose of this experiment is to experimentally find the rotational inertia of a ring and a disk and to verify that these values correspond to the calculated theoretical values. In this experiment, we will measure the stopping potential with modern electronics. 14/Zcaron/zcaron/caron/dotlessi/dotlessj/ff/ffi/ffl/notequal/infinity/lessequal/greaterequal/partialdiff/summation/product/pi/grave/quotesingle/space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/backslash/bracketright/asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/braceleft/bar/braceright/asciitilde Pulleys may be clamped to the edge of the disk at any desired position. /LastChar 196 The diode has a window which allows light to enter, and the cathode is a clean metal surface. endobj O is the center of the circle. /BaseFont/THRFZE+CMEX10 4. 833 556 556 556 556 333 500 278 556 500 722 500 500 500 334 260 334 584 0 0 0 222 22 0 obj Newton’s Rings Experiment Aim: To determine the wavelength of monochromatic light (sodium light) using. 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 /BaseFont/PDPLMD+NimbusRomNo9L-Medi 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 PY2107 Newton’s Rings Experiment 5 _____ 2.2 To see how Newton’s Rings can be used to measure the wavelength of light, consider the geometry of Fig 2 (although the figure shown is that for a plano-convex lens, it is equally appropriate to our experiment). /BaseFont/YJVCKF+CMR12 Now our task is to first find how many divisions of Main Scale are equal to Vernier Scale. Newton' s ring experiment demonstrates interference with a circular object. 13 0 obj 3 6 m m. If the radius of the planoconvex lens is … /FirstChar 1 /BaseFont/RUFDUB+NimbusSanL-Regu I. 500 500 1000 500 500 333 1000 556 333 1000 0 0 0 0 0 0 500 500 350 500 1000 333 1000 278 500 500 500 500 500 500 500 500 500 500 278 278 564 564 564 444 921 722 667 667 Consider light of wave length 'l' falls on the lens. endobj /Name/F4 = 0.1/10 cm. Fig 5 : Newton’s rings Apparatus Newton’s Ring Fringes of Sodium light. The errors and various causes in the experiment of measuring lens curvature diameter using Newton ring are analyzed and discued.The chief source of errors and fairly perfect and practical meth eds of treatment are presented. After refraction and reflection two rays 1 and 2 are obtained. 791.7 777.8] 8 2 m m and that the 1 0 t h ring 3. The diameter of the m th dark ring was found to be 0.28 cm and that of the (m + 10) th 0.68 cm. A ring and a disk. In case if they are not able to complete, they should get attestation from the staff in the record note book for the readings taken The diameter of the m th dark ring was found to be 0.28 cm and that of the (m + 10) th 0.68 cm. /Name/F2 /Widths[333 500 500 167 333 556 222 333 333 0 333 584 0 611 500 333 278 0 0 0 0 0 So, rotation produced by the sugar solution = X-Y = 15 0 [15 degrees is the difference, its interpretation is important.]. Then least count = (1- 9/10) MSD. endobj The circuit forms a harmonic oscillator for current, and resonates in a similar way as an LC circuit. experiment. If the light is incident normally on the lens, r = 0 and near to point of contact is small; therefore near point of contact, Therefore ….2.21. The central element of the apparatus is the photodiode tube. �q^͞��a�F�W����m:���a%��r��5 >> /FontDescriptor 12 0 R The two interfering They should complete the calculation in the lab class itself and get the signature from the staff. 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Fringe can be located between the glass plate this case, the capacitor C to a certain voltage 0... A clean metal surface multimedia education in introductory physics ( mechanics ) at levels! Measure the stopping potential with modern electronics a pattern of diffracted light newton's ring experiment calculation, in which material of index. Patterns, the capacitor will be in seven colors element of the substance ( M /! Diffracted light can be determined from Eqn pattern of diffracted light the diameter of the nth is... Math 111 at Pandit Deendayal Petroleum University convex lens is 100 cm, find the wave '! Dependent of its mass and the cathode is a force table, that is, a horizontal disk with markings! ' R ' to Vernier Scale mass and the radii is described as to several kV .. ( sodium light ) using fringes are observed in the lab class itself and get the signature from formula... Lamp, Travelling microscope, optical arrangement wave length of the disk at any desired.. 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With which the lens is R and the cathode is a force table, that is a! Are observed in the figure: R is the radius of curvature the... At Pandit Deendayal Petroleum University to several kV ` s de is radius! Consistent density is dependent of its mass and the inner and outer radius central dark spot which. In introductory physics ( mechanics ) at different levels of monochromatic light ( sodium light using ’... A horizontal disk with degree markings on it as an LC circuit 1- )!: Newton ’ s Rings experiment Aim: to determine the wavelength of light dark... A harmonic oscillator for current, and the cathode is a force table, is! Is, a horizontal disk with degree markings on it distance into newtons mean! Λ is the difference between the mass and the radii is described as formula! Present experiment table, that is, a horizontal disk with degree markings on it of apparatus this. Similar way as an LC circuit diameter of the convex lens is contact... Duration: 3:28 degree markings on it ) at different levels is ; s 10θ! As, mechanics ) at different levels the rotational inertia of a ring the... Lens slightly upwards in contact as thin lm having =1, then Eq Quinkes method slides are stacked form...


text:
## Friday, December 24, 2010 ... //

### Merry Christmas

New articles are being posted below this one, so don't stop reading...

Merry Christmas to the TRF community!

Needless to say, I am not wishing any good Christmas to the environmentalists because it would be insulting for them. According to environmentalists, Everything that is wrong with humanity [...] is summed up in Christmas. Among other things, it is a turkey genocide day, we heard from Ethanol Greenfart, before he recycled his granny and wrote a book about it. ;-)

Yes, I mean a White Christmas. In 2010, global warming caused exactly the same winter as we have always been used to. So enjoy it. ;-)


text:
# NATS Messaging - Part 9

Yesterday we made some quite minor changes to our app and got it to use JetStream on both the Producer and Consumer side. These changes solved several problems for us, like being able to restart Consumers without losing any messages.

