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| | Our take a look at topics had been common-sized thirteen yr-old boys. The wristband match comfortably on the wrist for both boys. After a couple of hours, one boy complained it was too unfastened and eventually made his personal gap further up the band together with his pocket knife. Both boys stated they noticed the yellow plastic grommet but neither have been bothered by it. The wristband match snugly on the boys' ankles and in both cases, got here loose whereas they performed within the lake. Each [http://Sfhyouthforum.org.uk/who-makes-the-best-throwing-knives-5/ boys stated] the band felt no different than any other gel bracelet they are accustomed to sporting. "It simply smelled better," one boy declared.<br><br><br><br>If you're trying to find the perfect folding knife in your need you then could be conscious that choosing one that's greatest is a tough activity and it becomes a mammoth activity to grab an ideal knife when numerous competing brands are out their. Although in case [http://Www.thebestpocketknifereviews.com/best-throwing-knives-top-recommendations/ Best Throwing Knives] you focus all of your consideration the belongings you gonna so with folding knife then choosing a best folding knife develop into easier. Likewise a pocketknife won't be a very good for many who loves tenting and face survival conditions. It needs an ideal knife that will match for survival conditions. Tactical knife may be useful.<br><br>I beforehand reviewed this knife , and my thoughts on it nonetheless stand. It is a nice trying knife with average blade steel. It isn’t my personal alternative for an everyday knife, due to the weight and dimension. It’s additionally not a knife I would wish to take out in the [http://www.thebestpocketknifereviews.com/best-throwing-knives-top-recommendations/ Best Throwing Knives] woods to depend on for survival. That said it's a strong around the home / sitting on your desk type of knife. This is the knife that I feel is wearing holes in my pockets. I really like this knife — it’s solid feeling, an excellent dimension, a pleasant blade.<br><br>After I purchased this knife I used to be anticipating not to prefer it, however I found it for an excellent worth. What shocked me is that I instantly favored the knife. The long sleek blade, the handle that fit in my hand perfectly, all the things seemed nice. I’d not worry one bit if this had been the knife I was caught in the woods with, despite the fact that it isn’t a large knife. The unlucky part is that I discovered it too bulky to hold in any pants lighter in weight than denims — which for me is a non-starter.<br><br>One of the first stuff you’ll notice once you hold the Sibert is that it is heavy! Weighing 12 ounces, it's a tank and, given its military design, is becoming. This knife is for tremendous heavy obligation users and military personnel solely. The weight makes it too heavy to be an EDC in my ebook. The Delica4, the fourth installment of the Delica sequence, is certainly one of Spyderco’s hottest and best selling knife. The knife is extremely gentle weight at 2.5oz and ideal for EDC. Different, extra heavy duty knives, can deal with harder duties but the Delica4′s VG-10 metal permits it to face up to harsh work if needed.<br><br>We all have a narrative like that. Many of us turn out to be fairly connected to our knives from our youth. I've turn into quite hooked up to a few of them and will never part with them. I really hope that your little one gets to have those self same experiences with his or her knife—and is without doubt [http://www.knifecenter.com custom knives] one of the lucky few who don’t lose their first, second, third or fifteenth knife like most of us. The characteristics of the metal will also be changed by the way in which the blade is rolled and heated within the finishing process. Some blade manufacturers additionally choose to coat the blade to further improve the finish. Vital blade properties<br><br>on-line reviewers stated that the knife was 4.5 out of 5 stars. Comments corresponding to “nice product,” “great knife,” and “great, light-weight utility knife,” have been common. KnifeUp recommends this knife if you're someone who is worried that you’ll lose a knife. At this value, the knife performs properly and, when you occur to lose it, it is no massive deal. #8 Benchmade 581 The knife has a locking lever that is tremendous powerful and durable. Once you buy the knife, the lock will be tough but, after time, the lock will break in. The lock makes the knife really feel very safe. |
| An '''arbitrarily varying channel (AVC)''' is a communication [[channel model]] used in [[coding theory]], and was first introduced by Blackwell, Breiman, and Thomasian. This particular [[Communication channel|channel]] has unknown parameters that can change over time and these changes may not have a uniform pattern during the transmission of a [[codeword]]. <math>\textstyle n</math> uses of this [[Channel model|channel]] can be described using a [[stochastic matrix]] <math>\textstyle W^n: X^n \times</math> <math>\textstyle S^n \rightarrow Y^n</math>, where <math>\textstyle X</math> is the input alphabet, <math>\textstyle Y</math> is the output alphabet, and <math>\textstyle W^n (y | x, s)</math> is the probability over a given set of states <math>\textstyle S</math>, that the transmitted input <math>\textstyle x = (x_1, \ldots, x_n)</math> leads to the received output <math>\textstyle y = (y_1, \ldots, y_n)</math>. The state <math>\textstyle s_i</math> in set <math>\textstyle S</math> can vary arbitrarily at each time unit <math>\textstyle i</math>. This [[Channel model|channel]] was developed as an alternative to [[Claude Shannon|Shannon's]] [[Binary symmetric channel|Binary Symmetric Channel]] (BSC), where the entire nature of the [[Channel model|channel]] is known, to be more realistic to actual [[Communication channel|network channel]] situations.
