studiot

The ear is insensitive to the phase of soundwaves. The sides of the bell will be launching spherical (not planar) soundwaves into the free air. But some parts of the sides will be travelling one way and some in the opposite direction, due to overtones. That is the vibration is segmented as in the picture. These spherical waves diffract round the bottom of the bell. So the soundfield underneath will depend upon its width. There are likely to be zones of cancallation and zones of reinforcement in your area A. Even with the finite element calculations shown in the google images bell calculations are far from an exact science.

I do not like the notion that you can 'learn maths quickly'. These books were written by real experts and are particularly suitable for self study or to enhance an existing course at your level. I usually seem to end up recommending Oxford University Press publications and this situation is no exception. Gardener : Discovering Modern Algebra : OUP Ferrar : Mathematics For Science :OUP Lambe : Applied Mthematics for Engineers and Scientists : English Universities Press (EUP) Finally you will find Howard Anton's "Elementary Linear Algebra" or "ELA, Applications Version" better (IMHO) than Strang for your purposes.

Sorry this is too simplistic. Note that I said that the vibrations are developed by the bell. The bell has various modes of vibration which you can see from the images segment the bell into zones. https://www.google.co.uk/search?q=vibration+modes+of+a+bell&tbm=isch&tbo=u&source=univ&sa=X&ei=2mQcU_q1OIbBhAeKs4FI&ved=0CEcQsAQ&biw=1280&bih=585

I will look out some better references for you. Wiki is a good place to start, but often rather abrupt. A good book for a mathematician to start with is Fractal Geometry Kenneth Falconer (Wiley) (He has written several others as well)

Here is a learned paper on the subject http://www.cs.cornell.edu/~dph/papers/HKR-TPAMI-93.pdf

You should look up the Hausdorff Dimension http://en.wikipedia.org/wiki/Hausdorff_dimension

I really like this thread, it shows some great scientific observation +1. First the experimental evidence. By using the stainless steel food mixer bowl I can confirm function's results. Firstly by ear I can definitely tell that the sound is louder outside the bowl than inside. So I repeated this using a sound level meter. The ambient was recording around 60 dB. Outside the bowl I was getting readings of 70 - 74 dB Inside the bowl the sound level read 64dB As to the explanation I do not think it is an interference effect. The bowl is a curved vibrating plate, more commonly known as a bell. Unlike wind instruments there is no vibrating volume (column) of air generating the sound. The sound is generated by direct vibrational modes of the bell and transferred to the air. Being contained in a nearly enclosed space the air inside the bell will offer a much stiffer response than the external air. So generation of the same pressure variation will take more energy inside than outside the bell. or the same energy will generate a smaller response. This will show up as different acoustic impedences inside and outside the bell.

When we derive the mechanical wave equation we consider the action of a restoring force and after some manipulation end up with the differential equation, known as the wave equation. This restoring force is provided by the medium of propagation. The initial energy is transferred temporarily to the medium and then returned to the wave by way of the restoring force. This process is repeated over and over and so the wave progresses. An electromagnetic wave has no medium and so has a different restoring force mechanism. It basically generates its own restoring force, if the initial disturbance is of the correct form. Mike, since you like pictures you might like to wade through this explanation. http://ocw.tufts.edu/data/30/365841.pdf

There is no requirement for either to be 'true', other possibilities exist.

I am not sure whether this post puts you for or against the idea that you can add up infinitely many mass points to make 5kg or any other value. It seems to me that the B-T decomposition allows one to carry out uncomfortably many such constructions, rather than bar you from them. Incidentally, whilst is does not detract from the mathematical flow about the B-T decomposition, attempting to apply it unqualified to the physical universe does. The B-T decomposition assumes that all 'points' form an equivalence class. That is any one point can stand in for any other. In particular it ignores the well ordered principle of the reals. This is OK in appropriate context, but definitely not OK in physical space. The real numbers 1.0 and 3.0 may be equivalent in one sense, but they are not the same. This does not mean that one is in superior or preferable either, so relativity is preserved. What the B-T decomposition is saying is that you can map all the points between 1 and 2 to all the points between 3 and 4 or 5 and 6 or even 0 and infinity on a one to one basis. So you can, in that sense, create a 'copy' of all the points in the interval [1, 2 ] within the interval [3,4] by such a mapping. But it does not say you can construct two (multiple) copies of all the points within the interval [1,2] within the interval [1,2]. You require another interval to perform this.

