Sea ice emissivity modelling: Difference between revisions

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'''Witsenhausen's [[counterexample]]''', shown in the figure below, arises in a deceptively simple [[toy problem]] in decentralized [[stochastic control]]. It was formulated by [[Hans witsenhausen|Hans Witsenhausen]] in 1968.<ref>A counterexample in stochastic optimum control. SIAM J. Control Volume 6, Issue 1, pp. 131–147 (February 1968)</ref> It is a counterexample to a natural conjecture that one can generalize a key result of centralized [[linear-quadratic-Gaussian control]] systems: that affine (linear) control laws are optimal. Witsenhausen showed that there exists a nonlinear control law that outperforms all linear laws. The problem of finding the optimal control law remains unsolved.<ref name=Ho>Review of the Witsenhausen problem. Yu-Chi Ho. Proceedings of the 47th IEEE Conference on Decision and Control (CDC), pp. 1611–1613, 2008.</ref>
<br><br>Some dwelling remedies can be applied to eliminate white bumps on face. The useless skin must be removed and for that you need to use a mild scrub to exfoliate the skin. You can too use a moisturizer which has a light consistency around these bumps. One of the main reasons for white bumps is the buildup of impurities, oil and makeup, so you must clear your pores and skin every morning and before you go to bed. Attempt avoiding make-up, particularly across the eyes, that don't enable oil to be launched into the surface of the pores and skin.<br><br>Many a time it's possible you'll discover white bumps or razor bumps on face after shaving which can be caused as a consequence of several components. One of the vital frequent causes for white bumps after shaving is ingrown hair, wherein some hair develops back into the hair follicle. This will cause the follicles to inflame thereby causing bumps. White bumps on account of ingrown hair is generally present in people with curly hair. White bumps could be prevented by not shaving close to the pores and skin. You can forestall inflammation by making use of lotion whose main composition is glycolic acid.<br><br>Like most acne, whiteheads are formed when extra oil from the sebaceous glands clogs up the pores or hair follicles on the skin. This sebum gets trapped together with other particles and a few micro organism which can be airborne or from the skin's surface so that when the trapped sebum and bacteria stay below the pores and skin's surface, a whitehead is formed. [http://cairo.Gspeedhost.com/ads/how-to-remove-blackheads-from-nose-with-extractor/ Whiteheads] might show up as tiny white spots, or they could be so small that they are invisible to the bare eye. As whiteheads are closed off from the skin air, they continue to be white. Whiteheads are often discovered hidden away such as on the brow below lengthy bangs.<br><br>Mainly, whiteheads break out when our physique tries to battle in opposition to a bacterial assault on the facial skin. Sebum is an oily substance released by sebaceous glands to maintain our pores and skin and hair lubricated. When there's an overproduction of sebum, it tends to clog up the facial skin pores fully. Presence of useless pores and skin cells inside the pores worsens the situation additional as it becomes a perfect breeding ground for the dangerous micro organism. Our body then sends white blood cells to the infected part of the pores and skin to struggle out the micro organism. This ends in inflammation and reddening of the skin. Thus whiteheads are shaped. <br><br><br>A darkening of the trapped particles happens when it's exposed to oxygen which makes the pores look black. Very often, bacteria will are available in contact with blackheads causing them to develop into purple, infected whiteheads. Some people are inclined to develop an ongoing downside and want a remedy for whiteheads that is both specific and effective. The easiest way to deal with the condition is to mix a routine of eating regimen and train with a pores and skin therapy protocol. [http://Ielaw.Uibe.Edu.cn/wiki/index.php/How_To_Get_Rid_Of_Blackheads_On_Nose_Naturally_Fast Dermatologists agree] that skin problems should not typically directly attributable to meals. Nevertheless, there's a link between an unbalanced food regimen and certain set off foods.<br><br>Because of the truth that pimples are a very common incidence, there are a selection of residence cures that you would attempt with the intention to assist relieve the situation initially after which fully eradicate it. Toothpaste is considered to be a very efficient house [http://Www.budzpon.com/removing-blackheads-from-nose-video/ remedy choice] when treating acne and all that is required is that you simply apply a small quantity of the paste excessive of a blemish and permit it to dry earlier than washing it off. Avoid leaving the paste on for too long because it might are inclined to irritate the skin. You can make a mix of equal quantities of lemon juice and groundnut oil.<br><br>A whitehead could be finest removed by steaming the face for 1-2 minutes. When the whiteheads within the pores have Softened, prick the apex with a clear sterilized needle. They very gently press it out along with your two fingers preserving them on both facet of the whitehead and gently putting stress on it. Another strategy to take away whiteheads is to take shavings of baby soap and blend in desk salt. Take a wet and tender face brush, dip in the mixture and very gently rub within the portions having white heads with the comb for a couple of minutes. Then wash off after a number of rubbings.<br><br>Sterilize any pin you employ. For stubborn blackheads, white heads, and especially facial Milia, you might need to use a pin/needle to help open up the skin sufficient so that the facial impurities will push up and out of your skin. If you happen to find this a obligatory process, you'll want to use alcohol to sterilize the tip of the pin/needle to keep away from infecting your pores and skin. If it's worthwhile [https://hairnet-paradise.com/how-to-remove-blackheads-from-nose/ How To Get Rid Of Blackheads On Nose Products] to extract an impurity close to your eye, don't attempt this yourself. If you have to use a pin/needle to open the pores and skin, you could possibly by accident stick your eye with the pin/needle. Search out medical help.
 
