Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

Atrium > Mathematics, Physics & Philosophy: latest changes

Bottom of Page

101 to 108 of 108

  1. <
  2. 1
  3. 2
  4. >
    • CommentRowNumber101.
    • CommentAuthorzskoda
    • CommentTimeOct 5th 2009
    I have been working a bit last night on nlab. I click on latest changes, get a link to forum, log in, respond to Urs's entry which concerns the item I added on, click back, get just the entry related to that entry, not the list of all recent udpates like in latest changes before. Before it was all in one, now it is many pages. I will not tire myself up at all with this, I was up to yesterday ready to move but I won't. I will not log my changes into new complicated system, it is messing me up and I do not want to study a complicated system whose purpose I do not understand at all.
    • CommentRowNumber102.
    • CommentAuthorJonAwbrey
    • CommentTimeOct 5th 2009
    • (edited Oct 5th 2009)

    Yes, the forum-based system of logging latest changes is turning into a regular blog-bog where you can't be sure whether you've seen what you need to see — like just yesterday in the cafe someone said something was "very clever" and I wasted a ½ hour cycling around between the find-on-page and chronological and thread views trying to figure out what it was that was "very clever" and I never did find it — and this will only get multiplicatively worse as the number of headings hydrafy.

    I think a better thing to think about is whether we can get section-local editing on the wiki — this would reduce the chances of lockouts and edit conflix globally, not just on the latest changes page.

    • CommentRowNumber103.
    • CommentAuthorMike Shulman
    • CommentTimeOct 5th 2009

    Actually, I don't know whether I would like the current setup if I weren't using a feed-reader. For purposes of replies, I think it would be better to have a threaded view of everything that's been written, rather than just a view of the first post in each discussion. What do others think?

    Jon, presumably you're referring to this post which was a reply to this one. I don't usually have any trouble finding the replied-to post on the cafe, but on the rare occasions when it is a problem, what I do is switch to thread view and scroll up until the vertical line immediately to the left of the reply first disappears. Possibly what the cafe could really use is a collapsible thread-view.

    • CommentRowNumber104.
    • CommentAuthorJonAwbrey
    • CommentTimeOct 5th 2009
    • (edited Oct 5th 2009)

    Thanks, Ariadne, that was just the clue! — but I have to say it was a mite further back than I normally pull.

    Theseus

    • CommentRowNumber105.
    • CommentAuthorUrs
    • CommentTimeOct 5th 2009

    Well, this is just a test run to see how we like it. We can still switch back.

    But I don't at the moment see why you find it harder to follow recent changes here. I find it easier, even without RSS. All the new links are at the top, easily visible. no?

    • CommentRowNumber106.
    • CommentAuthorTobyBartels
    • CommentTimeOct 5th 2009

    I find the new system a little easier, I think, because I can view the various updates more independently.

    This would not work if I didn't have an account. But since I do, and since I accept the cookie to stay logged in (until it disappears after a month or something and I need to log in again), the main page tells me where new posts are, and clicking on ‘New’ takes me right where I want to be.

    • CommentRowNumber107.
    • CommentAuthorAndrew Stacey
    • CommentTimeOct 6th 2009

    Zoran, I'm sorry that you feel that way. I understand that you don't like changing a system that you've gotten used to. I do feel that you make life a little complicated for yourself:

    I click on latest changes, get a link to forum, log in, respond to Urs's entry which concerns the item I added on, click back, get just the entry related to that entry, not the list of all recent udpates like in latest changes before.

    Why not have a tab open at the forum and keep yourself logged in, then it's no more clicks than the old system. And to get the full list, simply click on the words 'latest changes' wherever they appear (apart from the title of this discussion, of course), that takes you to the category view where you get a summary of all that's going on.

    Indeed, I'm sure that some smart cookie who knows a little about javascript could write a "smart bookmarklet" that took the title of the page that is being edited and started a new discussion here on that subject all with one click.

    However, I realise that your situation is somewhat unique so I don't want to make you feel hassled about this.

