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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

Mod 3

Stage: 4 Challenge Level:

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Modulus Arithmetic and a Solution to Dirisibly Yours

Stage: 5

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Plus or Minus

Stage: 5 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

Sixational

Stage: 4 and 5 Challenge Level:

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

More Sums of Squares

Stage: 5

Tom writes about expressing numbers as the sums of three squares.

Sums of Squares and Sums of Cubes

Stage: 5

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

A Biggy

Stage: 4 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Take Three from Five

Stage: 4 Challenge Level:

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Prime AP

Stage: 5 Challenge Level:

What can you say about the common difference of an AP where every term is prime?

Number Rules - OK

Stage: 4 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

N000ughty Thoughts

Stage: 4 Challenge Level:

How many noughts are at the end of these giant numbers?

Ordered Sums

Stage: 4 Challenge Level:

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

How Many Solutions?

Stage: 5 Challenge Level:

Find all the solutions to the this equation.

Perfectly Square

Stage: 4 Challenge Level:

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Big, Bigger, Biggest

Stage: 5 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Dalmatians

Stage: 4 and 5 Challenge Level:

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

Common Divisor

Stage: 4 Challenge Level:

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

The Root Cause

Stage: 5 Challenge Level:

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

For What?

Stage: 4 Challenge Level:

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

Telescoping Functions

Stage: 5

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Three Frogs

Stage: 4 Challenge Level:

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

Golden Eggs

Stage: 5 Challenge Level:

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

The Golden Ratio, Fibonacci Numbers and Continued Fractions.

Stage: 4

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

On the Importance of Pedantry

Stage: 3, 4 and 5

A introduction to how patterns can be deceiving, and what is and is not a proof.

Square Pair Circles

Stage: 5 Challenge Level:

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Modulus Arithmetic and a Solution to Differences

Stage: 5

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

Modular Fractions

Stage: 5 Challenge Level:

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

Why 24?

Stage: 4 Challenge Level:

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Archimedes and Numerical Roots

Stage: 4 Challenge Level:

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Polite Numbers

Stage: 5 Challenge Level:

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

Stage: 5 Challenge Level:

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

Rational Roots

Stage: 5 Challenge Level:

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Without Calculus

Stage: 5 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

Stage: 5 Short Challenge Level:

Can you work out where the blue-and-red brick roads end?

Similarly So

Stage: 4 Challenge Level:

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Mediant

Stage: 4 Challenge Level:

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

Mouhefanggai

Stage: 4

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Euler's Formula and Topology

Stage: 5

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

A Computer Program to Find Magic Squares

Stage: 5

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Composite Notions

Stage: 4 Challenge Level:

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Pent

Stage: 4 and 5 Challenge Level:

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Unit Interval

Stage: 4 and 5 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Circle Box

Stage: 4 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

Generally Geometric

Stage: 5 Challenge Level:

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Pareq Exists

Stage: 4 Challenge Level:

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Round and Round

Stage: 4 Challenge Level:

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Proofs with Pictures

Stage: 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Matter of Scale

Stage: 4 Challenge Level:

Prove Pythagoras' Theorem using enlargements and scale factors.

Rhombus in Rectangle

Stage: 4 Challenge Level:

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

The Triangle Game

Stage: 3 and 4 Challenge Level:

Can you discover whether this is a fair game?


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# The spectrum of a random operator is a random set

Research paper by Gunnar Taraldsen

Indexed on: 27 Nov '20Published on: 13 Sep '19Published in: arXiv - Mathematics - Probability

#### Abstract

The theory of random sets is demonstrated to prove useful for the theory of random operators. A random operator is here defined by requiring the graph to be a random set. It is proved that the spectrum and the set of eigenvalues of random operators are random sets. These results seem to be a novelty even in the case of random bounded operators. The main technical tools are given by the measurable selection theorem, the measurable projection theorem, and a characterisation of the spectrum by approximate eigenvalues of the operator and the adjoint operator. A discussion of some of the existing definitions of the concept of a random operator is included at the end of the paper. Keywords: Random operators; Set-valued functions; General topics in linear spectral theory; Random operators and equations; Stochastic integrals; Disordered systems

format: plain


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## singular or plural

3

I was reading an essay about the environment and how we can save the forests. But actually while reading I felt that the following sentence is not correct, it used the plural are instead of is as I think it should. Look the sentence:

...Is this a problem? Yes it certainly is.
First, rising CO2 levels are causing global warming.

I think the (rising) in CO2 levels is the cause of the global warming, then the sentence must use the singular is. Why did it use the plural? How can you explain this?

Sorry if the question is too basic.

2Rising is a Participle and you should actually consider the plural noun "Levels" which is referring to co2 emitters around the world. – None – 2014-06-08T15:54:25.910

The sentence would use the singular is if it was phrased like so: First, the rising of CO2 levels is causing global warming. – Pockets – 2014-06-08T17:51:18.167

2Contrast rising with raising. – snailplane – 2014-06-08T23:10:30.777

I don't know why you want to contrast rising with raising...Actually,this essay is taken from a curriculum i've studied before and it was written(rising).Why do you want to contrast? – kathrine – 2014-06-09T09:56:14.307

rising is an adjective here. @snailplane Am I missing something with your word contrast? – Maulik V – 2014-06-09T11:16:20.457

@Maulik V:Does(raising)matter the word order..?I don't think,raising can also be an adjective,then..Why does he want to contrast? – kathrine – 2014-06-09T11:24:41.680

1

You can say:

"The rising in CO2 levels is causing..."

"The rising CO2 levels are causing..."

"The rising CO2 level is causing..."

All of these are correct grammatically but are slightly different in what is being stated.

In the first "the rising" (a noun) is the singular subject. In the second "CO2 levels" is the plural subject (rising is an adjective). In the third "CO2 level" is the singular subject (rising is an adjective).

There is really only one global CO2 level since atmospheric mixing is quick and efficient on the scale of the release process, so a singular is technically correct (but "levels" is okay also).

1

The rising CO2 levels are causing

The subject in this sentence is CO2 levels.

rising describes the subject, like an adjective would.

Rising CO2 levels are causing

The jumping little children are hyper.

The falling trees are dangerous

Susie's singing, dancing, green frog is dying.

0

I would propose that one uses the plural when what one is observing can not be properly observed at just one location. The use of the singular might be taken as implying a local anomaly.


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# For all $n \in N$, ${10^n} + {3.4^{n + 2}} + 5$ is divisible byA.23B.3C.9D.207

Verified
122.4k+ views
Hint: In the given question, we have been given that there is an expression involving four constants. The constants raised to some variable power. We are asked when some value is put in the variable power, and the answer we get, by what number is it always divisible by. First, we need to look at the options here. There can be multiple answers depending on the options. If only the option A is correct, then only A is correct. If only B is correct, then only B is correct. If option C is correct, then C as well as B is correct (because B is a factor of C). And, if D is correct, then all options are correct, because each option is a factor of the D option. Now, to solve this question, we just need to put in some values in the given range and see what answer we get.

The given expression is $p\left( x \right) = {10^n} + {3.4^{n + 2}} + 5$ and its range is $N$.
So, we put in the smallest possible value $\left( 1 \right)$ and see what we get. Now, if we get any of the options, then it means that by putting in any other value, we are going to get the same result, as $1$ divides every other natural number.
So, $p\left( 1 \right) = {10^1} + {3.4^{1 + 2}} + 5 = 10 + 3 \times {4^3} + 5$
Now, we know ${4^3} = 64$, so
$p\left( 1 \right) = 10 + 192 + 15 = 207$
So, on the first attempt only we got the answer that each of the options is going to divide the value of the expression, as $207$ is a multiple of every other number – $3,9,23$.


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# the journey towards real time circle packing

a journey for aesthetics and speed

Order and chaos are often two sides of the same coin. We find satisfaction in order emerging from chaos and from order dissolving into chaos. Circle packing is a good example of something in between. There’s a lot of different version of circle packing out there, but specifically I like the look of almost randomly placed circles forming an image. So, I set out to implement an idea - a real time image processing program that would give me the look I wanted, random circles of different sizes forming the original image.

To achieve this look, first we need to ascribe order to the chaos. How are random circles going to form an image? I thought it would be interesting to only use outlines of circles and use the empty space to represent brighter sections of the original image, which led to the following goal: pick a distinct set of points from the image $$P$$, such that for every point $$p \in P$$ let $$c_p$$ be the circle of radius $$r(p)$$ centered at $$p$$, then:

$$\forall q \in P$$, $$q \neq p$$, $$c_p$$ does not intersect $$c_q$$

where $$r(p)$$ is some function that makes the radius proportional to the brightness of the pixel.

I think it would be extra satisfying if this could work in real time against a webcam stream, so the goal is to get this to work at the rate of at least 20FPS, which is totally arbitrary, but I think it’s a good enough framerate to look decent. Ideally we could get to 30+. With the problem defined, we can get to implementing!

## laying down a baseline

To start with, we’re going to need to come up with a basic approach that satisfies the problem statement. I don’t really know how to approach the problem, and I want this to be a challenge, so I’m not going to look up other popular methods to solve this until I feel like I’ve done the best I can. Since we just want to get something working so that we can iterate on an algorithm, we’ll want to start simple. To that end, I’m going to be using the simplest tools, no fancy WebGL today, just plain old canvas 2d rendering.

