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H: How many positive integers $n$ satisfy $n = P(n) + S(n)$
Let $P(n)$ denote the product of digits of $n$ and let $S(n)$ denote the sum of digits of $n$. Then how many positive integers $n$ satisfy
$$
n = P(n) + S(n)
$$
I think I solved it, but I need your input.
I first assumed that $n$ is a two digit number. The... |
H: Which mathematical tool or method should I use to compare two matrices most efficiently?
I have two matrices(the first one is mxm, while the second one is nxn, m>n). They store data pertaining to human speech. The second matrix contains a data segment that acts like an acoustic "signature". I need to find where thi... |
H: Meaning of symbol $\mathcal{P}$ in set theory article?
I am teaching myself real analysis, and in this particular set of lecture notes, the introductory chapter on set theory when explaining that not all sets are countable, states as follows:
If $S$ is a set, $\operatorname{card}(S) < \operatorname{card}(\mathcal{... |
H: Surface area element of an ellipsoid
I would like to evaluate an integral numerically over the surface of an ellipsoid. Take an $N \times N$ grid over the parameter space $(u, v) \in [0, 2\pi) \times [0, \pi) $. A simple approximation of the integral is
$$
\int_0^{2\pi}\int_0^\pi \mathbf{f}(u,v)\,dA \simeq \sum_{i,... |
H: Determine whether the function is a linear transformation:
Determine whether the function is a linear transformation. Justify your answer.
$T: P_2 → P_2$, where
$T(a_0+a_1x+a_2x^2) = a_0 + a_1(x+1) + a_2(x+1)^2$
My thought process on solving this:
I know that in order to be a linear transformation, the following 2... |
H: $A/ I \otimes_A A/J \cong A/(I+J)$
For commutative ring with unit $A$, ideals $I, J$ it holds
$$A/ I \otimes_A A/J \cong A/(I+J).$$
A proof can be found here (Problem 10.4.16) for example.
However, I'd be interested in a less explicit proof of this fact. Does anybody know a nice way to do this, for example with th... |
H: The completeness of the real numbers with respects to Cauchy sequences?
My Question: I am not sure about the very last inequality in the proof below; namely, where did we get
$\mid a_{n}-a_{N}\mid$ and $\mid a_{N}-b\mid$? I see that $\mid a_{n}-a_{N}\mid<\epsilon/2$ and that $\mid a_{N}-b\mid\leqslant\epsilon/2$,... |
H: Computing dimension over a field of rationals
I am looking to find the dimension the vector space $V$ over $\Bbb Q$, the field of rationals, where the vectors are real numbers of the form
$p + q\sqrt 2$, where $p$ and $q$ are rationals.
I'm thinking that the dimension is infinite, but having trouble with proving ... |
H: Are spinors, at least mathematically, representations of the universal cover of a lie group, that do not descend to the group?
Following on this question about how to characterise Spinors mathematically:
First, given a universal cover $\pi:G' \rightarrow G$ of a lie group $G$, is it correct to say we can always lif... |
H: $f:\mathbb{R} \to \mathbb{R}$ continuous, locally 1-1 implies $f$ globally 1- 1
Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous, and locally 1-1. I want to show it is globally 1-1 (without assuming the existence of $f'$).
The intermediate value theorem implies that $f$ is locally strictly monotonic. Intuitive... |
H: Check the convergence of $ \sum_{n=1}^{\infty} \ln(x)^n$
I`m trying to check the domain of $R$ for
$$ \sum_{n=1}^{\infty} \ln(x)^n$$
so what I did is to take $a_n$ and he is $1$ so $\rightarrow -1<x<1$
$$ -1<\ln(x)<1 \longrightarrow \frac{1}{e} < x < e$$ now lets check the left and right sides
for $e$ its Divergi... |
H: Topics of Group Theory Required to Understand Betti Numbers
I am studying Group Theory. I made sure I have a problem at hand, as part of the motivation for my study. I have chosen the problem as being able to understand as well as compute Betti Numbers where the complex is just a simplicial complex.