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| ==Capacities and associated proofs==
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| ===Capacity of deterministic AVCs===
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| An AVC's [[Channel capacity|capacity]] can vary depending on the certain parameters.
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| <math>\textstyle R</math> is an achievable [[Information theory#Rate|rate]] for a deterministic AVC [[Channel coding|code]] if it is larger than <math>\textstyle 0</math>, and if for every positive <math>\textstyle \varepsilon</math> and <math>\textstyle \delta</math>, and very large <math>\textstyle n</math>, length-<math>\textstyle n</math> [[block code]]s exist that satisfy the following equations: <math>\textstyle \frac{1}{n}\log N > R - \delta</math> and <math>\displaystyle \max_{s \in S^n} \bar{e}(s) \leq \varepsilon</math>, where <math>\textstyle N</math> is the highest value in <math>\textstyle Y</math> and where <math>\textstyle \bar{e}(s)</math> is the average probability of error for a state sequence <math>\textstyle s</math>. The largest [[Information theory#Rate|rate]] <math>\textstyle R</math> represents the [[Channel capacity|capacity]] of the AVC, denoted by <math>\textstyle c</math>.
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| As you can see, the only useful situations are when the [[Channel capacity|capacity]] of the AVC is greater than <math>\textstyle 0</math>, because then the [[Channel model|channel]] can transmit a guaranteed amount of data <math>\textstyle \leq c</math> without errors. So we start out with a [[theorem]] that shows when <math>\textstyle c</math> is positive in a AVC and the [[theorem]]s discussed afterward will narrow down the [[Range (mathematics)|range]] of <math>\textstyle c</math> for different circumstances.
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| Before stating Theorem 1, a few definitions need to be addressed:
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| * An AVC is ''symmetric'' if <math>\displaystyle \sum_{s \in S}W(y|x, s)U(s|x') = \sum_{s \in S}W(y|x', s)U(s|x)</math> for every <math>\textstyle (x, x', y,s)</math>, where <math>\textstyle x,x'\in X</math>, <math>\textstyle y \in Y</math>, and <math>\textstyle U(s|x)</math> is a channel function <math>\textstyle U: X \rightarrow S</math>.
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| * <math>\textstyle X_r</math>, <math>\textstyle S_r</math>, and <math>\textstyle Y_r</math> are all [[random variable]]s in sets <math>\textstyle X</math>, <math>\textstyle S</math>, and <math>\textstyle Y</math> respectively.
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| * <math>\textstyle P_{X_r}(x)</math> is equal to the probability that the [[random variable]] <math>\textstyle X_r</math> is equal to <math>\textstyle x</math>.
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| * <math>\textstyle P_{S_r}(s)</math> is equal to the probability that the [[random variable]] <math>\textstyle S_r</math> is equal to <math>\textstyle s</math>.
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| * <math>\textstyle P_{X_{r}S_{r}Y_{r}}</math> is the combined [[probability mass function]] (pmf) of <math>\textstyle P_{X_r}(x)</math>, <math>\textstyle P_{S_r}(s)</math>, and <math>\textstyle W(y|x,s)</math>. <math>\textstyle P_{X_{r}S_{r}Y_{r}}</math> is defined formally as <math>\textstyle P_{X_{r}S_{r}Y_{r}}(x,s,y) = P_{X_r}(x)P_{S_r}(s)W(y|x,s)</math>.
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| * <math>\textstyle H(X_r)</math> is the [[Information entropy|entropy]] of <math>\textstyle X_r</math>.