Since speed is not a concern for this implementation, we’ll just go with a brute force solution, we’ll generate a point, find it’s radius, if it intersects with any previous point we generated, discard it, otherwise draw a circle and save the point. Here’s a demo (open the browser console to see timing logs!):

Warning: Running this version may slow down your browser as it's computationally expensive.

Click this message to dismiss.
const previous = [];
const draw_limit = 100000;
const draw_batch_size = 250;
const draw_timeout = 0;
const attempt_limit = 10;

console.time("draw");
let draw_count = 0;
const draw = () => {
for (let batch_i = 0; batch_i < draw_batch_size; batch_i ++) {
draw_count++;
if (draw_count > draw_limit) {
console.timeEnd("draw");
return;
}

let num_attempts = 0;
while (true) {
const x = Math.floor(Math.random() * img.canvas.width);
const y = Math.floor(Math.random() * img.canvas.height);

const idx = 4 * (y * img.canvas.width + x);
const r = imgdata.data[idx];
const g = imgdata.data[idx + 1];
const b = imgdata.data[idx + 2];
const gray_value = 0.3 * r + 0.59 * g + 0.11 * b;
const radius = gray_value / 255 * 5;

let failed = false;
for (let prev of previous) {
const dx = prev[0] - x;
const dy = prev[1] - y;
const dist = Math.sqrt(dx * dx + dy * dy);

if (dist <= (prev[2] + radius)) {
failed = true;
break;
}
}

if (failed) {
num_attempts++;
if (num_attempts > attempt_limit) {
break;
}
continue;
}

const current = [x, y, radius];

ctx.beginPath();
ctx.strokeStyle = rgb(${r},${g}, \${b});
ctx.arc(...current, 0, Math.PI * 2);
ctx.stroke();

previous.push(current);
break;
}
}

console.log(draw_count, draw_limit);

if (draw_timeout) {
setTimeout(()=> { requestAnimationFrame(draw); }, draw_timeout);
} else {
requestAnimationFrame(draw);
}
}

omitted is the part that initializes the canvas and loads the original image.

Overall this method takes a while. Almost 2.5min to draw 100000 circles. Time to make it faster!

## implementing the first major optimization

The first thing to note is that the old method relies on an $$O(n^2)$$ algorithm. I thought about trying to use some clever datastructures (maybe a k-d tree?), but I came up with something much simpler. By drawing circles on the canvas, we’re already storing information about what space is occupied. The canvas 2d API is also pretty optimized, so we can think about drawing as a very fast operation.

The method I came up with was inspired by reading about how early version Gnome 3 tested for mouse click events. My hazy summary of it is that it worked by assigning each window ID a color (just cast the 32 bit window id into a 32 bit RGBA value), and then rendering all windows’ bounding rectangles in their unique color to an offscreen buffer in z-index order. Then the color underneath the mouse coordinates was looked up, letting the window manager know which window to forward the mouse event to. Here, we just draw every circle that in the set to an offscreen buffer, and for a new point with a given radius, just read the pixels that would be overwritten and check that no pixels are set. I did cheat a bit here and read the pixels from the bounding square of the new circle, but in practice that doesn’t affect the visual quality by much. Here’s the demo:

For best results please make sure the old version above isn't running. You can verify this either by checking the logs in your browser's console or by just refreshing the page.

Click this message to dismiss.

And here’s the code:

  // create offscreen canvas
// - don't parent it to a visible element!
const hitcanvas = document.createElement("canvas");
const hitctx = hitcanvas.getContext("2d");

...

const draw = () => {
...

const ra = Math.ceil(radius);
const d = 2 * ra;
const hitdata = hitctx.getImageData(
x - ra, y - ra, d, d);
for (let px = 0; px < hitdata.data.length; px += 4) {
const h_b = hitdata.data[px + 2];
if (h_b > 0) {
failed = true;
break;
}
}

...

hitctx.beginPath();
hitctx.arc(...current, 0, 2 * Math.PI, false);
hitctx.fill();

break;
...

With that we’ve taken the $$O(n^2)$$ algorithm and turned it into an $$O(1)$$ (since the maximum radius size is bounded by 5) algorithm at the cost of extra memory required to maintain the offscreen buffer. The runtime is now around 13s. This is a huge improvement, but it’s still not close to the requirements of running this in real time.

## Conclusion

There’s still a lot more that can be done! The next step I have in mind is to examine the random generation of points. If we have more information about where we’ve already generated points, maybe we can be more likely to select points that can be succesfully added to the set. Another possible approach can be change the way points are chose to do some kind of BFS-like exploration, building up a set of points by only looking at unexplored regions.

The first step to doing that is going to be to find out how many randomly select points are reject per run on average to find out if that really is problematic. Gathering data is really important!

Anyway for now, I’m celebrating the progress made so far! I hope this post was fun to read, and I hope I can add more exciting updates to this blog as I try new things!

Written on July 15, 2021


text:
Ghior.com 🏠 » Paragraphs » Paragraph on Renewable Energy and Non-Renewable Energy

# Paragraph on Renewable Energy and Non-Renewable Energy

Question: Write a paragraph on renewable and nonrenewable energy. Your paragraph should include the answers to the following questions.

1. Why is renewable energy better than non-renewable energy?
2. What are the sources of non-renewable energy?
3. On which energy sources do the world depend heavily?
4. What are of renewable energy?
5. What is your comment?

Answer: Renewable energy is better than non-renewable energy as far as our environment is concerned. Renewable energy sources do not exhaust and are environment-friendly. On the other hand, non-renewable energy sources pollute our environment and are limited in reserve. However, countries of the world rely heavily on non-renewable sources of energy such as petroleum, coal and natural gas, etc. We get energy by burning these gases. They pollute our environment by emitting smoke etc. But renewable energy does not pollute the environment because most renewable energy comes either directly or indirectly from the sun. The energy produced from the sunlight is known as solar energy. This can be used for heating and lighting homes, for generating electricity and for other commercial and industrial uses. There are also other renewable energy sources such as wind energy, ocean energy, geothermal energy, hydrogen energy, etc. All these sources of energy can be used to produce electricity. These sources are inexhaustible and do not pollute the environment. Countries of the world should pay more attention to utilize renewable sources of energy to meet the future energy crisis and make the world a less polluted place for humans.


text:
Corpus ID: 208006103

# $\mathbb{A}^{1}$-Local Degree via Stacks

@article{Kobin2019mathbbA1LocalDV,
title={\$\mathbb\{A\}^\{1\}\$-Local Degree via Stacks},
author={Andrew Kobin and Libby Taylor},
journal={arXiv: Algebraic Geometry},
year={2019}
}
• Published 2019
• Mathematics
• arXiv: Algebraic Geometry
We extend results of Kass--Wickelgren to define an Euler class for a non-orientable (or non-relatively orientable) vector bundle on a smooth scheme, valued in the Grothendieck--Witt group of the ground field. We use a root stack construction to produce this Euler class and discuss its relation to other versions of an Euler class in $\mathbb{A}^{1}$-homotopy theory. This allows one to apply Kass--Wickelgren's technique for arithmetic enrichments of enumerative geometry to a larger class of… Expand
2 Citations
The trace of the local $\mathbf{A}^1$-degree
• Mathematics
• 2019
We prove that the local $\mathbb{A}^1$-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local $\mathbb{A}^1$-degree over theExpand
An arithmetic enrichment of B\'ezout's Theorem
The classical version of Bezout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass andExpand

#### References

SHOWING 1-10 OF 32 REFERENCES
An enriched count of the bitangents to a smooth plane quartic curve
• Mathematics
• 2019
Recent work of Kass--Wickelgren gives an enriched count of the $27$ lines on a smooth cubic surface over arbitrary fields. Their approach using $\mathbb{A}^1$-enumerative geometry suggests that otherExpand
A "bottom up" characterization of smooth Deligne-Mumford stacks
• Mathematics
• 2015
In casual discussion, a stack is often described as a variety (the coarse space) together with stabilizer groups attached to some of its subvarieties. However, this description does not uniquelyExpand
An Arithmetic Count of the Lines on a Smooth Cubic Surface
• Mathematics
• 2017
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over $\mathbb{C}$ there are $27$ lines, and over $\mathbb{R}$ the numberExpand
Intrinsic signs and lower bounds in real algebraic geometry
• Mathematics
• 2011
A classical result due to Segre states that on a real cubic surface in ${\mathbb P}^3_\R$ there exists two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines areExpand
An arithmetic enrichment of Bézout’s Theorem
The classical version of Bézout’s Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass andExpand
3264 and All That: A Second Course in Algebraic Geometry
• Mathematics
• 2016
Introduction 1. Introducing the Chow ring 2. First examples 3. Introduction to Grassmannians and lines in P3 4. Grassmannians in general 5. Chern classes 6. Lines on hypersurfaces 7. SingularExpand
Artin-Schreier Root Stacks
We classify stacky curves in characteristic $p > 0$ with cyclic stabilizers of order $p$ using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve,Expand
The canonical ring of a stacky curve
• Mathematics
• 2015
Generalizing the classical theorems of Max Noether and Petri, we describe generators and relations for the canonical ring of a stacky curve, including an explicit Grobner basis. We work in a generalExpand
A1-algebraic topology
We present some recent results in A1-algebraic topology, which means both in A1-homotopy theory of schemes and its relationship with algebraic geometry. This refers to the classical relationshipExpand
THE CLASS OF EISENBUD–KHIMSHIASHVILI–LEVINE IS THE LOCAL A-BROUWER DEGREE
Given a polynomial function with an isolated zero at the origin, we prove that the local A-Brouwer degree equals the Eisenbud–Khimshiashvili–Levine class. This answers a question posed by DavidExpand


text:
# Tag Info

24

The vending machines have different product delivery box implementations. One of the most common ones -I've encountered- is the following: Basically you have a box which has an opening, as shown in the picture. The opening is big enough for the products to fall in. When the box pivots, then the closed section of the curved face "closes" the access ...

