Its a=0 that bothers me. x > 0 in my case.
In the code I use, the DerivativeStructure evaluates to NaN for a=0 when x
> 0 . I think we agree that in this condition the derivative should
evaluate to 0.
Perhaps I wrote something to mislead you on this detail.
-Ajo
On Wed, Aug 28, 2013 at 10:36 AM, Luc Maisonobe <luc.maison...@free.fr>wrote:
> Hi Ajo,
>
> Le 28/08/2013 16:56, Ajo Fod a écrit :
> > To define things precisely:
> > y = f(a,x) = |a|^x
> >
> > Can we agree that:
> > df(a,x)/dx -> 0 when a->0 and x > 0 :[ NOTE: x > 0]
>
> Yes, of course, it is perfectly true.
>
> >
> > If this is acceptable, we get this very useful property that df (a,x)/dx
> is
> > defined and continuous for all a provided x>0 because we use the modulus
> of
> > a in the function definition.
>
> Yes, as long as we don't have x = 0, we remain in a smooth, indefinitely
> differentiable domain.
>
> > In optimization, with this patch at |a|=0, I
> > can set an optimizer to search the whole real line without worrying about
> > a=0 otherwise I've to look out for a=0 explicitly. It seems unnecessary
> to
> > add a constraint to make |a|>0. I already have a constraint for x >0.
>
> I don't understand what you mean here. If you already know that x > 0,
> then you don't have to worry about a=0 or a>0 since in this case both
> approaches lead to the same result.
>
> If you look at the graph for df(a,x)/dx for a few values of a, you will
> see that we have:
>
> lim a->0+ df(a,x)/dx = 0 for x > 0
> lim a->0+ df(a,x)/dx = -infinity for x = 0
>
> and this despite df(a,x)/dx = ln(a) a^x is a continuous function,
> indefinitely differentiable. The limit of a continuous indefinitely
> differentiable function may be a non continuous function. It is a
> counter-intuitive result, I agree, but thre are many other examples of
> such strange behaviour in mathematics (if I remember well, Fourier
> transforms of step function exhibit the same paroble, backward).
>
> If you have x>0, you are already on the safe side of the singularity, so
> this is were I lose your tracks and don't understand how the singular
> point x=0 bothers you.
>
> best regards,
> Luc
>
> >
> > Cheers,
> > Ajo.
> >
> >
> >
> > On Tue, Aug 27, 2013 at 1:49 PM, Luc Maisonobe <luc.maison...@free.fr
> >wrote:
> >
> >> Hi Ajo,
> >>
> >> Le 27/08/2013 16:44, Ajo Fod a écrit :
> >>> Thanks for the constant structure.
> >>>
> >>> No. The limit value when x->0+ is 1, not O.
> >>>
> >>> I agree with this. I was just going for the derivatives = 0.
> >>>
> >>>
> >>>> The nth derivative of a^x can be computed analytically as ln(a)^n a^x,
> >>>> so the initial slope at x=0 is simply ln(a), positive for a > 1, zero
> >>>> for a = 1, negative for 0 < a < 1 with a limit at -inifnity when a ->
> >> 0+.
> >>>>
> >>>
> >>> Lets think about this for a sec:
> >>> Derivative of |a|^x wrt x at x=2.0 for various values of a
> >>> Derivative@0.031250=-0.003384
> >>> Derivative@0.015625=-0.001015
> >>> Derivative@0.007813=-0.000296
> >>> Derivative@0.003906=-0.000085
> >>> Derivative@0.001953=-0.000024
> >>> ... tends to 0
> >>
> >> yes, because 2.0 > 0.
> >>
> >>>
> >>> Derivative of |a|^x wrt x at x=0.5 for various values of a
> >>> Derivative@0.031250=-0.612555
> >>> Derivative@0.007813=-0.428759
> >>> Derivative@0.001953=-0.275612
> >>> Derivative@0.000488=-0.168418
> >>> Derivative@0.000122=-0.099513
> >>> Derivative@0.000031=-0.057407
> >>> Derivative@0.000008=-0.032528
> >>> Derivative@0.000002=-0.018176
> >>> ... tends to 0 when a->0
> >>
> >> yes because 0.5 > 0.