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| * <math>\textstyle H(X_r|Y_r)</math> is equal to the average probability that <math>\textstyle X_r</math> will be a certain value based on all the values <math>\textstyle Y_r</math> could possibly be equal to.
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| * <math>\textstyle I(X_r \land Y_r)</math> is the [[Conditional entropy|mutual information]] of <math>\textstyle X_r</math> and <math>\textstyle Y_r</math>, and is equal to <math>\textstyle H(X_r) - H(X_r|Y_r)</math>.
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| * <math>\displaystyle I(P) = \min_{Y_r} I(X_r \land Y_r)</math>, where the minimum is over all random variables <math>\textstyle Y_r</math> such that <math>\textstyle X_r</math>, <math>\textstyle S_r</math>, and <math>\textstyle Y_r</math> are distributed in the form of <math>\textstyle P_{X_{r}S_{r}Y_{r}}</math>.
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| '''Theorem 1:''' <math>\textstyle c > 0</math> if and only if the AVC is not symmetric. If <math>\textstyle c > 0</math>, then <math>\displaystyle c = \max_P I(P)</math>.
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| ''Proof of 1st part for symmetry:'' If we can prove that <math>\textstyle I(P)</math> is positive when the AVC is not symmetric, and then prove that <math>\textstyle c = \max_P I(P)</math>, we will be able to prove Theorem 1. Assume <math>\textstyle I(P)</math> were equal to <math>\textstyle 0</math>. From the definition of <math>\textstyle I(P)</math>, this would make <math>\textstyle X_r</math> and <math>\textstyle Y_r</math> [[Independence (probability theory)|independent]] [[random variable]]s, for some <math>\textstyle S_r</math>, because this would mean that neither [[random variable]]'s [[Information entropy|entropy]] would rely on the other [[random variable]]'s value. By using equation <math>\textstyle P_{X_{r}S_{r}Y_{r}}</math>, (and remembering <math>\textstyle P_{X_r} = P</math>,) we can get,
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| :<math>\displaystyle P_{Y_r}(y) = \sum_{x\in X} \sum_{s\in S} P(x)P_{S_r}(s)W(y|x,s)</math>
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| :<math>\textstyle \equiv (</math>since <math>\textstyle X_r</math> and <math>\textstyle Y_r</math> are [[Independence (probability theory)|independent]] [[random variable]]s, <math>\textstyle W(y|x, s) = W'(y|s)</math> for some <math>\textstyle W')</math>
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| :<math>\displaystyle P_{Y_r}(y) = \sum_{x\in X} \sum_{s\in S} P(x)P_{S_r}(s)W'(y|s)</math>
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| :<math>\textstyle \equiv (</math>because only <math>\textstyle P(x)</math> depends on <math>\textstyle x</math> now<math>\textstyle )</math>
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| :<math>\displaystyle P_{Y_r}(y) = \sum_{s\in S} P_{S_r}(s)W'(y|s) \left[\sum_{x\in X} P(x)\right]</math>
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| :<math>\textstyle \equiv (</math>because <math>\displaystyle \sum_{x\in X} P(x) = 1)</math>
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| :<math>\displaystyle P_{Y_r}(y) = \sum_{s\in S} P_{S_r}(s)W'(y|s)</math>
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| So now we have a [[probability distribution]] on <math>\textstyle Y_r</math> that is [[Independence (probability theory)|independent]] of <math>\textstyle X_r</math>. So now the definition of a symmetric AVC can be rewritten as follows: <math>\displaystyle \sum_{s \in S}W'(y|s)P_{S_r}(s) = \sum_{s \in S}W'(y|s)P_{S_r}(s)</math> since <math>\textstyle U(s|x)</math> and <math>\textstyle W(y|x, s)</math> are both functions based on <math>\textstyle x</math>, they have been replaced with functions based on <math>\textstyle s</math> and <math>\textstyle y</math> only. As you can see, both sides are now equal to the <math>\textstyle P_{Y_r}(y)</math> we calculated earlier, so the AVC is indeed symmetric when <math>\textstyle I(P)</math> is equal to <math>\textstyle 0</math>. Therefore <math>\textstyle I(P)</math> can only be positive if the AVC is not symmetric.
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| ''Proof of second part for capacity'': See the paper "The capacity of the arbitrarily varying channel revisited: positivity, constraints," referenced below for full proof.
