Finding optimal k0 values - Revision history

Xfxie at 23:14, 20 October 2013

2013-10-20T23:14:21Z

← Older revision		Revision as of 16:14, 20 October 2013
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	== Code and data ==		== Code and data ==

	* [http://www.cs.cmu.edu/~xfxie/software/K0Finder.zip Code for finding optimal <math> k_0</math> values~~] using a simple optimizer [http://www.cs.cmu.edu/~xfxie/software/depso.zip DEPSO~~]: Java, bash, and maple are needed.		* [http://www.cs.cmu.edu/~xfxie/software/K0Finder.zip Code for finding optimal <math> k_0</math> values]: Java, bash, and maple are needed.

Xfxie: /* Code and data */

2013-10-20T18:36:07Z

Code and data

← Older revision		Revision as of 11:36, 20 October 2013
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	== Code and data ==		== Code and data ==

	* [http://www.cs.cmu.edu/~xfxie/software/K0Finder.zip Code for finding an optimal <math> k_0</math>]: Java, bash, and maple are needed.		* [http://www.cs.cmu.edu/~xfxie/software/K0Finder.zip Code for finding optimal <math> k_0</math> values] using a simple optimizer [http://www.cs.cmu.edu/~xfxie/software/depso.zip DEPSO]: Java, bash, and maple are needed.

Xfxie: /* Benchmarks */

2013-10-20T13:15:54Z

Benchmarks

← Older revision		Revision as of 06:15, 20 October 2013
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	== Benchmarks ==		== Benchmarks ==

	The following table provides the results with ``clean'' parameter values for the best currently known instances of <math>c_\varpi, c_\delta, i</math> values, respectively without and with using Deligne's theorem.		The following table provides the results with clean parameter values for the best currently known instances of <math>c_\varpi, c_\delta, i</math> values, respectively without and with using Deligne's theorem.

	{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"		{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"

Xfxie: /* Benchmarks */

2013-10-19T23:33:12Z

Benchmarks

← Older revision		Revision as of 16:33, 19 October 2013
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	== Benchmarks ==		== Benchmarks ==

			The following table provides the results with ``clean'' parameter values for the best currently known instances of <math>c_\varpi, c_\delta, i</math> values, respectively without and with using Deligne's theorem.

	{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"		{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"
	\|+ '''~~Demonstration~~ results at <math>k_0 = k_0^{opt}</math> for the best currently known instances of <math>c_\varpi, c_\delta, i</math> values.'''		\|+ '''Clean results at <math>k_0 = k_0^{opt}</math> for the best currently known instances of <math>c_\varpi, c_\delta, i</math> values.'''
	\|-		\|-
	!colspan="3" \| Instance		!colspan="3" \| Instance
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	\| With Deligne's theorem		\| With Deligne's theorem
	\|}		\|}

			The following table provides the results with minimized parameter values at <math>k_0 = k_0^{opt}</math> for some instances of <math>c_\varpi, c_\delta, i</math> values.

	{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"		{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"
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	\| -5.0940E-06		\| -5.0940E-06
	\|}		\|}

			The following table provides the "unsatisfied" results with minimized parameter values at <math>k_0=(k_0^{opt}-1)</math> for some instances of <math>c_\varpi, c_\delta, i</math> values. Note that the results of these <math>k_0</math> values do not satisfy all constraints, as indicated from the positive objective values.

	{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"		{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"
	\|+ '''"~~Failure~~" results at <math>k_0=(k_0^{opt}-1)</math> for some instances of <math>c_\varpi, c_\delta, i</math> values.'''		\|+ '''"Unsatisfied" optimal results at <math>k_0=(k_0^{opt}-1)</math> for some instances of <math>c_\varpi, c_\delta, i</math> values.'''
	\|-		\|-
	!colspan="3" \| Instance		!colspan="3" \| Instance
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	\| 4.0614E-05		\| 4.0614E-05
	\|}		\|}

			The following table provides the results with minimized parameter values at <math>k_0=(k_0^{opt}-1)</math> for some instances of <math>c_\varpi, c_\delta, i</math> values, with an additional condition that <math>\left(1- \frac{j_{k_0-2}^2}{k_0(k_0-1)(1 + 4 \varpi)}\right) \ge 0</math>. Note that these <math>k_0</math> values are more unsatisfied, as indicated from the larger positive objective values that that in the previous table.


	{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"		{\| class="wikitable" border=1 style="margin: 1em auto 1em auto;"
	\|+ ~~Conservative~~ '''"~~Failure~~" results at <math>k_0=(k_0^{opt}-1)</math> for some instances of <math>c_\varpi, c_\delta, i</math> values.'''		\|+ '''More conservative "unsatisfied" results at <math>k_0=(k_0^{opt}-1)</math> for some instances of <math>c_\varpi, c_\delta, i</math> values.'''
	\|-		\|-
	!colspan="3" \| Instance		!colspan="3" \| Instance

Xfxie: /* Code and data */

2013-09-28T21:47:15Z

Code and data

← Older revision		Revision as of 14:47, 28 September 2013
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	== Code and data ==		== Code and data ==

			* [http://www.cs.cmu.edu/~xfxie/software/K0Finder.zip Code for finding an optimal <math> k_0</math>]: Java, bash, and maple are needed.