> >>
> >>>
> >>> The code I used for the print outs is:
> >>> static final double EPS = 0.0001d;
> >>>
> >>> public static void main(final String[] args) {
> >>> final double x = 0.5d;
> >>> int from = 5;
> >>> int to = 20;
> >>> System.out.println("Derivative of |a|^x wrt x at x=" + x);
> >>> for (int p = from; p < to; p+=2) {
> >>> double a = Math.pow(2d, -p);
> >>> final double calc = (Math.pow(a, x + EPS) - Math.pow(a,
> x)) /
> >>> EPS;
> >>> System.out.format("Derivative@%f=%f \n", a, calc);
> >>> }
> >>> }
> >>>
> >>> As for the x=0 case:
> >>> 1^0 = 1
> >>> 0.5^0 = 1
> >>> 0.0001^0 = 1
> >>> 0^0 is technically undefined, but 1 is a good definition:
> >>> http://www.math.hmc.edu/funfacts/ffiles/10005.3-5.shtml
> >>
> >> Yes.
> >>
> >>> ... so, a good value for the differential of da^x/dx limit x->0 and
> >> a->0 =
> >>> 0
> >>
> >> I don't agree. What you wrote in the lines above is another way to say
> >> what I wrote in my previous message: the value at x=0 is always y=1, and
> >> the value for x > 0 tends to 0 as a->0+.
> >>
> >> So the function always starts at 1 and dives more and more steeply as a
> >> becomes smaller, and the derivative at 0 becomes more and more negative,
> >> up to -infinity, *not* 0.
> >>
> >> The function is ill-behaved and the fact the derivative is infinite is
> >> consistent with this ill-behaviour.
> >>
> >> The definition of the derivative is :
> >>
> >> f'(x) = lim (f(x+h) - f(x))/h when h -> 0+
> >>
> >> when f(x) = 0^x and assuming 0^0 = 1 as you have agreed above, this
> gives:
> >>
> >> f'(0) = lim (0^(0+h) - 0^0)/h = lim (0 - 1)/h = -infinity
> >>
> >> which is exactly the same result as computing for a non-null a and then
> >> reducing it: d(a^x)/dx = ln(a) a^x = ln(a) when x=0, diverges to
> >> -infinity when a converges to 0.
> >>
> >>>
> >>>
> >>> As mentioned earlier, I think the cause for this is that log|a| ->
> >> infinity
> >>> slower than |a|^x -> 0 as |a|->0 .
> >>
> >> But a^x does *not* converge to 0 for x = 0! a^0 is always 1 (rigorously)
> >> regardless of the value of a as long as it is not 0, and then when we
> >> change a we can also consider the limit is 1 when a-> 0. This convention
> >> is well accepted. This convention is implemented in the Java standard
> >> Math.pow function, and we followed this trend. This is the reason why
> >> the functions becomes more and more steep as a becomes smaller. At the
> >> end, it is a discontinuous function (and hence should not be
> >> differentiable, or it is differentiable only if we use extended real
> >> numbers with infinity added).
> >>
> >> This is the heart of the ill-behaviour of 0^0. We want to compute it as
> >> a limit value for a^b when both parameters converge to 0, but we get a
> >> different result if we first set a fixed and converge b to 0, and later
> >> reduce a down to zero (your approach), and when we do the opposite. In
> >> one case we get 0, in the other case we get 1.
> >>
> >> Lets put it another way:
> >> If we consider the derivative f'(0) should be 0, then the value f(0)
> >> should also be considered equal to zero. This would mean as soon as we
> >> get a tiny non-zero a (say the smallest number that can be represented
> >> as a double), then f(0) would jump from 0 to 1 instantly, and f'(0)
> >> would jump from 0 to -infinity instantly. So we would have at a = 0 an
> >> initial null derivative, then a jump to a very negative derivative as a
> >> leaves 0, then the derivative would become less and less negative as a
> >> increase up to 1, at a=1 the derivative would again be 0, then the
> >> derivative would continue to increase and becode positive as a grows
> >> larger than 1 (all these derivatives are computed at x=0, and as written
> >> previously, they are simply equal to log(a)).