Now, I have st... |
H: Non-Locally Integrable fundamental solutions
Given a Linear pde $L$, a distribution $u$ is said to be a fundamental solution if $Lu=\delta$ where $\delta$ is the Dirac delta distribution. A common example is the Newtonian potential which is the fundamental solution of the Laplacian. Are there any examples of differ... |
H: Problem with simple laplacian equation
I would like to solve the following PDE:
$$
\partial_x^2 u + \partial_y^2 u = -\frac{2 x^2 (x^2-y^2)}{\left(x^2+y^2\right)^2}
$$
The right side comes from $ x^2 \partial_x^2 \log(x^2 +y^2) $. Switching the polar coordinates, the right side is deceptively simple:
$$
-2 \cos(\t... |
H: Extract and then insert back a sub-block of a picture using matlab
I have a picture as an input to matlab $(256\times256)$:
img=imread('cameraman.tif');
I want to embed a watermark using a spatial domain method. For this purpose i choose a random sub-block of the image and make my calculations:
%random selection... |
H: What are the two 'sides' of a decimal number called?
Is there a fancy name for the "left side" and "right side" of a decimal number?
(That is, the pre-decimal part and the post-decimal part.)
AI: We really do use "integer part" and "fractional part" respectively: see Wikipedia, e.g., on decimal fractions.
The in... |
H: Find equation of the plane through the origin with basis <1,2,-1> and <2,3,4>.
Find an equation of the plane through the origin with basis <1,2,-1> and <2,3,4>.
Could I get some advice on how to work this problem? I know how to find the basis given some plane, but not the other way around.
AI: Hint: Compute the cro... |
H: Construct a compact set of real numbers whose limit points form a countable set.
I searched and found out that the below is a compact set of real numbers whose limit points form a countable set.
I know the set in real number is compact if and only if it is bounded and closed.
It's obvious it is bounded since $\,d(1... |
H: If $f:\mathbb{R}^n \to \mathbb{R}^n$ is continuous with convex image, and locally 1-1, must it be globally 1-1?
For $f:\mathbb{R}\to \mathbb{R}$ which is continuous, being locally 1-1 implies being globally 1-1, see here. This is not true for a general mapping $f:\mathbb{R}^n\to \mathbb{R}^n$. My intuition as to ... |
H: Derangements basic practice question
practice questions not Homework
I have problems with this questions that I have answers for but cant understand how the answer was derived.
Q.1. In how many ways can the integer $1,2,3,...10$ be arranged in a line so that no even integer is in its natural position?
Answer) $10!-... |
H: Exponentials of a matrix
I just was working with matrix exponentials for solving problems in control theory. Suppose $A $ is a square matrix. How can we interpret $A_1 = e^ {\textstyle-A\log(t) }$, where $\log$ is natural logarithm? Is there a formula for extending the scalar case of $e^ {\textstyle-a\log(t) }$ whi... |
H: What does the notation $\epsilon(f(x))s$ mean?
I am very, very confused with the notion $\epsilon(f(x))s$.
To my understanding, $s$ is a map sends to $F(x,s)$, and $\epsilon$ is the distance function given a point $f(x)$. So what does $\epsilon(f(x))s$ means, when we just put them together? Multiplication of two fu... |
H: Integral of the product of $x^k$ and the upper half circle of radius 2.
As I was browsing through the introduction of a paper, I came across the following equality:
$\displaystyle\frac1{2\pi}\int_{-2}^2x^k\sqrt{4-x^2}~dx=\begin{cases}\frac1{ k/2+1}\binom{k}{k/2}&\text{if $k$ is even;}\\0&\text{if $k$ is odd.}\end{c... |
H: Ellipse circumference calculation method?
Actually I know how to calculate the circumference of an ellipse using two methods and each one of them giving me different result.
The first method is using the formula: $E_c=2\pi\sqrt{\dfrac{a^2+b^2}{2}}$
The second method is determining the arc length of the first quart ... |
H: Show that $f$ is continuous at 0.
EDIT: Fixed the limit.
This is a question from Spivak's Calculus, Ch.6, ex. 3.
Suppose that $f$ is a function satisfying $$|f(x)|\leq |x| \forall x$$
Show that $f$ is continuous at 0. (Notice that f(0) must equal 0.
I do not understand the solution at all. Could someone please ... |
H: Impossible identity? $ \tan{\frac{x}{2}}$
$$\text{Let}\;\;t = \tan\left(\frac{x}{2}\right). \;\;\text{Show that}\;\dfrac{dx}{dt} = \dfrac{2}{1 + t^2}$$
I am saying that this is false because that identity is equal to $2\sec^2 x$ and that can't be equal. Also if I take the derivative of an integral I get the functio... |
H: Find a metric space X and a subset K of X which is closed and bounded but not compact.
Find a metric space $X$ and a subset $K$ of $X$ which is closed and bounded but not compact.
I can find a metric space $X$ like the below.
