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H: Summation of cardinals is smaller than product of cardinals
How do i prove $\sum a_i$ is equipotent with a subset of $\prod a_i$ ??
I seems obviously true but its actually hard to prove it...
$\{a_i\mid i\in I\}$ is a set of cardinals and $a_i$ is a cardinal for each $i\in I$.
AI: I assume that $a_i\ge 2$ for each ... |
H: Finding a grammar for a formal language
I am looking for a grammar that describes the formal language
$L = \{ xyx^R \;|\; x,y \in \{a,b\}^*\}$
where $\{a,b\}^*$ corresponds to the regular expression [ab]*.
If there would be no y and the language would therefore contain all the words that are palindromes there would... |
H: Parabolic PDE existence/uniqueness
Consider the parabolic PDE:
$$\frac{\partial u}{\partial t} = u^2\frac{\partial^2u}{\partial x^2} + u^3$$
with some initial condition.
Apparently this is a straightforward parabolic PDE in which I can apply standard results to prove short term existence and uniquness. Can someone ... |
H: Finding all groups with given property
My problem is how to find all groups which have one exactly non-proper subgroup.
Thanks
AI: Groups with only one proper subgroup
A nontrivial group $G$ has no proper subgroups except the trivial group iff $G$ is finite and of prime order. |
H: Fitting of Closed Curve in the Polar Coordinate.
I know how to fit a curve when given some data points in the cartesian coordinate. Recently, I encountered a model that needs to fit a closed curve in the polar coordinate. I'm thinking of deducing a similar formula using Maximum Likelyhood, but the problem is I don'... |
H: Can % error in computed value be less than % error in one of its parameters
I am using an equation to compute energy as follows
$E= C_1\times t + C_2 \times a$
Here $C_1$ and $C_2$ are constants. $t$ shows the time-taken, $a$ shows the dynamic-activity.
Using a technique, I am estimating $t$ and $a$ and using them... |
H: Stone duality for ideals and filters (exercise)
In A Course in Universal Algebra (Burris, Sankapannavar), the exercise 4.4.7-8, p.158, says:
Let $A$ be a Boolean algebra. Denote $A^\ast:=\{\text{ultrafilters of }A\}$, and give $A^\ast$ the topology, defined by the basis of open sets $\{N_a; a\!\in\!A\}$, where $N_... |
H: $\lim\limits_{n \to\infty}x_n-x_{n+1}=0$, $x_{n_k}$ converges but $x_n$ does not converge
This is a counter example homework question that I can't seem to solve.
I need to find a sequence of real numbers $(x_n)_{n=1}^{\infty}$, and a monotonic increasing sequence of natural numbers $(n_k)_{k=1}^{\infty}$such that:
... |
H: Riemann's theorem on removable singularities
Theorem
Let $\Omega\subseteq \mathbb{C}$ open , $ a\in\Omega,\ f\in H(\Omega\backslash \{a\})$ and there is $r>0$ with
$f$ is bounded on $C(a,r)\backslash \{a\}$ ($C(a,r)$ is the circle with origin $a$ and radius $r$),
then $a$ is a removable singularity.
Proof
Let $h:\O... |
H: Casorati-Weiestrass theorem for essential singularities
Casorati-Weiestrass theorem for essential singularities
Let $\Omega\subseteq \mathbb{C}$ be open, $ a\in\Omega,\ f\in H(\Omega\backslash \{a\})$ ($f$ analytic on $\Omega\backslash \{a\})$. In the case of an essential singularity: If $ C(a,r)\subseteq\Omega$ ($... |
H: Is $f$ injective in $W$?
If
$\|f(x)-f(y)\|\geqslant \frac 1{2} \|x-y\|$ for any $x, y \in W$
then
$f$ is injective in $W$
How to prove this? If that inequality is right is it mean that the images are equal or not?
AI: Assume $f(x) = f(y)$. What is $\|f(x)-f(y)\|$? What can you conclude about $\|x-y\|$? |
H: Showing that a group of order $21$ (with certain conditions) is cyclic
How can i show that if $o(G)=21$ and if $G$ has only one subgroup of order $3$ and only one subgroup of order $7$, then show that $G$ is cyclic.
AI: Hint: Let $H$ and $K$ be the unique subgroups of order 3 and 7, respectively. Since they are the... |
H: An example for a homomorphism that is not an automorphism
Let $K/F$ be a field extension, I know that if $K/F$ is a finite extension then a simple argument from linear algebra shows that since every homomorphism of fields from $K$ to $K$ that fixes $F$ is 1-1 it is also onto, i.e. an automorphism of $K$.