> >>
> >> To summarize, the two choices are:
> >> 1) - first considering a fixed a, strictly positive,
> >> - then looking globally at the function a^x for all values x>=0,
> >> - then reducing a, noting that all functions start at the same
> >> point x=0, y=1 and the derivatives become more and more negative
> >> as the function becomes more and more ill-behaved
> >> 2) - first considering a fixed x, strictly positive,
> >> - then reducing a and identifying the limit values is 0 for all a,
> >> - then building a function by packing all the x>0, which is very
> >> smooth as it is identically 0 for all x>0
> >> - finally adding the limit value at x=0, which in this case would
> >> be 0 (and the derivative would also be 0).
> >>
> >> it seems well accepted to consider the value of 0^0 should be set to 1,
> >> and as a consequence the corresponding derivative with respect to x
> >> should be set to -infinity.
> >>
> >> I fully agree it is not a perfect solution, it is an arbitrary choice.
> >> However, this choice is consistent with what all implementations of the
> >> pow function I have seen (i.e. 0^0 set to 1 instead of 0).
> >>
> >> Your approach is not wrong, it is as valid as the other one. It is
> >> simply not the common choice.
> >>
> >> I would say an even better choice would have been to say 0^0 *is not*
> >> defined and even the value should be set to NaN (not even speaking of
> >> the derivative).
> >>
> >> Does this seem acceptable to you?
> >>
> >> best regards,
> >> Luc
> >>
> >>>
> >>> Cheers,
> >>> Ajo.
> >>>
> >>>
> >>>> The limit curve corresponding to a = 0 is therefore a singular
> function
> >>>> with f(0) = 1 and f(x) = 0 for all x > 0. The fact f(0) = 1 and not 0
> is
> >>>> consistent with the derivative being negative infinity, as by
> definition
> >>>> the derivative is the limit of [f(0+h) - f(0)] / h when h->0+, as the
> >>>> finite difference is -1/h.
> >>>>
> >>>>> }
> >>>>> }else{
> >>>>> for (int i = 0; i < function.length; ++i) {
> >>>>> function[i] = Double.NaN;
> >>>>> }
> >>>>
> >>>> This alternative case is a good improvement, thanks for it. I forgot
> to
> >>>> handle negative cases properly. I have therefore changed the code
> >>>> (committed as r1517788) with this improvement, together with several
> >>>> test cases.
> >>>>
> >>>>> }
> >>>>> } else {
> >>>>>
> >>>>>
> >>>>> in place of :
> >>>>>
> >>>>> if (a == 0) {
> >>>>> if (operand[operandOffset] == 0) {
> >>>>> function[0] = 1;
> >>>>> double infinity = Double.POSITIVE_INFINITY;
> >>>>> for (int i = 1; i < function.length; ++i) {
> >>>>> infinity = -infinity;
> >>>>> function[i] = infinity;
> >>>>> }
> >>>>> }
> >>>>> } else {
> >>>>>
> >>>>>
> >>>>> PS: I think you made a change to DSCompiler.pow too. If so, what
> >> happens
> >>>>> when a=0 & x!=0 in that function?
> >>>>
> >>>> No, I didn't change the other signatures of the pow function. So the
> >>>> value should be OK (i.e. 1) but all derivatives, including the first
> >>>> one, should be NaN. What the new function brings is a correct negetive
> >>>> infinity first derivative at singularity point, better accuracy for
> >>>> non-singular points, and possibly faster computation.
> >>>>
> >>>> best regards,
> >>>> Luc
> >>>>
> >>>>>
> >>>>>
> >>>>> On Mon, Aug 26, 2013 at 12:38 AM, Luc Maisonobe <l...@spaceroots.org>
> >>>> wrote:
> >>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>> Ajo Fod <ajo....@gmail.com> a écrit :
> >>>>>>> Are you saying patched the code? Can you provide the link?