11

I think the locking mechanism is more or less like this: Note, the actual setup could be different but the concept of locking through a barrier that prevents the slide plate been pushed up remains the same.

6

Strictly speaking, very few metals are "stable" in terms of the laws of thermodynamics. True chemical stability is when the atoms are in their lowest energy state. For most metallic elements, various oxides, sulfides, and chlorides are lower energy states than the pure or alloyed metal. This is why corrosion occurs in the first place--the atoms ...

5

We define reactions uniquely depending on the reactants and products. Here are examples related to your question. Formation: Na(s) + (1/2)Cl$_2$(g) $\rightarrow$ NaCl(s) Lattice Formation: Na$^+$(g) + Cl$^-$(g) $\rightarrow$ NaCl(s) Solution: NaCl(s) $\rightarrow$ NaCl(aq) Hydration: Na$^+$(g) $\rightarrow$ Na$^+$(aq) Atomization: Na(s) $\rightarrow$ Na(g) A ...

3

When you are raising the plate you are also raising the bearings and since they need to roll there is the frictional component pulling down on the bearings and pulling down on the plate. When you lower the system you are pulling the bearings down and since they again need to roll there is the frictional component pulling up on the bearings and pulling up on ...

3

TL;DR (Serious): The problem is that you don't into account that the bending moment equation changes at different section of the beam. TL;DR: (Tongue in cheek) Don't do this unless you a) really really don't have anything better to do, or b) you are getting paid to do it, c) you really like this sort of problems. :-P (I wasn't getting paid to do it, so I ...

3

You advance the date manually, if required. It can be done without disturbing the time setting. Figure 1. Image source: Watches under 500. Position I – for hand-winding of the main spring. Position II – position of the crown in which you will able to change the day and date. If your watch only has a date window, you can change the date by turning the crown ...

2

This sketch is traced off from yours, you may update any missing/incorrect details. However, my impression is you were calculating the forces on plane a-a with the force applied as shown. For such a case, the bolt acts as a shear lug/pin and subjects to shear force only, no bending will occur or only with a negligible amount of moment resulted from the force ...

2

This is a type of ellipsograph or Trammel of Archimededs. It's a mechanism that is widely documented in textbooks for both the kinematics and the dynamics. Regarding the second part of your question, it really depends on two things: the position of the rod (angle) the velocity at the point of calculation. In the simple case of velocity equal to zero, then ...

2

$EI\dfrac {d^2_y}{dx^2} = M$ Procedures: Write equations for the moment in segment AB and BC as a function of x. Integrate the moment of each segment to get the slope, $EI\dfrac{dy}{dx} = EI\theta_x$ Integrate the slope of each segment to get the deflection, $EIy = EI\Delta_x$ Apply boundary conditions to set up equations for the constants. Solve the ...

2

It's a tricky one. Figure 1. Your isometric is missing this line. Figure 2. A clue. The six-cornered shape is flat so it must be a plane going through these three points. Can you draw it? Figure 3. The correct isometric view. Figure 4. Slide 25 by Shelly Wilke on SlideShare. This and slide 9 may be instructive.

2

you also need to take into account that the complete burn for octane is: $$2C_8H_{18} + 25\;O_2 \rightarrow 16\;CO_2 +18\;H_2O$$ Therefore $a$ should be equal to $\frac{25}{2}$.

2

It depends on the environment , relative to corrosion. Sea water is relatively corrosive to most metals, but again ,it depends. Splash zone is vey aggressive, low oxygen at great depth is relatively benign. What do you want the answer to be ?

2

The cylinder is cast iron which is relatively soft, so a softer material is used for the head gasket. The soft gasket deforms to seal against the head. Because of the very large size there are more likely machining imperfections that the soft gasket accommodates and seals. Whatever the reason , washers are unusual on engine head bolts. For very large flanges ...

1

Use a pulley with spooled string on it dropping a weight on the input shaft for input torque, and a pulley spooling up a string with a weight on the output shaft for load torque. With pulley radius and mass you can calculate input and output torques. You then need an RPM sensor on one of the shafts and you can use the gear ratio to calculate the RPM on the ...

1

you need to have your acrylic material properties data sheet, such as its modulus of elasticity, E and its max stress, $\sigma s$, and poisons ratio. Usually the acceptable deflection, y ranges between $y<L/180, \ or \ y<L/360$ Then you try different thicknesses, t, till you get the smallest t with acceptable deflection. Referring to Roark’s Formulas ...

1

You did pretty well, but @Transistor is correct that the horizontal line needs to be extended. Maybe it looks like this. However, I don't think the inclined plane is correct as there will be another line (shown in red) to connect the corners.

1

I am writing this as an answer to the comment about the superposition method. I started writing it in my original answer, but it was too long and confusing, so I opted to do another answer. Superposition method. Let's assume you have this problem. In that case you only need the following equation. Essentially what you do is you apply the above equation ...

1

I would suggest separate channels to each set of holes with orifice plates to limit the flow to / from each. That way you can control each set individually. The other solution will be to over-specify the flow rate so that the poorest one is sufficient...

1

It will be a next to impossible task to try and analytically predict the number/position/size of the holes in order to have exactly the same flow. Even if you do for a temperature and a set humidity, at different temperatures you might have unacceptable results. If you are really interested in equalising the flows, a better alternative is to use a flow ...

1

It depends on the loading conditions. It's important to note that the stress concentrations can be intuitively understood as created by the load's need to deviate from the hole. The larger the width of the hole, the more the load needs to move to go around it. If it's a uniform bi-axial load (your part is under tension/compression in both the X and Y ...

1

A screw thread would work, use the sensor to drive a motor with suitable gearing.

1

It is less economical to harness this energy/materials. From the perspective of the company, money is better spent on their immediate business needs such as locating large resources or improving the thermodynamic efficiency of their refinery. Reasons it may be less economical include: Intermittent flow; flow may be zero for long periods, with intermittent ...

1

I think this is a situation of "the devil is in the details', and of course, cost. In the US, gas from a few wells would not likely be of sufficient volume; but several/many wells could be combined (detail = pipeline plan and control). There are wells, mostly Middle East that do have enough flare gas. As a point of reference the last local power gas ...

1

What's acceptable will depend on your application. For a chair, 1mm of deflection will not be noticeable. For a CNC machine not so much. If you really care about optimizing the design, I would start by designing the object with narrow extrusions, and then see if/were the deflection will be too great, then bulk it up as necessary. This calculator ...

1

In general the acceptable deflection has to do with perception. What I mean by that is, if it is possible to perceive the deflection of the structure. Therefore it is usually presented in building codes (usually that's where you most see it), as a percentage of the span of the beam. However, there is no universally accepted allowable deflection. For example: ...

1

The short answer, is that there are bending stresses in the configuration you are presenting. The big difference is that for such small aspect ratio beams the shear stress are comparable to the bending stresses. Because, the Euler-Bernoulli beam theory (which is a cornerstone of structural engineering) ignores shear stresses, the equations are not applicable....

1

Assuming that the gear box is on the left end of the shaft (i.e. no flexible shaft between motor and gearbox). The angular velocity on the left end of the gear box is $\omega_1$. The angular velocity of the shaft side of the gear box is assumed as $\omega_1' = \frac{\omega_1}{N}$. The angular velocity on the right end of the shaft is $\omega_2$. So the ...

1

The SN diagram below answers your question. Note "High Cycle Fatigue (HCF)" failure is characterized by high repetitive cycles (N), usually equal to or greater than $10^4$, and low-stress (S) in the elastic zone.

Only top voted, non community-wiki answers of a minimum length are eligible


text:
This function plots adjusted predictions (y-axis) against values of one or more predictors (x-axis and colors).

## Usage

plot_cap(
model,
condition = NULL,
type = "response",
vcov = NULL,
conf_level = 0.95,
transform_post = NULL,
draw = TRUE,
...
)

## Arguments

model

Model object

condition

String or vector of two strings. The first is a variable name to be displayed on the x-axis. The second is a variable whose values will be displayed in different colors. Other numeric variables are held at their means. Other categorical variables are held at their modes. Other numeric variables are held at their means.

type

string indicates the type (scale) of the predictions used to compute marginal effects or contrasts. This can differ based on the model type, but will typically be a string such as: "response", "link", "probs", or "zero". When an unsupported string is entered, the model-specific list of acceptable values is returned in an error message. When type is NULL, the default value is used. This default is the first model-related row in the marginaleffects:::type_dictionary dataframe.

vcov

Type of uncertainty estimates to report (e.g., for robust standard errors). Acceptable values:

• FALSE: Do not compute standard errors. This can speed up computation considerably.

