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Despite all my efforts trying to crack these, i haven't been able to do so. An approach that i've tried gives me somewhat of an asymptotic approximation, but still fails to produce the values near x=0.
The method i've tried is to express the sum as an integral and then integrate it to obtain it's value but apparently t... |
Suppose that $(e_n)_{n=1}^{\infty}$ is an orthonormal basis in a Hilbert space $H$, and let $(f_n)$ be an orthonormal sequence in $H$ such that
$$\sum_{n=1}^\infty \|e_n-f_n\|<\infty.$$
How can one show that $(f_n)_{n=1}^{\infty}$ is also a basis in $H$.
|
What is the total derivative of $f(x,y(x,z))$ with respect to $x$? Is it $$\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}?$$
If this is correct, what is $\frac{\partial f}{\partial x}$? It seems to me that partial $f$ partial $x$ is equal to the derivative of $f$ with respect t... |
I have the subset $\left[0,1\right] \backslash \mathbb{Q}$ in $\mathbb{R} \backslash \mathbb{Q}$.
Am I right in thinking that this set is open and not closed in the space given?
Also, how do I go about finding the interior, closure and boundary?
|
Suppose having a model $f(x)=y$ where $f$ is unkown. Moreover, suppose you have some data points for this model i.e. $(x_1,y_1), (x_2,y_2), \dots , (x_n,y_n)$.
If one can find an approximate of $f $ called $\tilde{f}$ using the given data points.
When such aproximation is called interpolation? should the approximati... |
Here is the question I am stuck on:
Consider $T \in \mathrm{Hom}(\Bbb{R}[x]_{\le 2} ,\Bbb{R}[x]_{\le 2} )$ given by
$$ (Tf)(x)=\int_{-1}^1(x-y)^2f(y)dy-2f(0)x^2$$
for all $f \in \Bbb{R}[x]_{\le 2}$. Find all eigenvalues and eigenvectors for T.
So far, I'm trying to apply T to the monomial basis of $\Bbb{R}[x]_{\le 2}$... |
So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications.
In particular, if:
$$Lf(x) = \int_{X} k(x,y) f(y) du(y)$$
So then, naturally, if $\mathbf{A}$ is an $m,n$ matrix with entries $a_{i... |
So, i'm solving last's year's exams in Mathematical Analysis and i've found one interesting. It says: The equation $e^{-4x}=5x^2$ has one root close to (nearby) 0. By approaching $e^{-4x}$(close to 0) with a second degree polynomial, find an approach of this root.
Now, my mind went to using Taylor's Theorem. So i've f... |
I have a $20$ sided die.
I roll it $5$ times. Each time I get a number between $1-5$.
What are the odds for or against this outcome?
(It would be great to have a formula. Intuitively, I don't think the answer could be merely $1/4$, but I don't know how to calculate it.)
|
Suppose that the sequence $(a_n)_{n \in \mathbb{N}}$ is bounded. Prove that the sequence $(c_n)_{n \in \mathbb{N}}$ defined by
$$
c_n = \frac{a_n^3+5n}{a_n^2+n}
$$
is convergent and find its limit.
Would appreciate advice as to how to approach this question.
|
Let $K \subset L$ an algebraic extension. I want to prove for $a,b \in L$ that
$$[K(b): K] \geq [K(a,b): K(a)],$$
Where for example $[K(b):K]$ is the degree of the field extension $K \subset K(b)$. My ideas so far are:
The degree of $[K(a):K]$ is equal to $\deg(f_K^a)$, where $f_K^a$ is the minimum polynomial. So
$$K... |
Suppose that $T$ is a linear transformation and that $T(u)$ and $T(v)$ are linearly independent. Prove that $u$ and $v$ are linearly independent.