> >>>>>>
> >>>>>> I committed it in the development version. You just have to update
> >> your
> >>>>>> checked out copy from either the official
> >>>>>> Apache subversion repository or the git mirror we talked about in a
> >>>>>> previous thread.
> >>>>>>
> >>>>>> The new method is a static one called pow and taking a and x as
> >>>> arguments
> >>>>>> and returning a^x. Not to
> >>>>>> Be confused with the non-static methods that take only the power as
> >>>>>> argument (either int, double or
> >>>>>> DerivativeStructure) and use the instance as the base to apply power
> >> on.
> >>>>>>
> >>>>>> Best regards,
> >>>>>> Luc
> >>>>>>
> >>>>>>>
> >>>>>>> -Ajo
> >>>>>>>
> >>>>>>>
> >>>>>>> On Sun, Aug 25, 2013 at 1:20 PM, Luc Maisonobe <l...@spaceroots.org
> >
> >>>>>>> wrote:
> >>>>>>>
> >>>>>>>> Le 24/08/2013 11:24, Luc Maisonobe a écrit :
> >>>>>>>>> Le 23/08/2013 19:20, Ajo Fod a écrit :
> >>>>>>>>>> Hello,
> >>>>>>>>>
> >>>>>>>>> Hi Ajo,
> >>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> This shows one way of interpreting the derivative for strictly
> +ve
> >>>>>>>> numbers.
> >>>>>>>>>>
> >>>>>>>>>> public static void main(final String[] args) {
> >>>>>>>>>> final double x = 1d;
> >>>>>>>>>> DerivativeStructure dsA = new DerivativeStructure(1, 1,
> 0,
> >>>>>>> x);
> >>>>>>>>>> System.out.println("Derivative of |a|^x wrt x");
> >>>>>>>>>> for (int p = 10; p < 21; p++) {
> >>>>>>>>>> double a;
> >>>>>>>>>> if (p < 20) {
> >>>>>>>>>> a = 1d / Math.pow(2d, p);
> >>>>>>>>>> } else {
> >>>>>>>>>> a = 0d;
> >>>>>>>>>> }
> >>>>>>>>>> final DerivativeStructure a_ds = new
> >>>>>>> DerivativeStructure(1,
> >>>>>>>> 1,
> >>>>>>>>>> a);
> >>>>>>>>>> final DerivativeStructure out = a_ds.pow(dsA);
> >>>>>>>>>> final double calc = (Math.pow(a, x + EPS) -
> >>>>>>> Math.pow(a, x))
> >>>>>>>> /
> >>>>>>>>>> EPS;
> >>>>>>>>>> System.out.format("Derivative@%f=%f %f\n", a,
> calc,
> >>>>>>>>>> out.getPartialDerivative(new int[]{1}));
> >>>>>>>>>> }
> >>>>>>>>>> }
> >>>>>>>>>>
> >>>>>>>>>> At this point I"m explicitly substituting the rule that
> >>>>>>>> derivative(|a|^x) =
> >>>>>>>>>> 0 for |a|=0.
> >>>>>>>>>
> >>>>>>>>> Yes, but this fails for x = 0, as the limit of the finite
> >>>>>>> difference is
> >>>>>>>>> -infinity and not 0.
> >>>>>>>>>
> >>>>>>>>> You can build your own function which explicitly assumes a is
> >>>>>>> constant
> >>>>>>>>> and takes care of special values as follows:
> >>>>>>>>>
> >>>>>>>>> public static DerivativeStructure aToX(final double a,
> >>>>>>>>> final DerivativeStructure
> >>>>>>> x) {
> >>>>>>>>> final double lnA = (a == 0 && x.getValue() == 0) ?
