D**u
发帖数: 204 | 1 A guy starts walking from position x_0 on earth (we treat earth as a sphere
with radius R). He randomly chooses a direction and walk 1 meter (this is
the spherical distance) and reaches position x_1. He repeats the walk n
times and arrives at position x_n.
Question:
(1) what is E((x_n - x_0)^2)? (here ((x_n - x_0)^2 is the square of the
Euclidean distance, not the spherical distance).
(2) does E((x_n - x_0)^2) converge when n --> \infty? |
|
BR
发帖数: 4151 | 2 能详细说说吗？在euclidean space 里面我能理解wchwarz inequality，但是rudin 的
书里shwarz inequality 也是用在复数上面的。
具体的，shwarz inequality 说 |sum_{1 to n} a_j b_j|^2 小于等于 sum_{1 to n}|a_j|^2 sum_{1 to n}|b_j|^2。a_j 和b_j 都是复数。如果n=1, 这不就等同于|zw|=|z||w| 吗？
谢谢指点。 |
|
BR
发帖数: 4151 | 3 对了，shwarz inequality 里面写的确实是a_j 乘以 \bar{b_j}。我其实不太明白为什
么要用这个b_j 的conjugate.
在复数域实际上相当于一个二维的euclidean space， 对不对？可是我看见说等号只限
于两个矢量proportional 的情况。
confused。 请谅解。 |
|
l******o
发帖数: 1550 | 4 the inner product is defined in that way.
the generalized inner product requires the conjugate under commute
对了，shwarz inequality 里面写的确实是a_j 乘以 \bar{b_j}。我其实不太明白为什
么要用这个b_j 的conjugate.
在复数域实际上相当于一个二维的euclidean space， 对不对？可是我看见说等号只限
于两个矢量proportional 的情况。
confused。 请谅解。 |
|
D**u
发帖数: 204 | 5 【 以下文字转载自 Quant 讨论区 】
发信人: DuGu (火工头陀), 信区: Quant
标 题: Partition R^3 into a union of circles
发信站: BBS 未名空间站 (Fri Dec 4 13:30:22 2009, 美东)
Question:
Can you partition a 3-dim Euclidean space R^3 into a union of a set of
circles. Namely, for every point A in R^3, A is on 1 and only 1 circle from
the set. |
|
m********t
发帖数: 80 | 6 In Euclidean space using only straight edge and compass, to place a straight
line equal to a given straight line with one end at a given point.
谢谢大牛的帮忙
埃 大学应该学数学专业的
一失足成千古恨
谢谢大家！ |
|
|
n***p
发帖数: 7668 | 8 The answer is no. Here is an counterexample.
Consider the 2D Euclidean space and let
X =(2,1) and E=(-a, 2a). Then X is perpendicular to E.
Then Y=(2-a, 1+2a). For p=4,
|Y|_4^4 = (2-a)^4 + (1+2a)^4
= 17 - 24a + C_1 a^2 +C_2 a^3 +C_3 a^4
<17 = |X|_4^4
if we choose a>0 to be small enough.
The same idea can be applied to construct counterexamples
for other spaces like L_p function spaces for p=4. I choose
p=4 because I can easily calculate the norms explicitly.