• TRUE: Unit-level standard errors using the default vcov(model) variance-covariance matrix.

• String which indicates the kind of uncertainty estimates to return.

• Heteroskedasticity-consistent: "HC", "HC0", "HC1", "HC2", "HC3", "HC4", "HC4m", "HC5". See ?sandwich::vcovHC

• Heteroskedasticity and autocorrelation consistent: "HAC"

• Mixed-Models degrees of freedom: "satterthwaite", "kenward-roger"

• Other: "NeweyWest", "KernHAC", "OPG". See the sandwich package documentation.

• One-sided formula which indicates the name of cluster variables (e.g., ~unit_id). This formula is passed to the cluster argument of the sandwich::vcovCL function.

• Square covariance matrix

• Function which returns a covariance matrix (e.g., stats::vcov(model))

conf_level

numeric value between 0 and 1. Confidence level to use to build a confidence interval.

transform_post

(experimental) A function applied to unit-level adjusted predictions and confidence intervals just before the function returns results. For bayesian models, this function is applied to individual draws from the posterior distribution, before computing summaries.

draw

TRUE returns a ggplot2 plot. FALSE returns a data.frame of the underlying data.

...

Additional arguments are passed to the predict() method supplied by the modeling package.These arguments are particularly useful for mixed-effects or bayesian models (see the online vignettes on the marginaleffects website). Available arguments can vary from model to model, depending on the range of supported arguments by each modeling package. See the "Model-Specific Arguments" section of the ?marginaleffects documentation for a non-exhaustive list of available arguments.

## Value

A ggplot2 object

Other plot: plot.marginaleffects(), plot_cco(), plot_cme()

## Examples

mod <- lm(mpg ~ hp + wt, data = mtcars)
plot_cap(mod, condition = "wt")

mod <- lm(mpg ~ hp * wt * am, data = mtcars)
plot_cap(mod, condition = c("hp", "wt"))


text:
# Are fundamental groups of aspherical manifolds Hopfian?

A group $G$ is Hopfian if every epimorphism $G\to G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth manifolds Hopfian?

Perhaps the manifold structure is irrelevant and makes examples harder to construct, so here is another variant that may be more sensible. Let $X$ be a finite CW-complex which is $K(\pi,1)$. If it helps, assume that its top homology is nontrivial. Is $\pi=\pi_1(X)$ Hopfian?

Motivation. Long ago I proved a theorem which is completely useless but sounds very nice: if a manifold $M$ has certain homotopy property, then the Riemannian volume, as a function of a Riemannian metric on $M$, is lower semi-continuous in the Gromov-Hausdorff topology. (And before you laugh at this conclusion, let me mention that it fails for $M=S^3$.)

The required homotopy property is the following: every continuous map $f:M\to M$ which induces an epimorphism of the fundamental group has nonzero (geometric) degree.

This does not sound that nice, and I tried to prove that some known classes of manifolds satisfy it. My best hope was that all essential (as in Gromov's "Filling Riemannian manifolds") manifolds do. I could not neither prove nor disprove this and the best approximation was that having a nonzero-degree map $M\to T^n$ or $M\to RP^n$ is sufficient. I never returned to the problem again but it is still interesting to me.

An affirmative answer to the title question would solve the problem for aspherical manifolds. A negative one would not, and in this case the next question may help (although it is probably stupid because I know nothing about the area):

Question 2. Let $G$ be a finitely presented group and $f:G\to G$ an epimorphism. It it true that $f$ induces epimorphism in (co)homology (over $\mathbb Z$, $\mathbb Q$ or $\mathbb Z/2$)?

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For question 2, $f$ induces an epi on abelianizations, i.e. on $H_1(G;Z)$. The universal coefficient theorem shows it remains epi over $Z/2$ or $Q$. –  Paul Apr 9 '10 at 12:30
I'm interested in higher homology groups. –  Sergei Ivanov Apr 9 '10 at 12:33
I think the question "do closed aspherical manifolds have Hopfian fundamental group?" is a well-known open problem. The difficulty in Question 2 is that non-Hopfian finitely presented groups are not so easy to construct; it is a good question! –  Igor Belegradek Apr 9 '10 at 16:44
I suspect that the closest known such group is the example of a non-Hopfian CAT(0) group that Wise constructed in his thesis. –  HJRW Apr 9 '10 at 16:55
One partial result: if a closed manifold X admits a metric of negative sectional curvature bounded away from 0 then it follows from a theorem of Sela that $\pi_1(X)$ is Hopfian. –  HJRW Apr 9 '10 at 17:28

The Baumslag-Solitar group $B(2,3)=\langle a,b\vert ba^2b^{-1}=a^3\rangle$ is not Hopfian. But it has a natural $K(\pi,1)$ given by the double mapping cylinder of $S^1 \rightrightarrows S^1$ where the maps are $z\mapsto z^2$ and $z\mapsto z^3$. This is a finite CW complex.

Edit: The double mapping cylinder can be constructed like this. Take a circle $S^1$ and a cylinder $S^1\times I$. Glue one end of the cylinder to the circle by the degree 2 map $z\mapsto z^2$, and glue the other end of the cylinder to the circle by the degree 3 map $z\mapsto z^3$. The $a$ in the presentation above is the loop around the circle, while the $b$ is the loop that goes along the cylinder (whose ends have been brought together, forming a loop).

To see that this is a $K(\pi,1)$, you can check that its universal cover is the product $T_5\times \mathbb{R}$ of the infinite $5$-regular tree $T_5$ with the line $\mathbb{R}$. (Think about what this CW complex looks like locally: away from the circle where we glued everything together, it's locally a $2$-manifold. At the circle, we have $2+3=5$ half-planes meeting along their edges.) There is a picture of this universal cover on page 3 of Farb-Mosher, "A rigidity theorem for the solvable Baumslag-Solitar groups", Inventiones 131 2 (1998), 419-451.

-
Thank you for explanation. –  Sergei Ivanov Apr 9 '10 at 13:54
Can you construct a manifold like that? –  Anton Petrunin Apr 15 '10 at 22:02

A couple of remarks on this question:

It is true for closed aspherical manifolds of dim $\leq 3$, since they have residually finite fundamental group.

If one could find an aspherical 4-manifold $W$ with boundary, such that $\pi_1 W$ is non-Hopfian, and a degree >1 map $f:W\to W$ which induces a self-covering $f: \partial W \to \partial W$, then one might be able to use a reflection trick of Davis to find a closed aspherical manifold with the same property. Since there exist 3-manifolds which are not co-Hopfian (in fact, which are finite-sheeted covers of themselves), this approach might work.

I attempted this approach by thickening up the Baumslag-Solitar group presentation complex to a 4-manifold with boundary, but the boundary wasn't co-Hopfian, so I couldn't get the self-mapping of the 2-skeleton to extend.

-
"Let $M$ be a closed oriented smooth $n$-manifold and let $f:M\to M$ be a map of degree $d$. The author addresses Hopf's conjecture that if $d=1$, or $d=-1$ then $f$ is a homotopy equivalence. He proves the conjecture for the case in which $M$ has virtually nilpotent fundamental group. He also gives partial answers to related questions. Example: For any $d$ is not $1$ or $-1$ and any $n\ge 6$ there exist $M$ and $f$ as above such that $f$ induces an epimorphism with nontrivial kernel on the fundamental groups."


text:
Article Text

Cigarette prohibition and the need for more prior testing of the WHO TobReg's global nicotine-reduction strategy
1. Lynn T Kozlowski
1. Correspondence to Dr Lynn T Kozlowski, Department of Community Health and Health Behavior, School of Public Health and Health Professions, University at Buffalo, 323 Kimball Tower, 3435 Main Street, Buffalo, NY 14214-8028, USA; lk22{at}buffalo.edu

## WHO Study Group on Tobacco Product Regulation (TobReg) Advisory Note recommends global nicotine-reduction strategy

One can expect that the WHO Framework Convention on Tobacco Control (FCTC) Conference of the Parties may well be influenced by the recent WHO TobReg ‘Advisory Note’1 that supports, albeit with a host of caveats, recommendations to implement product regulations requiring reduction of nicotine levels in cigarettes.2–4 Note that these cigarettes are not the same as conventional lower yield cigarettes that were subject to compensatory smoking that maintained tar and nicotine exposures to smokers.5 To quote from the report's conclusions, the first two ‘regulatory recommendations’ are:

• Mandated reductions in nicotine to minimally addictive levels should be supported by comprehensive regulation of all nicotine- and tobacco-containing products.

• Mandated reductions in nicotine to minimally addictive levels must be part of comprehensive tobacco control, including increased taxes on cigarettes, comprehensive smoking bans, antismoking educational campaigns and graphic warning labels or plain packaging (ref. 1, p. 28).

The next four regulatory recommendations concern precautions related to how to do this in the context of comprehensive tobacco control. The Advisory Note and its conclusions clearly appreciate and assume that (1) the nicotine-reduced cigarette will be challenging to traditional smokers and (2) alternative sources of nicotine are likely to be needed, used and monitored. The last recommendation warns: ‘A strategy to reduce the addictiveness of tobacco is not recommended in the absence of developed capacity for market surveillance and product testing.’ (ref. 1, p. 29).