I have no idea where to start in this case. Just need a nudge in the right direction.
|
When dealing with quadratics, completing the square is ubiquitous, and I can summarise my interpretation of it as the formula:
$$x^2-2ax=(x-a)^2-a^2$$
Likewise, when working with circles (and, more generally, conic sections) in $\mathbb{R}^2$, it's customary to complete the square in a similar way. Sometimes this passe... |
Let $c_{0}$ be the space of sequences which converge to $0$. Let $W$ be a subspace of $c_{0}$ defined as
$$W =\bigl\{ \{y_{n}\} \in c_{0} : \ y_{1}+y_{2}+y_{3}=0\bigr\}$$ How does one calculate the $\operatorname{dim}(c_{0}/W)$?
My attempts:
We know that $\operatorname{dim}(c_{0}/W) = \operatorname{dim}(c_{0})-\opera... |
So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance.
I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and constant density $\rho$. Now I want to study longitudinal waves in this rod when its end is attached to a wall. So first i de... |
Let $G$ be a group of $143$ elements acting on a set $X$ of $108$ elements. I need to prove that there is one element whose Stabilizer is the whole group.
I have tried doing it using the Orbits Equation, using that the possible indexes for the stabilizer are $1,11$ and $13$, but since I don't have any information on th... |
When I present partial fraction decomposition, there are typically two approaches to solve for the coefficients of the fractions:
Clear the fractions, multiply out the products, and equate coefficients (of the indeterminate). This leads to a system of linear equations.
Clear the fractions, and substitute several valu... |
Let $U= \{ p \in \mathscr P_4\mathbb{R} \ | \int_{-1}^1 p(x)dx=0\}$.
a.) Find a basis for $U$.
b.) Find a subspace $W$ of $\mathscr{P_4}(\mathbb{R})$ such that $\mathscr{P_4}(\mathbb{R})= U \oplus W$.
I am having trouble thinking of a set of polynomials which fulfill these requirements. Any thoughts would be appreci... |
Is there a way (there has to be), I can expand an expression like this?
$$\log_2 (3f(n)^n)$$
P.S. This part of an assignment I'm working on, please do not give solutions
|
Let $X$ be a locally compact space and consider $C_0(X)$.
We denote $b(X)$, by the set of all bounded functions on $X$. It is easy to be checked that $b(X)$ may be considered as a C*-sub algebra of $B(\ell^2(X))$ via:
$$\gamma:b(X)\to B(\ell^2(X)): \gamma(\phi)f=\phi f$$
$\gamma(b(X))$ is also a von Neumann algebra. I... |
Let $\mathbf{Top}$ be the category of topological spaces, $\mathbf{RS}$ the category of ringed spaces and $\mathbf{LRS}$ the category of locally ringed spaces.
There are forgetful functors
$$
U_{\mathbf{LRS}}: \mathbf{LRS} \to \mathbf{RS}
$$
(the inclusion of the (non-full) subcategory) and
$$
U_{\mathbf{RS}}: \mathbf{... |
Let $G_1$ be a non-compact topological space and let $G_2$ be a generic topological space. What are the compact sets in the product $G_1\times G_2$?
Surely we can take the sets of the form $K_1\times K_2$ where $K_1$ and $K_2$ are respectively compact sets of $G_1$ and $G_2$. Is this right? What are the others?
|
I have the following:
\begin{equation}
\sum_{j=0}^{\alpha_{2}}C\, e^{az}\, M(-\alpha_{2}+j,-\alpha_{3}+j,-\lambda z)\Bigg|_{z=-\infty}^{0}
\end{equation}
$C$ is a constant, $a,\alpha_{2},\alpha_{3}, \lambda$ are all positive. Also, in my case, the arguments of $M$ give me a polynomial of degree $\alpha_{2}$.
How can I... |
Given:
$X_1$ and $X_2$ are the roots of the equation $ax^2+bx+c = 0$ $a\neq 0$
expand $X_1^3 + X_2^3$ using the parameters a,b and c
Here's what I tried to do:
$X_1^3 + X_2^3 = $ $(X_1\cdot X_2)\cdot(X_1^2-X_1\cdot X_2+X_2^2)$
From Viet's formulas:
$(X_1\cdot X_2)(X_1^2-X_1\cdot X_2+X_2^2)$ = $\frac{-b}{a}\cdot(X_1^2+... |
Let $L$ be a field and $K$ be an extension of $L$ such that $[K:L]=2$.