> >>>>>>>>> Double.NEGATIVE_INFINITY :
> >>>>>>>>> FastMath.log(a);
> >>>>>>>>> final double[] function = new double[1 + x.getOrder()];
> >>>>>>>>> function[0] = FastMath.pow(a, x.getValue());
> >>>>>>>>> for (int i = 1; i < function.length; ++i) {
> >>>>>>>>> function[i] = lnA * function[i - 1];
> >>>>>>>>> }
> >>>>>>>>> return x.compose(function);
> >>>>>>>>> }
> >>>>>>>>>
> >>>>>>>>> This will work and provides derivatives to any order for almost
> any
> >>>>>>>>> values of a and x, including a=0, x=1 as in your exemple, but
> also
> >>>>>>>>> slightly better for a=0, x=0. However, it still has an important
> >>>>>>>>> drawback: it won't compute the n-th order derivative correctly
> for
> >>>>>>> a=0,
> >>>>>>>>> x=0 and n > 1. It will provide NaN for these higher order
> >>>>>>> derivatives
> >>>>>>>>> instead of +/-infinity according to parity of n.
> >>>>>>>>
> >>>>>>>> I have added a similar function to the DerivativeStructure class
> >>>>>>> (with
> >>>>>>>> some errors above corrected). The main interesting property of
> this
> >>>>>>>> function is that it is more accurate that converting a to a
> >>>>>>>> DerivativeStructure and using the general x^y function. It does
> its
> >>>>>>> best
> >>>>>>>> to handle the special case, but as written above, this does NOT
> work
> >>>>>>> for
> >>>>>>>> general combination (i.e. more than one variable or more than one
> >>>>>>>> order). As soon as there is a combination, the derivative will
> >>>>>>> involve
> >>>>>>>> something like df/dx * dg/dy and as infinities and zeros are
> >>>>>>> everywheren
> >>>>>>>> NaN appears immediately for these partial derivatives. This cannot
> >> be
> >>>>>>>> avoided.
> >>>>>>>>
> >>>>>>>> If you stay away from the singularity, the function behaves
> >>>>>>> correctly.
> >>>>>>>>
> >>>>>>>> best regards,
> >>>>>>>> Luc
> >>>>>>>>
> >>>>>>>>>
> >>>>>>>>> This is a known problem that we already encountered when dealing
> >>>>>>> with
> >>>>>>>>> rootN. Here is an extract of a comment in the test case
> >>>>>>>>> testRootNSingularity, where similar NaN appears instead of +/-
> >>>>>>> infinity.
> >>>>>>>>> The dsZero instance in the comment is simple the x parameter of
> the
> >>>>>>>>> function, as a derivativeStructure with value 0.0 and depending
> on
> >>>>>>>>> itself (dsZero = new DerivativeStructure(1, maxOrder, 0, 0.0)):
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>> // the following checks shows a LIMITATION of the current
> >>>>>>> implementation
> >>>>>>>>> // we have no way to tell dsZero is a pure linear variable x = 0
> >>>>>>>>> // we only say: "dsZero is a structure with value = 0.0,
> >>>>>>>>> // first derivative = 1.0, second and higher derivatives = 0.0".
> >>>>>>>>> // Function composition rule for second derivatives is:
> >>>>>>>>> // d2[f(g(x))]/dx2 = f''(g(x)) * [g'(x)]^2 + f'(g(x)) * g''(x)
> >>>>>>>>> // when function f is the nth root and x = 0 we have:
> >>>>>>>>> // f(0) = 0, f'(0) = +infinity, f''(0) = -infinity (and higher
> >>>>>>>>> // derivatives keep switching between +infinity and -infinity)
> >>>>>>>>> // so given that in our case dsZero represents g, we have g(x) =
> 0,
> >>>>>>>>> // g'(x) = 1 and g''(x) = 0
> >>>>>>>>> // applying the composition rules gives:
> >>>>>>>>> // d2[f(g(x))]/dx2 = f''(g(x)) * [g'(x)]^2 + f'(g(x)) * g''(x)
> >>>>>>>>> // = -infinity * 1^2 + +infinity * 0
> >>>>>>>>> // = -infinity + NaN
> >>>>>>>>> // = NaN
> >>>>>>>>> // if we knew dsZero is really the x variable and not the
> identity
> >>>>>>>>> // function applied to x, we would not have computed f'(g(x)) *
> >>>>>>> g''(x)
> >>>>>>>>> // and we would have found that the result was -infinity and not
> >>>>>>> NaN
> >>>>>>>>>
> >>>>>>>>> Hope this helps
> >>>>>>>>> Luc
> >>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> Thanks,
> >>>>>>>>>> Ajo.