Let $X$ be an infinite set. For $p,q\in X$, define $d(p,q)=\begin{cases}1,&\text{if $p\... |
H: Domain of composite functions
In "Introduction to Set Theory" we have the following:
Theorem Let $f$ and $g$ be functions. Then $g \circ f$ is a function. $g \circ f$ is defined at $x$ if and only if $f$ is defined at $x$ and $g$ is defined at $f(x)$, i.e.,
$$ \text{dom }(g \circ f)= \text{dom } f \cap f^{-1}[\t... |
H: Proving there exist an infinite number of real numbers satisfying an equality
Prove there exist infinitely many real numbers $x$ such that $2x-x^2 \gt \frac{999999}{1000000}$.
I'm not really sure of the thought process behind this, I know that $(0,1)$ is uncountable but I dont know how to apply that property to th... |
H: If A and B are disjoint open sets, prove that they are separated
I just proved this statement like the below.
Is this valid or solid proof?
Thank you!
AI: Your idea looks fine. The proof that $\overline A\cap B=\varnothing$ can be direct, and by symmetry (as you say), we also get $A\cap\overline B=\varnothing$.
Ta... |
H: Proof that for any interval (a,b) with a
Background: We are assuming that the elements of $\mathbb{R}\setminus\mathbb{Q}$ are irrational number. If $x$ is irrational and $r$ is rational then $y=x+r$ is irrational. Also, if $r\neq 0$ then $rx$ is irrational as well. Likewise, if a number is irrational then its recip... |
H: Inverse properties of inequalities
Take a simple inequality such as 1 >= 1/x. Just by looking at it we can see that x cannot be any number between 0 and 1-; the solution is 0 > x >= 1.
Now if we multiply both sides of the inequality by x:
(1)x >= (1/x)x
x >= 1
Great, but what happened to x < 0?
If we subtract 1/x ... |
H: Relation Between Two Homomorphisms
Let $f,g: \mathbb Z_5\to S_5$ be two non-trivial homomorphisms.
Prove that for every $x\in\mathbb Z_5\,$ there exists $\,\sigma \in S_5\,$ such that $\,f(x)=\sigma g(x)\sigma^{-1}.$
I have found that $f,g$ are injective, but cannot proceed any further.
Thanks for any help.
AI... |
H: Saturation of the Cauchy-Schwarz Inequality
Consider a vector space ${\cal S}$ with inner product $(\cdot, \cdot)$. The Cauchy-Schwarz Inequality reads
$$
(y_1, y_1) (y_2, y_2) \geq \left| (y_1, y_2) \right|^2~~\forall y_1, y_2 \in {\cal S}
$$
This inequality is saturated when $y_1 = \lambda y_2$. In particular, th... |
H: General solution for the system of PDEs from the curl of a vector field equaling another
In my vector calculus class, when we were introduced to the curl operator the professor gave us this example:
Is it possible to find a vector field $\mathbf{G}$ such that
$$\mathbf{F} = \nabla \times {\mathbf{G}}?$$
As a motiv... |
H: Help with simplifying with $a+bi$ format?
Can someone please explain to me why $\frac{3}{2+4i}$ simplified into $a+bi$ format is $\frac{3}{10}-\frac{3i}{5}$? I can't find any explanations of what $a+bi$ format is.
Thank you, and this is from precalculus.
AI: Hint: Multiplying both the numerator and denominator b... |
H: Which of the following vectors are in ker(T)?
Let T: R2→R2 be the linear operator given by the formula:
T(x,y) = (2x-y, -8x+4y)
Which of the following vectors are in ker(T)?
*Note that ker(T) is the kernel of T. The way I think I should approach this problem is to plug in the given vectors and see if I get 0 as an ... |
H: How to solve this simple integral with substitution and partial fraction decomposition
This is the question:
$$\int \frac{e^{4t}}{(e^{2t}-1)^3} dt$$
This is my solution:
$$\ln |e^{2t} - 1| - \frac{2}{e^{2t} - 1} - \frac{1}{2(e^{2t}-1)^2} $$
Which I got by first substituting $u = e^{2t}$ so $du = 2e^{2t}dt$ so
$$\fr... |
H: Differentiability only in isolated point
Do functions exist, which are differentiable in a point, but not in a neighborhood of this point?
Is $e^{\frac{1}{W(x)-2}}$, where W is the Weierstrass function, maybe an example of a such function?
AI: If we multiply the Weierstrass function by $x^2$, we get a function wh... |
H: notations of generators and relations
I need to understand the following about generators and relations notations:
Is $\langle a,b \mid a^kb^l\rangle =\langle a,b\mid a^k=b^l\rangle =\langle a,b\mid a^k,b^l\rangle$?
Is $ \langle a,b\mid a^k=b^l\rangle =\langle a,b\mid a^k=b^l=1\rangle$? (if not, why?)
Which of the... |
H: Slope of secant line vs slope at the mid-point
I am looking for broad sufficient conditions that allow comparison of $\displaystyle {{f(b)-f(a)}\over {b-a}}$ and $f'\left({{a+b}\over 2}\right)$. As in when $\displaystyle {{f(b)-f(a)}\over {b-a}} > f'\left({{a+b}\over 2}\right)$.
If there are generalizations to high... |
H: Which of the following are in R(T), when R(T) is the range of t.
Let T: P2→P3 be the linear transformation given by the formula:
T(p(x)) = xp(x)
Which of the following are in R(T)?
a) x+x^2
b) 1+x
c) 3-x^2
I just need some clarification on whether I'm doing this correctly. If I'm not doing this correctly, then p... |
H: Why is the property called continuity of measure?
If $\left\{A_k \right\}_{k=1}^{\infty}$ is an ascending collection of measurable sets then $m(\bigcup_{k=1}^{\infty} A_k) = \lim_{k\to\infty} m(A_k)$.
What does it have to do with continuity?
AI: Intuitively, we can think of continuity as a property of a mapping whe... |
H: True/False questions (from a real analysis course)
The source of these T/F problems is this.
http://www.math.drexel.edu/~rboyer/courses/math505/true_false1.pdf
(I am self-studying and found these problems by chance.)
Could you tell me if my answers are correct and give me some hints on the problem that I couldn't s... |
H: find area of dark part
let us consider following picture
we have following informations.we have circular sector,central angle is $90$,and in this sector there is inscribed small circle ,which touches arcs of sectors and radius,radius of this small circle is equal to $\sqrt{2}$,we should find area of dark pa... |
H: Equilibrium Points of ODE
I am given the following two equations as models for male and female population
$\dfrac {df} {dt}$ = $\dfrac {mf} {m+f}$$B_f-fD_f$
$\dfrac {dm} {dt}$ = $\dfrac {mf} {m+f}$$B_m-mD_m$
assume $D_m$=$D_f$=$D$
The equilibrium points two the previous two equations form a line in the $f-m$ plane ... |
H: Subspace over a line in a plane
Consider a line $L$ over a plane.Describe the topology $L$ inherits as a subspace of the followings.
$\mathbb{R}_k \times \mathbb{R}_k$ where $\mathbb{R}_k$ is $K$-topology on $\mathbb{R}$.
$(\mathbb{R} \times \mathbb{R})_d$ where is dictionary order topology over $\mathbb{R} \times... |
H: If $p(x) =x^4 - x^3 +px^2 -4x +q$, find $p$ and $q$ if $p(0)=3$ and $p(-1)=11$
If $p(x) =x^4 - x^3 +px^2 -4x +q$, find $p$ and $q$ if $p(0)=3$ and $p(-1)=11$
Can someone please teach me how to do this question? thanks
AI: Put the values x= 0 and x= -1 in p(x).
p(0) = q= 3.
Now put x=-1 then find another value. |
H: Designing very simple function
I don't have much mathematical background except for highschool and I'm struggling to design a very simple function.
I need a function f(x, y) that for the absolute difference of x and y would return a number between 0 and 4. If x, y are equal, it would return 4 lets say. If they have... |
H: If $p(x)=ax^3 -2x^2 +bx+c$, find $a, b$ and $c$ if $p(0)=12$, $p(-1)=3$ and $p(2)=36$
If $p(x)=ax^3 -2x^2 +bx+c$, find $a, b$ and $c$ if $p(0)=12$, $p(-1)=3$ and $p(2)=36$
Can someone please teach me how to do this question thanks!
AI: Use $p(2)=36$ , you'll have
$8a - 8 + 2b + 12 = 36$.
Simplify it, you'll get $4... |
H: A problem with the proof of a proposition
I have a problem with the proof of Proposition 5.1. of the article of Ito.(Noboru Itˆo. On finite groups with given conjugate types. I.
Nagoya Math. J., 6:17–28, 1953.).
I don't know what is "e" and "e-1" in the proof.