[[File:WitsenhausenCounterexample.jpg]]
 
The statement of the counterexample is simple: two controllers attempt to control the system by attempting to bring the state close to zero in exactly two time steps. There is a cost on the input <math>u_1</math> of the first controller, and the state <math>x_2</math> after the input of the second controller. The input <math>u_2</math> of the second controller is free, but its observations <math>y_2=x_1+z</math> are noisy. The objective is to minimize an average cost function, <math>k^2E[u_1^2]+E[x_2^2]</math>, where the average is over the randomness in the initial state <math>x_0</math> and the observation noise <math>z</math>, both of which are distributed independently and in a Gaussian manner.
 
==The significance of the problem==
The counterexample lies at the intersection of [[control theory]] and [[information theory]]. Due to its hardness, the problem of finding the optimal control law has also received attention from the [[theoretical computer science]] community. The importance of the problem was recently reflected upon in the 47th IEEE Conference on Decision and Control (CDC) 2008, Cancun, Mexico,<ref name=Ho/> where an entire session was dedicated to understanding the counterexample 40 years after it was first formulated.
 
The problem is of conceptual significance in decentralized control because it shows that it is important for the controllers to communicate<ref>Information and Control: Witsenhausen revisited. Mitter and Sahai. Learning, control and hybrid systems, 1999 – Springer.</ref> with each other implicitly in order to minimize the cost. This suggests that control actions in decentralized control may have a dual role: those of control and communication.
 
==The hardness of the problem==
The hardness of the problem is attributed to the fact that information of the second controller depends on the decisions of the first controller.<ref>Team Decision Theory and Information Structures. Yu-Chi Ho. Proceedings of the IEEE, Vol. 68, No.6, JUNE 1980</ref> Variations considered by [[Tamer Basar]] <ref>Variations on the Theme of the Witsenhausen Counterexample. Tamer Basar. 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9–11, 2008.</ref> show that the hardness is also because of the structure of the performance index and the coupling of different decision variables. It has also been shown that problems of the spirit of Witsenhausen's counterexample become simpler if the [[transmission delay]] along an external channel that connects the controllers is smaller than the [[propagation delay]] in the problem. However, this result requires the channels to be perfect and instantaneous,<ref>Rotkowitz, M.; Cogill, R.; Lall, S.; , "A Simple Condition for the Convexity of Optimal Control over Networks with Delays," Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on , vol., no., pp. 6686–6691, 12–15 December 2005</ref> and hence is of limited applicability. In practical situations, the channel is always imperfect, and thus one can not assume that decentralized control problems are simple in presence of external channels.
 
A justification of the failure of attempts that discretize the problem came from the computer science literature: [[Christos Papadimitriou]] and John Tsitsiklis showed that the discrete version of the counterexample is  [[NP-complete]].<ref>Intractable problems in control theory. [[Christos Papadimitriou]] and John Tsitsiklis. 24th IEEE Conference on Decision and Control, 1985</ref>
 
==Attempts at obtaining a solution==
A number of numerical attempts have been made to solve the counterexample. Focusing on a particular choice of problem parameters <math>(k=0.2,\;\sigma_0=5)</math>, researchers have obtained strategies by [[discretization]] and using [[neural networks]].<ref>Numerical solutions to the Witsenhausen counterexample by approximating networks. Baglietto, Parisini, Zoppoli IEEE Trans. Automatic Control. 2001.</ref> Further research (notably, the work of [[Yu-Chi Ho]],<ref>The Witsenhausen counterexample: A hierarchical search approach for nonconvex optimization problems. Lee, Lau, Ho. IEEE Trans. Automatic Control, 2001</ref> and the work of Li, Marden and [[Jeff S. Shamma|Shamma]] <ref>Learning approaches to the Witsenhausen counterexample from a view of potential games. Li, Marden, Shamma. IEEE Conference on Decision and Control, 2009</ref>) has obtained slightly improved costs for the same parameter choice. The first provably approximately optimal strategies appeared recently (Grover, Park, Sahai) <ref>The finite-dimensional Witsenhausen Counterexample. Grover, Sahai and Park. IEEE WiOpt 2010, ConCom workshop, Seoul, Korea.</ref> where [[information theory]] is used to understand the communication in the counterexample. The optimal solution of the counterexample is still an open problem.
 