Prove that $K$ is a normal extension.
What I have tried : Let $ f(x)$ be any irreducible polynomial in $L[x] $ having a root $\alpha$ in $K$ and let $\beta$ be another root. Then I have to show $\beta \in K$. I already have $L(\alpha)=K$, but n... |
$$\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$$
where V is the unit sphere.
No information is given about n but I assume it is an integral. All I could think to do was to convert to spherical co-ordinates and use reduction formulae, but I ended up with really messy answers. Any help would be brilliant.
|
if $ k \geq 0$ Prove that $\frac{(a+1)^k}{a^k+1} \leq 2^{k-1}$
Proof (induction)
let k = 1 then $$\frac{(a+1)^1}{a^1+1} \leq 2^{1-1}$$ , implies that $$ 1 \leq 1$$ so p(1) is true.
Inductive step
$p(l)$ : $\frac{(a+1)^l}{a^l+1} \leq 2^{l-1}$
$p(l+1)$ : $\frac{(a+1)^k{l+1}}{a^{l+1}+1}$ this is where i am having a dif... |
If I want to enlarge a shape, $A$, by scale factor $k$ about $\left(0,0\right) $ I multiply each point (in the form $\begin{bmatrix}x\\y\end{bmatrix}$) by $kI$.
However, I can't work out a general form for having the centre of enlargement at any other point. I managed to work out that
$$k\begin{bmatrix}-1 & 2\\2 & -1\... |
When I use trigo substitution to solve an integral I get an expression like that:
$$\frac{1}{4}\tan\left(\arcsin\left(\frac{x-2}{2}\right)\right)+C$$
How can I simplify it?
|
Let's consider real measurable functions defined in a bounded interval. As long as a function is bounded, oscillations at least cannot make the volume under the graph of the function infinite. But I'm curious that there is really no obstacles from very fast or irregular oscillations that make the function cannot be int... |
Let $X_0,X_1,...,X_n,...$ be an irreducible Markov chain with finite state space. Define $τ_{x,0}^+=0$, and $τ_{x,k}^+=\min\{t:t>τ_{x,k-1}^+,X_t=x\}$. In plain words, $τ_{x,k}^+$ is the time of the $k$th return to $x$. Let $Y_k=\sum_{t=\tau_{k-1}^+}^{\tau_k^+-1}X_t$, which is the sum of $X_t$ between the $(k-1)$th and ... |
Suppose $x_1,x_2,\ldots,x_n$ are unknowons satisfying the constraint
$a_1x_1 + \cdots + a_nx_n ≥ b$, where $a_1, \ldots , a_n, b ≥ 0$.
Then the minimum possible value of the expression $c_1x_1 + \cdots + c_nx_n$ is $c_i\cdot b/a_i$ where $i$ denotes the index for which
the ratio $c_i/a_i$ is minimum. How to prove thi... |
I am reading a piece of a physic paper where a function is mentioned without being given a name or reference - I guess it is a canonical one and that I should be familiar with.
The expression goes as:
$$|_2F_1(a,b;c;z)|$$
with $a,b,c \in {\mathbb R}$ and $z \in {\mathbb C}$. Also I am fairly sure the lowerscript 2 is a... |
It is well-known that the inclusion $f[\overline A] \subseteq \overline{f[A]}$ (for every subset $A$) characterizes continuous functions.1
Asking similar questions for other closure operators seems to be a rather natural question. So this leads me to the question:
Let $X$, $Y$ are vector spaces and $f\colon X\to Y$. W... |
Say I have
$$
f(x)=\frac{ax+b}{cx+d}
$$
I wish to assgin $f(x)=1$ and simplify to have
$$
ax+b-cx-d=0.
$$
How may I do this in Maple? I tried
simplify(f=1,x)
but it didn't work.
|
I Need to prove this by contradiction :
If $a$ is Irrational then $\frac{2a-3}{2a+3}$ is Irrational.