Can someon... |
H: sampling technique: lower error in estimation on using higher sampling rate
I am using sampling to estimate characteristics of a population. My sampling ratio is 32. I take N/32 samples, calculate their sum and multiply by 32. Then I compare it to actual value to get error. It is 3.67%.
Then I take sampling ratio a... |
H: Using the roots of polynomial finding the value of sum.
If $a,b$ and $c$ are the roots of $x^{3}+px^{2}+qx+r$, then how can we find the value of $\displaystyle \sum \frac{b^{2}+c^{2}}{bc}$.
AI: I think you are asking for
$$\frac{a^2+b^2}{ab}+\frac{b^2+c^2}{bc}+\frac{c^2+a^2}{ca},\tag{$1$}$$
or perhaps twice this qu... |
H: Open mapping theorem
I have some understanding problems with the open mapping theorem.
Lemma
Let $f\in H(D(z_{0},\ R))$ ($f$ analytical on the disc with the origin in $z_0$ and radius $R$) and
$$
|f(z_{0})|<\min_{|z-z_{0}|=r}|f(z)|
$$
for a $r\in(0,\ R)$. Then there is a $w\in D(z_{0},\ r)$ with $f(w)=0$.
Theorem
L... |
H: Why can't a polynomial fit infinitely many points of an exponential function?
Possible Duplicate:
Polynomial satisfying $p(x)=3^{x}$ for $ x \in \mathbb{N}$
I'm looking for an elementary solution to this question:
There is no polynomial $P$ such that $P(0)=1, P(1)=3, P(2)=9, P(3)=27, \dots$.
AI: Assume $P(X) =... |
H: How to "rotate" a polar equation?
Take a simple polar equation like r = θ/2 that graphs out to:
But, how would I achieve a rotation of the light-grey plot in this image (roughly 135 degrees)? Is there a way to easily shift the plot?
AI: Just put $\theta-135^\circ$ in place of $\theta$. Or if you're working in rad... |
H: Maximum principle
Theorem
Let $\Omega\subseteq \mathbb{C}$ be a region and $f\in H(\mathbb{C})$ ($f$ is analytic on $\mathbb{C}$). If $|f|$ has in $\Omega$ a local maximum, then $f$ is constant.
Proof. Let $ D(a,\ r)\subseteq\Omega$ be a disk with $|f(z)|\leq|f(a)|$ for all $z\in D(a,\ r)$ ($a$ is therefore a local... |
H: Products of Vector Bundles
Suppose that $E$ is a vector bundle over a compact, Hausdorff space $X$. Then $E^n$ is a vector bundle over $X^n$. If $D(E)$ is the disk bundle, there is a map on fibers $D(E^n)_x \rightarrow D(E)^n_x$. Does this induce a homotopy equivalence $D(E^n)\simeq D(E)^n$?
AI: Yes. For any ve... |
H: How to prove that $\mathrm{Fibonacci}(n) \leq n!$, for $n\geq 0$
I am trying to prove it by induction, but I'm stuck
$$\mathrm{fib}(0) = 0 < 0! = 1;$$
$$\mathrm{fib}(1) = 1 = 1! = 1;$$
Base case n = 2,
$$\mathrm{fib}(2) = 1 < 2! = 2;$$
Inductive case assume that it is true for (k+1) $k$
Try to prove that $\mathrm{f... |
H: Combinatorics: Selecting objects arranged in a circle
If $n$ distinct objects are arranged in a circle, I need to prove that the number of ways of selecting three of these $n$ things so that no two of them are next to each other is $\frac{1}{6}n(n-4)(n-5)$.
Initially I can select $1$ object in $n$ ways. Then its ne... |
H: Find the number of ways in which $6$ different toys may be distributed among $4$ children so that each child gets at least $1$ toy.
Find the number of ways in which $6$ different toys may be distributed among $4$ children so that each child gets at least $1$ toy.
The problem is simple enough, I only need to verif... |
H: Is there a general way to solve transcendental equations?