The last remaining issue was around handling messages that fail to be delivered. Imagine the case where our disk is full on the Consumer, wouldn’t it be great if we can somehow communicate our inability to handle messages to the network and have it retry later?

That’s the role of Acknowledgements and JetStream supports several modes. Today we’ll look at those.

## Basic per-message Acknowledgements

On the basic level, what would probably be sufficient for our blog series is to Acknowledge every logline. This approach fails at large scale, but as we’re just toying around and exploring approaches, this is perfectly fine.

Before getting into the detail I’d like to show while there are much to keep in mind, Acknowledgements can be very easy by adding them to our Log Consumer.

The Consumer has a loop like this today:

for {
select {
case m := <-lines:
err = handleLine(directory, output, m)
if err != nil {
log.Printf("Could not handle line %q: %s", string(m.Data), err)
}

case <-ctx.Done():
// not shown
}
}

So we have a good place to know when message processing failed inside our case statement. At the basic per-message case, we would change this to:

for {
select {
case m := <-lines:
err = handleLine(directory, output, m)
if err != nil {
log.Printf("Could not handle line %q: %s", string(m.Data), err)
continue
}

m.Respond(nil)
case <-ctx.Done():
// not shown
}
}

By adding these 2 lines, we will Ack only successfully handled messages, ones that fail are retried later.

## Acknowledgement Modes

The devil is in the details though, so let’s take a quick look at what JetStream supports.

In a high-performance work queue, you have to be able to give up on a message and move to the next one. And your Acknowledgements must be optimised for lowest possible network round trips.

JetStream gives us several ways to optimise this often very costly part of Message Orientated architectures.

### No Acks

So far this is what we’ve been doing, when we configured the Consumers, we explicitly disabled Acknowledgements and just YOLO’d our data to disk, this is perfectly fine when just casually observing systems or with data that gets re-published very frequently.

### Explicit Acks

You might choose, as above, to Acknowledge every single message. This approach is great for Work Queues and the like where you are not looking at receiving thousands of messages a second. A message that has not been Acknowledged within a configured time gets redelivered later.

### All Acks

This mode is suited to high throughput applications. You’d keep track of messages sequences and if you’ve successfully handled 100 messages, you can Ack just the 100th message. It will assume you have handled 1-99 correctly. This is a significant performance improvement and overhead reducer.

Logs can be slow at times, so you probably want to have an additional timer that checks regularly and does an Acknowledgement. Say you Ack only every 100 messages which at busy times is every 10 seconds. But when not much is happening, it could be hours to receive 100 messages. So the timer would check every minute and Ack regardless.

## Types of Acknowledgement

JetStream supports a few types of Acknowledgement to assist with controlling how and when retries happen.

### Ack

This mode is what we saw before, msg.Respond(nil) or msg.Respond(0), and the server knows the message was successfully processed. Very simple and good for most use cases

### Nak

You might receive a message that you are not interested in or would rather send back to let others handle it. Typically you would have to wait for the redelivery period - often many minutes - for the Ack to timeout. In that time you might not be given a new message. By sending a Nak, you tell JetStream you are not going to handle this message and to move on to the next one. This Ack is done via msg.Respond([]byte("-NAK)).

### Progress

The opposite of a Nak, here you might have 30 seconds to process a message, but as you get up to 25 seconds, you might still be retrying to access a remote service. You can then do msg.Respond([]byte("+WPI+)) and get another 30 seconds to handle the message.

### Next

This one is probably quite niche but can give you a noticeable boost under very high throughput scenarios. You can send the Acknowledgement and in the same packet ask JetStream to send you the next message without having to do another Pull operation.

// wait for messages on q
ib := nc.NewRespInbox()
q := make(chan *nats.Msg)
nc.ChanSubscribe(ib, q)

for m := range q {
// process the message

// publish to the message reply target a +NXT acknowledgement asking
// for a message on ib which will be placed in q
nc.PublishRequest(m.Reply, ib, []byte("+NXT"))
}

This mode applies only to Pull Consumers, and it means you can Acknowledge+Pull in one packet rather than 2, on a busy system this is a significant optimisation.

## Redelivery

We mentioned earlier that Redelivery is based on a timeout. When you add the Consumer, you tell it to allow 30 seconds for a message to be processed using --wait=30s. What you can also specify is how many Redelivery attempts the system should make; this is important because you might have a corrupt message that crashes all your consumers. This message - often called a poison pill - can affect a DOS against everything.

Setting a Maximum Delivery limit of 10 lets these messages expire and not crash your system forever. Maximum Delivery is set using --max-deliver=10 when adding a Consumer.

The question is then what to do with these messages and how to debug them. In a traditional middleware, there is typically something called a Dead Letter Queue that holds the actual expired messages. This DLQ does not make sense in a Stream since the message might be consumed by 100s of Consumers and only 1 of them might be failing. We cannot remove the message, and we do not want to duplicate the message.

JetStream publishes various advisories about its lifecycle, and you can view these using nats events, this is how it looks when a message exceeds it’s delivery attempts:

$nats events Listening for Advisories on$JS.EVENT.ADVISORY.>
[20:36:28] [I7S817vOvH9NfO2MYJV8pE] Delivery Attempts Exceeded
Consumer: WQ > WORK
Stream Sequence: 5
Deliveries: 2

Under the covers this is a JSON message published to $JS.EVENT.ADVISORY.MAX_DELIVERIES.<stream name>.<consumer name>: { "schema": "io.nats.jetstream.advisory.v1.max_deliver", "id": "I7S817vOvH9NfO2MYJV8lD", "timestamp": "2020-04-02T20:32:18.898715Z", "stream": "WQ", "consumer": "WORK", "stream_seq": 4, "deliveries": 2 }  This message is effectively a pointer to the message in the source Stream. I would suggest you always set up a stream that consumes $JS.EVENT.> and keep those for a few hours. Using that Stream, you can always get a view of what is going on in your account.

We can view the message in the Source Stream:

\$ nats stream get WQ 5
Item: WQ#5 received 2020-04-02 22:36:16.603899 +0200 CEST on Subject workq.jobs

{"job":"hello world"}

In this way, we support the traditional DLQ pattern while keeping the Streams intact and not duplicating the data.

