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    Does randomness really exist?

    No, but to me it suggests that it wouldn't be unreasonable to act on the basis that those phenomena are fundamentally deterministic at some level of awareness (unless doing so made one miserable.)
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    Does randomness really exist?

    I guess I'm inclined to assume determinism is more fundamental than randomness because almost every fundamental physical experiment to date has been consistent with the postulate of unitarity in quantum mechanics, and I interpret unitarity (i.e. constant, vanishing von Neumann entropy) as...
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    Is there a better name for "citizen science"?

    It just occurred to me that one potentially crucial distinction between Citizen Science and Amateur Science is that the word 'citizen' suggests a kind of civic connection that might not always exist in the amateur setting, and which could be understood in various ways (for example, that the...
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    Is there a better name for "citizen science"?

    The term Citizen Science has, at least according to wikipedia, existed for at least several decades. Is it time to come up with a hip new 21st century moniker? Or are we happy with "Citizen Science" as it is?
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    Does randomness really exist?

    It's been a while since I've posted here, but I like this general topic and so I couldn't resist getting involved. It sounds as though there are two schools of thought expressed in this thread, one of which centers on whether randomness exists in a fundamental sense, and the other on its...
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    B Coriolis problem - Point mass movement upon release from Earth

    This sounds a bit like a homework question, but here are a few hints: (i) What quantity (or quantities) are conserved? (ii) From a corotating reference frame, what fictitious force determines the magnitude of the longitudinal angular acceleration of the particle? (iii) At a more advanced level...
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    I Best models to describe molecules

    Sean Carroll has given several popular talks about the meaning of quantum mechanics that I would recommend. Part of the OP's confusion might be that the wave function is defined empirically to describe quantum statistics, and, strictly speaking, does not describe the empirical behavior of...
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    Misc. Ideas for DIY projects

    A set of balance scales (sometimes called 'scales of justice') would be extremely useful.
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    Challenge Math Experiment: Let's Prove Something!

    Also, not that this should be necessary, but I also acknowledge that the extent of my mathematical ability is quite meager compared to a large number of readers and members of PF.
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    Engineering Entry level Renewable energy engineering positions?

    There are some pretty cool greenhouse design companies out there now, with what seems to be a good variety of size and composition, ranging from smaller innovation-driven startups to larger and more established groups (though the latter are mainly to be found in Nordic countries.)
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    What are the benefits of STEM Learning Education for kids?

    'STEM education' can mean many things. Ensuring that children are provided the option to become scientists or mathematicians or engineers if they so wish is immeasurably valuable, both for the children and for the legitimacy of the whole enterprise of discovery.
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    Challenge Math Experiment: Let's Prove Something!

    36, 37 Apologies for the more-than-somewhat incomplete post. The idea is to construct ##f_{\frac{1}{2}}(x)## from a given ##f(x)## by solving the 'differential equation' ## f_\frac{1}{2}'(f_\frac{1}{2}(x))f_\frac{1}{2}'(x)=f'(x)## on ##[0,1]##, or whatever the existence interval turns out to...
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    Challenge Math Experiment: Let's Prove Something!

    (11, 12, 16, 17, 24, 25) Uniqueness of 'square roots' The functional ##\mathcal{F}[g](x)=g(1-x)## exchanges the spaces of increasing and decreasing bijections and is its own inverse, and so ##G## consists of two topologically equivalent connected components, say ##G=G_+\cup G_-## (a path from...
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    Challenge Math Experiment: Let's Prove Something!

    (11, 12, 16, 21) Prop: The function ##d(f,g) \equiv \frac{1}{2}(\|x-f^{-1}\circ g(x)\|_2+\|x-g^{-1}\circ f(x)\|_2)## is invariant under left multiplication. Proof: ##d(af,ag) = \frac{1}{2}(\|x-f^{-1}\circ (a^{-1}\circ a)\circ g(x)\|_2+\|x-g^{-1}\circ (a^{-1}\circ a)\circ f(x)\|_2)=d(f,g).##...
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    Challenge Math Experiment: Let's Prove Something!