To make things definite, let's narrow them and call transcendental equation of the form
$$f(x) = 0$$
where $f$ is a real elementary function in the usual sense. For example
$$\cos(\pi x) + x^2 = 0$$
or
$$a = x \tan x$$
Is there a general way to solve such eq... |
H: Coloring points in a cycle
I have a question that relates to the Widom-Rowlinson model of statistical physics. Take a cycle on $n$ vertices. How many ways are there to color the $n$ vertices with the colors $\{\text{Red, Yellow, Blue}\}$ with the only restriction being that Red vertices cannot be next to Blue ver... |
H: Unit speed geodesics
This might be a very trivial question so bare with me. If $(X,d)$ is a length space we define a unit speed geodesic to be a path $\gamma:[0,1]\to X$ for which
\begin{align*}
d(\gamma(s),\gamma(t))=|t-s|d(\gamma(0),\gamma(1))\,\,\mathrm{for}\,\,\mathrm{all}\,\,s,t\in[0,1].
\end{align*}
The autho... |
H: Solving problem: Area of Triangle
I have this data:
$a=6$
$b=3\sqrt2 -\sqrt6$
$\alpha = 120°$
How to calculate the area of this triangle?
there is picture:
AI: Because the angle at $A$ is obtuse, the given information uniquely determines a triangle. To find the area of a triangle, we might want:
the length of ... |
H: $K/E$, $E/F$ are separable field extensions $\implies$ $K/F $ is separable
I wish to prove that if $K/E$, $E/F$ are algebraic separable field extensions then $K/F$ is separable. I tried taking $a\in K$ and said that if $a\in E$ it is clear, otherwise I looked at the minimal polynomials over $E,F$ and called them $f... |
H: Proof of 2 Matrix identities (Traces, Logs, Determinants)
I am working through a derivation in someone's thesis at the moment to understand an important result, but I am more than a bit rusty on matrices. Could anyone give me some tips on these identities? They are stated without proof and I'm having a hard time fi... |
H: Nonexistence of a strongly multiplicative increasing function with $f(2)=3$
Show that there does not exist a strictly increasing function
$f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying
$$f(2)=3$$ $$f(mn)=f(m)f(n)\forall m,n\in\mathbb{N}$$
Progress:
Assume the function exists. Let $f(3)=k$
Since $2^3 < 3^2$,
$... |
H: Dirichlet Series and Average Values of Certain Arithmetic Functions
If an arithmetic function $f(n)$ has Dirichlet series $\zeta(s) \prod_{i,j = 1} \frac{\zeta(a_i s)}{\zeta(b_j s)}$, for which values of $a_{i}$ and $b_{j}$ is the following true? That
\begin{align}
\lim_{x \to \infty} \tfrac{1}{x} \sum_{n \leq x} f... |
H: Questions about the center of a group
I need help with this advanced algebra problem.
Let $G$ be a group. We call the set $C(G)= \{a \in G : ab=ba, \forall b \in G\}$ the center of $G$.
Prove that:
(a) $C(G)$ is normal subgroup of $G$.
(b) $C(G)=G$ if and only if $G$ is abelian group.
(c) If $a... |
H: Integral of $\int_{3}^{\infty} \frac{dx}{(x-2)^{3/2}}$
I am not sure how to evaluate this problem
$$\int_{3}^{\infty} \frac{dx}{(x-2)^\frac{3}{2}}$$
I do not know if it converges or diverges but I do know that it is undefined at $3$.
It is $3$ not $-3$.
AI: The integrand is defined for any $x\ge3$. The integral is ... |
H: $\mid\theta-\frac{a}{b}\mid< \frac{1}{b^{1.0000001}}$, question related to the dirichlet theorem
The question is:
A certain real number $\theta$ has the following property: There exist infinitely many rational numbers $\frac{a}{b}$(in reduced form) such that:
$$\mid\theta-\frac{a}{b}\mid< \frac{1}{b^{1.0000001}}$$
... |
H: Given an interval $I$, what does the notation $\overline{I}$ mean?
I have below a beginning of a theorem:
If a function $f:I \rightarrow \mathbb{C}$ defined on an interval $I$ of length $p$ can be expanded to a piecewise differentiable function on $\overline{I}$, then will...
What does $\overline{I}$ mean in this... |
H: Proving that a language build from two regular languages is regular as well
Let $L, M \subseteq \Sigma^*$ be regular languages. I need to prove that
$$N = \{x \in \Sigma^* \; | \; \exists y \in L : xy \in M\}$$ is as well a regular language.
My favored approach is to find a finite state machine that recognizes that... |
H: Smallest possible perimeter of a triangle if area is 135
What is the smallest possible perimeter of a (flat, regular) triangle if the area is 135?
I tried various equations but I always ended up with two unknown variables.
