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In [[mathematics]] &mdash; specifically, in [[probability theory]] &mdash; the '''concentration dimension''' of a [[Banach space]]-valued [[random variable]] is a numerical measure of how &ldquo;spread out&rdquo; the random variable is compared to the [[norm (mathematics)|norm]] on the space.
 
==Definition==
 
Let (''B'',&nbsp;||&nbsp;||) be a Banach space and let ''X'' be a [[Gaussian distribution|Gaussian random variable]] taking values in ''B''. That is, for every linear functional ''ℓ'' in the [[dual space]] ''B''<sup>&lowast;</sup>, the real-valued random variable &lang;''ℓ'',&nbsp;''X''&rang; has a [[normal distribution]]. Define
 
:<math>\sigma(X) = \sup \left\{ \left. \sqrt{\operatorname{E} [\langle \ell, X \rangle^{2}]} \,\right|\, \ell \in B^{\ast}, \| \ell \| \leq 1 \right\}.</math>
 
Then the '''concentration dimension''' ''d''(''X'') of ''X'' is defined by
 
:<math>d(X) = \frac{\operatorname{E} [\| X \|^{2}]}{\sigma(X)^{2}}.</math>
 
==Examples==
 
* If ''B'' is ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> with its usual Euclidean norm,{{Clarify|date=May 2012}} and ''X'' is a standard Gaussian random variable, then ''&sigma;''(''X'')&nbsp;=&nbsp;1 and E[||''X''||<sup>2</sup>]&nbsp;=&nbsp;''n'', so ''d''(''X'')&nbsp;=&nbsp;''n''.
* If ''B'' is '''R'''<sup>''n''</sup> with the [[supremum norm]], then ''&sigma;''(''X'')&nbsp;=&nbsp;1 but E[||''X''||<sup>2</sup>] (and hence ''d''(''X'')) is of the order of log(''n'').
 
==References==
 
* {{cite book
| last1 = Ledoux
| first1 = Michel
| last2 = Talagrand | first2 = Michel | author2-link = Michel Talagrand
| title = Probability in Banach spaces
| publisher = Springer-Verlag
| location = Berlin
| year = 1991
| pages = xii+480
| isbn = 3-540-52013-9
| MR=1102015}} (See chapter 9)
 
[[Category:Dimension]]
[[Category:Probability theory]]

Latest revision as of 10:56, 7 January 2015

Photo Credit Comstock Images/Comstock/Getty Images Middle school is a time of great change for students and parents.
Students have several teachers instead of one. This means that parents often need to meet with more than one teacher to get a clear understanding of the student’s performance. Communication between parents and teachers is vital for any student’s success. However, parents don’t always know what to ask or what to say.
Planning ahead can help you get the most from parent-teacher conferences and ultimately, help your child achieve.

Also, look for changes in your child's grades, especially if there is a dip. Make note of any changes and discuss them with the appropriate teacher.
In the Conference Go angielskie cytaty beyond just asking each teacher how your child is doing at the moment. Also inquire about grading. Ask the teacher how he grades assignments. Ask what percentage of the total grade is based on quizzes, exams and homework. Discuss any behavioral issues that your child might have and whether he is handing in his assignments on time.

Before coming up with your own questions, talk to your child. Discuss any concerns or worries he might have about school. If he has some, determine how you can present them to his teacher. For example, if he is worried about his math grade, you can explain your child's concern to his math teacher and ask if you should consider a tutor or if the teacher might have some extra time to spend with your child.
Also talk with your child about how he thinks school is going. Ask him If he could ask his teachers specific questions what they might be? His responses and questions can help you formulate your plan for meeting with his teachers and the kinds specific questions you might want to ask regarding your child's schoolwork or other issues at school.

Past Performance Look over your child’s report card as well as her tests and assignments. Look for patterns. For example, you might notice that your child gets passing grades on her papers, but consistently misuses verb tense. This keeps her from truly excelling.
You can mention this to her English teacher and ask if it's possible for the teacher to give your child a few extra assignments in verb tense that can help her get up to speed. As classes move forward, if your child missed a crucial step, it can make learning other skills more difficult.

Does your child respect teachers, administrators and peers? Does he take care of school property? Your child should feel that you and his teacher have similar expectations. Learn about the teacher's discipline techniques and try to employ some of those strategies at home to give your child familiar routines.
As Ron Clark, who wrote the book, "The End of Molasses Classes: Getting Our Kids Unstuck -- 101 Extraordinary Solutions for Parents and Teachers," writes at CNN.com, "Be a partner instead of a prosecutor."

Ask the teacher if he feels that your child is fully challenged -- and if he feels that your child is working hard. Talk about the plan for the rest of the year. Ask about the dates for upcoming assignments.
Planning for the Future If college is in your child’s future, talk to the teachers about whether the class is helping your child gain the skills he needs for higher education. Talk about what additional classes or resources the school offers for college-bound students.

Children who feel a strong connection between home and school are often more comfortable in their classrooms and might be more academically successful. Knowing that you and his teacher are on the same team is likely to make your child work harder.
What Will My Child Learn? Find out what your child will learn about in each subject and what he will be expected to master by the end of the year. Such information will help you prepare for the work ahead. The teacher should be able to give you an overview of the curriculum and the standards that your child must meet based on district goals and state frameworks.

How Can I Help You? An effective way to strengthen your relationship with your child's teacher is to find out how you can help. Perhaps the teacher needs a volunteer for an activity, a chaperone for a field trip or some extra supplies. Show your interest in making her life easier.

National Library of Medicine, "Parent involvement in a child's education is consistently found to be positively associated with a child's academic performance."
How is My Child Behaving? Your child's behavior can affect his academic achievement. Find out if your child makes occasional healthy, age-appropriate mistakes or if the teacher notices a pattern of poor behavior. According to a 2007 National Science Foundation press release called "Children's Bad Behavior Gives Insights to Academic Achievement and Later Career Success," "Early behavior problems don't impede children's school achievement, but persistent problems may impede educational and occupational outcomes." Inquire about his ability to follow directions and obey rules.