    I might try using the reflection property of inverse functions, and the fact that ##x<1\Rightarrow x^2<x## (so the ##L^2## norm is bounded by the ##L^1## norm in this case.) It could be there exists a more natural metric, however (one that is invariant with respect to the Lie group multiplication.)
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    Challenge Math Experiment: Let's Prove Something!

    (i) Consider derivatives of ##\sqrt{g}##. (ii) After (i), uniqueness of flows follows from continuity. (x) There are a large number of mutually commuting, distinct, one-parameter flows. (Are there more?) (xi) The 'center' of ##G## consists of just the identity. This can be seen by considering...
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    Challenge Math Experiment: Let's Prove Something!

    Here's a basic question: what's the smallest subset in ##G## that generates ##G##? (Or even more basic: what subsets in ##G## generate all of ##G##?) Here are a few simple conjectures: $$\text{(i) Existence of (e.g. square) roots: if }g\in G,\text{ then }\exists! \sqrt{g} \text{ s.t...
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    How to calculate the damping factor for a vibrating string?

    Here's my advice: (i) Model the local viscous dissipation of the string using two-dimensional Stokesian (non-inertial) flow around a disk. (ii) If you have one, estimate the total acoustic dissipation from a dB-reader placed near the string (where cylindrical symmetry is strongest.) Compare the...
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    Archived Oscillation with Green's Function

    The final position can be found by balancing the limiting force with Hook's law.
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    Manipulating differentials in thermodynamics

    I assume that the expression you are trying to evaluate has the form \begin{equation} \int_\gamma PdV \end{equation} for some path ##\gamma## that is specified by ##(P(\tau),T(\tau))##, where ##\tau## is an arbitrary parameter that runs from ##a## to ##b## say. The above expression is...
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    Introduction to Algebraic Geometry in String Theory?

    There is also `Mirror symmetry' by Hori et. al. that discusses applications of algebraic geometry in string theory.
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    Quasi-particles: how typical are they?

    More or less, but I think failure of perturbation theory is a little subjective since it depends on how the expansion is made. An alternative approach is to look for RG trajectories that avoid approaching fixed points as much as possible. The models or effective field theories associated with...
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    Quasi-particles: how typical are they?

    Yes, the lack of particle-like excitations at a given energy scale could be related to the failure of perturbation theory, in the following sense. Fixed points of RG transformations correspond to regions in parameter space where the model is stable under scale changes (either zooming out or...
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    Quasi-particles: how typical are they?

    My question above is extremely vague, and I appreciate your thoughtful answer. A more appropriate question (though unfortunately still vague) might be `How typical are particle-like excitations'. I will try to avoid using wrong terminology, but I don't have much experience in condensed matter...
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    Manipulating differentials in thermodynamics

    Yes, it's true that state functions typically depend on two or more variables. However, the notation ##\frac{dE}{d\tau}## is meaningful as long as a path ##(S(\tau),V(\tau))## is given as well. In shawnstrausser's example, the notation suggests a path given by...
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    Manipulating differentials in thermodynamics

    Yes, this is an application of the chain rule: you are taking the derivative of ##E(S,V)## along a path in ##S,V## space, where you have chosen to use ##V## as the parameter for the path (i.e. the path is given by ##(S(V),V)## for some function ##S(V)##).
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    Quasi-particles: how typical are they?

    It seems fascinating to me that quasi-particles appear everywhere, and can appear at multiple scales within the same system. Why is this? Is there a good intuitive reason for why entities behaving like free particles should exist at various scales in a system that is otherwise strongly...
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    Difference between Tensors and matrices

    What do you want to use tensors for? (e.g. general relativity, quantum mechanics, engineering/materials science, information theory/statistics)
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    Difference between Tensors and matrices

    Usually tensors are associated with a linear vector space ##V## and its dual space ##V^*##. A tensor of rank ##(p,q)## is then a multilinear function from ##p## copies of ##V## and ##q## copies of ##V^*## to some scalar field (usually ##\mathbb{R}## or ##\mathbb{C}##). In this sense, a tensor...
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