## Research to date

The recent, high-profile, randomised controlled trial is arguably the most substantial study until now and has been viewed as supportive, but one should look at it closely and critically.6 The report's introduction summarises the prior literature: ‘The results of several relatively small studies suggest [emphasis added] that cigarettes with very low nicotine content are associated with a desirable set of outcomes…’ (p. 1341). The article itself concludes: ‘This study provides preliminary short-term data suggesting (emphasis added) that as compared with the nicotine content of conventional cigarettes, a substantial reduction in nicotine content is associated with reductions in smoking, nicotine exposure, and nicotine dependence, with minimal evidence of nicotine withdrawal, compensatory smoking, or serious adverse events’ (p. 1349). The clinical significance of any of the statistically reliable effects ranges from unclear to doubtful. For example, changes of about 0.5 units in the Fagerström Test for Nicotine Dependence (FTND)7 would not be expected to have any clinical importance for cessation.8 ,9 The study was not designed to assess smoking cessation or the effects of use of alternative nicotine sources (as has been recognised as an overall component of the strategy).

The authors acknowledge that their sample was not representative of smokers in the USA, in part from excluding individuals with serious physical or mental disorders or who used any illicit drugs (except marijuana). Individuals with mental illness are more likely to be smokers, be more nicotine dependent and have a harder time quitting; these individuals account for 40–50% of the cigarettes consumed, despite being only 28% of the population.10 ,11 Analyses of more representative US data find that mental illness was associated with a much greater likelihood of nicotine withdrawal and estimate that 44% of nicotine withdrawal syndrome diagnoses are attributable to mental illness.12 It is, of course, not surprising or inappropriate that preliminary research would focus on healthy samples initially. However, given the recognised health disparities in smokers and the connections between smoking, mental illness, and use of alcohol and other substances, it should be important to assess the effects on such groups of restrictions to only very low-nicotine cigarettes, before recommending governmental regulations for all.

The authors note that ‘use of non-study cigarettes was common’, and explicitly acceptable, though discouraged, but also indicated as happening more often when on reduced-nicotine cigarettes. The up to US$835 payment for participation was structured so that if one dropped out after screening, one received only US$25, with increasing rewards for each type of session based on length. By completing all sessions and being timely, participants received a US$200 bonus (making the maximum payment US$835). Free cigarettes can be seen as an added financial incentive. Such an incrementally rewarded (a standard practice), non-mandatory, 6-week opportunity for healthy volunteers to try low-nicotine cigarettes does not really simulate what would happen if a representative group of smokers, some of them poor, had to pay themselves for only low-nicotine cigarettes with no legal access to their usual cigarettes.

No study that I know of has been done on how a representative sample of adult smokers experiencing mandatory low-nicotine cigarettes would respond, and the existing research on biased samples of well-compensated, healthy volunteers is described by the authors themselves as ‘preliminary’ and ‘suggesting’ good outcomes. On balance, it is very hard to see that we are close to having an evidence base that would support any government to embark on implementation of a mandatory regulation for all their smokers—no matter how advanced their tobacco control programming.

## Calling a cigarette prohibition a cigarette prohibition

The low-nicotine cigarette now being tested by the US Food and Drug Administration (FDA) was on the market in the USA for several years, and it has been judged a commercial failure despite extensive promotion (eg, US\$20 million in advertising).13 Nicotine-reduced cigarettes were not very appealing to consumers and were not overall a preferred product.14 ,15 Although not conclusive, this experience does raise questions about how such a product would be received as the only legal cigarette.

The terms of the US FDA tobacco law encourage the view that cigarette nicotine reduction is a way to regulate the addictiveness out of cigarettes without actually banning them.16 The law both forbids the banning of cigarettes and also appears to explicitly authorise the reduction of nicotine levels in cigarettes as long as they do not make it ‘zero’ nicotine (see ref. 17 for a discussion of the complexity of the FDA law). Outside the framework of the law, it may be easier to accept a more commonsensical perspective. If one subverts the functions of a drug-delivery system (the cigarette) by reducing the active drug dose to ineffective levels, this is fundamentally a banning of the consumer-preferred conventional cigarette. For those who seek the psychoactive effects of nicotine in cigarettes, the mandatory low-nicotine cigarette is a ban of their preferred product.

Popular recreational drug-delivery systems come in particular forms for which users can have very strong attachments. Imagine non-decaffeinated coffee lovers who were told that they must switch to tea (or vice versa) because of a newly discovered health risk if they wanted to continue safely having significant doses of caffeine. The coffee lovers might well consider themselves prohibited from buying ‘real’ coffee; or consider social drinkers who enjoy the psychoactive effects of wine or beer or liquor, preferring one form over the others. How would they react to regulations that forbid access to their preferred alcohol-containing beverage? Some drinkers might not care very much, but some would care a great deal and would see it as a banning of their preferred form of alcohol use. If the dose of the active ingredient in a preferred product is reduced to only ineffective levels, the preferred product has in effect been banned.

## Prohibition of conventional cigarettes would be a major step in tobacco control quite unlike the mainstays of tobacco control

While some tobacco control interventions have been implemented without any prior assessment of the impact on communities overall, these interventions did not involve a proposed degradation of a key active ingredient in cigarettes. For example, putting a cigarette in a plain package or putting a graphic warning label on it does not change the product itself. Even banning flavours leaves the drug-delivery system fundamentally unimpaired. The nicotine-reduction policy needs to be assessed in some actual societal context in a full range of smokers to be able to project the impact of such a regulatory policy that has never been tested in the real world. A recent assessment of the USA illicit tobacco market, which included some consideration of the nicotine-reduction research, concluded that the findings were ‘suggestive, not conclusive’.18 Many of the elements described in the second bullet point above from the Advisory Note have actually been assessed under real-world conditions.19 For example, there is a rich empirical sense of how taxes work to influence sales in actual communities,20 and that higher taxes can also promote black market sales.21 One of the lessons of tobacco taxation is that consumers will pay a great deal to obtain this desired product. One should expect that a number of consumers will pay the personal and financial costs for contraband conventional cigarettes.

## How would low-nicotine cigarette regulations actually work?

The TobReg report did indicate several important considerations for implementation (p. 26), including the need for complex surveillance systems with accompanying efforts to reduce contraband and smuggling as well as the importance of making available acceptable alternative forms of nicotine. It is understandable that their report would not go into great detail on implementation issues, but other questions are worth noting here. What would the law and its enforcement look like? Would sales, purchase or possession be prohibited? Would there be fines or incarceration? What priority would police or courts give to this issue? How hard would it be to distinguish legal cigarettes from illegal cigarettes? Would the legal tax base (that might support tobacco control) plummet? What sort of crime networks and economies might develop to supply smokers who prefer not to use reduced-nicotine cigarettes or switch to less harmful nicotine products (the market for which can differ greatly across countries)? The Advisory Note admits: ‘No published study has provided an estimate of the probable illicit sales of conventional cigarettes with a higher nicotine content in the context of a reduced-nicotine market.’ Some may view illicit trade as something that the tobacco industry has been willing to support. Be that as it may, one needs to consider the human predicament of the smoker who wants the product in the context of alternative tobacco/nicotine product markets that are likely to be quite different from country to country.

Countries differ in the array of legal nicotine delivery systems they offer. A reduced-nicotine cigarette in Sweden, for example, might spur the use of snus (a lower toxin tobacco product), but in most of the European Union, snus is banned. In India a push arising from reduced-nicotine cigarettes could encourage the increased use of indigenous, smokeless tobacco products that are more highly toxic than snus. The Advisory Note does ask for surveillance and epidemiological work, but these are likely to be limited tools in the face of a possibly fast-moving social experiment with economic, health and legal implications.

Consider two scenarios. In the first, you are in charge of tobacco control in a country that has instituted a plain packaging law, but your neighbouring countries have not done so. Would you be worried about cross-border issues arising from the smuggling of cigarettes in traditional branded packs (the exact same cigarette you sell in plain packs)? Probably not so much. However, would you answer the same if you were instituting mandatory reduced-nicotine cigarettes and your neighbours were not?

Before moving to make traditional cigarettes an illegal drug product, it could be instructive to study the international experience with cannabis, which has been described as the most widely used ‘illegal’ drug globally in 2012.22 If the nicotine-reduction model were widely adopted, traditional cigarettes could become one of the world's most widely used ‘illegal’ drug products. A recent analysis of the options of prohibition, decriminalisation and legalisation of cannabis (noting that prohibition is currently the dominant model globally) recommends that ‘legalisation with strict regulation’ is the superior option for public health in ‘high income countries in North America’.23 These authors also note that in addition to health harms caused by cannabis: ‘Under a system of prohibition, the enforcement of cannabis laws results in extensive costs, and in high levels of arrests and criminal records in the population.’(p. 42). One might suppose that no one is anticipating criminal records for possession of conventional cigarettes, but it does raise the question of the need to know what exactly would be proposed and how it would work (or be ignored) in a community.