AI: Hint: You want an equilateral triangle. That should give you a single equation. |
H: Arc length of $y = \frac{x^3}{3} + \frac{1}{4x}$
Arc length of $y = \frac{x^3}{3} + \frac{1}{4x}$ over $1 \leq x \leq 2$
I know that the first thing I need to do is take the derivative.
$$y' = x^2 - 4x^{-2}$$
Then I take the integral on that range using the arc length formula.
$$\int_1^2 \sqrt{1 + (x^2-4x^{-2})^2}$... |
H: Arc length of $y = 1/3 \sqrt{x} (x-3)$
$y = 1/3 \sqrt{x} (x-3)$ from 1-9
In my book it is actually $x = 1/3 \sqrt{y} (y-3)$ but I prefer to work with y so I just swap the two variables and I think everything should be the same.
The first thing I need to do is get an integral.
$y = 1/3 \sqrt{x} (x-3)$ = $\frac{x^\fr... |
H: Rotation group, altitude
Could someone give me a rigorous proof that the group of rotations each element of which is a composition of rotations around the altitudes of a tetrahedron that transform the tetrahedron into itself is isomorphic to the alternating group $A_4$.
AI: Jykri's solution is excellent, but it may... |
H: Example for changing the compactness of a manifold by considering another topology
If compactness depends on topology, what are good examples for "alternative" topologies of spaces which steal away their compactness, i.e. I ask for spaces which are usually considered together with a topology such that they are comp... |
H: Winding number on a simply connected region
Let $\Omega$ be a simply connected region in $\mathbb{C}$ and $\gamma$ a closed piecewise continuous differentiable path.
Is there an intuitive explanation why the winding number $\mathrm{ind}_\gamma(\alpha)=0$, $\alpha \in \mathbb{C}\backslash \Omega$, on simply connecte... |
H: Is the integral $\int_1^\infty\frac{x^{-a} - x^{-b}}{\log(x)}\,dx$ convergent?
Is the integral $$\int_1^\infty\frac{x^{-a} - x^{-b}}{\log(x)}\,dx$$ convergent, where $b>a>1$?
I think the answer lies in defining a double integral with $yx^{(-y-1)}$ and applying Tonelli's Theorem, but the integral of $\frac{x^{-a}}{\... |
H: Application residue theorem for improper integrals
Let $R(z)=\displaystyle \frac{P(z)}{Q(z)}$ be a rational function of order(Q) $\geq \mathrm{order}(P)+2$ and $Q(x)\neq 0$ for all $x\in \mathbb{R}$. Then we have:
$$
\int_{-\infty}^{\infty}R(x)\mathrm{d}x=2\pi i\sum_{z:\ \mathrm{Im} \ z>0}{\rm Res}(R,z)
$$
Why is
... |
H: Is $\mathbb{R}$ a vector space over $\mathbb{C}$?
Here is a problem so beautiful that I had to share it. I found it in Paul Halmos's autobiography. Everyone knows that $\mathbb{C}$ is a vector space over $\mathbb{R}$, but what about the other way around?
Problem: Prove or disprove: $\mathbb{R}$ can be written as ve... |
H: Simplifying $\sqrt {1+(x/2 - 1/(2x))^2}$
I am having trouble figuring this out.
$$\sqrt {1+\left(\frac{x}{2}- \frac{1}{2x}\right)^2}$$
I know that $$\left(\frac{x}{2} - \frac{1}{2x}\right)^2=\frac{x^2}{4} - \frac{1}{2} + \frac{1}{4x^2}$$ but I have no idea how to factor this since I have two x terms with vastly dif... |
H: is $\frac{\{1, \ldots ,n\}}{n+1}$ proper notation when $n\geq1$?
I am trying to explain R code:
(1:n)/(n+1)
such that:
> n <- 4
> (1:n)/(n+1)
[1] 0.2 0.4 0.6 0.8
I might use
$$\frac{\{1, \ldots ,n\}}{n+1}$$
Is that okay? It seems to imply $n\neq1$. Does it?
AI: A lot of people are going to be completely mystifi... |
H: What does it mean to take the splitting field of $f(x)\in F[x]$ over $K$ where $K/F$ is a field extension
Let $K/F$ be a field extension and let $f(x)\in F[x]$. I know $f(x)$ have a splitting field, i.e. a field $E$ that $f(x)$ splits in ($E/F$ and $f(x)$ doesn't split in any proper subfield of $E$).
I heard the te... |
H: Touch Typing Index - Speed and Accuracy
I am trying to determine the ability of my students to touch type.
I have data on their speed (in seconds) and their accuracy (number of errors). I also know the number of words in the test (50 words).