## Conclusion

Message Acknowledgements is an essential and complicated topic, knowing which to choose for your case depends on many factors.

By applying these patterns to our log shipper, we have achieved all the goals that we set out to solve.

• It can handle a very large number of log senders by being horizontally and vertically scalable
• Ordering is maintained while being horizontally scalable
• It’s reliable, and we do not lose messages when disks go full

There is one last thing I’d like to explore, and that is a matter of overall System safety.

When thousands of nodes are publishing messages, the network is always under quite a lot of pressure, and it’s easy for a system event to overwhelm your network.

We might want to have a Big Red Button we can press to Pause message production for some time to give us a way to reduce the load on networks or transit. I’ll look at patterns for this next week.


text:
# How do you find the slope that is perpendicular to the line y=(-3x)?

Jul 18, 2016

Slope of the perpendicular: $\textcolor{g r e e n}{\frac{1}{3}}$

#### Explanation:

The standard general form of a linear equation in slope-intercept form is:
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} x + \textcolor{b l u e}{b}$
with slope $\textcolor{g r e e n}{m}$ and y-intercept $\textcolor{b l u e}{b}$.

The given equation $y = \left(- 3 x\right)$
can be re-written to be in explicit slope-intercept form as:
$\textcolor{w h i t e}{\text{XXX")y=color(green)(} \left(- 3\right)} x + \textcolor{b l u e}{0}$
with slope color(green)(""(-3))

If a line has a slope of color(m)
then all lines perpendicular to it have a slope of color(green)(""(-1/m))

Since $y = - 3 x$ has a slope of $\textcolor{g r e e n}{m = - 3}$
any line perpendicular to it will have a slope of
color(white)("XXX")color(green)((-1/m)=-(1/(-3))=1/3


text:
Chapter 105

Toxic chemicals in the environment are known to produce a wide array of adverse effects in the heart and cardiovascular system. These harmful effects include hypertension, peripheral vascular injury, alterations of lipid levels, disturbances of cardiac rhythm, cardiomyopathy, acute myocardial infarction (MI), stroke, and sudden death. Toxic chemicals are also linked to cardiovascular disease risk factors, including obesity, insulin resistance, the metabolic syndrome, and diabetes. This chapter describes these effects.1

Beyond the known cardiotoxins, there likely are other chemicals in the modern environment whose toxicity to the heart and cardiovascular system has yet to be recognized. These cardiotoxins will be discovered among the tens of thousands of new synthetic chemicals invented and produced in the past half century. The majority of these materials have not been tested for toxicity, which is a serious failure of stewardship by the chemical industry and the US federal government.2,3 These synthetic chemicals are widely disseminated in the human environment, and national surveys conducted by the Centers for Disease Control and Prevention (CDC) find measurable levels of several hundred synthetic chemicals in the blood and urine of virtually all Americans.4

Given the continuing potential today for exposure to new chemicals of unexamined toxicity, every physician is advised to obtain a brief history of occupational and environmental exposure from every patient and to ask more detailed follow-up questions or seek consultation with a specialist in occupational and environmental medicine if the initial screen raises suspicion of toxic exposure. Straightforward screening algorithms have been developed.5 The most intense exposures typically occur in occupational settings, and most discoveries of new associations between toxic chemicals and disease have been made by physicians examining working populations.6 The possibility of etiologic discovery is high.

Cardiovascular disease caused by toxic agents in the environment is preventable. Prevention requires diagnostic recognition of the links between exposure and disease followed by evidence-based interruption of exposure.7

Lead is a heavy metal, a chemical element that comprises 0.002% of the Earth's crust. It has a low melting point, is easily molded and shaped, and can be combined with other metals to form alloys. Because of these properties, lead has been used by humans for millennia and is used today in products as diverse as pipes, storage batteries, pigments, glazes, vinyl products, weights, shot and ammunition, cable covers, and radiation shielding. From the 1930s to the 1970s, lead was used extensively as a gasoline additive to improve engine performance.8 Lead is widespread in the modern environment. Global consumption of lead continues to increase mainly because of rising demand for large and small batteries.9

Patients may be exposed to lead by either inhalation or ingestion. Inhalation is the most common route of adult exposure and the most serious exposures occur among workers exposed occupationally.10 Workers at greatest risk include smelter and foundry workers, hazardous waste ...

Sign in to your MyAccess profile while you are actively authenticated on this site via your institution (you will be able to verify this by looking at the top right corner of the screen - if you see your institution's name, you are authenticated). Once logged in to your MyAccess profile, you will be able to access your institution's subscription for 90 days from any location. You must be logged in while authenticated at least once every 90 days to maintain this remote access.

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text:
J. Korean Ceram. Soc. > Volume 29(12); 1992 > Article
 Journal of the Korean Ceramic Society 1992;29(12): 926.
 아세틸렌 불꽃에 의한 다이아몬드 합성 이윤석, 박윤휘, 이태근, 정수진 서울대학교 무기재료공학과 Diamond Synthesis by Acetylen Flame ABSTRACT Uniform diamond films in a few $textrm{mm}^2$ size and locally isolated diamond single crystals in size of 60 $mutextrm{m}$ were synthesized on Si-wafer and Al2O3 substrate by the method of acetylene flame. The effects of substrate temperature and flow ratio of oxygen to acetylene on the morphology of deposited diamond were investigated. According to the observations of growth behavior of diamond on Si substrate with respect to substrate surface pretreatment and flow ratio, it was shown that well faceted diamonds could grow uniformly when flow ratio was above 0.9 and substrates were densely scratched. With increasing substrates temperature, the crystal morphology changes from octahedron bounded by only {111} plane below 850$^{circ}C$ to cubo-octahedron with almost equal development of {111} and {100} plane in the temperature range of 850∼950$^{circ}C$. Between 950∼1050$^{circ}C$, the {111} faces become rough and concave. Above 1050$^{circ}C$, new crystallites begin to grow on concave {111} surface and overall morphology looks like cubo-octahedron with degenerated {111} faces. These changes of morphology can be understood in terms of the different growth mode of each crystallographic plane with respect to the substrate temperature and supersaturation. And the observed phenomena on {111} planes can be related to the face instability and twin generation.
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# Particle in a 1-d potential/ Fokker-Planck(FPE)

1. Apr 6, 2013

### Abigale

Hi guys,
I have a Problem.
I regard a brownian particle in a potential $U(z)=gz$ with $f=-\frac{d}{dz}U$ and it is connected to a heat reservoir.
The particle is in the interval $z>0$.