I did: Iff $p$ is Irrational, then $\frac{2a-3}{2a+3}$ is Rational and a Rational number can be written as $\frac{p}{q}$ where p and q are integers and q is different than zero so,
$\dfrac{2a-3}{2a+3}=\dfrac{p}{q}$ w... |
I am given this:
Find the number of elements in $A \cup B \cup C$ if there are 50 elements in $A$, 500 in $B$, and 5,000 in $C$ if:
$A \subseteq B$ and $B \subseteq C$
The sets are pairwise disjoint.
There are two elements common to each pair of sets and one element in all
three sets
For #1, would the answer simply b... |
If I got it right, given two planes in $N$-dimensional space ($N\gg2$), their 2 principal angles ($\theta_1$, $\theta_2$) and 4 vectors ($\vec{a}_1$, $\vec{a}_2$, $\vec{b}_1$, $\vec{b}_2$) can be found, such that the first plane is defined by the vectors $\vec{a}_1$ and $\vec{a}_2$, the second plane by the vectors $\ve... |
Is the category of monoids cartesian closed? Why?
I read Steve Awodey's "Category Theory", and could not solve the exercise in chapter 6, stated above.
Here I speak of the "category of monoids" as the category with objects monoids and arrows homomorphisms between monoids.
|
I would like some help sketching the critical points of a given function.
The function $f$ is given by
$f (x, y) = (\sin x + \sin y)^2$ for all $(x, y)$ in $\mathbb R^2$.
The critical points we have gotten so far are:
$(((π/2)+2πn),(π/2)+2πn)$, $((π/2)+ 2πn,(3π/2)+2πn)$,
$((π/2) + πn, (π/2) + 2πn)$, $((3π/2) + 2πn, (π/... |
For my bachelor project I'm studying the symmetry of the Platonic solids (as a start at least). I computed the symmetry group of the tetrahedron by labeling the vertices, and the cube by labeling the four diagonals through its center. For the dodecahedron and icosahedron by advisor suggested a labeling system based on ... |
I think I've found a proof of the following, but my proof is horrible, and I feel like I've made a mistake or that I've missed an important principle:
Theorem: Let $G$ be a finite group of odd order and suppose $g \in G$ is conjugate to $g^{-1}$. Then $g = e$.
Proof: Since $g$ is conjugate to its inverse, there is an e... |
Show that the orthogonal projection is linear.
Let $x_i=y_i+z_i$, where $x_i\in X$, $y_i\in Y$, $z_i\in Y^\perp$, and $\alpha,\beta$ be scalars. Then \begin{align}P(\alpha x_1+\beta x_2)&=P(\alpha(y_1+z_1)+\beta(y_2+z_2))\\&=P(\alpha y_1+\beta y_2+\alpha z_1+\beta z_2)\end{align}
Now I know $\alpha x_1 + \beta y_1 \in ... |
Supposer that $V = W_{1} \oplus W_{2}$, $f_{1}$ and $f_{2}$ are inner product at $W_{1}$ and $W_{2}$, respectively. Show that there is only one inner product $f$ in $V$ such that
i) $W_{2} = W_{1}^{\bot}$;
ii)$f(\alpha,\beta) = f_{k}(\alpha,\beta)$ when $\alpha,\beta \in W_{k}$, $k = 1,2.$
my first impression is that t... |
$\displaystyle \frac{\text{d}z}{\text{d}x} = \frac{2a(x-z)}{1-ax}$ where $a$ is a constant. $(x>0, a\ne 0)$
So after rewriting the above as $\displaystyle \frac{\text{d}z}{\text{d}x} + \frac{2az}{1-ax} = \frac{2ax}{1-ax}$ I tried using an integrating factor of $\displaystyle \frac{1}{(1-ax)^2}$ to solve this, but I en... |
Defn: A set $X$ is connected if there do not exist non-empty, disjoint open sets $U,V$ s.t $U$ $\cup$ $V$ $=X$.
I thought intuitively that this meant that this was like the English dictionary definition of connected- there is no gap in the set.