> >>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> On Fri, Aug 23, 2013 at 9:39 AM, Luc Maisonobe
> >>>>>>> <luc.maison...@free.fr
> >>>>>>>>> wrote:
> >>>>>>>>>>
> >>>>>>>>>>> Hi Ajo,
> >>>>>>>>>>>
> >>>>>>>>>>> Le 23/08/2013 17:48, Ajo Fod a écrit :
> >>>>>>>>>>>> Try this and I'm happy to explain if necessary:
> >>>>>>>>>>>>
> >>>>>>>>>>>> public class Derivative {
> >>>>>>>>>>>>
> >>>>>>>>>>>> public static void main(final String[] args) {
> >>>>>>>>>>>> DerivativeStructure dsA = new DerivativeStructure(1,
> 1,
> >>>>>>> 0,
> >>>>>>>> 1d);
> >>>>>>>>>>>> System.out.println("Derivative of constant^x wrt x");
> >>>>>>>>>>>> for (int a = -3; a < 3; a++) {
> >>>>>>>>>>>
> >>>>>>>>>>> We have chosen the classical definition which implies c^x is
> not
> >>>>>>>> defined
> >>>>>>>>>>> for real r and negative c.
> >>>>>>>>>>>
> >>>>>>>>>>> Our implementation is based on the decomposition c^r = exp(r *
> >>>>>>> ln(c)),
> >>>>>>>>>>> so the NaN comes from the logarithm when c <= 0.
> >>>>>>>>>>>
> >>>>>>>>>>> Noe also that as explained in the documentation here:
> >>>>>>>>>>> <
> >>>>>>>>>>>
> >>>>>>>>
> >>>>>>>
> >>>>>>
> >>>>
> >>
> http://commons.apache.org/proper/commons-math/userguide/analysis.html#a4.7_Differentiation
> >>>>>>>>>>>> ,
> >>>>>>>>>>> there are no concepts of "constants" and "variables" in this
> >>>>>>> framework,
> >>>>>>>>>>> so we cannot draw a line between c^r as seen as a univariate
> >>>>>>> function
> >>>>>>>> of
> >>>>>>>>>>> r, or as a univariate function of c, or as a bivariate function
> >>>>>>> of c
> >>>>>>>> and
> >>>>>>>>>>> r, or even as a pentavariate function of p1, p2, p3, p4, p5
> with
> >>>>>>> both c
> >>>>>>>>>>> and r being computed elsewhere from p1...p5. So we don't make
> >>>>>>> special
> >>>>>>>>>>> cases for the case c = 0 for example.
> >>>>>>>>>>>
> >>>>>>>>>>> Does this explanation make sense to you?
> >>>>>>>>>>>
> >>>>>>>>>>> best regards,
> >>>>>>>>>>> Luc
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>> final DerivativeStructure a_ds = new
> >>>>>>>> DerivativeStructure(1,
> >>>>>>>>>>> 1,
> >>>>>>>>>>>> a);
> >>>>>>>>>>>> final DerivativeStructure out = a_ds.pow(dsA);
> >>>>>>>>>>>> System.out.format("Derivative@%d=%f\n", a,
> >>>>>>>>>>>> out.getPartialDerivative(new int[]{1}));
> >>>>>>>>>>>> }
> >>>>>>>>>>>> }
> >>>>>>>>>>>> }
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>> On Fri, Aug 23, 2013 at 7:59 AM, Gilles
> >>>>>>> <gil...@harfang.homelinux.org
> >>>>>>>>>>>> wrote:
> >>>>>>>>>>>>
> >>>>>>>>>>>>> On Fri, 23 Aug 2013 07:17:35 -0700, Ajo Fod wrote:
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>> Seems like the DerivativeCompiler returns NaN.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> IMHO it should return 0.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> What should be 0? And Why?
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>> Is this worthy of an issue?
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> As is, no.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> Gilles
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>> Thanks,
> >>>>>>>>>>>>>> -Ajo
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>
> >>>>>>>
> >>>>
> >>
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> >>>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>
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