    Mike, things like that can be tweaked. This forum has themes and stylesheets that can be set up to give almost any presentation that is desired. However, I don't regard this system as The System. A much better one would be a forum-like set-up integrated into Instiki. However, I think that this is a necessary step on the way to that as it gives us a chance to see what we would like from such an integrated system.

    Jon, that applies to your comment as well (though I don't understand your comment about not knowing if you've seen what you needed to see, the inference is that that was better in the old system). It's all very well pipedreaming, but some things are easy to implement and others are hard. And it's a long road between saying "wouldn't it be great to have section-local editing" to an actual implementation - just look over the threads about redirects to see the sort of things that can go on. For example, you can have "section-local editing" right now: simply make a master page with lots of includes of sub-pages. But that wouldn't work for latest changes because everyone would have to keep editing the master page to add their new link to the subpage that they'd just created to record their latest change. Plus there'd be much more to remember and many more steps to remember.

    We are trying this out for a month. There will be no major changes during that month. After that month, we can discuss what the consensus is.

    • CommentRowNumber108.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 24th 2019
    • (edited Mar 24th 2019)

    Re #13: My apologies if they should not have been removed. Here is the content. The pages can be put back if requested. The source compiles correctly in the actual nLab, the differences are due to the fact that only the old renderer is used on the nForum; I have lightly edited some syntax to make it display reasonably here.

    Hyperanalytic functions

    Idea

    An hyperanalytic function is a function that is locally given by a converging power series with a speed that corresponds to tetration.

    Definitions

    Let VV and WW be complete Hausdorff topological vector spaces, let WW be locally convex, let cc be an element of VV, and let (a 0,a 1,a 2,…)(a_0,a_1,a_2,\ldots) be an infinite sequence of homogeneous operators from VV to WW with each a ka_k of degree kk.

    Given an element cc of VV, consider the infinite series

    ∑ ka k(x−c) k \sum_k a_k(x - c)^k

    (a power series). Let UU be the interior of the set of xx such that this series converges in WW; we call UU the domain of convergence of the power series. This series defines a function from UU to WW; we are really interested in the case where UU is inhabited, in which case it is a balanced neighbourhood of cc in VV (which is Proposition 5.3 of Bochnak–Siciak).

    Let DD be any subset of VV and ff any continuous function from DD to WW. This function ff is hyperanalytic if, for every c∈Dc \in D, there is a power series as above with inhabited domain of convergence UU such that

    f(x)=∑ ka k(x−c) k f(x) = \sum_k a_k(x - c)^k

    for every xx in both DD and UU and just rare a ka_k are not equal zero. (That ff is continuous follows automatically in many cases, including of course the finite-dimensional case.)

    Examples

    {#Examples}

    It is known that there is a fundamental connection between analyticity of the function and the convergence of its Fourier coefficients. The better the function, the faster its coefficients tend to zero, and vice versa. The power decrease of Fourier coefficients is inherent in functions of the C kC^{k} class while exponential to analytical functions. Here there is a possibility of existence of the hyperanalytic functions, for which the decrease of the Fourier coefficients corresponds to tetration.

    Natural hyperanalytic function occurs when considering reticulum with a step L, in which nodes there are not defined yet objects. The distribution of center’s objects can be described using the reticulum functions (RF). The definition of a one-dimensional RF is based on the following identity:

    1σ2π∫ −∞ ∞e −12(xσ) 2dx=1σ2π∫ −L2 L2∑ n=−∞ ∞e −12(x−nLσ) 2dx=1. \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(\frac{x}{\sigma})^{2}}dx=\frac{1}{\sigma\sqrt{2\pi}}\int_{-\frac{L}{2}}^{\frac{L}{2}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}(\frac{x-nL}{\sigma})^{2}}dx=1.
    Definition

    From here RF is

    ℝ(x)=1σ2π∑ n=−∞ ∞e −12(x−nLσ) 2. \mathbb{R}(x)=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}(\frac{x-nL}{\sigma})^{2}}.

