4, 3, 2, 1, 0.!!! Using the five countdown digits (in their same order) and the eight punctuation marks, create an expression which equals 2015. Only basic arithmetic operations [+, -, x, /] are allowed however you will need to substitute parentheses for the commas.
Pretending that spheres, balls, and Euclidean spaces can have real dimensions: * let d_Amax := the real dimension d where the formula A(d) = 2 pi^(d/2) / Gamma(d/2) for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum. -and- * let d_Vmax := the real dimension d where the formula V(d) = pi^(d/2) / Gamma(d/2 + 1) for the d-dimensional content of the unit d-ball in R^d takes its maximum Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+. In particular they have the same fractional part: upsilon := 0.256946404860576780132838388690769236619+ QUESTION: Is anything known about the number theoretic properties of upsilon? Is it known to be irrational or transcendental? Or related to other numbers, like Euler gamma, whose number-theoretic properties are unknown? --Dan
It has some rather large terms in the continued fraction: [0; 3, 1, 8, 4, 24, 2, 1, 4, 7, 2, 1, 1, 1, 3, 32, 1, 7, 1, 21, 3, 4, 1, 1, 2, 1, 3, 75, 4, 2, 4, 1, 1, 2, 2, 9, 1, ...] Truncating before the 24 gives 37/144. On Tue, Dec 23, 2014 at 1:54 PM, Daniel Asimov <asimov@msri.org> wrote:
Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
* let d_Amax := the real dimension d where the formula
A(d) = 2 pi^(d/2) / Gamma(d/2)
for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum.
-and-
* let d_Vmax := the real dimension d where the formula
V(d) = pi^(d/2) / Gamma(d/2 + 1)
for the d-dimensional content of the unit d-ball in R^d takes its maximum
Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+.
In particular they have the same fractional part:
upsilon := 0.256946404860576780132838388690769236619+
QUESTION: Is anything known about the number theoretic properties of upsilon?
Is it known to be irrational or transcendental? Or related to other numbers, like Euler gamma, whose number-theoretic properties are unknown?
--Dan
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On Tue, Dec 23, 2014 at 1:54 PM, Daniel Asimov <asimov@msri.org> wrote:
Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
Very delicate hypothesis. You would have to prove that there is a continuous family of geometrical objects, to whom volume and surface make sense, etc. and according precisely to these formulae.
upsilon := 0.256946404860576780132838388690769236619+
Is it known to be irrational or transcendental? Or related to other
numbers,
like Euler gamma, whose number-theoretic properties are unknown?
Simplifying the expression for the derivative of Vmax relative to d, you get that d_Vmax is such that EulerGamma + Log[Pi] == HarmonicNumber[d_Vmax / 2] in Mathematica notation, or equivalently Log[Pi] == PolyGamma[0, d_Amax / 2] Olivier Gérard
Where I suppose for x > 0 real we define HarmonicNumber[x] := Integral_{0,1} (1 - u^x) / (1 - x) dx ??? (Which is 1 + 1/2 + 1/3 + ... + 1/n for n in Z+.) --Dan
On Dec 23, 2014, at 3:03 PM, Olivier Gerard <olivier.gerard@gmail.com> wrote:
On Tue, Dec 23, 2014 at 1:54 PM, Daniel Asimov <asimov@msri.org> wrote:
Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
Very delicate hypothesis. You would have to prove that there is a continuous family of geometrical objects, to whom volume and surface make sense, etc. and according precisely to these formulae.
upsilon := 0.256946404860576780132838388690769236619+
Is it known to be irrational or transcendental? Or related to other
numbers,
like Euler gamma, whose number-theoretic properties are unknown?
Simplifying the expression for the derivative of Vmax relative to d, you get that d_Vmax is such that
EulerGamma + Log[Pi] == HarmonicNumber[d_Vmax / 2]
in Mathematica notation, or equivalently
Log[Pi] == PolyGamma[0, d_Amax / 2]
= Daniel Asimov <asimov@msri.org> wrote:
Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
="Olivier GERARD" <olivier.gerard@gmail.com> Very delicate hypothesis. You would have to prove that there is a continuous family of geometrical objects, to whom volume and surface make sense, etc. and according precisely to these formulae.
="Bill Gosper" <billgosper@gmail.com> This exposes a bug in the terminology "unit ball", which really ought to mean "unit diameter ball".
I don't know balls, but, prompted by this mysterious maximum, a number of years ago I asked this list if it was always possible to construct a fractal of any given real dimension (ie we're not just restricted to values of the "classical" type of log-mumble expressions), and y'all said "yes"--and I think moreover even gave an explicit method. Starting with such a fractal as the "space", can you then hand-wave up a suitably-constructed "unit subset" that degenerates to the Euclidean ball at integer dimensions? The content formulae over this continua of constructions will of course likewise degenerate to the expression under discussion here at integer dimensions, but it might have additional twiddle factors that vanish at those values--hence different maxima as it interpolates in-between. So another tack, symbolic instead of geometric: the volume formula can be constructed directly by iterated integration. I think the usual fractional iterated integration operator also just changes factorial to gamma. If that generalization is unique, it would seem that the generalized fractional dimensional geometric construction must also be. If not, the converse.