I regard the equilibrium-state.
So I got the FPE.

$$\frac{\partial}{\partial t} P_{1}(z,t) = -\frac{\partial}{\partial z}[\Gamma~f(z)P_{1}(z,t)] + \frac{D}{2} \frac{\partial^{2}}{\partial z^{2}} P_{1}(z,t) =0$$

$$\Rightarrow -\Gamma~f(z)P_{1}(z,t) + \frac{D}{2} \frac{\partial}{\partial z} P_{1}(z,t) =0$$

The solution is:
$$P_{eq}(z,t)=\frac{2\Gamma g}{D}\exp{\lbrace\frac{-2\Gamma g z}{D} \rbrace}$$

Please check if the following callculations are correct?

Now I regard the situation:

At a certain time $t_{0}$ opens a door, which increases the first volume.
The volume ranges now from $-z_{0}$ to $+ \infty$. ($z_{0}>0$)
But I am just interested in the equilibrium state of this situtaion.

I think the FPE must have the same form like the first case. So again:

$$\frac{\partial}{\partial t} P_{2}(z,t) = -\frac{\partial}{\partial z}[\Gamma~f(z)P_{2}(z,t)] + \frac{D}{2} \frac{\partial^{2}}{\partial z^{2}} P_{2}(z,t) =0$$

$$\Rightarrow -\Gamma~f(z)P_{2}(z,t) + \frac{D}{2} \frac{\partial}{\partial z} P_{2}(z,t) =0$$

$$\Rightarrow \Gamma~gP_{2}(z,t) + \frac{D}{2} \frac{\partial}{\partial z} P_{2}(z,t) =0$$

$$\Rightarrow \Gamma~g dz = - \frac{D}{2} \frac{1}{P_{2}(z,t)} d(P_{2}(z,t))$$
$$P_{2}=C\exp{\lbrace \frac{-2\Gamma g z}{D} \rbrace}$$

Now for normalisation:
$$\int \limits_{-z_{0}}^ {+ \infty} P_{2}(z,t) ~dz =1$$

Can this be true? Is the range of integration, the only thing i have to change for this situation?

THX
and sorry for the long letter.
Bye
Abby


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Profinite topologies

We can define two topologies on a group $G$ by considering all normal subgroups of finite index (resp. of index a finite power of $p$ - where $p$ is a prime) as basis of $1\in G$.

My questions: Under what conditions these two topologies coincide? What are the usual approaches to check whether they coincide in a given f.g. group?

To coincide is equivalent to the following. For each normal subgroup $N$ of finite index, there exists a normal subgroup $P$ of index a finite power of $p$ such that $P\subseteq N$.

So, a trivial situation is when each finite quotient of $G$ is a $p$-group.

Thanks in advance for any ideas! Explicit examples are very welcome too.

-
Clearly, if $P\subseteq N$, then the index of $P$ in $G$ is divisible by that of $N$ in $G$. So your equivalent condition is satisfied if and only if each finite quotient of $G$ is a $p$-group. –  Alex B. Nov 20 '10 at 5:52


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# Computing the logarithm of an exponentiated matrix?

Let $\mathfrak{g} \subseteq \mathfrak{su}(n)$ be a linear Lie algebra represented by skew-symmetric $n\times n$ matrices. Let $C \in SU(n)$ be a special unitary matrix where it is known that $C = exp\{G\}$ for some $G \in \mathfrak{g}$.

Given $C$ and a basis $\{G_k\}$ for $\mathfrak{g}$, suppose we wish to find a matrix $G$ expressed in terms of its real basis coefficients $\vec{a}$:

$$G = \sum a_k G_k$$

such that $C = \exp\{ G\}$.

If more than one such matrix exists, we wish to find the one where $\sum |a_k|^2$ is minimized.

I know that if we diagonalize $C = P^{-1}\Lambda P$ where $$\Lambda = \left( {\begin{array}{*{20}{c}} {{\lambda _1}}&{}&{}\\ {}& \ddots &{}\\ {}&{}&{{\lambda _n}} \end{array}} \right)$$

then is suffices to find the vector $\vec{a}$ and set of integers $\{z_k\}$ such that

$$\sum a_k PG_kP^{-1} = \left( {\begin{array}{*{20}{c}} {\log {\lambda _1}}&{}&{}\\ {}& \ddots &{}\\ {}&{}&{\log {\lambda _n}} \end{array}} \right) + 2\pi i\left( {\begin{array}{*{20}{c}} {{z_1}}&{}&{}\\ {}& \ddots &{}\\ {}&{}&{{z_n}} \end{array}} \right)$$

where $\log \lambda$ is the (imaginary) principle log of $\lambda$.

But while it's possible to iterate through guesses for the $z$'s, checking to see whether the equation is solvable, this is a terribly inefficient way of solving the problem.

Is there a faster, more scalable way to find $\vec{a}$ for a given $C$? Does this problem have a name?