For other values of p, we can use tay... 阅读全帖 |
|
g****a
发帖数: 1520 | 9 【 以下文字转载自 WaterWorld 讨论区 】
发信人: xiaoshushu (songshu), 信区: WaterWorld
标 题: 关于近期Fano流形上构造KE度量的工作(转载)
发信站: BBS 未名空间站 (Sat Aug 31 10:30:12 2013, 美东)
关于近期Fano流形上构造Kähler-Einstein度量的工作
最近公布的Fano流形上构造Kähler-Einstein度量的工作，是Kähler几
何近年来引人注目的进展，专家们正在验证。若验查无误，将证明丘成桐关于Fano流形
的构想与猜测是正确的。Donaldson的稳定性条件是其中的关键步骤，还需在代数几何
上把此概念搞清楚，这样丘猜测就为深刻理解Fano流形奠定了基础。由于近期发生了一
些混淆不清的事件，我们将相关工作的公开记录做了客观、学术的分析，望有助于澄清
事实。本文主要涉及文献的比较，阅读本文无需是专家，数学专业本科高年级学生或研
究生可读懂绝大部分。欢迎关于数学上的批评与指正。
本文分三个部分：
1) 陈-Don... 阅读全帖 |
|
g****a
发帖数: 1520 | 10 【 以下文字转载自 WaterWorld 讨论区 】
发信人: xiaoshushu (songshu), 信区: WaterWorld
标 题: 关于近期Fano流形上构造KE度量的工作(转载)
发信站: BBS 未名空间站 (Sat Aug 31 10:30:12 2013, 美东)
关于近期Fano流形上构造Kähler-Einstein度量的工作
最近公布的Fano流形上构造Kähler-Einstein度量的工作，是Kähler几
何近年来引人注目的进展，专家们正在验证。若验查无误，将证明丘成桐关于Fano流形
的构想与猜测是正确的。Donaldson的稳定性条件是其中的关键步骤，还需在代数几何
上把此概念搞清楚，这样丘猜测就为深刻理解Fano流形奠定了基础。由于近期发生了一
些混淆不清的事件，我们将相关工作的公开记录做了客观、学术的分析，望有助于澄清
事实。本文主要涉及文献的比较，阅读本文无需是专家，数学专业本科高年级学生或研
究生可读懂绝大部分。欢迎关于数学上的批评与指正。
本文分三个部分：
1) 陈-Don... 阅读全帖 |
|
f**********d
发帖数: 4960 | 11 d维euclidean空间的矢量x1=[n1,n2,...,nd], x2=[m1,m2,...,md]。
问题是比较它们的长度。
条件是不能直接计算||.||_2。因为复杂度太高O(d)。
我在考虑能否从n1-m1, n2-m2,..., nd-md从中推得。因为问题是比较x1,x2的长度。只
需要比较就行了，并不需要知道它们确切的大小。 |
|
w**********5
发帖数: 1741 | 12 把大象装进冰箱里！
一 数学家的办法
把大象放到冰箱里的分析学方法
1）先把大象微分，然后把它放到冰箱里，再在冰箱里把它积分。
2）重新定义冰箱或者大象的测度（如Radon测度）。
3）用Banach-Tarski定理。
把大象放到冰箱里的代数学方法
1）先证明大象的每一部分都可以放到冰箱里。
2）再证明冰箱对加法封闭。
把大象放到冰箱里的拓扑学方法
1）让大象把冰箱吞了，再把冰箱从里到外翻出来。
2）把冰箱做成Klein瓶。
把大象放到冰箱里的线性代数方法
1）把大象的基先放进去，再在冰箱里张成空间。
2）把大象作奇异值分解，去掉大于冰箱维数所对应的特征值，由剩余特征向量即可恢
复出一头可以放进冰箱之中的大象。
把大象放到冰箱里的集合论方法
1）冰箱 = {大象}
2）大象和冰箱的内部有相同的势。
把大象放到冰箱里的复分析方法
1）把冰箱放在原点，大象放在单位圆之外，作反演变换。
把大象放到冰箱里的数值计算方法
1）把大象的尾巴放进去，剩余部分当作余项处理。
2）用最快的Pentium解决这个问题。
把大象放到冰箱里的统计学方法
1）取大象的尾巴作样本。
糟糕的统计学方法
1）不停地... 阅读全帖 |
|
w**********5
发帖数: 1741 | 13 把大象装进冰箱里！
一 数学家的办法
把大象放到冰箱里的分析学方法
1）先把大象微分，然后把它放到冰箱里，再在冰箱里把它积分。
2）重新定义冰箱或者大象的测度（如Radon测度）。
3）用Banach-Tarski定理。
把大象放到冰箱里的代数学方法
1）先证明大象的每一部分都可以放到冰箱里。
2）再证明冰箱对加法封闭。
把大象放到冰箱里的拓扑学方法
1）让大象把冰箱吞了，再把冰箱从里到外翻出来。
2）把冰箱做成Klein瓶。
把大象放到冰箱里的线性代数方法
1）把大象的基先放进去，再在冰箱里张成空间。
2）把大象作奇异值分解，去掉大于冰箱维数所对应的特征值，由剩余特征向量即可恢
复出一头可以放进冰箱之中的大象。
把大象放到冰箱里的集合论方法
1）冰箱 = {大象}
2）大象和冰箱的内部有相同的势。
把大象放到冰箱里的复分析方法
1）把冰箱放在原点，大象放在单位圆之外，作反演变换。
把大象放到冰箱里的数值计算方法
1）把大象的尾巴放进去，剩余部分当作余项处理。
2）用最快的Pentium解决这个问题。
把大象放到冰箱里的统计学方法
1）取大象的尾巴作样本。
糟糕的统计学方法
1）不停地... 阅读全帖 |
|
c****e
发帖数: 2097 | 14 Ising model
very important, but equally annoying
point approximation.
unless in euclidean signature, the phase is just a phase. |
|
m********e
发帖数: 1156 | 15 我还算有知的，无论深度还是广度，奈何世界上的知识太多。
有人贴出老爱的几百篇论文，我一看大多数是德文，不知道，无法判断。
您瞅一眼，看看有多少是真正的科研论文？
1913 Einige Argumente für die Annahme einer molekular Agitation beim
absoluten Nullpunkt Annalen der Physik(ser. 4), 40, 551–560, link
Some Arguments for the Assumption of Molecular Agitation at Absolute
Zero
1913 Déduction thermodynamique de la loi de l'équivalence
photochimique Journal de physique (ser. 5), 3, 277–282
Thermodynamic Deduction of the Law of Photoche... 阅读全帖 |
|
a*******a
发帖数: 1240 | 16 Before embarking on a metaphysical critique of the famed "theory of general
relativity" (GR) amidst the fanfare of the recent observation by LIGO of
gravitational wave, an explorative discussion of mathematics is both helpful
and necessary.
In any theoretical investigations, mathematics plays a pivotal role. The
natural question arising herein is then, what is mathematics? And what kind
of roles can we possibly expect of mathematics in any theoretical enquiries
(i.e., by deductions) or in a prac... 阅读全帖 |
|
|
k***g
发帖数: 7244 | 18 nod, 其实就是指的 the set of feasible alternatives is some subset of the mul
ti-dimentional Euclidean space, R^k. 主要是 Spatial Model需要用这个（这个Spa
tial Model在经济学里称 Hotelling Model，政治学里称 Downsian Model，但是最早
是
Hotelling 发明的），而 Spatial Model是 Positive Political Theory 的基石之一
，通常都是假设 k=1 也就是一维的空间，k>1 时 在很多情况下 Preference 都是 循
环
的。
closer
实
到
重 |
|
b***k
发帖数: 2673 | 19 ☆─────────────────────────────────────☆
ValueBet (2009) 于 (Sat Jan 10 22:29:46 2009) 提到:
遇到一个面试题。请各位大侠赐教。
In the Euclidean plane are given a circle and a square that are disjoint but
have equal areas. Three points C1, C2, C3 are randomly chosen in the circle
, and three points S1, S2, S3 are randomly chosen in the square. Find the
probability that m(C1,C2,C3) > m(S1,S2,S3).
☆─────────────────────────────────────☆
daj (肉丝炒饭--小吵肉fan) 于 (Sat Jan 10 22:38:28 2009) 提到:
what is m(x,y,z) ?
b |
|
D**u
发帖数: 204 | 20 【 以下文字转载自 Mathematics 讨论区 】
发信人: DuGu (火工头陀), 信区: Mathematics
标 题: spherical random walk
发信站: BBS 未名空间站 (Wed Nov 4 15:15:16 2009, 美东)
A guy starts walking from position x_0 on earth (we treat earth as a sphere
with radius R). He randomly chooses a direction and walk 1 meter (this is
the spherical distance) and reaches position x_1. He repeats the walk n
times and arrives at position x_n.
