组合数学 (Fall 2015)/Problem Set 2 and Permeability (electromagnetism): Difference between pages

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(The value of µ0 is 4E-7\pi, this is how it is defined.)
 
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=要求 =
{{MagneticCircuitSegments}}
<font color=red size=5>解答要求有完整的过程。</font>
'''Permeability''' is a [[property]] of a material that describes how [[density|dense]] a [[magnetism|magnetic field]] would be if the same amount of [[electric current|current]] was passed through it. Permeability is measured in henries per [[metre]] (H/m) and its symbol is <math>\mu</math>.


== Problem 1==
Since [[vacuum|empty space]] has a [[constant]] permeability (called the '''permeability of free space''' or <math>\mu_{0}</math>) of exactly <math>0.0000004 \times \pi</math>, most materials are listed with a ''relative permeability'' (symbol <math>\mu_{r}</math>). Relative permeability is the permeability of the material divided by the permeability of free space (<math>\mu_{r} = \mu / \mu_{0}</math>). The permeability of most materials is very close to 1. That means that the permeability of most materials is close enough that we can typically ignore it and use the permeability of free space instead.<ref>Lines and Fields in Electronic Technology, Stanley and Harrington pg 13</ref> The biggest exceptions are materials called [[ferromagnetism|ferromagnetic materials]]. Some examples are [[iron]] (5000) and [[nickel]] (600). Some materials have been specially designed to have a permeability one million times larger than empty space.<ref>http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/#mur</ref>
假设我们班上有n+2个人，其中两个人是DNA完全相同的双胞胎。我们收上n+2份作业后，将这些作业打乱后发回给全班同学，每人一份。要求每个人不可以收到自己那一份作业或者与自己DNA相同的人的作业。令<math>T_n</math>表示满足这个要求的发回作业的方式，问：
* 计算<math>T_n</math>是多少；
* 在<math>n\to\infty</math>时，随机重排并发回作业后，满足上述要求的概率是多少。


==Problem 2==
== References ==
Let <math>\pi</math> be a permutation of <math>[n]</math>.
{{reflist}}
Recall that a cycle of permutation <math>\pi</math> of length <math>k</math> is a tuple <math>(a_1,a_2,\ldots,a_k)</math> such that <math>a_2=\pi(a_1), a_3=\pi(a_2),\ldots,a_k=\pi(a_{k-1})</math> and <math>a_1=\pi(a_k)\,</math>. Thus a fixed point of <math>\pi</math> is just a cycle of length 1.
* Fix <math>k\ge 1</math>. Let <math>f_k(n)</math> be the number of permutations of <math>[n]</math> having no cycle of length <math>k</math>. Compute this <math>f_k(n)</math> and the limit <math>\lim_{n\rightarrow\infty}\frac{f_k(n)}{n!}</math>.


==Problem 3 ==
[[Category:Electromagnetism]]
Let <math>N_m</math> denote the number of objects from a collection of <math>N</math> objects that possess ''exactly'' <math>m</math> of the properties <math>a_1,a_2,\ldots,a_r</math>. Generalize the principle of inclusion-exclusion by computing <math>N_m</math> as the following form
:<math>N_m=\sum_{k=m}^r(-1)^{k-m}{k\choose m}s_k</math>,
:and please explicitly give the <math>s_k</math>.
 
== Problem 4 ==
你要设计一个标志，以下形状中的12条线段可以分别又红、绿、蓝三色之一构成。要求考虑这个形状的“转动”和“反转”两种对称。
    __
  __|  |__
|__    __|
    |__|
 
*定义这个对称构成的群，可以通过生成元定义，也可以直接把元素都写出来；
*写出cycle index；
*如果要求三种颜色出现的次数一样多，写出这时的pattern inventory；
*如果有四种颜色红、绿、蓝、黄，并要求四种颜色出现的次数一样多，写出这时的pattern inventory；
整个过程中可以借助一些数学软件如Mathematica的帮助。

Latest revision as of 16:23, 28 June 2016

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Permeability is a property of a material that describes how dense a magnetic field would be if the same amount of current was passed through it. Permeability is measured in henries per metre (H/m) and its symbol is [math]\displaystyle{ \mu }[/math].

Since empty space has a constant permeability (called the permeability of free space or [math]\displaystyle{ \mu_{0} }[/math]) of exactly [math]\displaystyle{ 0.0000004 \times \pi }[/math], most materials are listed with a relative permeability (symbol [math]\displaystyle{ \mu_{r} }[/math]). Relative permeability is the permeability of the material divided by the permeability of free space ([math]\displaystyle{ \mu_{r} = \mu / \mu_{0} }[/math]). The permeability of most materials is very close to 1. That means that the permeability of most materials is close enough that we can typically ignore it and use the permeability of free space instead.[1] The biggest exceptions are materials called ferromagnetic materials. Some examples are iron (5000) and nickel (600). Some materials have been specially designed to have a permeability one million times larger than empty space.[2]

References

Template:Reflist

  1. ↑ Lines and Fields in Electronic Technology, Stanley and Harrington pg 13
  2. ↑ http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/#mur
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