Eg. Name: Bob, Speed: 113s, Errors: 19
Eg. Name: Jane, Speed: 831s, Errors... |
H: Indefinite integral of secant cubed $\int \sec^3 x\>dx$
I need to calculate the following indefinite integral:
$$I=\int \frac{1}{\cos^3(x)}dx$$
I know what the result is (from Mathematica):
$$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$
but I don't know how to integrate it myself. I have been trying some substitut... |
H: Is this the free abelian group functor?
Let $\mathbb{Z}(.) : \mathbf{Set} \to \mathbf{Ab}$ be the functor that assigns to any set $S$ the set of maps
$\mathbb{Z}(S) := \{ z: S \to \mathbb{Z} \; | \; z(s)=0 \mbox{ for almost all } s \in S \}$
and to any set map $f: S \to T$ the morphism $\mathbb{Z} f :\mathbb{Z}(... |
H: Arc length of $y^3 = x^2$
I am trying to find the arc length of $y^3 = x^2$ and I am suppose to use two formulas, one for in terms of x and one for in terms of y.
At first I need to find (0,0) (1,1) and I start with in terms of x
$$y\prime = \frac{2}{3}x^\frac{-1}{3}$$
$$\left(\frac{2}{3}x^\frac{-1}{3}\right)^2 = \... |
H: Degrees of freedom vs. cardinality of tuples
Sometimes it is said that the number of DoF of a system means how many real numbers have to be used at least to describe the system.
But we know from set theory that the cardinality of any tuple of reals is the same as the reals, so all the information that is in a 3-tup... |
H: The Nearest Points
Given a set $R$ of $N$ points $R={(x_1, y_1, z_1), (x_2, y_2, z_2),....., (x_n, y_n, z_n)}$ and set $S$ of $M$ points $S={ ((a_1, b_1, c_1), (a_2, b_2, c_2),...(a_m, b_m, c_m))}$.
for each point $p_i(i=1\space \text{to}\space N)$ in Set $R$, find the point $q_j(j=1\space \text{to}\space m)$ in s... |
H: Is there really no way to integrate $e^{-x^2}$?
Today in my calculus class, we encountered the function $e^{-x^2}$, and I was told that it was not integrable.
I was very surprised. Is there really no way to find the integral of $e^{-x^2}$? Graphing $e^{-x^2}$, it appears as though it should be.
A Wikipedia page on... |
H: Elliptic Curves over Noncommutative rings
It is known that we can define elliptic curves over commutative rings. However can we define an elliptic curve over a noncommutative ring?
This question is considered to some extent in this thesis (Section 4.4) but reaches no conclusion on the matter.
NOTE: By an elliptic ... |
H: Identifying compactness and connectedness of subspace P = $\{(x, y, z)\in \mathbb{R}^3 : x^2+y^2+z^2 = 1 ,~ x^2+y^2\neq 0\}$
I have to check for compactness and connectedness of subspace P = $\{(x, y, z)\in \mathbb{R}^3 : x^2+y^2+z^2 = 1 ,~ x^2+y^2\neq 0\}$
Intuitively it is clear to me that subspace P is not comp... |
H: Laplace transform problem RL circuit
I'm having trouble solving an RL circuit using the Laplace Transform.
There's just a 2H inductor in series with a 5M ohm resistor. The inductor is initially charged to 1.25A.
So... Here's what I've done:
$$
v_L(t) = v_R(t)\\
2i_L'(t) = 5Mi_L(t)
$$
Now taking the Laplace Transfo... |
H: How to prove such a function doesn't exist?
I was wondering (or mind-wandering) about a function described as:
For any given $x_1$, $x_2$, $x_m = \frac {x_1+x_2} {2}$,
$$f(x_m) = f(x_1) + a (f(x_2) - f(x_1)) , a \in ]0;1[$$
For example, $f(x_m)$ is $\frac 23$ of the distance between $f(x_1)$ and $f(x_2)$.
I can i... |
H: Find the remainder when $ 12!^{14!} +1 $ is divided by $13$
Find the remainder when $ 12!^{14!} +1 $ is divided by $13$
I faced this problem in one of my recent exam. It is reminiscent of Wilson's theorem. So, I was convinced that $12! \equiv -1 \pmod {13} $ after this I did some test on the exponent and it seems l... |
H: Do $\omega^\omega=2^{\aleph_0}=\aleph_1$?