It is easy to see $(0,1) \cup (2,3)$ is disconnected. Intuitively it seems ... |
Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum:
$S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$
Suppose $S$ is constant for all $\alpha \in R$. Does it necessarily follow that $r_1=r_2=...=r_n=0$?
|
There is a set of points on the Cartesian coordinate system. How do I group points into several circles with a given radius and meeting the following requirements:
each point belongs to a group at least， specifically,
circles can intersect
a point can belong to several circles
points can be placed on the boundray
... |
Prove: If $x$ is an isolated point of a set $S$, then $x \in \mathrm{bd} \, S$.
I have two ways to solving this problem, but I believe the first one has a logical issue which I will explain below. Thanks for any help.
Solution 1 Let $x$ be an isolated point of a set $S$, and let $N$ be an arbitrary neighborhood of $x$.... |
I want to find the expression for the following integral
$$\int_0^\infty\text{d}x\frac{e^{i k x}}{x}$$
I have tried deriving with respect to $k$, transforming into an integral over the whole real line... with no luck. Would anyone give me a hint or the actual steps to take? Thank you so much.
EDIT: Following @Dr. MV's ... |
A factory has $3$ production lines $A, B$ and $C$ contributing $20$%, $30$% and $50$%, respectively, to its total output. The percentages of substandard items produced by lines $A, B$ and $C$ are $10, 5$, and $2$, respectively.
a) If an item is chosen at random from the total output, what is the probability that it wil... |
How to prove the following:
For any complex numbers $z_1,...,z_n$, there exists a subset $E$ of $\{1,...,n\}$ s.t.
$$\left| {\sum\limits_{j \in E} {{z_j}} } \right| \geqslant \frac{1}{6}\sum\limits_{j = 1}^n {\left| {{z_j}} \right|} $$
|
Find the inverse of $f$ if $f(x)=$
$$
\begin{cases}
\sqrt{2-x}, &\text{for $x<0$}\\
1-x^2, &\text{for $x \ge 0$} \\
\end{cases}
$$
My effort
For $y=\sqrt{2-x}$ ,we find
\begin{array}{c}
y=\sqrt{2-x} \\
y^2=2-x &\text{(I can do this since $\sqrt{2-x}>0$ ,for $x<0$)} \\
x=2-y^2
\end{array}
So for this piece of funct... |
This question is inspired by an answer to Nitrogen's answer to my Are the path connected components of $\Omega S_1$ contractible? .
Here we are asked whether the space of paths in $\mathbb{R}$ sending $0$ to $0$, $1$ to $n$, with $[0,n] \subset \mathbb{R}$, is contractible? The answer is yes because $\mathbb{R}$ is co... |
Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix
$$X:=\left[\begin{array}{cc}
A & B \\
C & D \end{array}
\right],$$
where $A \in \mathbb F^{r_1 \times s_1}$, $B\in \mathbb F^{r_1\times s_2}$, $C \in \mathbb F^{r_2\times s_1}$ and $D \in \mathbb F^{r_2\times s_2}$.
Q... |
Short version for people who don't like reading:
Let $f\colon\mathbb{R}\to\mathbb{R}$ be $1$-periodic, measurable and bounded. Is it true that, for almost all $x$, the average of $f(x)$, $f(x+\frac{1}{n})$, $f(x+\frac{2}{n})$, …, $f(x-\frac{1}{n})$ tends to $\int_0^1 f$ when $n\to+\infty$?
And now for a more detaile... |
$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$
$2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$
$3.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R})\mid A \text{ is invertible}\}$
For the first o... |
We have the next two quadratic functions:
$ ax^2 + bx + c = 0 $
$ mx^2 + nx + p = 0 $
If $ a/m = b/n = c/p $ then they have the same roots.
What is the intuition behind this statement?
|
Let $f:\mathbb R\to\mathbb R$ be continuous such that $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}~\forall~x,y\in\mathbb R.$ How to show that $f$ is a straight line?
|
I'm having trouble to understand a step of a proof.
Let $S$ be a subset of $\mathbb{R}$. Prove that $S$ is null (Lebesgue measure).