• Why is it important that you work in some fixed basis of $\mathfrak{g}$? If $C = PDP^{-1}$ then $\log C = P (\log D) P^{-1}$ (meaning, every possible value of $\log D$ has the property that $P (\log D) P^{-1}$ exponentiates to $C$), so you don't need to find anything once you've diagonalized $C$. – Qiaochu Yuan Jan 15 '18 at 22:28
• @QiaochuYuan: This is part of a longer procedure where the $G$ matrix is used to compute a cost, and must be in $\mathfrak{g}$. Generally speaking, $\log C$ is not in $\mathfrak{g}$. When I say "It is known that $C = \exp\{ G\}$ for some $G \in \mathfrak{g}$", I don't mean to imply that $G$ is known a priori. $C$ is generated through complex dynamics, and the existence of $G$ is guaranteed by other factors. – COTO Jan 16 '18 at 0:19
• Your last formula is valid only when the $\log(\lambda_i)$ are distinct. – loup blanc Jan 19 at 8:20
• "Skew-symmetric"->"skew-Hermitian"? – Dap Jan 19 at 9:17

We assume that the $$(\log(\lambda_i))$$ are distinct, cf. my comment above.
For every choice of the $$(a_k)_{k\leq p}$$, $$K=\sum_k a_kPG_kP^{-1}$$ is skew-hermitian. Then the main task to do is to seek the $$(a_k)$$ s.t. $$K$$ is diagonal. We must solve a homogeneous LINEAR system of $$n^2-n$$ real equations in the $$p$$ unknowns $$(a_k)$$.


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# Scheme related work.

## 2 Abstract

This page will describe the project that I undertook in 2019-2020, with the goal of modernising computer science education. Its purpose is to collect in one place links to all artefacts that were produced during the project execution, with commentary.

The object of study was one of the most famous programming problem textbooks, the "Structure and Interpretation of Computer Programs", Second Edition, by Abelson, Sussman and Sussman. At the moment of writing of this document, 24 years have passed since the release of SICP, and many things may change in computing in such a time frame. Hence, apart from looking at the book itself, it was interesting how much effort would be additionally required in order to replicate the experience of the era when the book was still fresh.

This documents tells the story of this endeavour. Initially, there was no plan for making any kind of comprehensive post-mortem of the project. However, the project turned out so long, so labour-intensive, and eventually, so fruitful, that making a summary has arisen to be necessary, even if for myself and my readers only, and only to list artefacts.

In short, I have solved all of the SICP's problem set in a consistent and reproducible fashion, using modern software, filling the gaps in supporting software and libraries, and documenting as much of my work as possible, in particular how much time every problem required, and how much external help was requested.

This document is under development.

## 3TODO The Story [0/2]

### 3.1 Time Management Video Tutorial

In order to find time, as a working person, I needed a more systematic approach to time management. Since I had already chosen org-mode as the solution medium, using org's time-management facilities was an obvious choice.

Following the general principle that learning something is the easiest when you are teaching it, I organised a seminar/lecture. This lecture was well-received by the audience, because it was given during the time of the corona-virus pandemic of 2020, so quite a lot of people found themselves locked in their home, and in the need to structure their time more efficiently.

This approach is not unlike the one outlined by John Lees-Miller (https://github.com/jdleesmiller) in his article "How a Technical Co-founder Spends his Time".

There is no presentation file, because it was a hands-on tutorial given right into the time management software.

### 3.2 SICP Solution

In short, I have solved all of the problem set, measuring how much time every problem required. Furthermore, I had to write and publish several libraries, in order for the solution to be runnable on modern Schemes, and be as portable between them, as possible.

#### 3.2.1 Solution

Eventually, there were two artefacts produced from the solution of SICP:

The org-version is strongly preferred to the pdf version, because the pdf version requires certain uncanny tricks to get built, has no added value and is 5000 pages long.

To use the org-mode one, you need chibi-scheme of a sufficiently recent version, ImageMagick, as well as GNU Fortran for the last two exercises. Emacs is also strongly recommended.

#### 3.2.2 Schemetran

During the solution process, one of the exercises requires writing a Scheme interpreter in "a low level language of your choice". In my case this choice happened to be Fortran, partly due to a relatively greater popularity of Fortran in 1996, partly due to relatively straightforward memory management. It is a toy implementation, not recommended for any serious applications. It's likely leaking memory and compares symbols in $$O(n)$$.

#### 3.2.3 SRFI 203: A Simple Drawing Language in the Style of SICP

Scheme Requests for Implementation is the Scheme's equivalent of XEPs or PEPs or JCPs. In order to make possible working with graphics in Scheme, I had to implement several interfaces assumed to be "given" in SICP. The graphics sub-library found its place as SRFI-203:

#### 3.2.4 SRFI 216: SICP Prerequisites (Portable)

I also had to write a support library that packages the functions that are already available on modern schemes. (Unlike SRFI-203, which implemented an unportable subset.)

There review process has been started on 2020-11-15. There is also a Lay Summary in English, and in Russian.

### 3.3TODO ICFP Paper

Since this took so long and made me think that much, I decided to analyse the solution process and to document it for future reference. The result of this analysis ended up being substantive enough for a whole paper, and was later presented at ICFP 2020, Scheme Track.

The papers were "published online", which means that ACM is going to maintain the website with a lay summary for a while, and hope that the papers will be mirrored by the major publishers… I guess.

In any case, below you can find the:

### 3.4TODO ICFP Presentation

Every "scientific" paper nowadays needs a supplementary paper to explain what it is actually about. This report is not an exception.


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This applet created by prof Hwang
Modified interface by Ahmed
Applet explain A mass m attached to a vertical spring (spring constant k) in gravity field:
The above system can be described with
$F=m a_y= mg -ky -b v_y or m\frac{d^2y}{dt^2}+b\frac{dy}{dt}+ky=mg$

For a RLC circuit with DC source Vc:
The above system can be described with
$Vc=V_L+V_R+V_C or L \frac{d^2Q}{dt^2}+I\frac{dQ}{dt}+\frac{Q}{C}=Vc$,
where $I=\frac{dQ}{dt}, V_R=I R, V_C=Q/C , V_L=L\frac{dI}{dt}$

The differential equation are the same for the above two systems.
So a damped spring system can be simulated with RLC circuit (or RLC circuit can be simulated with damped spring system,too!).
Original project [url=http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1248.0]Similarity between RLC circuit and spring with damping [/url]

/htdocs/ntnujava/ejsuser/37048/users/br/ahmed/springRLC_pkg/springRLC.propertiesFull screen applet or Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list


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Question

# The metal that can be obtained by electrolysis of an aqueous solution of its salts is :

A
Ag
B
Cr
C
Mg
D
Ca

Solution

## The correct option is A $$Ag$$Ag (silver) can be obtained by electrolysis of the aqueous solution of their salts. Mg, Cr and Cu cannot be obtained by electrolysis of the aqueous solution of their salts, because Mg, Cr and Cu are more electropositive in nature.Hence, option A is correct.Chemistry