I'd be really greatfull if someone help me.You can find... |
H: If $A^2+2A+I_n=O_n$ then $A$ is invertible
Let $A$ a matrix of $n\times n$ and $I_n, O_n$ the identity and nule matrix respectively. How to prove that if $A^2+2A+I_n=O_n$ then $A$ is invertible?
AI: Hint: Notice that $A(A+2I_n)=-I_n$.
More generally, you can prove in the same way that if $P(A)=0$ for some polynomia... |
H: Proof: Invariant angle measure - same result for any circle drawn.
Below I have quoted Wikipedia.
I am particular interested in the statement:
The value of $\theta$ thus defined is independent of the size of the
circle: if the length of the radius is changed then the arc length
changes in the same proportion, ... |
H: Confused with Eigenvalues and Eigenvectors and Vector transformations
Hello fellow mathematicians, I am studding " Eigenvalues and Eigenvectors " at this point and I need to understand something here:
I am actually performing automatic operations on finding them, but I don't really understand what they are and wha... |
H: why the sigma algebra generated by null set and Brownian Motion is right continuous?
I mean why the generated one satisfies the definition of right continuous?
AI: There is the following result, I quote from Karatzas/Shreve:
For a $d$-dimensional strong Markov Process $X=\{X_t;\mathcal{F}_t^X;t\ge 0\}$ with initia... |
H: Bounded sequence with divergent Cesaro means
Is there a bounded real-valued sequence with divergent Cesaro means (i.e. not Cesaro summable)?
More specifically, is there a bounded sequence $\{w_k\}\in l^\infty$ such that $$\lim_{M\rightarrow\infty} \frac{\sum_{k=1}^M w_k}{M}$$ does not exist?
I encountered this prob... |
H: Finding the total differential for the matrix function $F(A) = A^T A - \mathbb{1}_{n \times n}$
I am having difficulties understanding how to find the total differential for the function
$F: Mat_{n\times n}(\mathbb{R}) \rightarrow S_{n\times n}(\mathbb{R})$
where $S_{n\times n}(\mathbb{R})$ is the set of all symme... |
H: Do maps between topological spaces somehow induce maps between Banach spaces?
If $X,Y$ are topological spaces and $h:X\rightarrow Y$ is a continuous map, is there some sort of induced map
\begin{align*}
h':C_b(X)\rightarrow C_b(Y)
\end{align*}
(or in the other direction) where $C_b(X)$ is the Banach space of bounde... |
H: How to show that $f(x,y)$ is continuous.
How to show that $f(x,y)$ is continuous.
$$f(x,y)=\frac{4y^3(x^2+y^2)-(x^4+y^4)2x\alpha}{(x^2+y^2)^{\alpha +1}}$$ for $\alpha <3/2$.
Please show me Thanks :)
AI: Notice that $|x|,|y|\leq (x^2+y^2)^{1/2}=||(x,y)||$ so we have
\begin{align}|f(x,y)|&=\frac{|4y^3(x^2+y^2)-(x... |
H: How to define a function that gives us the number of pentagons formed in between two or more hexagons?
I have been trying to make a general formula/function that helps in calculating the number of pentagons that may be formed using 2 or more hexagons. Like it is shown in the picture below:
In Fig. 1 there are 2 he... |
H: Proving existence of $T$-invariant subspace
Let $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ be a linear transformation. I'm trying to prove that there exists a T-invariant subspace $W\subset \mathbb{R}^3$ so that $\dim W=2$.
How can I prove it? Any advice?
AI: Hint:
Consider the minimal polynomial $\mu_T$ of $T$. ... |
H: Problem on convergence of sequences
Given that $\lim f_n=1>0$, Show that there exists a positive integer $m$ such that $f_n\ge 0 \\ \forall n \ge m $
AI: Aman, simply employ the usual definition of convergence. If $f_n \to 1$, then for all $\epsilon > 0$, there exists $m \in \mathbb N$ such that for all $n \geq m... |
H: Finding all ordered tuples
Suppose $a+b+c+d+e=t$ and $a,b,c,d,e \geq r$ where all the given variables are positive integers.
How do you calculate all ordered tuple of $a,b,c,d,e$ such that the above equation holds.
The stars and bars formula can't be applied here, since we have a restriction namely $t$, for all the... |
H: Three Dimensional Vectors Question
Let $\ell_1 , \ell_2 $ be two lines passing through $M_0= (1,1,0) $ that lie on the hyperboloid $x^2+y^2-3z^2 =2 $ .
Calculate the cosine of angle the between the two lines.