    It is obvious that the RF can not be laid out in the Fourier series because it does not have antiderivative that can be expressed as elementary functions. By virtue of this RF cannot be decomposed into even and odd functions, while an arbitrary analytic function ff can be only presented in the form of sum of odd and even functions in the interval [a,b][a,b]:

    f(x)=g(x)+h(x), f\left(x\right)=g\left(x\right)+h\left(x\right),

    where

    g(x)=f(x−a)−f(b−x)2, g\left(x\right)=\frac{f\left(x-a\right)-f\left(b-x\right)}{2}, h(x)=f(x−a)+f(b−x)2. h\left(x\right)=\frac{f\left(x-a\right)+f\left(b-x\right)}{2}.

    Due to this the RF can be laid out in an endless row of two primitive hyperanalytic functions by sequential attempts to decompose on even and odd functions. Thus, the RF can be decomposed by the simplest way, but such a series is not one like the orthonormal basis of Fourier series.

    Decomposition of RF

    The mean value of RF

    As it follows from (1.1) the mean value of RT is 1. However as will be seen from the further, it is expedient to choose the greater value of the decomposition’s constant member. Introduce the following definitions:

    Definition

    ℝ(0)\mathbb{R}\left(0\right) is

    ℝ(0)=ℝ max=1σ2π∑ n=−∞ ∞e −12(−nσ) 2, \mathbb{R}\left(0\right)=\mathbb{R}_{max}=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}\left(\frac{-n}{\sigma}\right)^{2}},

    ℝ(1/2)\mathbb{R}\left(1/2\right) is

    ℝ(1/2)=ℝ min=1σ2π∑ n=−∞ ∞e −12(1/2−nσ) 2. \mathbb{R}\left(1/2\right)=\mathbb{R}_{min}=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}\left(\frac{1/2-n}{\sigma}\right)^{2}}.

    Then A 0A_{0} is the mean value of RF:

    A 0=ℝ max+ℝ min2. A_{0}=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}.

    First difference

    One can approximate first difference by the following way:

    A 1(x)=ℝ max−ℝ min2cos(2πx). A_{1}\left(x\right)=\frac{\mathbb{R}_{max}-\mathbb{R}_{min}}{2}cos\left(2\pi x\right).
    Definition

    Let introduce parameter of the fine structure α\alpha as function of σ\sigma:

    α(σ)=12ℝ max−ℝ minℝ max+ℝ min. \alpha\left(\sigma\right)=\frac{1}{2}\frac{\mathbb{R}_{max}-\mathbb{R}_{min}}{\mathbb{R}_{max}+\mathbb{R}_{min}}.

    Now A 1(x)A_{1}\left(x\right) can be expressed:

    A 1(x)=ℝ max+ℝ min2(2α(σ)cos(2πx)). A_{1}\left(x\right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}\left(2\alpha\left(\sigma\right)cos\left(2\pi x\right)\right).

    The choice of the name and symbol of this parameter is due to the fact that

    α(0.4992619105929628)=α=e 24πε 0ℏc \alpha\left(0.4992619105929628\right)=\alpha=\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}

    is the value known in physics as a fine structure constant.

    Even differences

    Even differences are a primitive hyperanalytic function 𝕍¯(2i×2πx)\overline{\mathbb{V}}(2i\times2\pi x), which is quasisymmetric relative to the point x=0.25\text{x=0.25}.

    Its symmetrical part approximated in the following way:

    A 2i(x)=c 2i(cos(2i×2πx)−1) A_{2i}\left(x\right)=c_{2i}\left(cos\left(2i\times2\pi x\right)-1\right)

    and

    ∑ i=1 ∞c 2i=ℝ max+ℝ min2−1=2*∑ k=1 ∞α 4 k. \sum_{i=1}^{\infty}c_{2i}=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}-1=2 * \sum_{k=1}^{\infty} \alpha^{4^{k}}.