Electric current is not energy either.

Actually you have this backwards. The density of the earth is inferred by dividing the mass, deduced from astronomical measurements of interactions with other heavenly bodies by the measured dimensions. This is then used to refine theories about composition.

Tony, a quantity called mass appears in two unconnected areas of mechanics. Gravitational mass Inertial mass Perhaps you would like to consider the triumph of science in showing that these are in fact equivalent.

Operational Research and Game Theory flowered in the 1940s and attracted many of the era's big names in mathematics Kantarovich Mathematical Methods in The Organisation and Planning of Production (Russian 1939, English 1960) VonNeuman The Theory of Games and Economic Behaviour Nash's work appeared in papers in the early 1950s Nash The Bargaining problem Econometrica 18 (1950) Non-Cooperative Games Annals of Mathematics 54 (1951) Two-person Cooperative Games Econometrica 21 (1951)

In another current thread Unity+ is asking about the ancient Greeks. The ancient Greeks took a largely geometric view of maths and they offered a geometric construction for 'division'. It is instructive to try to apply this to both division of a nozero line (=division of a real number by zero) and a zero line (=0/0).

The number axioms are suprisingly few and do not include division. In advanced algebra there is a formal entity called a division ring. http://en.wikipedia.org/wiki/Division_ring Part of the reason we have rings, fields, groups and so on (carefully) defined is to overcome the difficulties that appear in less detailed treatments.

Since there is evidence that other creatures have rudimentary counting and measurement skills, maths cannot be a purely manmade construct.

In our 3D world there are actually 3 distinguishable axes. There are many beautiful an interesting phenomena without trying to invent some by guesswork. The equations everyone are referring to are known as Euler's Equations. These can give rise to chaotic action as can be seen and tested in this simple example. Take a brick shaped object eg a book, a matchbox, a box of chocolates, or even a small brick, so that the length, breadth and depth are all different. Take the block by a pair of opposite faces and and toss it up the into the air, spinning it at the same time. You will find that the block spins readily and stably about two of the three axes, defined by the faces, but starts to wobble very quickly about the third.

A good introductory text to this side of the subject of business mathematics, up to and including the Axioms of Nash, is to be found in "An Introduction to Linear Programming and Game Theory" by Paul R Thie, published by Wiley.

What's nice about them? I only wish my wallet obeyed the rules for manipulating infinities, rather than the rules for manipulating real numbers, so If I take out £10 from my wallet I still have the same amount of money in it.

You must have had some bad fish for dinner. The OP was very wide ranging and I'm simply trying to offer up a selection of points for consideration. I make no claims that the my list is exhaustive. As I understand game theory it is about the balance between the possible losses and the possible gains and maximising the function gains minus losses, allowing that the maximum may be zero. But the seller can do 'better', although that would mean the buyer doing worse. Value, of course, in an imprecise term.

You would need to elaborate on that and explain why it is superior (or at least no worse than) to more conventional arrangements.

Consider these examples. 1) I buy 10 fish for $1 each and offer them for sale at a fish market, where all unsold fish must be disposed of at the end of the day's trading at 5c per fish. What price should I charge? As I understand game theory it says that I should start offering them cheaply and increase the price as I begin to sell them. This minimises the potential loss if I do not sell any. However this is the obverse of the practice enacted by most market traders I have seen who start dear and cut prices at the end of the day. 2) I buy 10 fancy boxed dolls for $1 each and sell them at one of those 'one day only' auctions. Here traders habitually offer the first below cost and bid each one up to hope to sell the last one at $15 or $20.

Agreed, but what was you question here? Flexible small bore pipes and pulsed fluid action (unsteady flow) introduce many new variables.

One thing to observe is the relationship between time and money. The 'value' of money depends, in part, on when it occurs. You should look up concepts like Net Present Value (NPV) Discounted Cash Flow (DCF) Pricing policy may dpend upon game theory but also upon time and perishability of the product. So it's OK to play games with real estate that will still have a value next week or next month if unsold, but not with fresh fish that will have no value, or even negative value (disposal costs) next week if unsold.