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| ===Capacity of AVCs with input and state constraints===
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| The next [[theorem]] will deal with the [[Channel capacity|capacity]] for AVCs with input and/or state constraints. These constraints help to decrease the very large [[Range (mathematics)|range]] of possibilities for transmission and error on an AVC, making it a bit easier to see how the AVC behaves. | |
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| Before we go on to Theorem 2, we need to define a few definitions and [[Lemma (mathematics)|lemmas]]:
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| For such AVCs, there exists:<br>
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| :- An input constraint <math>\textstyle \Gamma</math> based on the equation <math>\displaystyle g(x) = \frac{1}{n}\sum_{i=1}^n g(x_i)</math>, where <math>\textstyle x \in X</math> and <math>\textstyle x = (x_1,\dots,x_n)</math>.
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| :- A state constraint <math>\textstyle \Lambda</math>, based on the equation <math>\displaystyle l(s) = \frac{1}{n}\sum_{i=1}^n l(s_i)</math>, where <math>\textstyle s \in X</math> and <math>\textstyle s = (s_1,\dots,s_n)</math>.
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| :- <math>\displaystyle \Lambda_0(P) = \min \sum_{x \in X, s \in S}P(x)l(s)</math>
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| :- <math>\textstyle I(P, \Lambda)</math> is very similar to <math>\textstyle I(P)</math> equation mentioned previously, <math>\displaystyle I(P, \Lambda) = \min_{Y_r} I(X_r \land Y_r)</math>, but now any state <math>\textstyle s</math> or <math>\textstyle S_r</math> in the equation must follow the <math>\textstyle l(s) \leq \Lambda</math> state restriction.
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| Assume <math>\textstyle g(x)</math> is a given non-negative-valued function on <math>\textstyle X</math> and <math>\textstyle l(s)</math> is a given non-negative-valued function on <math>\textstyle S</math> and that the minimum values for both is <math>\textstyle 0</math>. In the literature I have read on this subject, the exact definitions of both <math>\textstyle g(x)</math> and <math>\textstyle l(s)</math> (for one variable <math>\textstyle x_i</math>,) is never described formally. The usefulness of the input constraint <math>\textstyle \Gamma</math> and the state constraint <math>\textstyle \Lambda</math> will be based on these equations.
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| For AVCs with input and/or state constraints, the [[Information_theory#Rate|rate]] <math>\textstyle R</math> is now limited to [[codeword]]s of format <math>\textstyle x_1,\dots,x_N</math> that satisfy <math>\textstyle g(x_i) \leq \Gamma</math>, and now the state <math>\textstyle s</math> is limited to all states that satisfy <math>\textstyle l(s) \leq \Lambda</math>. The largest [[Information_theory#Rate|rate]] is still considered the [[Channel capacity|capacity]] of the AVC, and is now denoted as <math>\textstyle c(\Gamma, \Lambda)</math>.
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| ''Lemma 1:'' Any [[Channel coding|codes]] where <math>\textstyle \Lambda</math> is greater than <math>\textstyle \Lambda_0(P)</math> cannot be considered "good" [[Channel coding|codes]], because those kinds of [[Channel coding|codes]] have a maximum average probability of error greater than or equal to <math>\textstyle \frac{N-1}{2N} - \frac{1}{n}\frac{l_{max}^2}{n(\Lambda - \Lambda_0(P))^2}</math>, where <math>\textstyle l_{max}</math> is the maximum value of <math>\textstyle l(s)</math>. This isn't a good maximum average error probability because it is fairly large, <math>\textstyle \frac{N-1}{2N}</math> is close to <math>\textstyle \frac{1}{2}</math>, and the other part of the equation will be very small since the <math>\textstyle (\Lambda - \Lambda_0(P))</math> value is squared, and <math>\textstyle \Lambda</math> is set to be larger than <math>\textstyle \Lambda_0(P)</math>. Therefore it would be very unlikely to receive a [[codeword]] without error. This is why the <math>\textstyle \Lambda_0(P)</math> condition is present in Theorem 2.
