studiot

Nice short input arc that hits the nail. +1

Like Mordred, I feel there may be a germ of an idea in your post. But I also feel there is an implied contradiction in your opening post. It runs like this. Your explanation states "The interference pattern has peaks and valleys of wave energy density which influence the path of the particle between the slits and the detector". This seems to me to be at variance with the statement of simultaneity, " Individual wave-particles can display both their wave and particle nature at that same time", because in order for the peaks and valleys pattern to influence anything it must be present ie it must be there when the 'particle' arrives ie fractionally before the particle. Please explain how this apparent discrepancy is overcome?

Let me just say I find Wiki disingenuous. All the laws? Too many things are undefined or omitted in the gloss over treatment. Without this most of useful mathematics falls apart. Well I'm sorry I can't even quote from your Wiki link here I just had some stupid error message. I've just goT through changing their stupid image types for ordinary letters. So much for standard. Talking of standards, perhaps you would like to write to Professor G H Hardy, Cambridge University and tell him to amend his book, A Course in Pure Mathematics To conform to you standards.

OK so let us try to solve my question using the extended real number system. Would you agree with me that the symbol [math]\infty [/math] must always stand for the same thing, just as say the symbol 9 must always stand for the same real number? So using this terminology [math]\mathop {\lim }\limits_{n \to \infty } {x_n} = \mathop {\lim }\limits_{n \to \infty } {n^2} = \infty [/math] and [math]\mathop {\lim }\limits_{n \to \infty } {y_n} = \mathop {\lim }\limits_{n \to \infty } n = \infty [/math] Substituting [math]\mathop {\lim }\limits_{n \to \infty } \left( {{x_n} - {y_n}} \right) = \mathop {\lim }\limits_{n \to \infty } {x_n} - \mathop {\lim }\limits_{n \to \infty } y = \infty - \infty = [/math] What is the infinite symbol take away the infinite symbol ie what is (the same thing) take away (the same thing) ? And why is it not zero? Because I think the 'limit' should actually be infinity. The penalty you pay for using this extended number system is that normal arithmetic no longer works. You cannot guarantee that sum/product/difference/quotient will yield a sensible result You cannot guarantee the axiom of associativity. Worse. The point/power of convergence is to determine if there are any solutions to differential equations/complex integration/transformations etc and once you let indeterminate elements into your fold you loose this ability. Yes I said somewhere back that some authors do this (Professor Thurston for instance), but associating limits only with convergence gains you far more than it looses.

First statement Yes, of course a sequence can diverge to infinity. But no I do not agree that infinity is a limit. Perhaps you would like to offer a solution to the following proposed limit (or ask Bruce) Let xn = n2 and yn = n Does the limit exist, and if so what is it? [math]\mathop {\lim }\limits_{n \to \infty } \left( {{x_n} - {y_n}} \right) = ?[/math] Second statement. I think you are confusing unbounded and infinite. They have different meanings. An infinite sequence can be unbounded eg 0,1,2,3........ or bounded eg 0,1,0,1,0,1..................

Please note there is no reason to suggest that the basis vectors or axes have to be orthogonal. Calculation convenience often dictates this but skew axes are also sometimes useful as in shear transformations.

Posts 472 and 473 here http://www.scienceforums.net/topic/29763-bannedsuspended-users/page-24#entry968109 I was trying to help other members who are also struggling with elementary limits as many do. Help in such a venture is always appreciated.

Far from being annoyed, perhaps you should read the earlier post questions and replies more thoroughly. I was trying to develop a consistent presentation starting from pretty elementary stuff. I had not even reached the stage of epsilon-delta. I was also avoiding the fact you seem to me to have implied that a 'limit' can never be reached When of course some limits are accessible and some are not. Both of these are in the next stage of the development.

Yes it is an interesting question and not yet fully resolved. You need to study the Continuum Hypothesis from Cantor to Cohen for this. https://www.google.co.uk/?gws_rd=ssl#q=cohen+and+the+continuum+hypothesis

The total interaction force is a combination of the electrostatic (zeta) and the hydrostatic pressure force. Water has a high surface tension, which can affect the latter, which is why surfactants are used. These are also called plasticisers in the concrete and mortar industry. The addition of such a compound may help with your flow and levelling issues.

Perhaps some clarification of terminology is in order. Some authors use the term 'functional analysis' to mean the theory of functions of a (real or complex) variable. That is the word 'functional' is used as an adjective. I think this is the meaning intended here. Unfortunately the word functional is also used in mathematics as a noun with a particular meaning. This meaning was introduced by Kantorovich, Banach and Kerysig A functional is a map from a space of test functions, [math]\Im [/math], (ie functions of interest) to its underlying field. In your case I think the underlying field is the field of Real Numbers, R So a definite integral is such a map from the space of integrable functions to the reals and outputs a real number for each definite integral. Functionals can be non linear or linear. A linear real functional, F, is a map [math]\Im \mapsto R[/math] such that for any two functions, [math]\left( {\varphi \,and\,\psi \in \Im } \right)[/math] and scalars (real numbers) [math]\left( {a\,and\,b \in R} \right)[/math] [math]F\left( {a\varphi \, + b\,\psi } \right) = aF\left( \varphi \right) + bF\left( \psi \right)[/math] Some texts on the subject are Kantorovich Functional Analysis Kreysig Introductory Functional Analysis with Applications Griffel Applied functional Analysis Functional Analysis grew out of Dirac's version of the theory of 'generalised functions' called Distribution Theory.