As we know, $2^{\aleph_0}$ is a cardinal number, so it is a limit ordinal number. However, it must not be $2^\omega$, since $2^\omega=\sup\{2^\alpha|\alpha<\omega\}=\omega=\aleph_0<2^{\aleph_0}$, and even not be $\sum_{i = n<\omega}^{0}\omega^i\cdot a_i$ where $\forall i \l... |
H: What is the difference between the terms "classical solutions" and "smooth solutions" in the PDE theory?
What is the difference between the terms "classical solutions" and "smooth solutions" in the PDE theory? Especially，the difference for the evolution equations? If a solution is in $C^k(0,T;H^m(\Omega))$，can I ca... |
H: Evaluate $(\overline{z}_3)^4$ given that $z_3 = -\frac{1}{2}+j\frac{\sqrt{3}}{2}$
Given that:
$$z_3 = -\frac{1}{2}+j\frac{\sqrt{3}}{2}$$
evaluate the following:
$(\overline{z}_3)^4$
Solution:
$$(\overline{z}_3)^4 = [-\frac{1}{2}-j\frac{\sqrt{3}}{2}]^4$$
$$=[1\angle(-\frac{2\pi}{3})]^4$$
$$=1\angle(-\frac{8\pi}{3})$... |
H: Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
Can we impose such condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that
$(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$?
This question came into my mind... |
H: Intensity of sound wave question
The question is:
The intensity of sound wave A is 100 times weaker than that of sound wave B. Relative to wave B the sound level of wave A is?
The answer is -2db
I tried doing (10dB)Log(1/100) but that equals -20dB
thanks
AI: Your answer of $-20$ dB is correct.
I suspect that you’re... |
H: Partial Derivation: $\lim_{(x,y)\to(0,0)}\frac{x^2+\sin^2y}{2x^2+y}$
$$\lim_{(x,y)\to(0,0)}\frac{x^2+\sin^2y}{2x^2+y}$$
Above problem is in my textbook. it's different from others because I don't know how to process with trigometric element: in this example is $\sin^2y$
Thanks :)
AI: HINT: Don’t let the trig functi... |
H: How to write a combinations formula for this?
I have 8 distinct elements. Each set has 4 pairs from the 8 elements above.
How many such distinct sets are possible?
e.g.
8 elements - 1,2,3,4,5,6,7,8
example set - 1,2;3,4;5,6;7,8 the ordering of elements in a pair is not important
AI: I’m assuming that the order of ... |
H: Common eigenvectors of two special commuting matrices
Suppose you have a symmetric real 3x3 Matrix $S$ and an orthogonal matrix $O$ such that $O$ commutes with $S$, i.e. $OS = SO$. Suppose that $O$ is a nontrivial rotation about an axis in direction of $n \in \mathbb{R}^3$, i.e. $On = n$ and $O \neq \mathrm{id}$.
I... |
H: Eigenvalues of $A+B$
$A,B$ are symmetric matrices, $A$ has eigenvalues in $[a,b]$ and $B$ has eigenvalues in $[c,d]$ then we need to show that eigenvalues of $A+B$ lie in $[a+c,b+d]$, I am really not getting where to start. What I know $A,B$ have real eigenvalues, they are diagonalizable also.
AI: We can use Rayl... |
H: Proving a function satisfies the binomial recurrence relation and that it equals $\binom{n-k+1}{k}$
I have the recurrence relation $g(n,k)=g(n-2,k-1) + g(n-1,k)$ for all $k\geq1$
with the boundary conditions $g(n,k)=0$ if $n<2k-1$ and $g(2k-1,k)=1$
What I'm trying to do is define a new function by the equation $ h... |
H: NP-hardness reduction
Although I know the notion of polynomial time reduction since many years, I am currently confused about the following problem.
In a reduction from 3-SAT to 3-Coloring, one constructs (in polytime) a graph $G_F$ out of a given 3-CNF formula $F$, s.t. $F$ is satisfiable if and only if $G_F$ is 3... |
H: Prove that the product of four consecutive positive integers plus one is a perfect square
I need to prove the following, but I am not able to do it. This is not homework, nor something related to research, but rather something that came up in preparation for an exam.
If $n = 1 + m$, where $m$ is the product of fou... |
H: The language that contains no proper prefixes of all words of a regular language is regular
Let $L$ be a regular language. I need to prove that the language
$$M_L = \{w \in L \; | \forall x \in L \; \forall y \in \Sigma^+ : w \neq xy \}$$
that contains all words of L that do not have a related proper prefix in L i... |
H: Counting matrices over $\mathbb{Z}/2\mathbb{Z}$ with conditions on rows and columns
I want to solve the following seemingly combinatorial problem, but I don't know where to start.