The book says the following:
"It is clear that we can restrict ourselves to subsets of the unit interval; thus assume that $S\subset [0,1]$"
I don't understand why that is true. I've trie... |
The following problem is from A First Course in Probability by Sheldon Ross, and it was assigned as homework by my professor. I was wondering if you guys could help me find a answer to the problem.
$A$ and $B$ flip coins. $A$ starts and continues flipping until a tails occurs, at which point $B$ starts flipping and co... |
$$\int \frac{dx}{(9+x^2)^2}$$
$x=3\tan\theta$
$dx=\frac{3}{\cos^2\theta}d\theta$
$$\int\frac{\frac{3}{\cos^2\theta}}{(9[1+\tan^2\theta])^2} \,d\theta = \int\frac{\frac{3}{\cos^2\theta}}{(\frac{9}{\cos^2\theta})^2} \, d\theta =\frac{3}{81} \int \cos^2\theta \, d\theta=\frac{3}{162}\int (\cos2\theta+1) \, d\theta$$
$u=... |
Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., x^c_n$. How to show that with these two operation $V$ is a subspace?
In my opinion, $x+'(-1\cdot 'x)=x+'(x^{-1}_1, ..., x^... |
What is the domain of this function
$F(x)=\frac{1}{x}|x^2 - 1|$
Can someone please tell me how to find it ?
|
Are the functions $f, g, h$ given below linearly independent? If they are not linearly independent, find a nontrivial solutions to the equations below:
$$f(x)=e^{2x}- \cos(9x), \quad g(x)=e^{2x}+ \cos(9x), \quad h(x)= \cos(9x)$$
My take so far is that, I know these functions are not linearly independent (meaning this... |
In Farb and Margalit's "A Primer on Mapping Class Groups" we have the following
Proposition 1.7: Two transverse simple closed curves in a surface $S$ are in minimal position iff they do not form a bigon.
The proof for the reverse direction for the case $\chi(S) < 0$ starts as follows:
Assume simple closed curves $\a... |
Which change of variable should I do to solve this PDE?
$u_{xy}(x, y) + au_x(x, y) + bu_y(x, y) + abu(x, y) = 0$
|
In dealing a $13$-hand card that with at least $1$ suit that does not appear, I came up with this:
We can choose $3$ of the $4$ suits, as in $3 \choose 4$, and then $13$ cards out of the $39$ cards (from the $3$ suits), which is $39 \choose 13$. Multiply them together to get $3 \choose 4$$39 \choose 13$, and there we h... |
I want to find the extreme values of the function $f(x,y,z) = 2x + 2y + z$ under the constraints $x^2+y^2+z^2 \le 2$ and $x^2 + y^2 \le z$
The task is to use a parametrization of the two constraints/conditions to find stationary points of f on the boundary.
I don't even know where to begin with this - I can solve it ... |
I want to integrate a function as $f(x)=\sin^{-1}x$. What should be the proper method of doing it?
|
Let $A,B,C,D$ be sets. Consider $A\times B$ and $X\subseteq A\times B$. Is it true that $X$ has the form $A'\times B'$ where $A'\subseteq A$ and $B'\subseteq B$ ?
At the same time is it true that $(A\times B)\cup (C\times D)=(A\cup C)\times (B\cup D)$?
For me are both false but I don't know how to prove them.
thank you... |
There is a well-known theorem in Euclidean geometry (Eucl. VI-19) that says that the ratio of the areas of any two similar polygons is equal to the square of the corresponding ratio of similarity.
In many texts treating the notion of area in the sense of Jordan one can find the analogue of the above result for Jordan ... |
I was trying to understand the union of a chain of cardinalities and I found this equation
$$\kappa=\bigcup_{\alpha<\kappa} \alpha$$ for any cardinal $\kappa$ in the answers to this question.
Can every set be expressed as the union of a chain of sets of lesser cardinality?
I dont understand how this could be possible ... |
Let the subspaces $W_1=\{\begin{pmatrix}a&b\\-b&a \end{pmatrix}|a, b\in \mathbb{R}\}$ and $W_2=\{\begin{pmatrix}c&d\\d&-c \end{pmatrix}|c, d\in \mathbb{R}\}$ of $M_{2\times 2}(\mathbb{R})$,how to show that $M_{2\times 2}(\mathbb{R})=W_1\oplus W_2$?