I take it this question is related to your other one. The sort of proof you are looking for is known as an existence and uniqueness theorem, of which there are several each pertaining to a particular area of mathematics. It looks as though you are studying what is known as the functions of a real variable. Even in this limited area the E&U theorems involve some highly abstract maths which fills large textbooks. A few 'standards' are Hurewicz Dimension Theory Hobson The Theory of Functions of a Real Variable (2 vols) Titchmarsh The Theory of Functions Burkhill A second Course in Mathematical Analysis Graves Theory of Functions of Real Variables Verblunsky An introduction to the Theory of Functions of a Real Variable Littlewood The Elements of the Theory of Real Functions Some of these are rather old.

So if in the series configuration each bulb has 60 volts across it and produces 60 watts and, as you say, has 1 amp current through it, What is its resistance? What will then be the wattage it produces if connected to 240 volts? (assuming it does not blow)

A good idea would have been to post the exact words from your textbook. However if the power is to be the same, by definition, the power rating must be the same ie 4 x 60W (assuming all the new bulbs are the same). Note that the power rating of a mains electric device is only correct at its rated (ie mains) voltage. In the UK both voltage and current are printed on the bulb. Since the bulbs are now in series, what do you think the voltage across each one now is?

Let's consider the power question first, since it doesn't make sense as it stands. "to produce the same amount of power?" "If these lights were to be placed in series" You can't place the same lights in series to produce the same amount of power. Either you place different lights of a different power rating in series to produce the same amount of power as 4 x 60W bulbs in parallel or You place the same lights in series and obtain a different power.

Well numerical integration will just give you a numerical area, and I suppose that you want a plot of the primitive? If you differentiate your equation that will turn it into a differential equation. You can then create the plots by one of several numerical methods starting from the (presumably known) boundary conditions. Edit A thought occurred to me When you wrote this Did you mean the formula is the derivative of the function? That is did you mean f'(x)=cos(-Asin(x)) ? In which case you already have a differential equation to work on.

I'll await the sketch before offering further comment then.

I've noted a few things about limits they don't tell you at the beginning but I think help understanding. There's quite a lot more to pull it all together if you want to go on. When you can see the whole picture and how it ties in with the rest of analysis, it may suddenly make much more sense.

No, it fails on several criteria. We were discussing numbers and infinity is not a number. A limit has to be of the same type of mathematical object as that being 'limited' Note it does not have to be a member of the set of elements but it does have to be of the same type. I already noted this in post#4. Further the meaning of the word is 'without limit' or 'limitless' or endless. Another way to look at it might be to think of a limit as a stop but there could be more elements (numbers) beyond the limit, if it is a member of the set of elements considered. So [math]\mathop {\lim }\limits_{x \to 0} x:x \in N = 0[/math] That is if we approach the limit of a sequence of negative integers it is zero, but there are all the positive integers beyond zero. What do you think is beyond infinity?

If your question was How does the balance of opposing forces work for electrons in molecules then consider this picture. Sails on sailing boats and pitched tents are subject to a large number of opposing pulls by the controlling ropes. That is exactly what gives the tent or sail its shape and position. Electron orbitals are very similar, although the source of forces are different.

Returning to the OP What you consider a minimum night time temperature below which you would want to reheat anyway will also affect this. Where I live the weather is not particularly severe so an outside temperature of -5oC will lead to an indoor temperature of around 6 to10oC by morning. So some reduced running during some nights may be desirable, unless you want to sit and shiver. Of course with my heat pump, the efficiency is lower at the colder night temperatures so extracting energy during the day may be more attractive.

So? We don't know which sort of system Alfred has or if he is operating it correctly. That is why I asked for more information. Further, if his system is on a low night rate tariff, it makes sense to operate it as much as practicable at night and store the heat somehow.

Sorry, but I am not sure what you mean. There are also good engineering circumstances when leaving a tightly controlled system permanently on can save money over cruder on/off systems. The 'heating experience' will be different when the systems are compared, but again that depends on circumstances.

Like I said, it depends upon circumstances. Economics is not always about pure physics or engineering. I recently changed my heating from a gas fired boiler to an electric heat pump feeding a heat store. My electricity supplier offers a tariff (for politico/economic reasons) where the night time rate is less than half the day time rate. The economics of the heat pump itself are heavily influenced by politics from both the EU and UK governments and the so called RHI initiative. The RHI brought down the Northern Ireland government last week)

I'm with CharonY here. The answer depends upon circumstances. You need to supply a great deal more information to answer this. @swansont The question was about saving money not energy.