Xfxie at 13:10, 25 September 2013

2013-09-25T13:10:14Z

← Older revision		Revision as of 06:10, 25 September 2013
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	\| 19		\| 19
	\|}		\|}

			== Code and data ==

Xfxie at 02:51, 16 September 2013

2013-09-16T02:51:16Z

← Older revision		Revision as of 19:51, 15 September 2013
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	This is a sub-page for the Polymath8 project "[[bounded gaps between primes]]".		This is a sub-page for the Polymath8 project "[[bounded gaps between primes]]".

	* <math>~k_0~</math> (<math>~k_0 \ge 2, k_0 \in \Z~</math>) is a quantity such that every admissible <math>~k_0</math>-tuple has infinitely many translates which each contain at least two primes. ~~Would like to be as small as possible~~.		* <math>~k_0~</math> (<math>~k_0 \ge 2, k_0 \in \Z~</math>) is a quantity such that every admissible <math>~k_0</math>-tuple has infinitely many translates which each contain at least two primes. It is then used for [[finding narrow admissible tuples]].

	* <math>\text{MPZ}^{(i)}[\varpi,\delta]</math> holds for some combinations of <math>c_\varpi, c_\delta</math>, and <math>~i~</math> values, where <math>i \ge 1</math> means <math>~i</math>-tuply densely divisible, <math>c_\varpi > 0</math> and <math>~c_\delta > 0~</math> are constants in the constraint on <math>\varpi</math> and <math>~\delta~</math>, such that <math>c_{\varpi}\varpi+c_{\delta}\delta<1</math>.		* <math>\text{MPZ}^{(i)}[\varpi,\delta]</math> holds for some combinations of <math>c_\varpi, c_\delta</math>, and <math>~i~</math> values, where <math>i \ge 1</math> means <math>~i</math>-tuply densely divisible, <math>c_\varpi > 0</math> and <math>~c_\delta > 0~</math> are constants in the constraint on <math>\varpi</math> and <math>~\delta~</math>, such that <math>c_{\varpi}\varpi+c_{\delta}\delta<1</math>.

Xfxie: /* Optimization Model */

2013-09-15T20:27:43Z

Optimization Model

← Older revision		Revision as of 13:27, 15 September 2013
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	and <math>\varpi, \delta, \delta', A</math> are parameters to be optimized.		and <math>\varpi, \delta, \delta', A</math> are parameters to be optimized.

			More details are described in the page on [[Dickson-Hardy-Littlewood theorems]].

	== Benchmarks ==		== Benchmarks ==

Xfxie at 20:24, 15 September 2013

2013-09-15T20:24:32Z

← Older revision		Revision as of 13:24, 15 September 2013
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	<math>\text{minimize}~~k_0~</math>		<math>\text{minimize}~~k_0~</math>


	<math>\text{subject to:}~</math>		<math>\text{subject to:}~</math>
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	:<math>A \ge 0,</math>		:<math>A \ge 0,</math>


	<math>\text{where}~</math>		<math>\text{where}~</math>
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	:<math> \tilde \delta := \frac{\delta}{1/4 + \varpi}</math>		:<math> \tilde \delta := \frac{\delta}{1/4 + \varpi}</math>

	and <math>\varpi, \delta, \delta', A</math> are parameters ~~one is free~~ to ~~optimize over~~.		and <math>\varpi, \delta, \delta', A</math> are parameters to be optimized.

	== Benchmarks ==		== Benchmarks ==

Xfxie at 20:23, 15 September 2013

2013-09-15T20:23:48Z

← Older revision		Revision as of 13:23, 15 September 2013
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	This is a sub-page for the Polymath8 project "[[bounded gaps between primes]]".		This is a sub-page for the Polymath8 project "[[bounded gaps between primes]]".

	* <math>~k_0~</math> is a quantity such that every admissible <math>~k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible.		* <math>~k_0~</math> (<math>~k_0 \ge 2, k_0 \in \Z~</math>) is a quantity such that every admissible <math>~k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible.

	* <math>\text{MPZ}^{(i)}[\varpi,\delta]</math> holds for some combinations of <math>c_\varpi, c_\delta</math>, and <math>~i~</math> values, where <math>i \ge 1</math> means <math>~i</math>-tuply densely divisible, <math>c_\varpi > 0</math> and <math>~c_\delta > 0~</math> are constants in the constraint on <math>\varpi</math> and <math>~\delta~</math>, such that <math>c_{\varpi}\varpi+c_{\delta}\delta<1</math>.		* <math>\text{MPZ}^{(i)}[\varpi,\delta]</math> holds for some combinations of <math>c_\varpi, c_\delta</math>, and <math>~i~</math> values, where <math>i \ge 1</math> means <math>~i</math>-tuply densely divisible, <math>c_\varpi > 0</math> and <math>~c_\delta > 0~</math> are constants in the constraint on <math>\varpi</math> and <math>~\delta~</math>, such that <math>c_{\varpi}\varpi+c_{\delta}\delta<1</math>.
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	== Optimization Model ==		== Optimization Model ==

	For ~~<math>MPZ^{(i)}</math> with~~ given <math>c_\varpi, c_\delta, i</math>, the basic optimization model is defined as:		For a given set of <math>c_\varpi, c_\delta, i</math>, the basic optimization model is defined as: