L*******g
发帖数: 913 | 1 方程为 dP/dt=-div(V*P)-F*P,
where P(t,x,y,q) and F(x,y,q) are scalar functions, V(x,y,q) is a vector
function, div=d/dx+d/dy, q is a parameter.
我用numerical recipe的Lax格式算这个二维PDE，最后找出dP/dt=0时P的稳恒分布。定
义r=sqrt(x^2+y^2)，我画了一个P(r,q)的图，发现P在q的边界有振动。我的理解是Lax
格式对t和x,y做差分时做一定的平滑，但是没有办法对参数q平滑。有没有人知道如何
解决对参数q的稳定性？谢谢。 |
|
c*******h
发帖数: 1096 | 2 【 以下文字转载自 Mathematics 讨论区 】
发信人: cockroach (冬冬), 信区: Mathematics
标 题: a question on brownian motion
发信站: BBS 未名空间站 (Thu Feb 10 20:22:29 2011, 美东)
Let W(t) be the standard Brownian motion. It is known that the covariance
matrix K has entries K(i,j)=min{i,j}. Now, if t is a vector instead of a
scalar (I even don't know the name of this random process), what does the
covariance matrix look like? |
|
j*******e
发帖数: 586 | 3 can any optimization guru here give me some hints please? I don't know if my
question is just a trivial one in expert's eyes. But here it is:
my objective function f(x) is a scalar function of a vector x of length n. I
know that if x* is a solution, then cx* must also be a solution, where c is
a real constant. This actually reduces the n-dimensional problem to a n-1
dimension one. Is there any difference between the following two strategies:
(1) treat the problem as an unconstrained optimization |
|
h******9
发帖数: 84 | 4 K个正定矩阵, A_k
如何证明 (sum_k A_k^-1 )^-1 \leq sum_k A_k /(K^2) ???
上面这个不定式, scalar的特殊情况就是Cauchy不等式. 对于矩阵形式, 当K=2时候也
可以用svd来证明. 在下不知道对general K怎么证明. 恳请高手指点. 另外, 我已经用
matlab验证了很多把, 这个不等式应该是对的:-) 如果其实不成立, 也请指点下. 多谢
多谢. |
|
a******e
发帖数: 80 | 5 我仿真一个PLL （cadence schematic simulation），里面的VCO的频率取决于它的前
一级的电压。
因为我做的是时域的仿真，所以当我画VCO的输出电压时，得到一个振荡的信号，当然
，频率不是非常固定，
我现在想画出这个振荡信号的频率，可是当我选了Calculator中Special function里的
frequency时，我得到的是一个single value （scalar data），而不是一个随着时间
变化的值。
请问应该怎么做，谢谢。 |
|
X*****r
发帖数: 2521 | 6 著名无赖算法啊
好像scalar 的情况下，nonconvex用branchbound也可以找到global，就是慢点就是了
不知道matrix情况下能不能BB |
|
k**f
发帖数: 372 | 7
variable.
No wonder. Your function can only take a scalar as parameter. Call it with a
vector doesn't work for z(i) = k*x(i).
You will have to run a loop like this to get the intended result:
for x=0:0.1:1
y=value(x);
% now use y for this x value
end |
|
z*****n
发帖数: 7639 | 8 tau is a vector of path delays, each specified in seconds. If you set it
as a scalar 0, then you have only 1 path with delay 0.
pdb is a vector of average path gains, each specified in dB.
Normalizedpathgains is the absolute magnitude of your signal,
if you set it 1, then the total power of all the paths input signal is 1. |
|
d*******d
发帖数: 3382 | 9 这是实际碰到的问题，还是textbook quiz?
给了vector B和scalar roh，除了B(+-*/^...)roh或还能怎么用？
correlation要是为零，两者之间是没什么关系，没什么必要用B表达A了 |
|
a*****m
发帖数: 4745 | 10 Anyone how to deal with this problem?
When I try to specify the NSR matrix in deconvwnr(blurred, psf, NSR) to
perform
deconvolution, according to the help file, NSR can be a scalar or an array of
the
same size as blurred image. However, when i tried to use NSR as an array of
the same
size as blurred image, there poped up an error:
??? Error using ==> unknown
Matrix dimensions must agree.
Error in ==> deconvwnr at 120
Denom = abs(otf(nojunk(:))).^2 + K;
I tried that with my image file and the exa |
|
q*****g
发帖数: 1568 | 11 给你个trivial的例子：f(x)==0, 呵呵。
不过我估计那个不是那样理解的，是把它拆成两维实数空间而言。
根据scalar构成的域不同，同样的集合可以是完全不同的线性空间。
C可以看作是over C的线性空间，那么它是一维的；也可以看作是
over R的VS, 那么是2维的；还可以看作是over Q（有理数）的，
那么它居然就变成一个无穷维线性空间了。这些搞分析的人往往
不大care，这个习惯很不好。 |
|
f******k
发帖数: 297 | 12 来自主题: Mathematics版 - 不等式一题 Let a,b,c be vectors in R^n and e be a positive scalar with e<1.
Suppose we have |a-b| <= e|a| and <= -e|b||c| where |.| is the
Euclidean norm and <.> is the inner product, then prove the following
inequality: <= e|b||c|.
I proved it by some messy calculations, but is there a simple way
to do it? Thanks a lot! |
|
t**g
发帖数: 1164 | 13 谢谢
不过现在手头有一个方程我无法理解
curl(curl(E))-E=F
如果它是定义在2维上的
那么E, F均为2维向量函数
可是curl(curl(E))只是一个1维scalar
那这个方程如何make sense呢？ |
|
c******m
发帖数: 599 | 14 curl再维上面有2个定义
一个是作用在 vector 上 , 一个是作用在 scalar 上 |
|
|
B********e
发帖数: 10014 | 16 他的d应该只是一个scalar，x应该是D维向量 |
|
a*****e
发帖数: 176 | 17 我需要知道 解简单ODE的方法，原来学过的好像都是关于scalar的，有些通解的公式
现在碰到一些ODE方程组，包含vector，matrix,我就晕菜了，
其实都不是很复杂的方程组。
有没有什么这方面好的入门书？
谢谢！！ |
|
a***n
发帖数: 3633 | 18 其实道理很简单。x_dot=Ax如果你把变量重新做一次线性组合，使得新的x_dot=Ax中的
A
变成对角阵。这样就是n个独立的scalar ODE。 然后你在反解回去就可以了。基本
上就是这么回事了。实际运用的时候可以直接解exp(At).关键一点的是，除非At是对角
阵，否则exp(At)并非是把它各个元素放到指数上那么简单。这个计算需要解特征值
所以一般必须需要借助数学软件。 |
|
j******e
发帖数: 64 | 19 一般情况下我用newton raphson (因为只知道这个），
对于f(x), 当x是scalar或者vector的时候效果很好。
但是当x是matrix的时候，dimensionality很高的时候
就不太行了。请问有什么其他的gradient based method
比较好用， for matrix? Thanks. |
|
d****r
发帖数: 135 | 20 independent 的话一定是normal。你可以随便设出两个normal distribution r.v.，然
后利用independence得到两个r.v.的乘积density function，进一步用积分计算他们的
和的density function，你会发现还是一个normal的density function。这个计算不难。
楼上给出的例子不是一个反例。因为nomarl distribution 有一个极限情况，可以被认
为是variance趋向无穷的情况。而那个反例恰好就是这样子的，它的density function
是在0点blow up，其它点值为0，这个在一些讲广义函数的书里被叫做delta 函数。因
为正规normal distribution实际上是热方程在一维情况下，初值为delta函数的基本解
的图像（似乎差个scalar），因此delta 函数恰好就是一个极限状态的normal
distribution。 |
|
h******9
发帖数: 84 | 21 K个正定矩阵, A_k
如何证明 (sum_k A_k^-1 )^-1 \leq sum_k A_k /(K^2) ???