Question:
(1) what is E((x_n - x_0)^2)? (here ((x_n - x_0)^2 is the square of the
Euclidean distance, not t |
|
D**u
发帖数: 204 | 21 Question:
Can you partition a 3-dim Euclidean space R^3 into a union of a set of
circles. Namely, for every point A in R^3, A is on 1 and only 1 circle from
the set. |
|
p*****k
发帖数: 318 | 22 note 11 and 60 are co-prime, so the general solution is:
X = 121 (mod 660), or 660*m+121, where m is a non-negative integer.
but i think OP actually meant X modulo all numbers from 2 to 10
results 1, i.e., X=1(mod 2520) and X=0(mod 11).
again 11 and 2520 are coprime. by Euclidean algorithm,
the general solution is: X = 25201 (mod 27720) |
|
m********t
发帖数: 80 | 23 In Euclidean space using only straight edge and compass, to place a straight
line equal to a given straight line with one end at a given point.
完全没看懂，虚心向大家请教。谢谢大家的帮助！ |
|
D**u
发帖数: 204 | 24 Randomly (also uniformly) choose 3 points A,B,C on a 2-dimensional sphere.
Let x be the spherical area of the spherical triangle ABC;
and y be the volume of the (Euclidean) 四面体 formed by A,B,C and O
(the center of the sphere).
If we treat x and y as 2 random varibles,
then what is the correlation between x and y? |
|
r**a
发帖数: 536 | 25
time
You missed the point. If Hagan or Labordere's method is only small time
expansion, how can you imagine the formulas can be applied to 30Y swaption?
You should read paper and books more carefully and think more deeply. Hagan'
s method is operator perturbation. But you need to find what operator he is
perturbing and why the perturbation works.
The heat kernel expansion heavily depends on what heat equation you are
talking about. Based on your words, I may naively guess that you are talking
a... 阅读全帖 |
|
w**********5
发帖数: 1741 | 26 职位是model vetting
1 求 Bs d(Bs)的伊藤积分 （挂了）
-逆向运用伊藤公式就行
2 Banach, Hilbert, Euclidean空间那个最一般 （实变）
3 implied volatilty曲面怎么用离散点构造？需要什么constraint?
-瞎答一气。应该是 no arbitrage
4 外汇GK模型中的f_r对应BS模型的啥？
-dividend |
|
h***u
发帖数: 25 | 27 First of all, I don't think the AdS/CFT is a theory of the Universe.
But we can use the GR and SuperGravity(SUGRA) on any kind of Einstein
maniford.
AdS can be discribed by
-X_{-1}^2-X_0^2+X_i X^i=0 in a D+1 Euclidean space( with 2 time directions
and D-1 space directions). The metric of it can be induced from the flat
metric of R^{D-1,2}.
AdS/CFT correspondence said, The String theory (or is low energy limit: SUGRA)
is correlate to some CFT on the "boundary" of the AdS.
Although there is no pro |
|
l**n
发帖数: 67 | 28 We'd better keep a clear mind even we are facing numerous jargons
in physics and maths. A few comment on what we are talking
about in comso.8 could help.
Geometry analogy
Let's look at a point P on the flat plane in the 3-D Euclidean
space. And we just consider the local property of the P.
First we notice the distance relation is given by the familiar
law of Pythagoras,
ds^2 = dx^2 + dy^2 (1)
where x, y are our local Cartesian coordinates. Then we find we
can map the plane |
|
y**t
发帖数: 50 | 29 usually,the norm in a vector space(euclidean norm)
means the sqrt of the sum of squares of components
and the norm of a matrix is defined as the Sup_{|v|!=0}
(|Av|/|v|)and this is in fact the eigenvalue which has the largest absolute
value among all evs.so if given 2 vectors a and b, a
means each component of a is less than b,then what you
stated is true.I guess the necessary condition is
very dependent on the vector v. |
|
H****h
发帖数: 1037 | 30 如果有相交的两个线段。四个端点组成凸四边形。
两线段是该四边形的对角线。它们的长度和一定
大于任一对边的长度和(三角不等式)。用其中一
对对边替换这两条对角线，使整体路径连通。
你也无需假设所有点组成凸多边形，因为上述推
理中的凸四边形来源于相交线段。 |
|
F******n
发帖数: 160 | 31 【 以下文字转载自 Mathematics 讨论区 】
发信人: Feynmann (DG), 信区: Mathematics
标 题: A question about the distance measure of two matrices
发信站: BBS 未名空间站 (Sat Jan 6 16:07:01 2007)
Hello all,
I am looking for the references on the distance measure of two matrices,
both of which are positive definite and symmetric. For the unknown space (e
.g., non-Euclidean), I heard that the general mathematical treatment is
somehow related to Riemannian manifolds, and the problem could be further
formulated and boiled down |
|
s******h
发帖数: 539 | 32 Good question. I agree with you that least favorable prior is to compare
priors on 'Proper Priors', i.e. the priors are probability distributions.