Here's another approach to fractional dimensions. Suppose you have a finite set X of states that a physical system can be in. Each state has a particular energy. The partition function Z(T) = sum_{x in X} exp(-H(x)/kT) gives us the normalization factor at a given temperature. The probability of being in the state x is described by the Gibbs distribution p(x) = exp(-H(x)/kT) / Z(T). When the temperature is infinite, every state is equally likely. If there are n states, the system behaves something like a particle in n dimensions: Z(∞) is n. When the temperature cools, high-energy states become exponentially unlikely, reducing the effective dimensionality of the system. Thin films allow low-energy electrons to move in two dimensions, but high-energy electrons can fly right off the film. Z(T) decreases continuously from n down to 1 (only the ground state is occupied) as T goes down from infinity to zero. On Wed, Dec 24, 2014 at 9:40 AM, Marc LeBrun <mlb@well.com> wrote:
= Daniel Asimov <asimov@msri.org> wrote:
Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
="Olivier GERARD" <olivier.gerard@gmail.com> Very delicate hypothesis. You would have to prove that there is a continuous family of geometrical objects, to whom volume and surface make sense, etc. and according precisely to these formulae.
="Bill Gosper" <billgosper@gmail.com> This exposes a bug in the terminology "unit ball", which really ought to mean "unit diameter ball".
I don't know balls, but, prompted by this mysterious maximum, a number of years ago I asked this list if it was always possible to construct a fractal of any given real dimension (ie we're not just restricted to values of the "classical" type of log-mumble expressions), and y'all said "yes"--and I think moreover even gave an explicit method.
Starting with such a fractal as the "space", can you then hand-wave up a suitably-constructed "unit subset" that degenerates to the Euclidean ball at integer dimensions?
The content formulae over this continua of constructions will of course likewise degenerate to the expression under discussion here at integer dimensions, but it might have additional twiddle factors that vanish at those values--hence different maxima as it interpolates in-between.
So another tack, symbolic instead of geometric: the volume formula can be constructed directly by iterated integration. I think the usual fractional iterated integration operator also just changes factorial to gamma. If that generalization is unique, it would seem that the generalized fractional dimensional geometric construction must also be. If not, the converse.
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Here's another approach I thought of a very long time ago, fwiw: Let C = R/Z be the circle group. For any d in Z+, the element [1/d] in C generates a subgroup <[1/d]> of C. Then, R^d can be thought of as the set of all functions f: <[1/d]> -> R . Since the above makes sense for any d in R, not just Z+, we can define R^d for any d in R as the set of all functions f: <[1/d]> -> R once again. Only now if d is irrational then <[1/d]> is a countable dense subgroup of C. --Dan
On Dec 24, 2014, at 2:48 PM, Mike Stay <metaweta@gmail.com> wrote:
Here's another approach to fractional dimensions.
Suppose you have a finite set X of states that a physical system can be in. Each state has a particular energy. The partition function Z(T) = sum_{x in X} exp(-H(x)/kT) gives us the normalization factor at a given temperature. The probability of being in the state x is described by the Gibbs distribution p(x) = exp(-H(x)/kT) / Z(T). When the temperature is infinite, every state is equally likely. If there are n states, the system behaves something like a particle in n dimensions: Z(∞) is n. When the temperature cools, high-energy states become exponentially unlikely, reducing the effective dimensionality of the system. Thin films allow low-energy electrons to move in two dimensions, but high-energy electrons can fly right off the film. Z(T) decreases continuously from n down to 1 (only the ground state is occupied) as T goes down from infinity to zero.
Hello, the 2 numbers are in the OEIS database as well as mine here, and nothing else : I tried the 'try hard' function and nothing came out. the 2 entries are A074455 and A074454. I am afraid that these 2 numbers are solution of trans. equations which are in general not related to anything we know apart from Pi and the LambertW function. I tried most of the tools I know on it : nothing. Best regards, Simon Plouffe Le 23/12/2014 22:54, Daniel Asimov a écrit :
Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
* let d_Amax := the real dimension d where the formula
A(d) = 2 pi^(d/2) / Gamma(d/2)
for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum.
-and-
* let d_Vmax := the real dimension d where the formula
V(d) = pi^(d/2) / Gamma(d/2 + 1)
for the d-dimensional content of the unit d-ball in R^d takes its maximum
Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+.
In particular they have the same fractional part:
upsilon := 0.256946404860576780132838388690769236619+
QUESTION: Is anything known about the number theoretic properties of upsilon?
Is it known to be irrational or transcendental? Or related to other numbers, like Euler gamma, whose number-theoretic properties are unknown?
--Dan
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participants (6)
-
Daniel Asimov -
Marc LeBrun -
Mike Stay -
Olivier Gerard -
Simon Plouffe -
Wainwright, Robert