studiot

Generally a well mixed life/career probably with lots of interesting stories along the way. As for trigonometry and mathematics in general, we ar estill teaching maths as a series of procedures leading to formulae that most folks then 'plug and chug'. It follows that they need the manupilative skills to chug. In actual fact most folks never need to do this as the outcome is already has already been worked out or in most modern times is available in the form of online calculators. What they actually need is a simple appreciation of what the result means and what they could do with it. This applies to calculus as well as trigonometry. They can very quickly get the hang of sin and cos or dy/dx etc just as they can get the hang of a car accelerator, brake and clutch pedal, without all the fluffy mech eng gear and transmission theory.

I get that on any browser in WinXP. I am unable to type anything into that thin input editor box. yet I can type into the login box

See also the longer BBC science article. https://www.bbc.co.uk/news/science-environment-66407099

A couple of associated observations. First one of my own on a recent visit to the National Aquarium in Plymouth. Here they have many viewing 'portholes' into various tanks. Some of these portholes have a hemispherical window to give a fisheye view of the interior. So the setup is similar to the glass ball @JeffJo posted and earlier picture to. However the lighting is inside the tank so on the opposite side from the viewer. I looked very carefully but could discern no rainbow halo. This is in accordance with Met Office stipulation that the viewer has to be between the light source and the reflecting/refracting ball. Secondly here is an interesting associated phenomenon as recorded by the joint French - Spanish survey to measure the shape of the Earth.

I'm sure we are all well aware of at least the basics of how insurance works and that it can work for both many socialist (in the broadest sense) and non socialist societies. Actually I'm trying to find some solid foundation to this thread which has the makings of a good topic, but seems to me to be floundering around all over the place. It's your thread. Perhaps we could examine societies for which 'socialism' presents special difficulties ? For instance take the Modern day Outback or the pioneering North America. Self reliance was /is possibly the greatest virtue where folks live hundreds of miles from the nearest neighbour. Or societies that relied on slavery. Is there any point insuring your 'asset' if there is a ready supply of replacements ?

What does a discussion about insurance have to do with socialism ? Also I don't know how young you are, but have you heard of the 'National Socialists' ?

Physics ? Maybe you are looking at the wrong Science ? I am not a social scientist but I understand that (early) Christianity is scientifically classed as "Primitive Socialism". How dows that play out with your definition of Socialism? +1 by the way for actually offering a definition of a term you are intending using. This necessary courtesy is all too often omitted.

So far I have refrained from commenting on this thread because it already involves some rather juvenile mud slinging, as all too many global warming threads seem to attract. So +1 for bringing some SCIENCE to the table. I agree totally that the statistics so far presented are wholly inadequate for the job, especially when the theory of time series analysis goes back to the early 20th century and even a bit beyond. I don'r think LOWESS analysis is the way to go though, it is very short term. I would suggest a properly thought out moving average analysis , based on my initial guess of using a 10 year average, would definiteley show the relatives sizes of Mistermack's smaller rise to the later possibly accelerarting rise in the posted graphs. I would observe that including error bars on these plots has more to do with the fancy graphics abilities available nowadays at the push of a button, than the statistical purpose of the graphs. In my opinion such stuff actually obscures the trend lines. If ssomeone has a table of values I would be happy to turn them into a proper time series analysis.

I rather thought that I was doing a pretty good job of ignoring. 🙂

And what did you make of "Riemannian Calculus" ?

I wish. Unless, by using the plural of definition you mean that science has multiple meanings for many important words and concepts.

Yes I agree but as I have discovered, there is much more to this than that observation. https://opendatascience.com/ai-sentience-friend-or-foe/

You seem to be wanna-ing ? the aspects of philosophy that are not really amenable to scientific analysis. Remember this is a scientific site and we are supposed to be discussing the scientific aspects in subjects that are not directly scientific. You might be better off asking for these in an arts forum for instance, take Love. In Suits there are many scenes where the 'love' Donna has for Harvey is explored. In one she says " I don't love him like that , I love him like a brother" So remember many words do multiple duties so it is a good idea to specify which one you mean at the outset of any serious work.