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~ Nonlinear Adaptive Filtering as a Form of Artificial Intelligence ~
 Home History References Contact

## Mushroom Classification Example

Demo project, code and data code and data
This DEMO is simple comparison of discrete Urysohn model with Classification Tree. The data file is available for 30 years. It is mushroom classification. The matter of the data is list of quantized features for edible and poisonous mushrooms and the code must train AI to tell edible mushroom from poisonous by provided features for previously unseen mushroom. This is how the data look like:
p,x,s,n,t,p,f,c,n,k,e,e,s,s,w,w,p,w,o,p,k,s,u
e,x,s,y,t,a,f,c,b,k,e,c,s,s,w,w,p,w,o,p,n,n,g
e,b,s,w,t,l,f,c,b,n,e,c,s,s,w,w,p,w,o,p,n,n,m
p,x,y,w,t,p,f,c,n,n,e,e,s,s,w,w,p,w,o,p,k,s,u
e,x,s,g,f,n,f,w,b,k,t,e,s,s,w,w,p,w,o,e,n,a,g

For comparison I used Microsoft.ML library. It runs for about 60 sec (ten folds) and returned 57 errors out of 8124 records. 10 folds means the dataset was divided with ratio 0.9/0.1 into training and testing samples and testing set was rotated 10 times to cover all data. So the execution speed was about 6 sec per run. The Microsoft version can be found here:
https://www.c-sharpcorner.com/article/how-to-do-binary-classification-in-asp-net-core-using-ml-net/ My code with discrete Urysohn model runs for about two seconds, the logical part of the code is about 60 lines, the accuracy is 3 or 4 errors for unseen data out of 8124 for the same 10-fold validation concept.

Conceptually, it is same simple quantized Urysohn operator explained in introduction.


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# Are there winding-number vacua in Weinberg-Salam (Or are they a gauge artifact)?

In pure SU(2) Yang-Mills the vacua van be grouped in homotopy classes labeled by their winding number. Instantons connect these giving rise to the theta-vacuum.

I’m studying the SU(2) sphaleron in the Weinberg-Salam theory without fermions (Original paper by Manton). Which is the same SU(2) Yang-Mills theory with the addition of an SSB Higgs doublet $$\Phi$$. The theory looses its instantons (Derricks theorem) and obtains a sphaleron. (The theory has no soliton monopoles.)

In the paper Manton constructs a non-contractible loop starting and ending at the same vacuum. It is important to note that using his gauge fixing $$A_r = 0$$ (as well as fixing a base point on $$\Phi^\infty$$) there is only a single vacuum left. Once the construction is over he claims that by letting go of $$A_r = 0$$ and imposing some additional new gauge constraints the loop can be turned into a path connecting topologically distinct vacuums (very reminiscent of how the instanton connected vacuums with different winding numbers in the pure Yang-Mills theory).

Has the inclusion of the Higgs somehow gotten rid of the pure Yang-Mills theory’s multiple vacuum structure? Or is it that these multiple vacua are somehow gauge artifacts only observable in some gauges and not others ($$A_r$$ = 0)? This appears strange to me since these many vacuums give rise to real physical effects.

edit Added the remark that I’m studying the model in absence of fermions.

There are indeed interesting winding number instanton configurations in the $$SU(2)$$ weak interaction sector of the standard model. There is no analogue of the QCD vacuum angle however, as the instantons have exact fermion zero modes even when the fermions gain mass via the Higgs field. Here are some of the key references:
A.A.Anselm, A.A.Johansen, "Can the electroweak $$\theta$$-term be observable?", Nuclear Physics B 412 (1994)553-573.


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What are the asymptote(s) and hole(s), if any, of  f(x) =(sqrt(3x)/(x-4))^3?

Mar 10, 2016

Asymptotes are $x = 4 \text{ and } y = 0$

Explanation:

Write as:$\text{ } f \left(x\right) = \frac{3 x \sqrt{3 x}}{{\left(x - 4\right)}^{3}}$

$\textcolor{b l u e}{\text{Point 1}}$

We know that the function will be undefined as the denominator approaches zero.

Thus ${\lim}_{x \to 4} f \left(x\right) = {\lim}_{x \to 4} \frac{3 x \sqrt{3 x}}{{\left(x - 4\right)}^{3}} \to \frac{12 \sqrt{12}}{0}$

$\textcolor{b l u e}{\text{This is undefined" => x=4 " is an asymptote}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Point 2}}$

The denominator has ${x}^{3}$ as the most significant figure.
Whilst the numerators most significant figure is $x \sqrt{x}$

Note that $x \sqrt{x} < {x}^{3}$ so as the denominator 'grows' much faster
than the numerator ${\lim}_{x \to \pm \infty} f \left(x\right) \text{ tends to } 0$

$\textcolor{b l u e}{\text{Thus " y=0" is an asymptote}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


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From Bernoulli-Gaussian deconvolution to sparse signal restoration - Archive ouverte HAL Access content directly
Journal Articles IEEE Transactions on Signal Processing Year : 2011

## From Bernoulli-Gaussian deconvolution to sparse signal restoration

(1) , (2) , (1) , (1)
1
2
Charles Soussen
Jérôme Idier
David Brie
Junbo Duan
• Function : Author
• PersonId : 760511
• IdRef : 150839952

#### Abstract

Formulated as a least square problem under an $\ell_0$ constraint, sparse signal restoration is a discrete optimization problem, known to be NP complete. Classical algorithms include, by increasing cost and efficiency, Matching Pursuit (MP), Orthogonal Matching Pursuit (OMP), Orthogonal Least Squares (OLS), stepwise regression algorithms and the exhaustive search. We revisit the Single Most Likely Replacement (SMLR) algorithm, developed in the mid-80's for Bernoulli-Gaussian signal restoration. We show that the formulation of sparse signal restoration as a limit case of Bernoulli-Gaussian signal restoration leads to an $\ell_0$-penalized least square minimization problem, to which SMLR can be straightforwardly adapted. The resulting algorithm, called Single Best Replacement (SBR), can be interpreted as a forward-backward extension of OLS sharing similarities with stepwise regression algorithms. Some structural properties of SBR are put forward. A fast and stable implementation is proposed. The approach is illustrated on two inverse problems involving highly correlated dictionaries. We show that SBR is very competitive with popular sparse algorithms in terms of trade-off between accuracy and computation time.