## Lessons from alcohol prohibition in the USA

Experts have drawn several and sometimes conflicting conclusions from the experience of alcohol prohibition (and its repeal) in the USA.24 ,25 Whether one judges alcohol prohibition in some or all respects a success or failure, it is clear that implementation of prohibition created surprises for many of those involved.26

Those who moved to ban alcohol in the USA did so with no good sense of the societal consequences. Similarly, those recommending that FCTC countries adopt a nicotine-reduction strategy demonstrate relatively little concern for the possible negative societal consequences. Hall25 provides a scholarly review and concludes that the extensive prohibition of alcohol may have caused more harm than good, but that ‘limited prohibitions’ are probably very valuable. His example of a ‘limited prohibition’ is placing prohibitions on the legal age for purchasing alcohol. Indeed, such a ‘limited prohibition’ has been used, studied and found to be a valuable tobacco control tool.27 ,28

## First, if any country does it, a country with strong tobacco controls should try it and evaluate it

The Advisory Note cautions that the nicotine-reduction strategy is ‘not recommended’ if a country's tobacco control system is not advanced. This caution needs to be much stronger. Even an established, well-resourced, state-of-the-art tobacco control system needs to be prepared for a prohibition on conventional cigarettes that could have serious societal consequences, possibly resulting in a costly battle over and withdrawal of the law, either because of public objections or because of net negative effects on public health, especially in those portions of society facing the greatest health disparities. Until such time as a country or two with extensive, effective and comprehensive tobacco control programming has evaluated its real-world experience with the costs and benefits of such a strategy, other countries should be advised against attempting this regulation which as yet has been tried in no community.

Medical and scientific research is one element of policy development, especially if done on representative samples, but it can be critical to see the impact of policies in the complex systems that are communities. In a recent review of cannabis policy, it was noted that the circumstances of countries can differ greatly and that (1) policies should be developed that can ‘learn’ and change depending on how they are working and (2) one should be systematically learning from the experience of early adopters of a policy.29

## Going beyond the appearance of conflict of interest

The TobReg committee members who are likely to be the most knowledgeable on nicotine reduction have been long-standing supporters of the strategy,2 and their positive recommendation for dissemination may be unsurprising as well as would be some possible deference by other committee members or reviewers to the opinions of these experts. These members include Jack Henningfield (the proposal's cocreator),2–4 Dorothy Hatsukami,6 ,30 David Ashley (an FDA scientist/administrator involved with its tobacco law that explicitly requires assessment of this proposal) and the late Nigel Gray.3 ,31 Given that I have also published on what I consider to be unlikely prospects for and advisability of mandatory nicotine-reduction regulation in the USA, despite its apparently privileged position in FDA tobacco law,17 it might be equally unsurprising that I raise issues. None of this background for any one of us should be used to discount the positions taken, but readers should be mindful of the need to think carefully for themselves about the issues.32

## Conclusions

The reduced-nicotine cigarette strategy is an idea that is more elegant and attractive to some than others17 ,33 ,34 and on which the USA is investing millions of dollars in research. Some of the attraction most likely arises from the prospect of ‘banning’ cigarettes (without, according to the FDA framework, actually banning them) and from the elimination of the much feared ‘gateway effect’ by removing addictive cigarettes as an option for youth or adults.17 Nevertheless, a prior condition for disseminating even an appealing regulation that would act as a prohibition of the most popular tobacco product should be the support of a persuasive evidence base and assessment in communities of free-roaming individuals. The research on the low nicotine-reduction recommendation is until now preliminary and suggestive and not on representative samples of smokers. We have no direct evidence showing that it works to promote public health in any community anywhere. Such an untested community-level product prohibition is inappropriate for widespread dissemination by evidence-based health organisations. The amount of money for tobacco control is always limited, and it is premature to seek a formal place for the nicotine-reduction regulation within comprehensive tobacco control. Standard tobacco control measures (such as age restrictions, taxes, cessation supports, public smoking bans) have been tried in a number of places and have a persuasive evidence base. Even lower harm products such as vaping products and snus are providing some evidence of how they work in communities.35–37 Programmes to ‘do good’ have regularly arisen from simplified models of what a community needs and major mistakes can result.38 If no high-functioning comprehensive tobacco control programme is prepared to assess the nicotine-reduction strategy as regulatory policy, especially in the light of the currently only preliminary evidence in limited samples, then it should sit on the shelves and not be disseminated as a call for action.

## Footnotes

• Competing interests None declared.

• Provenance and peer review Not commissioned; externally peer reviewed.

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 Physics , 2002, Abstract: The spacetime dependent lagrangian formalism of references [1-2] is used to obtain a classical solution of Yang-Mills theory. This is then used to obtain an estimate of the vacuum expectation value of the Higgs field, viz. $\phi_{a}=A/e$, where $A$ is a constant and $e$ is the Yang-Mills coupling (related to the usual electric charge). The solution can also accommodate non-commuting coordinates on the boundary of the theory which may be used to construct D-brane actions. PACS:11.15.-q ; 11.27+d ; 11.10.Ef
 Physics , 1998, Abstract: Starting from lagrangian field theory and the variational principle, we show that duality in equations of motion can also be obtained by introducing explicit spacetime dependence of the lagrangian. Poincare invariance is achieved precisely when the duality conditions are satisfied in a particular way. The same analysis and criteria are valid for both abelian and nonabelian dualities. We illustrate how (1)Dirac string solution (2)Dirac quantisation condition (3)t'Hooft-Polyakov monopole solutions and (4)a procedure emerges for obtaining {\it new} classical solutions of Yang-Mills (Y-M) theory. Moreover, these results occur in a way that is strongly reminiscent of the {\it holographic principle}.
 Mark Hale Physics , 2000, DOI: 10.1016/S0393-0440(01)00064-X Abstract: We present a path integral formalism for quantising gravity in the form of the spectral action. Our basic principle is to sum over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries: the two-point space, and the matrix geometry M_2(C). On the first, the graviton is described by a Higgs field, and on the second, it is described by a gauge field. We start with the partition function and calculate the propagator and Greens functions for the gravitons. The expectation values of distances are evaluated, and we discover that distances shrink with increasing graviton excitations. We find that adding fermions reduces the effects of the gravitational field. A comparison is also made with Rovelli's canonical quantisation approach, and with his idea of spectral path integrals. We include a brief discussion on the quantisation of a Riemannian manifold.
 Douglas Singleton Physics , 1996, Abstract: Three dyon solutions to the SU(2) Yang-Mills-Higgs system are presented. These solutions are obtained from the BPS dyon by allowing the gauge fields to be complex, or by letting the free parameter of the BPS solution become imaginary. In all cases however the physically measurable quantities connected with these new solutions are entirely real. Although the new solutions are mathematically simple variations of the BPS solution, they have one or more physically distinct characteristics.
 Physics , 1997, DOI: 10.1142/S0217751X99001214 Abstract: Utilizing the conformal-flatness nature of 3-dim. Anti-de Sitter (AdS_3) black hole solution of Banados, Teitelboim and Zanelli, the quantisation of conformally-coupled scalar and spinor fields in this background spacetime is explicitly carried out. In particular, mode expansion forms and propagators of the fields are obtained in closed forms. The vacuum in this conformally-coupled field theories in AdS_3 black hole spacetime, which is conformally-flat, is the conformal vacuum which is unique and has global meaning. This point particularly suggests that now the particle production by AdS_3 black hole spacetime should be absent. General argument establishing the absence of real particle creation by AdS_3 black hole spacetime for this case of conformal triviality is provided. Then next, using the explicit mode expansion forms for conformally-coupled scalar and spinor fields, the bosonic and fermionic superradiances are examined and found to be absent confirming the expectation.
 Physics , 2004, DOI: 10.1088/0264-9381/22/7/004 Abstract: In this paper we develop the nonrelativistic quantum analysis of the charged particle-dyon system in the spacetime produced by an idealized cosmic string. In order to do that, we assume that the dyon is superposed to the cosmic string. Considering this peculiar configuration {\it conical} monopole harmonics are constructed, which are a generalizations of previous monopole harmonics obtained by Wu and Yang(1976 {\it Nucl. Phys. B} {\bf 107} 365) defined on a conical three-geometry. Bound and scattering wave functions are explicitly derived. As to bound states, we present the energy spectrum of the system, and analyze how the presence of the topological defect modifies obtained result. We also analyze this system admitting the presence of an extra isotropic harmonic potential acting on the particle. We show that the presence of this potential produces significant changes in the energy spectrum of the system.
 Physics Research International , 2008, DOI: 10.1155/2008/890916 Abstract: We construct a dyon solution for the noncommutative version of the Yang-Mills-Higgs model with a -term. Extending the Noether method to the case of a noncommutative gauge theory, we analyze the effect of CP violation induced both by the -term and by noncommutativity proving that the Witten effect formula for the dyon charge remains the same as in ordinary space.
 Physics , 2004, DOI: 10.1016/j.physletb.2004.11.016 Abstract: We consider vortex-type solutions in $d=5$ dimensions of the Einstein gravity coupled to a nonabelian SU(2) field posessing a nonzero electric part. After the dimensional reduction, this corresponds to a $d=4$ Einstein-Yang-Mills-Higgs-U(1)-dilaton model. A general axially symmetric ansatz is presented, and the properties of the spherically symmetric solutions are analysed.
 Physics , 1996, DOI: 10.1142/S0217751X97000347 Abstract: The problem of separation of variables in a dyon--dyon system is discussed. A linear transformation is obtained between fundamental bases of this system. Comparison of the dyon--dyon system with a 4D isotropic oscillator is carried out.
 Physics , 2001, DOI: 10.1142/S021773230200631X Abstract: The formalism of spacetime dependent lagrangians developed in Ref.1 is applied to the Sine Gordon and massive Thirring models.It is shown that the well-known equivalence of these models (in the context of weak-strong duality) can be understood in this approach from the same considerations as described in [1] for electromagnetic duality. A further new result is that all these can be naturally linked to the fact that the holographic principle has analogues at length scales much larger than quantum gravity. There is also the possibility of {\it noncommuting coodinates} residing on the boundaries. PACS: 11.15.-q: 11.10/Ef
 Page 1 /100 Display every page 5 10 20 Item


text:
# How much real and how much relative is our universe?