Many thanks for this thread and introducing me to a whole new area of concepts. I had never heard of the mirror test before, but looking into the subject I see that much thought has already gone into the subject area and I have much to look into. +1

Hello, Steve and welcome. I read your article and I must say I was pleasantly suprised as it addressed several thoughts I had when I initially saw your thread. I particularly liked the idea of defining what you mean by a rocket. So +1 for a good start. Now for some comment. Unfortunately it is difficult to say much more without any context as to what the article is meant for. [ Like TheVat I question terminology a bit it is hardly in the format of a (scientific) paper which is why I refer to it as an article. ] You have posted this in Homework Help so I assume you have read the rules for this section of ScienceForums. But you haven't indicated what his article is meant for and at what academic level ? Tell us a bit more so interesed members can offer better help.

12 Area units is correct Here is my derivation, avoiding equations one way or another. I use the property of rectilinear similar figures that the areas are in proportion to the squares of corresponding sides. So firstly complete the rectangle ANIM by adding 3 extra squares, making 6 in all. DH is a right bisector of two sides so passes through the centre of the rectangle at K AI is also a diagonal of the rectangle and more importantly bisects CE. Thus CK is equal to KE. Call this s. CK is one side of the triangle CKJ which we are given has having 1 square area unit. Triangle CKJ contains the same angles as triangle AMI since AN is parallel to MI. Hence triangle CKJ is similar to triangle AMI., with side CK corresponding to side MI Since the length of each side of a square is 2s and side MI occupies 3 squares its length is 6s. So the area of triangle AMI is (6)2 times the area of triangle CKJ or 36 area units. Triangle AMI is exactly half the rectangle ANIM. So the total area of the rectangle is 72 area units. The rectangle is comprised of 6 squares so each square has an area od 12 area units as required.

So AI is the diagonal of a rectangle 3 squares wide by 2 squares high and K must be the centre point of this rectangle. So by no equations I assume that I can use formulae you mean no equations need solving ? So I can say the area of this rectangle is 3 squares x 2 squares which makes 6 squares.

Let KI intersect EF in M. The triangle ABJ is similar to JCK is sim to KEM is sim to FMI. That is the triangles along the diagonal all have the same angles and areas in proportion. I take it ABCD, CEFG and EHIF are squares.

Since there is some interest in this subject I will post the outline mathematics. I am sorry I can no longer do the LaTex since SF will not let me post from WinXP. So if anyone can help with that I can expand. Basically we start with Laplace's equation in 3 dimensions, in polar format, and apply a potential , V. This can be colour value /intensity, or other quantity to produce the 'ripple' across the object. Successive radial (r) values of V are calculated for various values of two sectional angles theta and phi., where it intersect the object from geometrical model This is my first equation. The general subject is called spherical harmonics, which is concerned with solutions to this equation. A homogeneous algebraic equation separating r and the angles in the form of my second expression, rnf(theta, phi) gives values of V satisfying the first equation, in polar coordinates. These are known as solid spherical harmonics of the nth degree. The function f(theta phi) is known as a surface spherical harmonic of the nth degree. The equation can be simplified by symmetry when V is independent of phi. This lead to my second equation whcih upon the transformation indicated becomes Legendre's equation The solutions are known as Surface Zonal Harmonics, as indicated by John Cuthber. These may result in 'patterning'. This is not only used in CGI but also in computer tomography (CT scanning) via a finite element mesh. The equations are normally solved numerically by substituting suitable simple function such as' hat' functions on the mesh, with the computer doing lots of calculations.

You have shown it to this Mathematician. And I thought there were too many misunderstndings in it to be worth detailed consideration. So +1 to Genady for the patience to do this. +1 also to PhiforAll who got in before me when I was about to complain. A word of advice. You can expect ridicule as answers when you make such wildly and obviously incorrect statements. Such as claiming the non existence of zero. Consider this sequence of integers ...-4, -3, -2, -1, 1, 2, 3, 4... ...Even, odd, even, odd, odd, even, odd, even... Do you notice anything missing ? It implies that there are more odd numbers than even numbers. By the way do you understand the ... symbol? It is called an ellipsis. You have also been arguing wrongly about the meaning of the equals sign. It stands for two different properties - Identity, and simple equality. Sometimes the identity symbol with three bars not two is used instead for this.

