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| Raising a number to a non-integer power. |
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| SiliconWizard:
--- Quote from: RoGeorge on May 30, 2020, 10:42:51 am ---I'm not familiar with the Lambert W function, but calling that a function feels like cheating. By definition, a function can not have more than one output for a single input. Maybe in math we can change or extend the definition for what a function is, but then it will be out of this world. --- End quote --- This is called a multivalued function, and this isn't an ordinary function indeed. https://en.wikipedia.org/wiki/Multivalued_function --- Quote ---In the physical world, when for a single input we observe multiple results, it means we are missing something, like an extra input variable we disregarded, or an extra function/phenomenon we don't realize it's there. --- End quote --- Quoting Wikipedia: --- Quote ---In physics, multivalued functions play an increasingly important role. They form the mathematical basis for Dirac's magnetic monopoles, for the theory of defects in crystals and the resulting plasticity of materials, for vortices in superfluids and superconductors, and for phase transitions in these systems, for instance melting and quark confinement. They are the origin of gauge field structures in many branches of physics. --- End quote --- |
| Nominal Animal:
Yup! I used Lambert W as an example of a "completely unintuitive function-like thing" that is very useful but almost impossible to find real world analogs or useful intuitive reasoning for, to show why we must not expect to have good analogs or intuitive explanations for all our mathematical tools. When I first started learning calculus, I had a hard time giving up wondering "what value" δ(0) is. δ(x) is the Dirac delta function. δ(x)=0 for all x except x=0, but its integral over all real x is 1. I internally settled for something like "infinitesimal infinity" (I don't think I've ever said it aloud), before I realized the entire question is nonsense. δ(x) is a mathematical tool with specific rules and behaviour, and does not necessarily have that property at all! Indeed, if you find that to solve a problem you need the value of δ(0) outside any differential equation or similar context, you can be sure that you went wrong somewhere. Also consider dimensional analysis. In dimensional analysis, you drop all quantities (numbers) and apply your model (equation or formulae) to the units only, and see what kind of unit pops out. If you get e.g. m/s where you expect time units, you know that either you made a mistake, or the model is b0rked. But working out a formula or equation with units, and completely without numbers or variables, is pretty counter-intuitive to most people. Many learn to just ignore the units and work with the numbers only, and that sometimes leads to really bad mistakes. |
| T3sl4co1l:
--- Quote from: Nominal Animal on May 30, 2020, 04:10:08 pm ---I used Lambert W as an example of a "completely unintuitive function-like thing" that is very useful but almost impossible to find real world analogs or useful intuitive reasoning for, to show why we must not expect to have good analogs or intuitive explanations for all our mathematical tools. --- End quote --- Another way to express it could be as returning a vector, i.e., a function \$ \mathbb{R} \rightarrow \mathbb{R}^n \$, and more generally than that, functions of arbitrary dimension in the domain and range. This requires more formalism to pack up (i.e., are we using linear algebra, set theory, etc.?), but the basic idea can be seen. Applied in another domain, we might think of functions in computer science as some dimension in the parameters, and the return values (assuming pure functions, i.e. no side effects, no internal state). We introduce type theory (the vectors need to be not just the same number of elements, but the same types as well). And then we can think about data and its typing in various useful ways. --- Quote ---When I first started learning calculus, I had a hard time giving up wondering "what value" δ(0) is. δ(x) is the Dirac delta function. δ(x)=0 for all x except x=0, but its integral over all real x is 1. I internally settled for something like "infinitesimal infinity" (I don't think I've ever said it aloud), before I realized the entire question is nonsense. δ(x) is a mathematical tool with specific rules and behaviour, and does not necessarily have that property at all! Indeed, if you find that to solve a problem you need the value of δ(0) outside any differential equation or similar context, you can be sure that you went wrong somewhere. --- End quote --- Indeed, when a continuous function is desired, a common implementation is a Gaussian with lim width-->0 and height-->inf, taking the joint limit such that the integral property is preserved. This gives a rigorous basis and allows some analyses which would otherwise fail on the more basic definition. But it is indeed a not-function, in its basic form, and I think most instructors are careful to point that out when it's introduced (at least, mine were). --- Quote ---Also consider dimensional analysis. In dimensional analysis, you drop all quantities (numbers) and apply your model (equation or formulae) to the units only, and see what kind of unit pops out. If you get e.g. m/s where you expect time units, you know that either you made a mistake, or the model is b0rked. But working out a formula or equation with units, and completely without numbers or variables, is pretty counter-intuitive to most people. Many learn to just ignore the units and work with the numbers only, and that sometimes leads to really bad mistakes. --- End quote --- Yup, and since dimensions are conserved under arithmetic operations, you can assign dummy dimensions to mathematical variables (which might otherwise be dimensionless), and track down some algebraic errors in the process (when you're working large equations by hand). And it's not just a physicist's hand-waving tool, it has rigorous application! Using a change of variables, units can be extracted from otherwise very difficult expressions. Feynman integration is a famous case, where the units are extracted from an integral, so that the integral is over an abstract mathematical function, which then merely returns a dimensionless geometric constant. All the dimensional relationships are basic arithmetic, so that qualitatively speaking, we have everything we need to know about the problem, and exact results are merely proportional to that. And we can then compute that function through any experiment or dynamical system which is equivalent to it, not just the problem in question. Tim |
| TimFox:
The rigorous definition of the Dirac delta is not a function, but a “distribution”, sometimes called a “generalized function”. Although the original concept came before him, it was developed and tidied up by Laurent Schwartz midway through the last century. See Wikipedia for a summary. There is no difference between his formulation and the results obtained with common concepts involving shrinking tall Gaussian curves to the limit, but it is a good example of mathematics catching up with a useful bit of physics. (In my youth, we studied the history of the Einstein Summation Convention that was held discretely on the Kronecker Delta.) |
| Nominal Animal:
--- Quote from: TimFox on May 30, 2020, 08:04:59 pm ---The rigorous definition of the Dirac delta is not a function, but a “distribution”, sometimes called a “generalized function”. --- End quote --- In the context of this thread, "generalized function" would be key ;) (Because the mathematical concept of "distribution" doesn't really match what a layman might think a distribution is, you see, and this thread is all about our human need for intuitive real-life analogs.) From generalized functions or "distributions" (in the mathematical, not traditional statistical sense) one gets to functionals and vector spaces and linear algebra and whatnot, all very useful tools. Me, I was never particularly good at math, and what little I know, is all hard-fought. I didn't really grok differential calculus until I started working with interatomic potentials. Before, I could solve problems I recognized, sure; but it was just applying methods I knew, like treating each problem as a perp and looking them up in my mental records, then applying the suggested methods therein. No true insight, no creativity, just straightforward grunt work. Now, it feels different. Enjoyable. Like a very nice, well-worn, reliable tool. Or I might just be going bonkers in my middle age, and sliding waaay back on the D-K scale! |
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