Apologies in advance if my question appears more philosophical than practical. It is indeed philosophy, but a philosophy relating to our scientific understanding of the universe as a whole.

Special Relativity states that as an object increases in velocity, it gets more dense (mass increases, volume shrinks). An object undergoing a decrease in velocity would begin to diffuse (mass decreases, volume inflates).

Now let us consider the fact that since Big Bang, all the material objects in our universe are in a state of constant, reckless motion (galaxies moving apart, solar systems orbiting around their galaxies, planets around their stars etc).

1. Keeping this in mind, how "real" is our universe? As in, if all objects in our universe come to rest (provided they don't collapse into a horrible universal black hole), how much would they diffuse? How much of their mass (as we know it now, in the state of reckless motion) would remain and how much of it would be lost? Of course the volume would expand 3-fold than mass (with a decrease in velocity, length increases, and volume is a 3rd degree function of length).

2. We know that the rest mass of a photon is zero (a photon's physical existence is only as long as it moves?). What would be the "rest mass" of our universe?

• I think this question has potential, but I was about to write it off until you got to your point. Perhaps rephrase it. – Gödel Dec 2 '14 at 20:47
• I added a long introduction just to show in what context I am asking the question. I could cut it short, but then fellow member's would answer it in different contexts, incoherently from one another. – Youstay Igo Dec 2 '14 at 20:51
• This question seems to be based on the faulty/outdated notion of relativistic mass, $m_{rel}=\gamma m$. – Kyle Kanos Dec 2 '14 at 20:52
• Kyle, do you intend to mean that mass does not increase with velocity? – Youstay Igo Dec 2 '14 at 20:59
• @YoustayIgo: Correct. See also this, this, this, and probably many more here on Physics.SE. So yes, relativistic mass is an obsolete concept. – Kyle Kanos Dec 2 '14 at 21:05

Special Relativity states that as an object increases in velocity, it gets more dense (mass increases, volume shrinks).

No, no, no. Absolutely not.

But I know why you think this. Back many decades ago, before I was born, lots of physicists wrote physics papers and textbooks without really understanding relativity. They made mistakes. Not necessarily errors, but certainly pedagogical mistakes. And these mistakes have proven most insidious and hard to weed out. The most egregious of these is stating that mass increases with velocity.

As understood today by any and all who deal with relativity, objects have an invariant rest mass $m$. This is the same for all observers. As a Lorentz scalar, it does not change when going into different frames of reference.

Old papers -- and remember, we're talking the early to middle 20th century, when no one know what DNA was, computers didn't exist, and we hadn't even coined the term "Big Bang" -- would refer to the combined quantity $\gamma m$ as relativistic mass. Sometimes they would write it as just $m$, where my rest mass above was written $m_0$: $m = \gamma m_0$.

It is this second $m$, this "relativistic mass," that changes with an object's relative velocity. The true $m$, denoted $m_0$ back in the dark ages, never changes.

An object undergoing a decrease in velocity would begin to diffuse (mass decreases, volume inflates).

Even if we agree we're talking about $\gamma m$ decreasing, there is a lower limit. The factor $\gamma$ is given by $1/\sqrt{1-v^2}$, where $v$ is the relative velocity between reference frames. You can check for yourself that for any $v$ bounded by the speed of light, $-c < v < c$, we have $\gamma \geq 1$. The minimum value of $\gamma m$ is $m$, and it occurs when $\gamma = 1$, i.e. when you the observer are comoving with the object, i.e. when the object appears to be at rest. Even the accursed relativistic mass can't go to $0$.

How much would some sum over $\gamma m$ for all objects change if they stopped moving from our point of view? Not much, actually. For cosmological purposes, ordinary matter is modeled very well by what cosmologists call dust. Basically, this makes the approximation that $\gamma \approx 1$, because most matter really is moving slow compared to the speed of light: $v \ll c$.

Note that I'm only talking about peculiar motion here -- the motion in addition to the uniform expansion of the universe. This is because the apparent "velocities" with which galaxies are receding can't really be analyzed in the same special relativistic framework. (For one, recession velocities are often greater than $c$!)

• What I gather is that unlike my assumption, there IS a base of reality (invariant mass and invariant length) on which all relativity changes play about. So if the galaxies are moving faster than c, it would imply that special relativity can only be applied to local phenomena (only in a closed system of 2 objects, the subject and the observer) and not universal phenomena? :S I'm feeling lost here. – Youstay Igo Dec 2 '14 at 21:32
• On the scale of the universe, you really do need general relativity. Unfortunately, astronomers tend to convert redshifts, which are the fundamental physical observables, to velocities (by multiplying by $c$), as though they were caused by special relativistic Doppler shifts. There is no physical reason for this, and thinking of redshifts as velocities causes as much trouble as it solves, but it's done nonetheless. You're not the only one to feel lost. – user10851 Dec 2 '14 at 23:39


text:
author nipkow Wed, 10 Feb 2016 15:49:05 +0100 changeset 62224 9343649abb09 parent 62223 c82c7b78b509 child 62284 1fd4831e9f93
tuned text
--- a/src/Doc/Prog_Prove/document/intro-isabelle.tex	Tue Feb 09 16:38:43 2016 +0100
+++ b/src/Doc/Prog_Prove/document/intro-isabelle.tex	Wed Feb 10 15:49:05 2016 +0100
@@ -73,9 +73,10 @@
by clicking on that phrase or by hovering over underlined text.

\begin{warn}\label{proof-state}
-By default Isabelle/jEdit does not show the proof state but this book
-refers to it frequently. You should tick the Proof state'' box
-to see the proof state in the output window.
+Part I frequently refers to the proof state.
+You can see the proof state combined with other system output if you
+press the Output'' button to open the output area and tick the
+Proof state'' box to see the proof state in the output area.
\end{warn}

This should suffice to get started with the jEdit interface.


text:
## User: rafa

rafa60
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United States
Last seen:
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Email:
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#### Posts by rafa

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... I would start by reading the Details section of the bumphunter help file.  Regarding the table, each row is a candidate DMR. value - is the average of the estimated coefficient (usually representing the difference between two groups) in the region area - is the absolute value of the sum of the es ...
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... Andrew Jaffe found and fixed a bug in bumphunter's bootstrap method. The coefficient for which inference was performed was fixed at 2 (the default). This is now changed to the coefficient passed along as coef This has been fixed in version 1.9.3. ...
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... The function matchGenes in the bumphunter package found the nearest gene/transcript to each entry in a GRanges and annotated with information such as if it is in an intron, exon, which exon, if it overlaps, covers or is inside, as well as with gene symbols, and refseq IDs. For historical reasons, th ...
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... Hi I updated all this. I will make a general announcement. ...
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... Jessica, I've made some major changes to bumphunter to accommodate this. If you install the latest devel version 1.7.3 (still being tested) the matchGenes and annotateNearest will no longer assume hg19. Here is an example: library(bumphunter) islands=makeGRangesFromDataFrame(read.delim("http://raf ...
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# Tangent - Equation of Tangent and Normal

The applications of derivatives are:

• determining the rate of change of quantities
• finding the equations of tangent and normal to a curve at a point
• finding turning points on the graph of a function which in turn will help us to locate points at which largest or smallest value (locally) of a function occurs.

In this article, we will learn to use differentiation to find the equation of the tangent line and the normal line to a curve at a given point.

## Tangent and Normal Equation

We know that the equation of the straight line that passes through the point (x0, y0) with finite slope “m” is given as

y – y0 = m (x – x0)

It is noted that the slope of the tangent line to the curve f(x)=y at the point (x0, y0) is given by

$$\begin{array}{l}\frac{dy}{dx}]_{(x_{0}, y_{0})} (=f'(x_{0}))\end{array}$$

Therefore, the equation of the tangent (x0, y0) to the curve y=f(x) is

y – y0 = f ′(x0)(x – x0)

Also, we know that normal is the perpendicular to the tangent line. Hence, the slope of the normal to the curve f(x)=y at the point (x0, y0) is given by -1/f’(x0), if f’(x0) ≠ 0.

Hence, the equation of the normal to the curve y=f(x) at the point (x0, y0) is given as:

y-y0 = [-1/f’(x0)] (x-x0)

The above expression can also be written as

(y-y0) f’(x0) + (x-x0) = 0

### Points to Remember

• If a tangent line to the curve y = f (x) makes an angle θ with x-axis in the positive direction, then dy/dx = slope of the tangent = tan = θ.
• If the slope of the tangent line is zero, then tan θ = 0 and so θ = 0 which means the tangent line is parallel to the x-axis. In this case, the equation of the tangent at the point (x0, y0) is given by y = y0
• If θ →π/2, then tan θ → ∞, which means the tangent line is perpendicular to the x-axis, i.e., parallel to the y-axis. In this case, the equation of the tangent at (x0, y0) is given by x = x0

### Equation of Tangent and Normal Problems

Go through the below tangent and normal problems:

Example 1:

Find the equation of a tangent to the curve y = (x-7)/[(x-2)(x-3)] at the point where it cuts the x-axis.