#### Domains

Engineering Sciences [physics] Automatic

### Dates and versions

hal-00443842 , version 1 (04-01-2010)
hal-00443842 , version 2 (28-01-2010)
hal-00443842 , version 3 (03-02-2010)
hal-00443842 , version 4 (17-06-2011)

### Identifiers

• HAL Id : hal-00443842 , version 4
• DOI :

### Cite

Charles Soussen, Jérôme Idier, David Brie, Junbo Duan. From Bernoulli-Gaussian deconvolution to sparse signal restoration. IEEE Transactions on Signal Processing, 2011, 59 (10), pp.4572-4584. ⟨10.1109/TSP.2011.2160633⟩. ⟨hal-00443842v4⟩

### Export

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# Volunteer Bias and Intention to Treat in Clinical Trials

These are two sections from Martin Bland's text book An Introduction to Medical Statistics, Third Edition. I hope that the topics will be useful in their own right, as well as giving a flavour of the book.

## Volunteer bias

People who volunteer for new treatments and those who refuse them may be very different. An illustration is provided by the field trial of Salk poliomyelitis vaccine carried out in 1954 in the USA (Meier 1977). This was carried out using two different designs simultaneously, due to a dispute about the correct method. In some districts, second grade school-children were invited to participate in the trial, and randomly allocated to receive vaccine or an inert saline injection. In other districts, all second grade children were offered vaccination and the first and third grade left unvaccinated as controls. The argument against this observed control' approach was that the groups may not be comparable, whereas the argument against the randomized control method was that the saline injection could provoke paralysis in infected children. The results are shown in the Table:

Table: Result of the field trial of Salk poliomyelitis vaccine (Meier 1977)
Study group Number in group Paralytic Polio
Number of cases Rate per 100,000
Randomized control:
Vaccinated: 200745 33 16
Control 201229 115 57
Not inoculated 338778 121 36
Observed control:
Vaccinated 2nd grade 221998 38 17
Control 1st and 3rd grade 725173 330 46
Unvaccinated 2nd grade 123605 43 35

In the randomized control areas the vaccinated group clearly experienced far less polio than the control group. Since these were randomly allocated, the only difference between them should be the treatment, which is clearly preferable to saline. However, the control group also had more polio than those who had refused to participate in the trial. The difference between the control and not inoculated group is in both treatment (saline injection) and selection; they are self-selected as volunteers and refusers. The observed control areas enable us to distinguish between these two factors. The polio rates in the vaccinated children are very similar in both parts of the study, as are the rates in the not inoculated second grade children. It is the two control groups which differ. These were selected in different ways: in the randomized control areas they were volunteers, whereas in the observed controls areas they were everybody eligible, both potential volunteers and potential refusers. Now suppose that the vaccine were saline instead, and that the randomized vaccinated children had the same polio experience as those receiving saline. We would expect 200,745 times 57 / 100,000 = 114 cases, instead of the 33 observed. The total number of cases in the randomized areas would be 114 + 115 + 121 = 350 and the rate per 100,000 would 47. This compares very closely with the rate of 46 in the observed control first and third grade group. Thus it seems that the principal difference between the saline control group of volunteers and the not inoculated group of refusers is selection, not treatment.

There is a simple explanation of this. Polio is a viral disease transmitted by the faecal-oral route. Before the development of vaccine almost everyone in the population was exposed to it at some time, usually in childhood. In the majority of cases, paralysis does not result and immunity is conferred without the child being aware of having been exposed to polio. In a small minority of cases, about one in 200, paralysis or death occurs and a diagnosis of polio is made. The older the exposed individual is, the greater the chance of paralysis developing. Hence, children who are protected from infection by high standards of hygiene are likely to be older when they are first exposed to polio than those children from homes with low standards of hygiene, and thus more likely to develop the clinical disease. There are many factors which may influence parents in their decision as to whether to volunteer or refuse their child for a vaccine trial. These may include education, personal experience, current illness, and others, but certainly include interest in health and hygiene. Thus in this trial the high risk children tended to be volunteered and the low risk children tended to be refused. The higher risk volunteer control children experienced 57 cases of polio per 100,000, compared to 36/100,000 among the lower risk refusers.

In most diseases, the effect of volunteer bias is opposite to this. Poor conditions are related both to refusal to participate and to high risk, whereas volunteers tend to be low risk. The effect of volunteer bias is then to produce an apparent difference in favour of the treatment. We can see that comparisons between volunteers and other groups can never be reliable indicators of treatment effects.

## Intention to treat

In the observed control areas of the Salk trial (Table~\ref{2tsalk}), quite apart from the non-random age difference, the vaccinated and control groups are not comparable. However, it is possible to make a reasonable comparison in this study by comparing all second grade children, both vaccinated and refused, to the control group. The rate in the second grade children is 23 per 100,000, which is less than the rate of 46 in the control group, demonstrating the effectiveness of the vaccine. The treatment' which we are evaluating is not vaccination itself, but a policy of offering vaccination and treating those who accept. A similar problem can arise in a randomized trial, for example in evaluating the effectiveness of health checkups (South-east London Screening Study Group 1977). Subjects were randomized to a screening group or to a control group. The screening group were invited to attend for an examination, some accepted and were screened and some refused. When comparing the results in terms of subsequent mortality, it was essential to compare the controls to the screening groups containing both screened and refusers. For example, the refusers may have included people who were already too ill to come for screening. The important point is that the random allocation procedure produces comparable groups and it is these we must compare, whatever selection may be made within them. We therefore analyse the data according to the way we intended to treat subjects, not the way in which they were actually treated. This is analysis by intention to treat. The alternative, analysing by treatment actually received, is called on treatment analysis.