上面这个不定式, scalar的特殊情况就是Cauchy不等式. 对于矩阵形式, 当K=2时候也
可以用svd来证明. 在下不知道对general K怎么证明. 恳请高手指点. 另外, 我已经用
matlab验证了很多把, 这个不等式应该是对的:-) 如果其实不成立, 也请指点下. 多谢
多谢. |
|
s******h
发帖数: 539 | 22 Excellent method for the dude on the 2nd floor, what I want to comment is
how I might come up with this method:
1. In scalar version,
(\sum_i=1^n 1)^2
=[\sum_i=1^n \sqrt{a_i}*(1/\sqrt(a_i))]^2
<=(\sum_i=1^n a_i^2)(\sum_i=1^n 1/a_i^2)
As the author mentioned, it's Cauchy-Schwartz Inequality;
2. Remember how we prove Cauchy-Schwartz Inequality for different
versions, one common method is to consider 2nd moment as follows
\sum_{i=1}^n(\sqrt{a_i}+1/\sqrt{a_i}*x)^2>=0, for all x\in |
|
H****h
发帖数: 1037 | 23 For example, n=1, X is a plane; m=1, M is a line passing through 0.
And we assume that M is the x-axis.
Now you decompose the plane X into horizontal lines, one of which is M.
You view each line as a single point. You can define sum of two lines, which
is gives another horizontal line. Likewise, you have scalar product of a
line.
So you get a linear space, which is called X/M. And M is the "0" of X/M. |
|
h***l
发帖数: 3048 | 24 有一个scalar函数f在[0, \infty)上定义，并且总是正值，
那么从0到\infty对f^2积分，是否总小于或等于积分后
再平方？ |
|
c*******h
发帖数: 1096 | 25 that may be possible. consider the case when A and B are scalars.
the difference between exp(A+B) and (1+B)exp(A) w.r.t. exp(A)
grows exponentially as the magnitude of B grows, even if the ratio
of A and B is kept fixed.
the fact is that first order approximation to exponentials is far
below satisfatory unless the magnitude of the variable is small. |
|
p********y
发帖数: 111 | 26 在看一个作图形分析的论文,有几处没看懂
1,any scalar field can be distorted into any other scaler field
如何理解?
2,为了克服这个困难,论文中用了cut low-density region 方法:
Since the space now has holes in it-precisely because we cut then out and
threw them away-the topology of the remaining space is nontrivial and can be
studied.
我的理解:
For example, the equation x + 5y = 0 has the trivial solution (0, 0).
Nontrivial solutions include (5, –1) and (–2, 0.4). 我们扔掉的low-density
区域就是那些含有trivial solution (0, 0)的区域? 我对这个对应过程转不过弯.
谢谢 |
|
c****w
发帖数: 565 | 27 which one of following is true? A is scalar, a and b are 2nd order tensors. |
|
m********8
发帖数: 123 | 28 搞不懂的是张量？
那是因为写那些书得作者：
1。可能他们自己没有搞懂
2。可能他们思维不严谨
3。可能他们的表达能力有限
4。可能他们不明智：以为跳着写书可以节省读者的时间
其实，再怎么地，张量也比抽象数学简单10倍100倍；张量无非是把几个连续或离散的
数或函数比较好地那么放在一起，操作这些东西的时候还不是：
1。either elementary operation
2。or calculus
为什么要“张量”呢
这个用“力”和“应力”来说明比较明了：
力，是个滑移矢量，这个不错；可惜不是平移矢量，不是旋转矢量；所以，力和坐标有关
应力张量，是个2界张量，却跟坐标无关；这就有利于做很多数学操作；需要从应力张
量获得力的时候，点乘一个方向矢量就行了；
一阶张量，是scalar，跟坐标无关；
二阶张量，是矩阵，跟坐标无关；
高阶张量，是些中间、暂时的产物，跟坐标无关； |
|
e********e
发帖数: 35 | 29 Hi everyone,
I have a series of 2D surfaces over time, and wish to find each pair's
similarities. You may imagine the surface
as a rectangle patch of a sea surface, and wondering how this part of the
sea surface is changed over time, in
terms of a similarity or dissimilarity measure ( a scalar value).
Any suggestions appreciated. Many thanks. |
|
l******e
发帖数: 4 | 30 Consider the conjugation action of GL(n) on the space M(n) of n by n matrices. This defines a representation of GL(n) in M(n) whose kernel is just the scalars in GL(n). |
|
f******h
发帖数: 104 | 31 Thank you. I have one more question:
how can you prove that the induced map
on regular functions
k[GL(n^2)] --> k[GL(n)] is surjective?