It's not we don't want to compare all 'priors including improper ones',
while I think it's very hard for us to compare them, for if we consider all
'priors+improper priors', 'least favorable' may not be that useful or
meaningful.
Consider
parameter space: \Theta={0,1}
In this case the risk set is a convex set on 2-dim space (Euclidean), say S.
For any |
|
s*****n
发帖数: 2174 | 33 The reason why many people have this confusion is the definition of
similarity. Actually, correlation is never is measurement of similarity, but
a measurement of (linear) relationship. In other words, it measures the
similarity of change of X1 and change of X2, we can call it "relative
similarity".
Other things, such as Euclidean distance, could be a measure of similarity (
or "absolute similarity" in my words.) |
|
w*****g
发帖数: 798 | 34 有什么好的package?
我要做的事情是 min|Ax-Y| over x\in {0,1,2,...,n}^p
where A is a p by p matrix， Y is a p-dimentional vector and |.| is any type
of norm. Typically the choice of Euclidean norm reduces the problem to
integer quadratic programming.
我知道有个package叫Rcplex,但是安装不上。大家知道有什么办法解决这个问题，或者
别的package,或者怎么安装这个package.
多谢。 |
|
A*******s
发帖数: 3942 | 35 I recently read the book because i need to write some internal technical
reviews for missing data handling and reject inference. Mr. Siddiqi covers
some standard industry approaches but he usually gives incorrect
explanations and interpretations of those methods. And he has confusion
about some statistical terminologies.
Some examples I can remember:
1. whenever he talks about Euclidean distance, it should be Mahalanobis;
2. whenever he says nearest neighbor, he means actually discriminant
analy... 阅读全帖 |
|
I*****a
发帖数: 5425 | 36 u can search for some often used distance measures for clustering. wikipedia
should be enough for this. for instance, clusyering by euclidean distance,
or that after standardization, or correlation/pattern are different, but all
valid. |
|
h**t
发帖数: 1678 | 37 k-means is baed on Euclidean distance calculations. So what ever the data is
, it still calculates the distance. |
|
h***x
发帖数: 586 | 38 1) As catforfish said, the data point is not necessary to be a scalar, a
vector is fine. All my work on clustering are for multi dimension instead of
one dimension. I suggest you spending some time to learn clustering first.
2)In your example, v1 and v2 has strong correlation. If you want to take
this into account in clustering, you should not use euclidean distance as
the statistical measure, you can use other measures with the features you
like for your task.
3)For clustering, result explanati... 阅读全帖 |
|
a******1
发帖数: 201 | 39 He does not have 10,000 data point from one sample, what he has is actually
one data point of 10,000 dimensions. Although I said that he is the only one
who can determine the "correlation coefficient" he calculated is the way to
characterize the distribution of his data. I have come to the conclusion
that his "correlation coefficient" does not mean much, or he just
misunderstood the concept of correlation coefficient. As a simple example
for him to understand, let's say we have two people, and w... 阅读全帖 |
|
t**********y
发帖数: 374 | 40 The data were from RNAseq. The numbers are the gene counts (each gene has
one specific count in a specific sample)
I feel that correlation coefficient seem to be acceptable in term of
evaluating sample reproducibilities even though we all know the pitfalls.
I am interested in what you mentioned: euclidean distance. Could you please
know how to test those distances? Or any paper for recommendation?
Thanks a lot.
assay
are
research
those |
|
g**a
发帖数: 2129 | 41 If it is RNA-seq data, all we talked about won't make any sense. RNA-seq is
count data. Euclidean distance, Pearson regression or any multivariate
methods are all based on continuous variable. Don't use them for RNA-seq. If
you can clarify what is the biological question you want to answer, I may
be able to help.
please |
|
t**********y
发帖数: 374 | 42 euclidean distance 也不行吗？
is
If |
|
w*****n
发帖数: 375 | 43 也说一下为什么需要normalization, 对什么normalization |
|
i**********n
发帖数: 217 | 44 what is your purpose?