How many matrices in $\mathrm{Mat}_{M,N}(\mathbb{Z}_2)$ are there such that the sum of entries in each row and the sum of entries in eac... |
H: Computing probabilities involving committees
A committee consisting of 6 members is randomly selected from 25 students,
5 teachers, and 10 parents. I wish to find the following:
(i) the probability of having no teacher on the committee
(ii) the probability of having neither students nor parents on the committee.
... |
H: Why metric defined on $\mathbb{R}^2\times \mathbb{R}^2$ by $(a,b)\mapsto | a_1 - b_1| +| a_2 - b_2| $ is known as taxicab metric?
$\mathbb{R}^2$ with the function defined on $\mathbb{R}^2\times \mathbb{R}^2$ by $(a,b)\mapsto | a_1 - b_1| +| a_2 - b_2| $ where $a = (a_1, a_2)$ and $b = (b_1, b_2)$ is a metric. I wo... |
H: Where does complex exponential come from?
The complex exponential function is defined as : $$e^{ix} = \cos x + i\sin x$$ It shares most of its properties with real exponential and it allows a lot of trigonometric calculations such as de Moivre's formula : $$(\cos x+i\sin x)^n = \cos{nx}+i\sin{nx}$$
But where does t... |
H: Is there any sense in zero-padding a matrix to make it $n\times n$ and find its eigenvalues?
I am debuging my Kalman filter and the Jacobian matrix of partial derivatives of h(measurement function) with respect to x(state) is not n×n, it is 13×16.
$\displaystyle \quad\ \bf H_{[i,j]}$ = $\bf \frac{\partial h_{[i]}}... |
H: Can we get uncountable ordinal numbers through constructive method?
As we know, $2^{\aleph_0}$ is a limit ordinal number, however, it is greater than $\omega$, $\omega+\omega$, $\omega \cdot \omega$, $\omega^\omega$, $\omega\uparrow\uparrow\omega$, and even $\omega \uparrow^{\omega} \omega$.
My question is can we g... |
H: Closed-form for eigenvectors of rotation matrix
For matrices that are elements of $SO(3)$ is there a formula for the eigenvectors corresponding to the eigenvalue $1$ in terms of the entries of the matrix?
AI: Let $A \in SO(3)$. The matrix $A-A^T$ is skew-symmetric, hence of the form
$$
\begin{pmatrix}
0 & a & b \\
... |
H: Limit on a topological vector space
in the Wikipedia article on Gâteaux derivative , the limit of a function between two topological vector spaces is taken. How is the limit defined on a topological space for a function ? I find articles on net and filters for corresponding notions on topological spaces, but those ... |
H: Weak-* convergence on subinterval
If $U_{n} \to U$ weakly* in $BV[0,1]$ is it true that $U_{n}\mid_{[0,\frac{1}{2}]} \to U\mid_{[0,\frac{1}{2}]}$ weakly* in $BV[0,\frac{1}{2}]$?
I'll give some explication. Let $U_{n} \in BV[0,1]$ and $U_n \to U$ weakly*, i.e. for any $y \in C[0,1]$
$$
\int\limits_{0}^{1} y(t)dU_... |
H: Parabolic PDE local and global existence
If you have a local solution to a parabolic PDE (say we know it exists (weakly anyway) from time 0 to T), then if the solution is bounded in an appropriate way (in which norms?) then we can apparently extend the solution globally. Can someone refer me to these results or exp... |
H: Visualize a difference equation with Matlab
I have a difference equation for a Single Pole Infinite Impulse Response Filter, defined on a discrete time-series:
$y[n]-(1-\alpha)*y[n-1]=\alpha*x_n$
While the []s brackets refer to a position n within the series. I'm looking for a visual way to represent this in order ... |
H: How to prove that the sum and product of two algebraic numbers is algebraic?
Suppose $E/F$ is a field extension and $\alpha, \beta \in E$ are algebraic over $F$. Then it is not too hard to see that when $\alpha$ is nonzero, $1/\alpha$ is also algebraic. If $a_0 + a_1\alpha + \cdots + a_n \alpha^n = 0$, then dividin... |
H: Principal ideal ring analytic functions
Could someone sketch a proof and explain me in words, why the set of analytic functions on $\mathbb{C}$ does not form form a principal ideal ring?
Thank you!
AI: While I am not very familiar with analytic functions themselves, I would imagine that one could construct an infin... |
H: Is this computation of the tensor product correct?