By the definition of direct sum, $V= W_1\oplus W_2 \iff V=W_1+W_2 $ an... |
I am beginning to study dynamical systems. We are given $U \subset \mathbb{R}^n$ open, a vector field $f: U \to \mathbb{R}^n$, and an associated evolution operator for fixed $t \in \mathbb{R}$ $\Phi_t : U \to \mathbb{R}^n$ given by $\Phi_t(x_0) = x(t)$, where $x$ is the solution to the ODE
\begin{equation}
\dot x = ... |
This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it.
$$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$
How would you go about this?
|
(Edit: I need to revise this question with my original intent. Pls do not answer it yet. Thanks.)
Given the regular $n$-gon formed by the $n$-th roots of unity. For some $n$, how do we find $\sqrt{n}$ using the sum/difference of line segments?
$n=5:$
It is enough to use one line segment: If $x^5=1$, then it can be ... |
I'm trying to come up with $x'=Ax$, which is a system of linear differential equations, whose flow satisfies $\lim\limits_{t\to\infty} \lvert e^{tA}x\lvert = 0$ for all $x\in \mathbb{R}^n$, but $\lvert e^{tA} x\lvert$ is not monotonically decreasing.
I thought that in order to come up with an example of such a system, ... |
Here's where I'm at, not sure where to go from here.
Two vectors are orthogonal if their dot product is $0$. Knowing that;
$$u \cdot (v - \operatorname{proj} u(v)) = 0$$
$$u \cdot \left(v - \frac{u \cdot v}{u \cdot u} u\right) = 0$$
$$u \cdot v - \frac{u \cdot v}{u \cdot u} u = 0$$
I'm not sure where to go from here, I... |
I have been reading a number of "network science" papers where the authors perform transformations on networks that seem to preserve the topology of those networks. By "topology", I mean a collection of heuristics commonly used to probe the network: modularity, clustering coefficient, and mean path length. Is there a... |
This is the theorem about in Royden's real analysis book. And in the book there is a footnote I am confusing:
Can anyone help me understanding it with examples~~~
|
Let $R$ be an infinite set of complex numbers $\rho$ with $0<\Re(\rho)<1$, $x$ be a nonzero real number and consider the summation $\sum_{\rho} x^{\rho}$. Is this sum differentiable with respect to $x$ ?
|
So I am self-studying Complex Analysis. The idea behind the proof is clear, and I understand the intuition etc. There are a few statements that are really giving me trouble accepting this proof though. I also understand there are many ways of showing this result, but I would like to see it this way.
Definition of Limit... |
My textbook (RCA, Rudin) asserts (in the proof of theorem 5.15) that
$$
\lim_{n\to\infty}\lVert{\sum_{k=-n}^n}e^{ikt}\rVert_1=\infty.
$$
Why is this true?
I tried using Euler's formula to reduce the above to
$$
\lim_{n\to\infty}\int_{0}^{\pi}\lvert 1+2\sum_{k=1}^n\cos(kt)\rvert~dt=\infty,
$$
but I don't know how to sho... |
I am trying to understand the definition of the sign of a permutation $\pi$.
My textbook only mentions that $\operatorname{sgn}(\pi) = (-1)^k$ , where $k$ is the number of transpositions .
But I have also found online that $\operatorname{sgn}(\pi) = (-1)^{\operatorname{inv}(\pi)}$, where $\operatorname{inv}(\pi)$ is ... |
Consider a random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ such that $X:\Omega\rightarrow \mathbb{R}$.
Suppose that $X\sim N(\mu, \sigma^2)$.
Consider a random function $T(X):\mathbb{R}\rightarrow \mathbb{R}$.