You can either normalized variables first,
or use cos like distance. |
|
|
w**********5
发帖数: 1741 | 46 把大象装进冰箱里！
一 数学家的办法
把大象放到冰箱里的分析学方法
1）先把大象微分，然后把它放到冰箱里，再在冰箱里把它积分。
2）重新定义冰箱或者大象的测度（如Radon测度）。
3）用Banach-Tarski定理。
把大象放到冰箱里的代数学方法
1）先证明大象的每一部分都可以放到冰箱里。
2）再证明冰箱对加法封闭。
把大象放到冰箱里的拓扑学方法
1）让大象把冰箱吞了，再把冰箱从里到外翻出来。
2）把冰箱做成Klein瓶。
把大象放到冰箱里的线性代数方法
1）把大象的基先放进去，再在冰箱里张成空间。
2）把大象作奇异值分解，去掉大于冰箱维数所对应的特征值，由剩余特征向量即可恢
复出一头可以放进冰箱之中的大象。
把大象放到冰箱里的集合论方法
1）冰箱 = {大象}
2）大象和冰箱的内部有相同的势。
把大象放到冰箱里的复分析方法
1）把冰箱放在原点，大象放在单位圆之外，作反演变换。
把大象放到冰箱里的数值计算方法
1）把大象的尾巴放进去，剩余部分当作余项处理。
2）用最快的Pentium解决这个问题。
把大象放到冰箱里的统计学方法
1）取大象的尾巴作样本。
糟糕的统计学方法
1）不停地... 阅读全帖 |
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w**********5
发帖数: 1741 | 47 把大象装进冰箱里！
一 数学家的办法
把大象放到冰箱里的分析学方法
1）先把大象微分，然后把它放到冰箱里，再在冰箱里把它积分。
2）重新定义冰箱或者大象的测度（如Radon测度）。
3）用Banach-Tarski定理。
把大象放到冰箱里的代数学方法
1）先证明大象的每一部分都可以放到冰箱里。
2）再证明冰箱对加法封闭。
把大象放到冰箱里的拓扑学方法
1）让大象把冰箱吞了，再把冰箱从里到外翻出来。
2）把冰箱做成Klein瓶。
把大象放到冰箱里的线性代数方法
1）把大象的基先放进去，再在冰箱里张成空间。
2）把大象作奇异值分解，去掉大于冰箱维数所对应的特征值，由剩余特征向量即可恢
复出一头可以放进冰箱之中的大象。
把大象放到冰箱里的集合论方法
1）冰箱 = {大象}
2）大象和冰箱的内部有相同的势。
把大象放到冰箱里的复分析方法
1）把冰箱放在原点，大象放在单位圆之外，作反演变换。
把大象放到冰箱里的数值计算方法
1）把大象的尾巴放进去，剩余部分当作余项处理。
2）用最快的Pentium解决这个问题。
把大象放到冰箱里的统计学方法
1）取大象的尾巴作样本。
糟糕的统计学方法
1）不停地... 阅读全帖 |
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c********1
发帖数: 60 | 48 On top of my head，要实现这个功能并没有现成的package或者function可以调用。
如果要自己写代码来实现的话感觉还颇有难度。如果采用euclidean distance to
measure similarity, the distance would be dominated by the distance of
numerical covariates. 换句话说categorical covariates is somewhat ignored in
the similarity metrics.
谷歌了下，关于similarity metrics of categorical variables没有简单现成的答案
，基本都是paper。不知版上诸多大牛是否有好的解决方法？ |
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d******e
发帖数: 7844 | 49 我说的和manifold learning还是差个十万八千里的... ...
而且对于所有依赖于使用Euclidean distance和Local Linearity来做的manifold
learning方法来说，Curse of dimensionality都无法避免。 |
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l*******s
发帖数: 1258 | 50 有不少方法，比如Euclidean distance，比如Cosine，比如Kullback–Leibler
建议用cosine，各种结果表明，是个不错的方法
另外可以试试KL，虽然是个asymmetric，但是不妨试试。 |
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