I have no idea about it...
I guess it has something to do with the gradient of the function $F(x,y,z)=x^2+... |
H: Solving the inequality: $1\leq \cot^2(x)\leq 3$
I want to solve the following inequality:
$$1\leq \cot^2(x)\leq 3.$$
but I'm unsure of how to handle the positive and negative square roots.
If I take the square of the inequality, just focusing on the posive square roots, I get:
$$1\leq \cot^2(x)\leq 3 \Leftrightarro... |
H: $F(x,y)$ is continuous.
Prove that
$$ f(x,y)=\begin{cases}\frac{x^3-xy^2}{x^2+y^2}&\text{if }(x,y)\ne(0,0)\\0&\text{if }(x,y)=(0,0)\end{cases}$$
is continuous on $\mathbb R^2$ and has first partial derivatives everywhere on $\mathbb R^2$, but is not differentiable at $(0,0)$.
I want to show that the function ... |
H: Write $100$ as sum of $n$ numbers, such that each number is twice as big as its predecessor.
I don't quite know where to start on this one.
lets say we have a value 100.
and we want to split it in two parts where one is twice as big as the other.
That would be $v_1 = 66.666$ and $v_2= 33.333$ (sum $100$)
If we want... |
H: Find domain of $ \sin ^ {-1} [\log_2(\frac{x}{2})]$
Problem: Find domain of $ \sin ^ {-1} [\log_2(\frac{x}{2})]$
Solution: $\log_2(\frac{x}{2})$ is defined for $\frac{x}{2} > 0$
$\log_2(\frac{x}{2})$ is defined for $x > 0$
Also domain of $\sin ^ {-1}x$ is $[-1,1]$
When $x=1$ ,then $\log_2(\frac{x}{2})$ becomes $-... |
H: Is there a $f_n$ with two local maxima converges to f only one local maxima?
Is there a {$f_n$} with two local maxima converges(pointwise/uniform or other) to $f$ only one local maximum?
AI: Consider the function
$$
f:x\mapsto\begin{cases}1-|x| & \text{if } x\in[-1,1]\\0&\text{if not}\end{cases}
$$
For $n\in\mathb... |
H: Intersection of topologies
Is my proof that the intersection of any family of topologies on a set $X$ is a topology on $X$ correct?
Proof. We are required to show that the intersection satisfies the topology axioms. Let $\tau$ be an arbitrary intersection of topologies on $X$.
$\emptyset$ and $X$ are in every topo... |
H: Every element in a ring can be written as a product of non-units elements
I'm trying to understand a little detail in this proof:
I didn't understand why in a ring we can always write an element as a products of non-units elements.
I need help.
Thanks in advance
AI: The question has been answered by the comments.... |
H: Prove that $1+\sum_{n\geq1}\sum^n_{k=1}\frac{n!}{k!}{n-1\choose k-1}x^k\frac{u^n}{n!}=\exp\frac{xu}{1-u}$
Let $k,n\in\mathbb{N}_{>0}$. How do I get started to prove that"
$$1+\left(\sum_{n\geq1}\sum^n_{k=1}\frac{n!}{k!}{n-1\choose k-1}x^k \frac{u^n}{n!}\right) = \exp\frac{xu}{1-u}$$
Hints and help greatly apprecia... |
H: What's the intuition with partitions of unity?
I've been studying Spivak's Calculus on Manifolds and I'm really not getting what's behind partitions of unity. Spivak introduces the topic with the following theorem:
Let $A\subset \Bbb R^n$ and let $\mathcal{O}$ be an open cover of $A$. Then there is a collection $\... |
H: Is a finite group determined by the family of all its 2-generated subgroups?
At the last week I meet my old coauthor, Oleg Verbitsky who proposed me the following question. I think that here should be an easy counterexample, but I am not a pure group theorist and I am usually interested in infinite groups, so I dec... |
H: The number of irreducible representations
I am reading a textbook "Representation theory" by Fulton and Harris and I have a question.
They proved the following theorem on page 16.
With an Hermitian inner product on a set of class function, the characters of the irreducible representation of a finite group $G$ are o... |
H: complexity of matrix multiplication
For $n\times n$ dimensional matrices, it is known that calculating $\operatorname{tr}\{AB\}$ needs $n^2$ scalar multiplications. How many scalar multiplications are needed to calculate $\operatorname{tr}\{ABCD\}$?
Note that $\operatorname{tr}$ means the trace of a matrix.