Solution:

As the point cut at the x-axis, then y=0. Hence, the equation of the curve, if y=0, then the value of x is 7. (i.e., x=7). Hence, the curve cuts the x-axis at (7,0)

Now, differentiate the equation of the curve with respect to x, we get

dy/dx = [(1-y)(2x-5)] / [(x-2)((x-3)]

dy/dx](7, 0) = (1-0)/[(5)(4)] = 1/20

Hence, the slope of the tangent line at (7, 0) is 1/20.

Therefore, the equation of the tangent at (7, 0) is

Y-0 = (1/20)(x-7)

20y-x+7 = 0.

Example 2:

Find the equation of tangent and normal to the curve x(⅔)+ y(⅔) = 2 at (1, 1)

Solution:

Given curve: x(⅔)+ y(⅔) = 2

Finding Equation of Tangent:

Now, differentiate the curve with respect to x, we get

(⅔)x(-⅓) + (⅔)y(-⅓) dy/dx = 0

The above equation can be written as:

dy/dx = -[y/x]

Hence, the slope of the tangent at the point (1, 1) is dy/dx](1,1) = -1

Now, substituting the slope value in the tangent equation, we get

Equation of tangent at (1, 1) is

y-1 = -1(x-1)

y+x-2 = 0

Thus, the equation of tangent to the curve at (1, 1) is y+x-2 =0

Finding Equation of Normal:

The slope of the normal at the point (1, 1) is

= -1/slope of the tangent at (1, 1)

= -1/ -1

=1

Therefore, the slope of the normal is 1.

Hence, the equation of the normal is

y-1 = 1(x-1)

y-x = 0

Therefore, the equation of the normal to the curve at (1, 1) is y-x =0

### Practice problems

Solve the following problems:

1. Calculate the slope of the tangent to the curve y=x3 -x at x=2.
2. Determine the slope of the tangent to the curve y=x3-3x+2 at the point whose x-coordinate is 3.
3. Find the equation of tangent and normal to the curve y = x3 at (1, 1).
4. Find the equation of normal at the point (am2, am3) for the curve ay2=x3.
5. The slope of the normal to the curve y=2x2 + 3 sin x at x=0 is

(a)3  (b)  -3 (c)⅓  (d) -⅓


text:
TeXnik - Tips TeX and LaTeX

Welcome to the TeXnik website (under construction)

TOC-faq
 Main page Index UK FAQ Documentation Software CTAN Search CTAN: Statistics Google c.t.t. Google d.c.t.t.
Bibliography in TOC |  chapter*/section* into TOC  |  Continuing TOC (more than one page) |  Dotted lines for chapter |  Format of TOC |  Headerline |  Labelwidth |  Labelwidth with tocloft |  pagenumber |  TOC-Style |  TOC-Style with tocloft package |  Unnumbered Chapters/Sections into TOC |  Vertical Space  |

Vertical Space
If you do not like the extra vertical space in toc, lof, lot, ... when a new chapter begins, then write just before you want to place the different lists:
\begingroup
\listoffigures
\listoftables
...
\endgroup
This decreases only the addtional space to the default one. To get rid of any vertical space between chapter entries write into the preamble:
\renewcommand*\l@chapter[2]{%
\ifnum \c@tocdepth >\m@ne
%    \vskip 1.0em \@plus\p@% <--- the vertical space
\setlength\@tempdima{1.5em}%
\begingroup
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
\leavevmode \bfseries
\hskip -\leftskip
#1\nobreak\hfil \nobreak\hb@xt@\@pnumwidth{\hss #2}\par
\penalty\@highpenalty
\endgroup
\fi}
Changing the vertical space between the title line and the first item is possible for the KOMA classes with

For all other classes write into your doc, where the toc should appear:
\begingroup
\makeatletter
%  \vspace*{50\p@}% <---  space above TOC header
{\parindent \z@ \raggedright
\normalfont
\interlinepenalty\@M
\Huge \bfseries  #1\par\nobreak
%    \vskip 40\p@% <---- space below TOC header
}}
\makeatother
\tableofcontents
\endgroup
The above example decreases all vertical space to zero!

Dotted lines between Title and Pagenumber

By default the TOC has only dotted lines between title and pagenumber for section and lower.

CHAPTER TITLE                                                       2
Section Title ...................................................2
If you want to have this for chapters or part too,
CHAPTER TITLE ......................................................2
Section Title ...................................................2
write in LateX preamble (only for classes with chapters):
\makeatletter
\renewcommand*\l@chapter[2]{%
\ifnum \c@tocdepth >\m@ne
\vskip 1.0em \@plus\p@
\setlength\@tempdima{1.5em}%
\begingroup
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
\leavevmode \bfseries
\hskip -\leftskip
#1\nobreak\
\leaders\hbox{$\m@th \mkern \@dotsep mu\hbox{.}\mkern \@dotsep mu$}\hfil\nobreak\hb@xt@\@pnumwidth{\hss #2}\par
\penalty\@highpenalty
\endgroup
\fi}
\makeatother
For classes without a chapter write
\makeatletter
\renewcommand\l@section[2]{%
\ifnum \c@tocdepth >\z@
\setlength\@tempdima{1.5em}%
\begingroup
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
\leavevmode {\bfseries
\hskip -\leftskip
#1}\nobreak\
\leaders\hbox{$\m@th\mkern \@dotsep mu\hbox{.}\mkern \@dotsep mu$}
\hfil \nobreak\hb@xt@\@pnumwidth{\hss #2}\par
\endgroup
\fi}
\makeatother

pagenumber at the TOC-pages

with \thispagestyle{empty} you can suppress the pagenumbering for a single page. If you want use this for the TOC-pages it doesn't work if your TOC is longer than one page. Than you'll get a numbered page with the beginning of the second TOC-page. To prevend this write just at the position in your text, where the TOC shall be:

\tableofcontents
\addtocontents{toc}{\protect\thispagestyle{empty}}
If you have installed the fancyhdr package, look at this page for redefining the plain-style for headers and footers.

If you have a TOC of several pages, than try this in the preamble

\makeatletter
\let\myTOC\tableofcontents
\renewcommand{\tableofcontents}{%
\begingroup
\let\ps@plain\ps@empty
\pagestyle{empty}
\myTOC
\clearpage
\endgroup%
}
\makeatother
Insert the TOC in your doc as usual. You can delete the \clearpage statement if you do not need a new page.

Bibliography in TOC

Just before your command to insert your Bibfile in the document write

\addcontentsline{toc}{chapter}{Bibliography}
% \addcontentsline{toc}{section}{Bibliography}% alternative for article class
for book style put the cursor before this command and than layout->paragraph->pagebreak before! For twocolumnmode you can also try to write these latex commands just before you insert the bib:
\let\myCDP\cleardoublepage
\renewcommand\cleardoublepage{%
\myCDP
}
< insert here the bibfile as usual >

for chapter you can choose any other level like part, section, subsection ,... in this case the name Bibliography appear in the TOC just like a chapter, means bold characters.
Bibliography is the name which appears in the TOC, you can change it to your own language, like Literaturverzeichnis in german
with package hyperref choose \phantomsection

with \addcontensline a pagenumber is printed in the TOC. with

\addtocontents{toc}{Bibliography}
you'll get only the word Bibliography in the TOC (with standard-font).

Labelwidth in TOC without tocloft
If you don't want to use the package tocloft look in your desired classfile for lines like these one:

\makeatletter
\newcommand*\l@section{\@dottedtocline{1}{1.5em}{2.3em}}
\newcommand*\l@subsection{\@dottedtocline{2}{3.8em}{3.2em}}
\newcommand*\l@subsubsection{\@dottedtocline{3}{7.0em}{4.1em}}
\newcommand*\l@paragraph{\@dottedtocline{4}{10em}{5em}}
\newcommand*\l@subparagraph{\@dottedtocline{5}{12em}{6em}}

\newcommand*\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\makeatother
and the same for list of tables (lot) with
\makeatletter
\let\l@table\l@figure
\makeatother
the first value is the left indent and the second one the labelwidth. you change them just you like it. for example try a
\renewcommand{\l@section}{\@dottedtocline{1}{1.5em}{6em}}
in LaTeX-preamble for sections, which should give enough space ... ;-)

If you want to change the labelwidth only for parts of the TOC, you have to do it on this way (for example: koma-script):

\makeatletter
\newlength{\myLabelWidth}
\setlength{\myLabelWidth}{1.5em}% <--- the standard value
\renewcommand*\l@chapter[2]{%
\ifnum \c@tocdepth >\m@ne
\vskip 1.0em \@plus\p@
\setlength\@tempdima{\myLabelWidth}%
\begingroup
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
\leavevmode \sectfont
\hskip -\leftskip
#1\nobreak\hfil \nobreak\hb@xt@\@pnumwidth{\hss #2}\par
\penalty\@highpenalty
\endgroup
\fi}
\makeatother
For changing the labelwidth for the appendix write in the text just behind the command \appendix or \begin{appendix}