I feel that otherwise the adjoint
representation may be not a closed
immersion.(although the image is closed)
Sorry for my stupidity:)
matrices. This defines a representation of GL(n) in M(n) whose kernel is
just the scalars in GL(n). |
|
c******m
发帖数: 41 | 32 Let v be a non-zero column vector in R^(n)( i.e n维列向量空间） and v ^(T) (
i.e v 的转置）be its transpose.
(a) Show that the n*n matrix A defined by v*v^(T) ( i.e v乘以v的转置） is of
rank one and is a diagonalizable matrix.
(b) If u is a unit vector in R^(n), show that the matrix H=I-2u*u^(T) is an
invertible matrix. Here I is the identity matrix. ( Hint: Note that u^(T)*v
is a scalar for any column vector v. You may show that the null space of H
is trivial)
(c) If v and w are any two nonzero orthogonal vecto |
|
r*******y
发帖数: 1081 | 33 But it is not the way I want. I just define a scalar function of two
variables.
So no way to apply surf to this kind of defined funciton? Thanks. |
|
p**u
发帖数: 55 | 34 请各位数学系的朋友帮忙看看这个
现有一个n维实数空间中的向量场，\dot(x)=f(x)，
还有同一空间中的一个平滑参数曲线 x=g(r). r 是一个real scalar
问题是这样的，如果在曲线上任意一点x，曲线的切线都和f(x)平行，也就是说，存在
一个实数k(x)， 使得x点处的曲线切向量T(x)=k(x)*g(r), 那么从x点处出发在\dot(x)
=f(x) 作用下的轨迹是否与g(r)重合？
感觉似乎是重合的，但还没有想出怎么证明，该如何证这个？关于这个问题是否有现成
的定理？有什么参考文献吗？ 肯定有人做过这个，但我孤陋寡闻，不知道，要是能有
现成的参考文献就好了 |
|
c*******h
发帖数: 1096 | 35 Let W(t) be the standard Brownian motion. It is known that the covariance
matrix K has entries K(i,j)=min{i,j}. Now, if t is a vector instead of a
scalar (I even don't know the name of this random process), what does the
covariance matrix look like? |
|
l********e
发帖数: 3632 | 36 对应叉乘的那个？
inner product
or
scalar product |
|
s*****e
发帖数: 115 | 37 Still not able to see the point of the question. For the boundary value problem you have, associated with each eigenvalue there are infinitely many eigenfunctions, one differing from another just by a scalar multiple. So for each $\lambda$ you can always pick $g_{\lambda}$ such that its sup-norm is exactly 1, unless you have to pick $g_{\lambda}$ in curtain special way that is not described here. |
|
l******r
发帖数: 18699 | 38 对了，这些eigenfunctions都是normalized,使得它们的L2-norm都是1.
在这种normalization之下，是否可以证明它们一致有界？请看一楼更正，谢谢。
当然不同的eigenvalue对应的eigenfunctions都是正交的，所以它们构成L2[0,1]空间
里的标准正交基。只不过eigenfunctions的Sobolev norm不是1，而是一个依赖于特征
值的常数，而且没有上界。
不好意思，这个问题一开始设计的不严谨。这貌似是微分方程里很重要的问题，可
是文献里却找不到。谢谢热心帮助！
problem you have, associated with each eigenvalue there are infinitely many
eigenfunctions, one differing from another just by a scalar multiple. So for
each $\lambda$ you can always pick $g_{\lambda}$ such that its sup-norm is
exactl... 阅读全帖 |
|
g****a
发帖数: 1520 | 39 田肯定是抄了。田是听说了Donaldson他们已经接近于证明Yau猜想，才突然杀出来的，
Donaldson与合作者近几年来一直在推进这个问题，try了各种不同方法。田根本毫无贡
献，上一次有贡献是1997年那篇positive scalar curvature的文章。
如果搞一个committee来评判一下，田肯定完蛋。其实Donaldson都已经鉴定过了。
10 |
|
g****a
发帖数: 1520 | 40 田肯定是抄了。田是听说了Donaldson他们已经接近于证明Yau猜想，才突然杀出来的，
Donaldson与合作者近几年来一直在推进这个问题，try了各种不同方法。田根本毫无贡
献，上一次有贡献是1997年那篇positive scalar curvature的文章。
如果搞一个committee来评判一下，田肯定完蛋。其实Donaldson都已经鉴定过了。
10 |
|
q********e
发帖数: 1255 | 41 Murakami, Satoru(J-OKSC-AM)
Exponential asymptotic stability for scalar linear Volterra equations.