AI: As ... |
H: continuously averaging a stream of numbers
Ok, so I'd like a little help with the mathematical theory behind something I'm doing in a program. I've got a stream of numeric inputs, and I need to find the average of all the numbers in this stream of data. I'm going to output it once the stream ends.
In my code, my pr... |
H: characterization of potential
I have a Field $F(x,y):= \left(
\begin{array}{c}
ay\\
0\\
\end{array}
\right)$ and I have to find out whether this field is potential.
Well despite the terminology( wich is not completely clear to me) i thought that the answer is affermative iff there exist a $U$ such that $F=-\nabl... |
H: basic integral inequality
Consider $0 < r <R $. I have a function $u \in C_{0}^{\infty}(B(x_0,R))$ such that $u=1$ on $\overline{B(x_0,r)}$ . Consider $y \in \partial B(0,1)$ (fixed) . My book says :
$$ 1 \leq \int_{r}^{R} |\frac{d}{ds}u(sy)| \ ds$$
I tried to begin with $\frac{d}{ds}u(sy) = \nabla u(sy) . y$ and... |
H: Prove that if $a^x = b^y = (ab)^{xy}$, then $x + y = 1$
The question is prove that if $a^x = b^y = (ab)^{xy}$, then
$$x + y = 1$$
I've tried:
$$a^x = (ab)^{xy}$$
$$\log_aa^x = \log_a(ab)^{xy}$$
$$x = xy \log_ab $$
$$y^{-1} = \log_ab$$
but then I get stuck and I'm not sure if this is the right path. What is an el... |
H: Intersection of a hyperplane and a curve
Let us fix a projective curve X over a field k. With nt, I mean a variety with all irreducible components of dimension 1. Let us suppose that there is a smooth rational point $x \in X$. My question is:
Is it possible to find a hyperplane such that the intersection of X and t... |
H: Choosing Firearms (With Replacement)
In a shooting gallery there are 4 types of firearms. A practice shooter can choose 1 firearm
at each shooting (with replacement). In a given day the shooter practices 8 shootings. What is the
probability that in a given day he uses each of the 4 type of firearms at least once?... |
H: Strange expression for limit
What do these limits $\psi(x+0), \psi(x-0)$ mean?. I did calculus but have never come across this.
AI: These notations mean
$\displaystyle\psi(x+0)=\lim_{t\to x^+}\psi(t)$ and $\displaystyle\psi(x-0)=\lim_{t\to x^-}\psi(t)$ and these notations $\psi(x+),\psi(x^+)$ are also used for the ... |
H: Examples of logical propositions that are not functions
I'm trying to understand the axiom of reeplacement in set theory. To my understanding, please tell me if I'm wrong, if there is a logical proposition $\varphi(x,y,w_{1},...,w_{n})$ and an arbitrary set $A$, then we can have the set $B=\{y:\exists x\in A(\varph... |
H: Probability of $n$ times a $\frac1n$ event
I never studied probability at school and this problem has been bothering me for a long time:
Let's say I have a perfectly fair die. If I roll it, the odds of it landing on $6$ are $\frac{1}{6}$. If I roll two dice, the odds of at least one of them landing on 6 are $\frac{... |
H: Weibull Scale Parameter Meaning and Estimation
Wikipedia: http://en.wikipedia.org/wiki/Weibull_distribution gives a nice description on what the shape parameter (they call it k) means in the Weibull distribution, but I can't find anywhere what the scale parameter (denoted as λ on Wikipedia) means or how to estimate... |
H: Good texts in Complex numbers?
I have asked some members on chat about good text to study complex numbers , they recommended for example , "Visual Complex Analysis" by Needham and "complex analysis" by Steins.
But, I look for a text for complex numbers not complex analysis (I don't even know what is complex analysi... |
H: Representing $\mathbb{R}/\mathbb{Z}$ as a matrix group.
It was told to me that $G = \mathbb{R}/\mathbb{Z}$ is a real matrix group.
Can someone help me understand how to represent $G$ in $Gl_n(\mathbb{R})$ for some $n$?
(Supposedly, $n = 1$? But that's confusing because then G is [0,1) with a modular addition grou... |
H: Finite field extension over $\mathbb F_2$
I don't see why $[L:K]=4$, where
$L = \mathbb{F}_2(x,y) = \operatorname{Quot}(\mathbb{F}_2[x,y])$
and
$K = \mathbb{F}_2(x^2,y^2) = \operatorname{Quot}(\mathbb{F}_2[x^2,y^2])$
Let $p(X) = X^2-x^2 \in K[X]$. $p$ has leading coefficient $1$.