    Using the value

    ℝ(1/4)=ℝ 1/4=1σ2π∑ n=−∞ ∞e −12(1/4−nσ) 2 \mathbb{R}\left(1/4\right)=\mathbb{R}_{1/4}=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}\left(\frac{1/4-n}{\sigma}\right)^{2}}

    define the amplitude for c 2c_{2}: 12(ℝ max+ℝ min2−ℝ 1/4)=2α 4\frac{1}{2}\left(\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}-\mathbb{R}_{1/4}\right)=2\alpha^{4}. This definition allows to select approximation A(x)A\left(x\right) in the form:

    A(x)=ℝ max+ℝ min2(1+2α(σ)cos(2πx))+2α 4(cos(2×2πx)−1). A\left(x\right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}\left(1+2\alpha\left(\sigma\right)cos\left(2\pi x\right)\right)+2\alpha^{4}\left(cos\left(2\times2\pi x\right)-1\right).

    Odd differences

    Odd differences are a primitive hyperanalytic function 𝕎((2i−1)×2πx)\mathbb{W}\left((2i-1)\times2\pi x\right), which is quasiantisymmetric relative to the point x=0.25\text{x=0.25}.

    Quasiantisymmetry of 𝕎(2πx)\mathbb{W}\left(2\pi x\right) follows from the fact that the integral of A(x)A\left(x\right) differs from 1:

    ∫ −1/2 1/2A(x)dx−1=14(ℝ max+ℝ min)+12ℝ 1/4−1≃1.02E−34. \int_{-1/2}^{1/2}\text{A}\left(x\right)\text{dx}-1=\frac{1}{4}\left(\mathbb{R}_{max}+\mathbb{R}_{min}\right)+\frac{1}{2}\mathbb{R}_{1/4}-1\simeq1.02E-34.

    Thus function 𝕎((2i−1)×2πx)\mathbb{W}\left((2i-1)\times2\pi x\right) should be decomposed in the even and odd function. Its even part is:

    𝕎 qs((2i−1)×2πx)=𝕎((2i−1)×2πx)+𝕎((2i−1)×2π(0.5−x))2=𝕍¯(2(i+1)×2πx). \mathbb{W}^{\text{qs}}\left((2i-1)\times2\pi x\right)=\frac{\mathbb{W}\left((2i-1)\times2\pi x\right)+\mathbb{W}\left((2i-1)\times2\pi\left(0.5-x\right)\right)}{2}=\overline{\mathbb{V}}(2(i+1)\times2\pi x).

    However, as shown above, 𝕍¯(2i×2πx)\overline{\mathbb{V}}(2i\times2\pi x) is not an even function.

    The odd part of 𝕎((2i−1)×2πx)\mathbb{W}\left((2i-1)\times2\pi x\right) is no longer a hyperanalytic function and is equal to:

    W qa((2i−1)×2πx)=𝕎((2i−1)×2πx)−𝕎((2i−1)×2π(0.5−x))2. W^{\text{qa}}\left((2i-1)\times2\pi x\right)=\frac{\mathbb{W}\left((2i-1)\times2\pi x\right)-\mathbb{W}\left((2i-1)\times2\pi\left(0.5-x\right)\right)}{2}.

    It can be approximated with any degree of accuracy following way:

    A(W qa((2i−1)×2πx))=β(cos(3(2i−1)×2πx)−cos((2i−1)×2πx)), A(W^{\text{qa}}\left((2i-1)\times2\pi x\right))=\beta(cos\left(3(2i-1)\times2\pi x\right)-cos\left((2i-1)\times2\pi x\right)),

    where β\beta is a normalizing multiplier.

    Thus, the approximation of ℝ(x)\mathbb{R}(x) is:

    A(x)=ℝ max+ℝ min2(1+2αcos(2πx))+2∑ i=1 ∞α 4 i(cos(2i×2πx)−1)+2𝕎 max∑ i=1 ∞α 9i 2(cos(3×(2i−1)×2πx)−cos((2i−1)×2πx)), A\left(x\right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}(1+2\alpha cos\left(2\pi x\right)) +2\sum_{i=1}^{\infty}\alpha^{4^{i}}\left(cos\left(2i\times 2\pi x\right)-1\right)+\frac{2}{\mathbb{W}_{max}}\sum_{i=1}^{\infty}\alpha^{9{i}^2}\left(cos\left(3 \times (2i-1)\times 2\pi x\right)-cos\left((2i-1) \times 2\pi x\right)\right),

    where 𝕎 max\mathbb{W}_{max} is a normalizing multiplier.