Differential Integral Equations 4 (1991), no. 3, 519–525.
d**********[email protected]
3x a lot! |
|
L*m
发帖数: 235 | 42 最近十年在annals of mathematics上发表或合作发表文章的华人全统计（不包括
terrence tao和一位mit本科毕业的abc华人），单位统计以现在作者单位为准
annals
2015年
A proof of Demailly’s strong openness conjecture
关启安（北京大学） 周向宇（中科院）
A solution of an L2 extension problem with an optimal estimate and
applications
关启安（北京大学） 周向宇（中科院）
Finsler metrics and Kobayashi hyperbolicity of the moduli spaces of
canonically polarized manifolds
杨世琪（普渡大学） Wing-Keung To（新加坡国立大学）
Construction of Cauchy data of vacuum Einstein field equations evolving to
black holes
黎俊彬（中山大学）... 阅读全帖 |
|
L*m
发帖数: 235 | 43 统计了近十余年来中国大陆高校在四大刊物上的发文，有些是挂名的，但不管如何，还
是都统计了。全名单如下
Annals of Mathematics
A proof of Demailly’s strong openness conjecture
Qi'an Guan（关启安 北京大学） Xiangyu Zhou（周向宇 中科院）
A solution of an L2 extension problem with an optimal estimate and
applications
Qi'an Guan（关启安 北京大学） Xiangyu Zhou（周向宇 中科院）
Construction of Cauchy data of vacuum Einstein field equations evolving to
black holes
Junbin Li（黎俊彬 中山大学） Pin Yu（于品 清华大学）
Special test configuration and K-stability of Fano varieties
Chi Li(李驰 普林斯顿大学 现stony broo... 阅读全帖 |
|
x********i
发帖数: 905 | 44 2016 Breakthrough Prize in Mathematics
The Breakthrough Prize in Mathematics honors the world’s best
mathematicians who have contributed to major advances in the field.
Ian Agol, University of California at Berkeley and Institute for Advanced
Study: for spectacular contributions to low dimensional topology and
geometric group theory, including work on the solutions of the tameness,
virtual Haken and virtual fibering conjectures.
2016 New Horizons in Mathematics Prize
The New Horizons in Mathemat... 阅读全帖 |
|
a*****m
发帖数: 4745 | 45 Can anyone help me with this problem?
When I try to specify the NSR matrix in deconvwnr(blurred, psf, NSR) to
perform
deconvolution, according to the help file, NSR can be a scalar or an array of
the
same size as blurred image. However, when i tried to use NSR as an array of
the same
size as blurred image, there poped up an error:
??? Error using ==> unknown
Matrix dimensions must agree.
Error in ==> deconvwnr at 120
Denom = abs(otf(nojunk(:))).^2 + K;
I tried that with my image file and the exa |
|
s***h
发帖数: 592 | 46 你说的那个分解不是唯一的，
只不过分解成静水部分和偏量是最简单明了的
a |
|
s***h
发帖数: 592 | 47 不知道你说的分量平方和是不是指张量的度量
如果是的话，那是与坐标系无关的
a |
|
m**********e
发帖数: 12525 | 48 这里的标量是不变量,要求与具体坐标描述无关，也就是说，对具体某一坐标
描述下的张量uij作坐标变换，都必须保持标量的值比不变。用矩阵的语言来
说，就是对uij做相似变换，产生标量值不变，你说，有多少种结果？
还不明白的话，找本线性代数课本（或者群论课本）看一看
a |
|
q*d
发帖数: 22178 | 49 你老说了我老想说的
开的
be
formed
as
出2个 |
|
n********s
发帖数: 150 | 50 I am reading a paper "Class of scalar-field soliton solutions in three space
dimensions" (Phys.Rew.D 13 2739) by T.D.Lee. When the authors try to
quantize the soliton, they said "....Q is the total charge operator. Because
x is a cyclic variable with a period = 2 pi, the eigenvalue of its
conjugate momentum Q are all integers: 0, 1,2....".
I cannot understand this point. What does it means "cyclic variable"? why
its eigenvalue must be integers?
Thanks a lot. |
|