Let $\{Y_n\}$ be a sequence of random variables all defined on the probabil... |
Let $A^{\mathbb N}=\prod_{m\in\mathbb N} A_m$ be the infinite countably cartesian product of the sets $A_m$. Let $A_i'$ be a subset of $A_i$ for $i=1,...,n$. Is it true that $A_1'\times A_2'\times...\times A_n'\times A^{\mathbb N\setminus\{1,...,n\}}$ is equal to $A_1'\times A_2'\times...\times A_n'\times A^{\mathbb N}... |
Let $(\Omega,\mathcal{F})$ be a measure space and $X$ mapping from $\Omega$ to $\mathbb{R}$. Assume that $X^{-1}((a,b])\in \mathcal{F}$ for all intervals. Prove that $X$ is a random variable.
First, I have shown that $\mathcal{B}(\mathcal{R})$ is the $\sigma$-algebra generated by all intervals of the form $(a,b]$. Us... |
Lets say we have a limit $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = +\infty$, then is it safe to assume that $\lim_{n \rightarrow \infty} \frac{b_n}{a_n} = 0$?
|
$$y''+ay'+by=0,\quad y(0)=y(1)$$
where $a$ and $b$ are positive real numbers.
Let $V$ be the set of all the solutions of this equation. Then the dimension of $V$?
The answer to it that I think is $1$ but still want to know the exact logic behind this. Help from someone would be highly appreciated.
|
I have an optimization problem where I need to calculate the maximum of the following function
$$
f(x,y,z) = (2^{xy}{z \choose y})^{z+1}
$$
where
$$
(z+1)(a+y(\lceil{\log_2{(z+1)}}\rceil+x))\leq C
$$
$a,C$ are constants.
$a,x,y,z$ are all positive integers.
Using approximations is allowed. Just need to get close.
I ha... |
Let $k$ be a field of characteristic zero and let $A$ be a finitely generated $k$-algebra. Let $B=A[x_1,\ldots,x_n]$ be the polynomial ring over $A$ and let $I \subseteq B$ be an ideal such that $B/I$ is flat over $A$. Let $f_1, \ldots, f_r \in k[x_1,\ldots,x_n]$ and let $J \subseteq B$ be the ideal generated by $I$ an... |
Is the answer:
{w : w ∈ {0*,1*} and w contains at least 3 zero}
correct?
|
I need to prove that the subset relation “⊆” on all subsets of ℤ is a partial order but not a total order. I'm not experienced in these kind of proofs and was hoping to see an example of an easier one so I could attempts other ones.
|
Consider this infinite sum of matrices. Is there any closed form to express this sum?
$S=B+ABA^T +A^2B({A^T})^2+A^3B({A^T})^3+...$
And B is diagonal.
Thanks
|
Find a probability density function $f$ on $[0,6] \subset \mathbb{R}$, such that $\mathbb{P}([1,3]) = 0.5$
That is we need to find an $f$, such that $\int_{[0,6]} f(x)dx = 1$ and $\int_{1}^{3} f(x)dx = 0.5$. I already tried functions of the form $f(x) = -cx$ and $-cx^2$ just by plugging it into the integral for $[1,3... |
A random variable, $M$, is Poisson distributed with $\lambda=2$. ${X_1,X_2,\dots}$ are independently identically distributed random variables with $\mu=3$ and $\sigma=.2$.
Introduce a new random variable, $Z$, which is defined as:
$Z=\sum_{k=1}^{M}X_k$.
We are also given that $Z=0$ if $M=0$ and are asked to calculate t... |
What is this difference in notation between $F_X(x)$ and $F(x)$? (where $F(x) = P\{X \leq x\}$ Thanks.
|
The category of "just arrow" categories is equivalent to Cat as we see here. If we have a "just arrow" category $C$, I think we can also have the set of equations, $EQ_C$, over the (compositions of) the arrows of $C$. These equations are extra data that define $C$. I am not sure if this is possible. We see our "jus... |
I am trying to solve the initial value problem: $$ y'= \frac{10}{3}xy^{2/5}, \qquad y(0)=0 \qquad \qquad (1) $$
where $ x\in \mathbb{R} $.
The first order equation is not linear in the form:
$$ y'=f(x,y)$$ and some considerations can be done on the existence of possible solutions.
First we note that $f(x,y) \text{ is... |
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