I'm reading the proof of the existence of the tensor product. If $M,N$ are two $R$-modules then we can construct the tensor product $T$ as the quotient $C/D$ where $C$ is the free module over $M \times N$ and $D$ is the submodule generated by the set of all element... |
H: Why (finite) Blaschke products are actually rational fractions?
I have found in several books the following affirmation :
Let $f: \Delta \rightarrow \Delta$ be a non constant holomorphic function that extends continuously to $\overline{\Delta}$, $\Delta$ being the open unit disk. Then $f$ is a finite Blaschke prod... |
H: Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
In the book "Zero: The Biography of a Dangerous Idea", author Charles Seife claims that a dart thrown at the real number line would never hit a rational number. He doesn't say that it's only "unlikely" or that t... |
H: What is Gal($\mathbb{F}_{q^k}/\mathbb{F}_q)$?
I know that if $q=p$ (where $p$ is prime) then Gal($\mathbb{F}_{p^k}/\mathbb{F}_p)$ is cyclic of order $k$.
I heard that in general (for $q=p^m$) the galois group is cyclic of the order of the extension (i.e. that :Gal($\mathbb{F}_{q^k}/\mathbb{F}_q)=C_{[\mathbb{F}_{q^k... |
H: Rational embedded irreducible curves in a complex surface.
Given a complex surface $X$ and an embedded irreducible compact curve $C$ with its arithmetical genus $g(C) = 0$, how can one show that $C$ is non-singular ?
Thanks for your answers!
AI: Given an irreducible complete curve $C$ and its normalization $\tilde... |
H: How to induce a connection on a submanifold?
Suppose an affine connection is given on a smooth manifold $M$ and let $N\subset M$ be an embedded submanifold. Is there a canonical way of defining an induced connection on $N$?
In classical differential geometry of smooth surfaces in Euclidean 3-space, the correspondi... |
H: combinatorics: The pigeonhole principle
Assume that in every group of 9 people, there are 3 in the same height.
Prove that in a group of 25 people there are 7 in the same height.
I started by defining:
pigeonhole- heights.
pigeons-people.
I do not know how to use the assumption.
thanx.
AI: Let $n_i$ be the number o... |
H: can the statement "a simplicial set is the nerve of a category if and only if it satisfies a horn-filling condition" be tweaked for groupoids?
For some reason I convinced myself that a simplicial set (or maybe I mean directly Kan complex) is homotopy equivalent to the nerve of a groupoid if and only if it has no hi... |
H: Weak a.s. convergence VS a.s.weak convergence
Let's consider a sequence $(\mu_n)_n$ of random probability measures on $\mathbb R$, and let $C_b$ be the Banach space of bounded continuous functions on $\mathbb R$. I am considering the following types of convergence, where $\mu$ is some non-random probability measure... |
H: Calculate $\ln(2)$ using Riemann sum.
Possible Duplicate:
Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?
Show that
$$\ln(2) = \lim_{n\rightarrow\infty}\left( \frac{1}{n + 1} + \frac{1}{n + 2} + ... +
\frac{1}{2n}\right)$$
by considering the lower Riemann sum of $f$ where $f(x) = \frac{... |
H: Primes of the form $p=a^2-2b^2$.
I've stumbled upon this and I was wondering if anyone here could come up with a simple proof:
Let $p$ be a prime such that $p\equiv 1 \bmod 8$, and let $a,b\geq 1$ such that
$$p=a^2-2b^2.$$
Question: Is $b$ necessarily a square modulo $p$?
I have plenty of numerical data to suppor... |
H: Prove the derivative of $(a-x) / x$ by definition
Hello and thanks in advance for any help!!
I currently have to get to the derivative function of $\frac{a-x}{x}$ by definition.. that is
$$\lim_{h\to0} \frac{\frac{a - (x+h)}{x+h}- \frac{a-x}{x}}{h}$$
So it's kind of a little mess for a newbie in algebra like me. I'... |
H: The Affine Property of Connections on Vector Bundles
Given any two connections $\nabla_1, \nabla_2: \Omega^0 (V) \to \Omega^1 (V)$ on a vector bundle $V \to M$, their difference $\nabla_1 - \nabla_2$ is a $C^\infty (M)$-linear map $\Omega^0 (V) \to \Omega^1 (V)$.
Question: I have difficulties swallowing the implic... |
H: Variation Tolerance
I came across a statement in my course book that 3$\sigma$ is considered as a means of tolerance. Can anyone explain it to me. I understand that +3$\sigma$ to -3$\sigma$ constitutes 99% of the samples. But how is tolerance related to this. Is it like a threshold we are putting on the sample's va... |
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