It is $p(X) = (X-x)(X+x)$ where $\... |
H: How does Liu intend his readers to compute $H^0(X,\Omega^1_{X/k})$ of this scheme?
In the chapter on duality theory in Liu's Algebraic Geometry and Arithmetic Curves, there is the following exercise. I'm having trouble seeing how one can apply the duality theory developed in Liu to solve it.
$4.2$. Let $k$ be a f... |
H: Help with difficult integral
According to my textbook,
$$\int \left( 2 \cot^2{x} - 3 \tan^2{x} \right)dx = -2 \cot{x} - 3 \tan{x} + C$$
I am unable to arrive at this answer. Is this correct? If so, please help me with the integral.
AI: Here is a hint:
Use $$
\cot^2(x) = \csc^2(x) - 1\\
\frac{d}{dx} (-\cot(x)) = \cs... |
H: Proving $\|f(x)-f(a)\|\le M\| x-a\|$
Theorem(1): (M.V.T for real valued functions)
Let $V \subseteq \Bbb R^n$ be open. Suppose $ f: V \to \Bbb R$ is differentiable on $V$. If $x, a\in V$ then there is $c\in L(x;a)$ such that $f(x)-f(a)=\nabla f(c)\cdot (x-a)$
Theorem(2): (M.V.T for vector valued functions)
Let $V ... |
H: Pi Estimation using Integers
I ran across this problem in a high school math competition:
"You must use the integers $1$ to $9$ and only addition, subtraction, multiplication, division, and exponentiation to approximate the number $\pi$ as accurately as possible. Each integer must be used at most one time. Parenthe... |
H: boolean algebra: simplify $ a* b *d + \tilde a *\tilde c*d + b* \tilde c* d$
Simplify the following function(algebraically):
$$y = a*b*d + \tilde a *\tilde c*d + b *\tilde c *d$$
the solution is: $$a*b*d + \tilde a * \tilde c * d$$ which i checked via karnaugh and also wolfram.
my "solution" so far: $b*d*(a + \til... |
H: Given an operator and its' representation in a non-orthogonal basis. Is it normal?
Given T, an operator in $V = \mathbb {C^2}$ and a basis $B = \{ (1,1), (1,0) \}$. Is $T$ a normal operator if $[T]_B = \begin{pmatrix} 1 & i \\ 2 & \frac{1}{2} \end{pmatrix}$ and $[T^{*}]_B = \begin{pmatrix} 1 & 2 \\ -i & \frac{1}{2... |
H: For what values of $r$ and $b$ is $X ∩ Y$ a smooth manifold? When it is a manifold, what is its dimension?
Consider the hyperbolic paraboloid $X$ contained in $\mathbb{R}^3$, and the sphere $Y$ in $\mathbb{R}^3$ given by the equations $x^2 −y^2 =z$ and $x^2+y^2+(z−1)2=r^2.$
For what values of $r$ and $b$ is $X ∩ Y$... |
H: Maximize $(a-1)(b-1)(c-1)$ knowing that : $a+b+c=abc$.
If : $a,b,c>0$, and : $a+b+c=abc$, then find the maximum of $(a-1)(b-1)(c-1)$.
I noted that : $a+b+c\geq 3\sqrt{3}$, I believe that the maximum is at : $a=b=c=\sqrt{3}$. (Can you give hints).
AI: Go for Lagrange Multipliers! Let $f(a,b,c)=(a-1)(b-1)(c-1)$, an... |
H: The space of sequences of integers, and an analog of topological space for classes?
Is the collection of integer sequences a set or a class? If it's not a set, then is there an analog of topological spaces for classes? Thank you!
AI: Assuming that you are talking about sequences of elements of $\Bbb N$ indexed by $... |
H: commutator subgroup and semidirect product
Suppose $G$ is a solvable group such that $G = N \rtimes H$. Then I can show that $G' = M \rtimes H'$, where $G'$ is the commutator subgroup of $G$ and $M = N \cap G'$, $H' = H \cap G'$. I can also show that $H'$ is indeed the commutator subgroup of $H$. So $H$ and $H'$ ar... |
H: $20$ hats problem
I've seen this tricky problem, where $20$ prisoners are told that the next day they will be lined up, and a red or black hat will be place on each persons head. The prisoners will have to guess the hat color they are wearing, if they get it right the go free. The person in the back can see every h... |
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