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| '''Theorem 2:''' Given a positive <math>\textstyle \Lambda</math> and arbitrarily small <math>\textstyle \alpha > 0</math>, <math>\textstyle \beta > 0</math>, <math>\textstyle \delta > 0</math>, for any block length <math>\textstyle n \geq n_0</math> and for any type <math>\textstyle P</math> with conditions <math>\textstyle \Lambda_0(P) \geq \Lambda + \alpha</math> and <math>\displaystyle \min_{x \in X}P(x) \geq \beta</math>, and where <math>\textstyle P_{X_r} = P</math>, there exists a [[Channel coding|code]] with [[codeword]]s <math>\textstyle x_1,\dots,x_N</math>, each of type <math>\textstyle P</math>, that satisfy the following equations: <math>\textstyle \frac{1}{n}\log N > I(P,\Lambda) - \delta</math>, <math>\displaystyle \max_{l(s) \leq \Lambda} \bar{e}(s) \leq \exp(-n\gamma)</math>, and where positive <math>\textstyle n_0</math> and <math>\textstyle \gamma</math> depend only on <math>\textstyle \alpha</math>, <math>\textstyle \beta</math>, <math>\textstyle \delta</math>, and the given AVC.
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| ''Proof of Theorem 2'': See the paper "The capacity of the arbitrarily varying channel revisited: positivity, constraints," referenced below for full proof.
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| ===Capacity of randomized AVCs===
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| The next [[theorem]] will be for AVCs with [[Information entropy|randomized]] [[Channel coding|code]]. For such AVCs the [[Channel coding|code]] is a [[random variable]] with values from a family of length-n [[block code]]s, and these [[Channel coding|code]]s are not allowed to depend/rely on the actual value of the [[codeword]]. These [[Channel coding|codes]] have the same maximum and average error probability value for any [[Channel model|channel]] because of its random nature. These types of [[Channel coding|codes]] also help to make certain properties of the AVC more clear.
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| Before we go on to Theorem 3, we need to define a couple important terms first:
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| <math>\displaystyle W_{\zeta}(y|x) = \sum_{s \in S} W(y|x, s)P_{S_r}(s)</math><br> | |
| <math>\textstyle I(P, \zeta)</math> is very similar to the <math>\textstyle I(P)</math> equation mentioned previously, <math>\displaystyle I(P, \zeta) = \min_{Y_r} I(X_r \land Y_r)</math>, but now the [[/Probability mass function|pmf]] <math>\textstyle P_{S_r}(s)</math> is added to the equation, making the minimum of <math>\textstyle I(P, \zeta)</math> based a new form of <math>\textstyle P_{X_{r}S_{r}Y_{r}}</math>, where <math>\textstyle W_{\zeta}(y|x)</math> replaces <math>\textstyle W(y|x, s)</math>.
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| '''Theorem 3:''' The [[Channel capacity|capacity]] for [[Information entropy|randomized]] [[Channel coding|codes]] of the AVC is <math>\displaystyle c = max_P I(P, \zeta)</math>.
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| ''Proof of Theorem 3'': See paper "The Capacities of Certain Channel Classes Under Random Coding" referenced below for full proof.
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| ==See also==
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| * [[Binary symmetric channel]]
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| * [[Binary erasure channel]]
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| * [[Z-channel (information theory)]]
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| * [[Channel model]]
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| * [[Information theory]]
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| * [[Coding theory]]
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| | |
| == References ==
| |
| <!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
| |
| * Ahlswede, Rudolf and Blinovsky, Vladimir, "Classical Capacity of Classical-Quantum Arbitrarily Varying Channels," http://ieeexplore.ieee.org.gate.lib.buffalo.edu/stamp/stamp.jsp?tp=&arnumber=4069128
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| * Blackwell, David, Breiman, Leo, and Thomasian, A. J., "The Capacities of Certain Channel Classes Under Random Coding," http://www.jstor.org/stable/2237566
| |
| * Csiszar, I. and Narayan, P., "Arbitrarily varying channels with constrained inputs and states," http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=2598&isnumber=154
| |
| * Csiszar, I. and Narayan, P., "Capacity and Decoding Rules for Classes of Arbitrarily Varying Channels," http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=32153&isnumber=139
| |
| * Csiszar, I. and Narayan, P., "The capacity of the arbitrarily varying channel revisited: positivity, constraints," http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=2627&isnumber=155
| |
| * Lapidoth, A. and Narayan, P., "Reliable communication under channel uncertainty," http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=720535&isnumber=15554
| |
| | |
| [[Category:Coding theory]]
| |
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