==References==
{{Reflist}}
 
{{Use dmy dates|date=September 2010}}
 
{{DEFAULTSORT:Witsenhausen's Counterexample}}
[[Category:Control theory]]
[[Category:Stochastic control]]

Latest revision as of 22:28, 25 June 2014



Some dwelling remedies can be applied to eliminate white bumps on face. The useless skin must be removed and for that you need to use a mild scrub to exfoliate the skin. You can too use a moisturizer which has a light consistency around these bumps. One of the main reasons for white bumps is the buildup of impurities, oil and makeup, so you must clear your pores and skin every morning and before you go to bed. Attempt avoiding make-up, particularly across the eyes, that don't enable oil to be launched into the surface of the pores and skin.

Many a time it's possible you'll discover white bumps or razor bumps on face after shaving which can be caused as a consequence of several components. One of the vital frequent causes for white bumps after shaving is ingrown hair, wherein some hair develops back into the hair follicle. This will cause the follicles to inflame thereby causing bumps. White bumps on account of ingrown hair is generally present in people with curly hair. White bumps could be prevented by not shaving close to the pores and skin. You can forestall inflammation by making use of lotion whose main composition is glycolic acid.

Like most acne, whiteheads are formed when extra oil from the sebaceous glands clogs up the pores or hair follicles on the skin. This sebum gets trapped together with other particles and a few micro organism which can be airborne or from the skin's surface so that when the trapped sebum and bacteria stay below the pores and skin's surface, a whitehead is formed. Whiteheads might show up as tiny white spots, or they could be so small that they are invisible to the bare eye. As whiteheads are closed off from the skin air, they continue to be white. Whiteheads are often discovered hidden away such as on the brow below lengthy bangs.

Mainly, whiteheads break out when our physique tries to battle in opposition to a bacterial assault on the facial skin. Sebum is an oily substance released by sebaceous glands to maintain our pores and skin and hair lubricated. When there's an overproduction of sebum, it tends to clog up the facial skin pores fully. Presence of useless pores and skin cells inside the pores worsens the situation additional as it becomes a perfect breeding ground for the dangerous micro organism. Our body then sends white blood cells to the infected part of the pores and skin to struggle out the micro organism. This ends in inflammation and reddening of the skin. Thus whiteheads are shaped.


A darkening of the trapped particles happens when it's exposed to oxygen which makes the pores look black. Very often, bacteria will are available in contact with blackheads causing them to develop into purple, infected whiteheads. Some people are inclined to develop an ongoing downside and want a remedy for whiteheads that is both specific and effective. The easiest way to deal with the condition is to mix a routine of eating regimen and train with a pores and skin therapy protocol. Dermatologists agree that skin problems should not typically directly attributable to meals. Nevertheless, there's a link between an unbalanced food regimen and certain set off foods.

Because of the truth that pimples are a very common incidence, there are a selection of residence cures that you would attempt with the intention to assist relieve the situation initially after which fully eradicate it. Toothpaste is considered to be a very efficient house remedy choice when treating acne and all that is required is that you simply apply a small quantity of the paste excessive of a blemish and permit it to dry earlier than washing it off. Avoid leaving the paste on for too long because it might are inclined to irritate the skin. You can make a mix of equal quantities of lemon juice and groundnut oil.

A whitehead could be finest removed by steaming the face for 1-2 minutes. When the whiteheads within the pores have Softened, prick the apex with a clear sterilized needle. They very gently press it out along with your two fingers preserving them on both facet of the whitehead and gently putting stress on it. Another strategy to take away whiteheads is to take shavings of baby soap and blend in desk salt. Take a wet and tender face brush, dip in the mixture and very gently rub within the portions having white heads with the comb for a couple of minutes. Then wash off after a number of rubbings.

Sterilize any pin you employ. For stubborn blackheads, white heads, and especially facial Milia, you might need to use a pin/needle to help open up the skin sufficient so that the facial impurities will push up and out of your skin. If you happen to find this a obligatory process, you'll want to use alcohol to sterilize the tip of the pin/needle to keep away from infecting your pores and skin. If it's worthwhile How To Get Rid Of Blackheads On Nose Products to extract an impurity close to your eye, don't attempt this yourself. If you have to use a pin/needle to open the pores and skin, you could possibly by accident stick your eye with the pin/needle. Search out medical help.

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