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Old 09-30-2012, 10:50 PM
mileschen mileschen is offline
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Default Problem 2.14(c)

For Problem 2.14(c), to determine the min value, the way I think would be try to solve the equation in (b) and get L. Maybe L is the second part of the min. However, how to solve the equation is a really hard question. Thus, could anyone tell me how to solve the equation or give me a hint on how to get the right answer?
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Old 10-01-2012, 04:46 AM
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magdon magdon is offline
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Default Re: Problem 2.14(c)

Yes, solving the equation is really hard. It is simpler to show that if \ell takes on the value in the second part of the min, the condition in (b) is satisfied.
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For Problem 2.14(c), to determine the min value, the way I think would be try to solve the equation in (b) and get L. Maybe L is the second part of the min. However, how to solve the equation is a really hard question. Thus, could anyone tell me how to solve the equation or give me a hint on how to get the right answer?
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Old 07-14-2014, 05:29 PM
BojanVujatovic BojanVujatovic is offline
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Default Re: Problem 2.14(c)

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It is simpler to show that if \ell takes on the value in the second part of the min, the condition in (b) is satisfied.
I have difficulties solving this problem. If I assume that \ell=d_{VC} \log_2 d_{VC} + (d_{VC}+1) \log_2 K, then the condition in (b) 2^\ell>K\ell^{d_{VC}+1} is not satisfied.
(e.g. when d_{VC}=2, K=2, then \ell=5 and 2^5=32 \ngtr 2 \cdot 5^3=250).

I believe the right thing to do would be to assume that \ell \geq d_{VC} \log_2 d_{VC} + (d_{VC}+1) \log_2 K because the min bound will still hold and I believe the condition in (b) is then satisfied? But how do I prove that?

I appreciate any help.
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Old 07-17-2014, 08:26 AM
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magdon magdon is offline
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Default Re: Problem 2.14(c)

There is a typo in the equation, sorry.

The second term in the minimum should be

7(d_{VC}+K)\log_2(d_{VC}K).

Rather than solve the inequality in (b) to get this bound, you may rather just verify that this is a bound by showing that if \ell=7(d_{VC}+K)\log_2(d_{VC}K), then the inequality in (b) is satisfied, namely 2^\ell>2K\ell^{d_{VC}}.

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Originally Posted by mileschen View Post
For Problem 2.14(c), to determine the min value, the way I think would be try to solve the equation in (b) and get L. Maybe L is the second part of the min. However, how to solve the equation is a really hard question. Thus, could anyone tell me how to solve the equation or give me a hint on how to get the right answer?
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Old 07-20-2014, 08:48 AM
BojanVujatovic BojanVujatovic is offline
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Default Re: Problem 2.14(c)

Thank you for your reply!
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  #6  
Old 10-01-2015, 07:46 AM
zhaozb15 zhaozb15 is offline
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Default Re: Problem 2.14(c)

Quote:
Originally Posted by magdon View Post
There is a typo in the equation, sorry.

The second term in the minimum should be

7(d_{VC}+K)\log_2(d_{VC}K).

Rather than solve the inequality in (b) to get this bound, you may rather just verify that this is a bound by showing that if \ell=7(d_{VC}+K)\log_2(d_{VC}K), then the inequality in (b) is satisfied, namely 2^\ell>2K\ell^{d_{VC}}.
If d_{VC}=K=1, then 7(d_{VC}+K)\log_2(d_{VC}K)=0. Seems not correct
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Old 10-06-2015, 09:15 PM
ilson ilson is offline
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Default Re: Problem 2.14(c)

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If d_{VC}=K=1, then 7(d_{VC}+K)\log_2(d_{VC}K)=0. Seems not correct
I came here to say exactly this. Also, if K=1 then d_{\text{vc}}(\mathcal{H}) = d_{\text{vc}} trivially, so can we assume that K\geq 2?
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Old 10-07-2015, 06:05 AM
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magdon magdon is offline
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Default Re: Problem 2.14(c)

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I came here to say exactly this. Also, if K=1 then d_{\text{vc}}(\mathcal{H}) = d_{\text{vc}} trivially, so can we assume that K\geq 2?
Yes, the problem should state that K>1, otherwise the problem is trivial.
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Old 07-12-2017, 10:59 AM
RicLouRiv RicLouRiv is offline
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Default Re: Problem 2.14(c)

I'm pretty stuck on this one -- any hints?
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