Embarrased or not, your teacher was not so far wrong. Car Tyres are indeed an entirely different case. Not only that but there are further complications I will try to elaborate. The friction theory they teach at elementary level only refers to the contact of dry solids. Some textbooks and teachers make this point, some do not. A car tyre is in general neither solid nor dry ! The study of forces involved between bodies in contact is called tribology. Furthermore a car is a dynamical system, with constantly changing velocities, directions and accelerations. More of this later. Here is a well produced table by Professor Sir Charles Inglis of Cambridge University. Note very carefully what he says about reactions and contact area. You were taught only the left hand column. Once again car tyres are neither solid nor dry. They are actually a mixture of all three types (type 3 = partially lubricated) - about as complicated as you can get. Why do I say all this ? Well the forces of interest for car tyres and the road surface are collectivelly known as the grip. Here is a simple summary from https://www.racecar-engineering.com/tech-explained/tyre-grip/ Note that they describe tyres as having viscoscity. Solids don't have viscoscity, liquids do. Viscoelastic refers to a mixture of characteristics. But even this does not tell the complete story. You are asking why early racing cars and modern road cars have thinner tyres. Well road cars have a dry coefficient of friction of 0.8 dry and zero to 0.6 wet. Modern racing cars enjoy a coefficient of 1.4 to 1.7, on their fatter tyres. Cars are not like your diagram of a solid block with full or nearly full contact area and a simple C of G load point. The loads are distributed to the four extreme points. They are dynamical systems (when running) which means that the loads are also constantly changing with the overall motion of the car as it goes not only forwards but twists and turns as well. So the parts suffer accelerations not only due to the car's overall motion, but also due to load transfer. With modern cars this transfer is modified by being transferred through springs. For example a racing car's coeffiecient of friction can drop to 1.2 when also suffering sideways loads from cornering. As noted in the linked article the tyres themselves have an internal dynamical structure which also modifies the friction laws. There is even more to the subject as the tyres distort and recover and use energy, heating up in the process, leading to the so called rolling resistance or coefficient of rolling friction. Also the wider tyres give a safety margin against slipping and locking which is needed for the higher speeds and stresses involved in F1.

Here is your first error. It is incorrect to write an equation with infinity on one side. That is it is incorrect both mathematically and physically. In fact your whole thesis ls built on a very shaky understanding of ' infinity'. Things are a great deal more complicated in both maths and physics and indeed other sciences. Please tell me what the tangenrt of Pi/2 is ? There are many equations in Science employing the tangent function. A simple example would be to do with friction. Physical Chemists use a totally different form of infinity called infinite dilution. In fluid mechanics a property called specific energy tends to infinity in a fluid structure called a hydraulic jump. I have already told you about what is probably the simplest one - density, which is handled by a French discovery L'Hopital's rule. Another French infinity is the formation of a square wave from and infinite series of sinusiodal waves. Note I am not saying that Man can reach 'infinity, just that it can be valid in Physics or other sciences and must be handled accordingly. There are as many ways of handling infinities as there are examples.

Are you being deliberately obtuse ? What was unclear about my first and most important question? There were no spelling mistakes and it was written in textbook correct English. Since I placed 3 question marks in my first post I referred to 3 questions in my second post. So why do you reply in the singular ? Note I spotted and enquired about a contradiction in your original post. It was also unclear to me whether you main interest concerned the mathematics and or physics of infinity, as the title suggests, as against a particular effect in Physics.

I asked you 3 questions. All of which you ignored. One of those questions actually contains the answer to your reply question. Do you call this a discussion ? I can assure you the rules here do not.

Surely your title is contradicted by your first line. If infinity is not defined in Physics how could it be permissible to use it ? Of course it is well defined. And why to you need such an exotic example as the Casimir Effect ? What is wrong with schoolboy Physics. Density = mass/volume So what is the density at a point, which has zero volume ?