    Three-dimensional RF

    Three-dimensional RF ℝ(x,y,z)\mathbb{R}\left(x,y,z\right) can be obtained from the definition (1.2)\left(1.2\right):

    ℝ(x,y,z)=ℝ max 2ℝ(x). \mathbb{R}\left(x,y,z\right)=\mathbb{R}_{max}^{2}\mathbb{R}\left(x\right).

    Thus, the approximation of the three-dimensional RF is also the series of the fine structure constant α\alpha along any axis of the reticulum three-dimensional space, and the constant itself is a function of the dimensionless parameter σ\sigma, which is equal to quotient of the “diameter” of some physical object, located in each cell, to the grid step L.

    Quantum derivative with respect to time

    To quantize the time the direct use of the lattice idea is too formal. It is therefore appropriate to use a definition of derivative with respect to time but without moving to the limit. Let ℝ(t)\mathbb{R}\left(t\right) is RF on a unit interval [−T/2,T/2]\left[-\text{T/2},\text{T/2}\right] and τ=σ\tau=\sigma и T=1\text{T}=1:

    ℝ(t)=1τ2π∑ i=−∞ ∞[exp(−12(t+T/4−iτ) 2)−exp(−12(t−T/4−iτ) 2)]. \mathbb{R}\left(t\right)=\frac{1}{\tau\sqrt{2\pi}}\sum_{i=-\infty}^{\infty}\left[\exp\left(-\frac{1}{2}\left(\frac{t+\text{T/4}-i}{\tau}\right)^{2}\right)-\exp\left(-\frac{1}{2}\left(\frac{t-\text{T/4}-i}{\tau}\right)^{2}\right)\right].

    By consistently subtracting sinuses, one can show that the approximation of the ℝ(t)\mathbb{R}\left(t\right) has the following form:

    A(t)=∑ k=0 ∞(−1) k+1a ksin(2π(2k+1)t). A\left(t\right)=\sum_{k=0}^{\infty}\left(-1\right)^{k+1}a_{k}sin\left(2\pi\left(2k+1\right)t\right).

    Let use k+1k+1 equations with different values of ll to determine the coefficient’s values a ka_{k}:

    ∑ i=0 k(−1) ia isin(2i+12l+12π4)=ℝ(14(2l+1)). \sum_{i=0}^{k}\left(-1\right)^{i}a_{i}sin\left(\frac{2i+1}{2l+1}\frac{2\pi}{4}\right)=\mathbb{R}\left(\frac{1}{4\left(2l+1\right)}\right).

    Given that A(1/4)A\left(1/4\right) is numerically equal to 2(ℝ max(τ)+ℝ min(τ))α(τ)2\left(\mathbb{R}_{max}\left(\tau\right)+\mathbb{R}_{min}\left(\tau\right)\right)\alpha\left(\tau\right), equation can be written as follows:

    α eff(t,τ)=12(ℝ max(τ)+ℝ min(τ))∑ k=0 ∞(−1) k+1a ksin(2π(2k+1)t). \alpha_{eff}\left(t,\tau\right)=\frac{1}{2\left(\mathbb{R}_{max}\left(\tau\right)+\mathbb{R}_{min}\left(\tau\right)\right)}\sum_{k=0}^{\infty}\left(-1\right)^{k+1}a_{k}sin\left(2\pi\left(2k+1\right)t\right).

    ℝ(t)\mathbb{R}\left(t\right) is also a hyperanalytic function, as the next approximation takes place:

    α eff(t,τ)=∑ k=0 ∞(−1) k+1α (2k+1) 2sin(2π(2k+1)t). \alpha_{eff}\left(t,\tau\right)=\sum_{k=0}^{\infty}\left(-1\right)^{k+1}\alpha^{(2k+1)^{2}}sin\left(2\pi\left(2k+1\right)t\right).

    Related concepts

    References

    The theory of hyperanalytic function was constructed to some extent by A. Rybnikov (2014) http://www.gaussianfunction.com/.

101 to 108 of 108

  1. <
  2. 1
  3. 2
  4. >