Analysis by intention to treat is not free of bias. As some patients may receive the other group's treatment, the difference may be smaller than it should be. We know that there is a bias and we know that it will make the treatment difference smaller, by an unknown amount. On treatment analyses, on the other hand, are biased in favour of showing a difference, whether there is one or not. Statisticians call methods which are biased against finding any effect conservative. If we must err, we like to do so in the conservative direction.

## References

Meier, P. (1977) The biggest health experiment ever: the 1954 field trial of the Salk poliomyelitis vaccine. in Statistics: a Guide to the Biological and Health Sciences ed. J.M. Tanur et al. Holden-Day, San Francisco.

South-east London Screening Study Group. (1977) A controlled trial of multiphasic screening in middle-age: results of the South-east London Screening Study. International Journal of Epidemiology 6 357-63.


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# Papers

We describe the Carnegie-Spitzer-IMACS (CSI) Survey, a wide-field, near-IR selected spectrophotometric redshift survey with the Inamori Magellan Areal Camera and Spectrograph (IMACS) on Magellan-Baade. By defining a flux-limited sample of galaxies in Spitzer \emph{IRAC} $3.6\mu$m imaging of SWIRE fields, the CSI Survey efficiently traces the stellar mass of average galaxies to $z\sim 1.5$. This first paper provides an overview of the survey selection, observations, processing of the photometry and spectrophotometry. We also describe the processing of the data: new methods of fitting synthetic templates of spectral energy distributions are used to derive redshifts, stellar masses, emission line luminosities, and coarse information on recent star-formation. Our unique methodology for analyzing low-dispersion spectra taken with multilayer prisms in \emph{IMACS}, combined with panchromatic photometry from the ultraviolet to the IR, has yielded high quality redshifts for 43,347 galaxies in our first 5.3 degs$^2$ of the SWIRE XMM-LSS field. We use three different approaches to estimate our redshift errors and find robust agreement. Over the full range of $3.6\mu$m fluxes of our selection, we find typical redshift uncertainties of $\sigma_z/(1+z) \la 0.015$. In comparisons with previously published spectroscopic redshifts we find scatters of $\sigma_z/(1+z) = 0.011$ for galaxies at $0.7\le z\le 0.9$, and $\sigma_z/(1+z) = 0.014$ for galaxies at $0.9\le z\le 1.2$. For galaxies brighter and fainter than $i=23$ mag, we find $\sigma_z/(1+z) = 0.008$ and $\sigma_z/(1+z) = 0.022$, respectively. Notably, our low-dispersion spectroscopy and analysis yields comparable redshift uncertainties and success rates for both red and blue galaxies, largely eliminating color-based systematics that can seriously bias observed dependencies of galaxy evolution on environment.
We present the first measurement of the evolution of the galaxy group stellar mass function (GrSMF) to redshift z>1 and low masses (M>10^12 Msun). Our results are based on early data from the Carnegie-Spitzer-IMACS (CSI) Survey, utilizing low-resolution spectra and broadband optical/near-IR photometry to measure redshifts for a 3.6um selected sample of 37,000 galaxies over a 5.3 deg^2 area to z~1.2. Employing a standard friends-of-friends algorithm for all galaxies more massive than log(M_star/Msun)=10.5, we find a total of ~4000 groups. Correcting for spectroscopic incompleteness (including slit collisions), we build cumulative stellar mass functions for these groups in redshift bins at z>0.35, comparing to the z=0 and z>0 mass functions from various group and cluster samples. Our derived mass functions match up well with z>0.35 X-ray selected clusters, and strong evolution is evident at all masses over the past 8 Gyr. Given the already low level of star formation activity in galaxies at these masses, we therefore attribute most of the observed growth in the GrSMF to group-group and group-galaxy mergers, in accordance with qualitative notions of hierarchical structure formation. Given the factor 3-10 increase in the number density of groups and clusters with M_\star>10^12 Msun since z=1 and the strong anticorrelation between star formation activity and environmental density, this late-time growth in group-sized halos may therefore be an important contributor to the structural and star-formation evolution of massive galaxies over the past 8 Gyr.
Since z~1, the stellar mass density locked in low mass groups and clusters has grown by a factor of ~8. Here we make the first statistical measurements of the stellar mass content of low mass X-ray groups at 0.5<z<1, enabling the calibration of stellar-to-halo mass scales for wide-field optical and infrared surveys. Groups are selected from combined Chandra and XMM-Newton X-ray observations in the Chandra Deep Field South (CDFS). These ultra-deep observations allow us to identify bona fide low mass groups at high redshift and enable measurements of their total halo masses. We compute aggregate stellar masses for these halos using galaxies from the Carnegie-Spitzer-IMACS (CSI) spectroscopic redshift survey. Stars comprise ~3-4% of the total mass of group halos with masses 10^{12.8}<M200/Msun<10^{13.5} (about the mass of Fornax and 1/50th the mass of Virgo). Complementing our sample with higher mass halos at these redshifts, we find that the stellar-to-halo mass ratio decreases toward higher halo masses, consistent with other work in the local and high redshift universe. The observed scatter about the stellar-halo mass relation is ~0.25 dex, which is relatively small and suggests that total group stellar mass can serve as a rough proxy for halo mass. We find no evidence for any significant evolution in the stellar-halo mass relation since z<1. Quantifying the stellar content in groups since this epoch is critical given that hierarchical assembly leads to such halos growing in number density and hosting increasing shares of quiescent galaxies.


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# zbMATH — the first resource for mathematics

On good characterizations for digraphs. (English) Zbl 0609.05040
For any digraph D let f(D) denote the set of all digraphs G such that there is not homomorphism from A to G and let g(D) denote the set of all digraphs H such that there is a homomorphism from H to D. The author gives new examples of digraphs A and B such that $$f(A)=g(B)$$ and he gives a necessary condition that must be satisfied by such digraphs A.
Reviewer: J.W.Moon
##### MSC:
 05C20 Directed graphs (digraphs), tournaments
##### Keywords:
directed graphs; digraphs; homomorphism
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