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How to represent A depends on B while B doesn't depend on A? |
I'm having some problems solving this integral:
$$ I = \mathcal{P} \int_{-\infty}^{+\infty} \frac{1-e^{2ix}}{x^2} \ dx$$
where $\mathcal{P}$ is the Cauchy principal value. The exercise suggests to use the fact that:
$$I_* = \frac{1}{2} \operatorname{Re} \left[I\right]=\mathcal{P} \int_{-\infty}^{+\infty} \frac{\sin... |
I am required to find the range of the following trignometric function
$$2 \sin^2(x) + 2 \sin(x)$$
My approach to solving the above question has been as follows
The range of $2 \sin^2(x)$ must be $[0,2]$ and the range for $2 \sin(x)$ is from [-2, 2] so therefore the range of the function $2 \sin^2(x) + 2 \sin(x)... |
When I was in high school, I discovered in an exercise that $$\boxed{\pi=12\int_{0}^{2-\sqrt3}\frac{dt}{1+t^2}}$$since $\frac{\pi}{12}=\arctan( 2-\sqrt3)$, what we were made to demonstrate with formulas with half angles.
I tried the following
$$\forall |t|<1, \frac{1}{1+t^2}=1-t^2+t^4-t^6+...$$So $$\pi\approx12... |
For simplicity, I assume that $f$ and $g$ are real-valued, Lebesgue measurable functions.
In [this question](https://math.stackexchange.com/questions/2490594/product-of-measurable-functions-is-measurable) the following is asked:
> Prove that if $f$
and $g$
are measurable then $fg$
is also measurable (expres... |
So the formal definition:
$(\forall \epsilon > 0)(\exists \delta > 0)(\forall x \in \mathbb{R})(0 < \mid x - x_0 \mid < \delta \implies \mid f(x) - L \mid < \epsilon)$
If this is true then we say $\lim_{x \to x_0} f(x) = L$. In my opinion, this really only makes sense if the function is defined for all values of... |
How does the formal definition of limit deal with functions that are not defined at multiple points, maybe even entire intervals? |
$$\int\frac{1}{y}\frac{1}{\ln y}dy$$
$$\int\frac{1}{y}\frac{1}{\ln^2y}dy$$
I know that i should use some substitution, but i don't understand how.Is there any other way than substitution?
I've tried to understand the integral-calculator.com output, but no luck.
Can anyone explain to me how to solve these ... |
How do you prove that a function is independent of a variable in an interval? |
Can you prove this law? without using vector properties |
**Problem**: Assume $ a^2 + b^2 + c^2 = 1 $. Calculate the improper integral $\int_{B(\mathbf{0}, 1)} \frac{\mathrm{d} x \mathrm{~d} y \mathrm{~d} z}{1-a x-b y-c z}$ where $B(\mathbf{0}, 1)=\left\{x^2+y^2+z^2 \leq 1\right\}$ is the unit ball in $ \mathbb{R^3}$.
**Attempt**: Assuming $ a,b,c \neq 0 $ ( I had difficu... |
The first quote is regarding the definition of a first-order property, not a universal prohibition. The sentences of the axiom schema are not first-order sentences in this sense, but the formulas $\varphi$ that the schema ranges order are all the first-order formulas.
So in other words, what they mean is that for e... |
Using the formula of the sum of the sinus and cosinus:
$$
\sin(a) + \sin(b) = 2\sin(\frac{a+b}{2})\cos(\frac{a-b}{2})
$$
$$
\cos(a) + \cos(b) = 2\cos(\frac{a+b}{2})\cos(\frac{a-b}{2})
$$
With this in mind, we can simplify $F(t)$:
$$
F(t) = \frac{2\sin(\frac{t+t+a}{2})\cos(\frac{t-t-a}{2})}{2\cos(\frac{t+t+... |
A group of *2n* individuals consisting of *n* couples, are randomly arranged at a round table. You are required to find an upper bound for the probability that none of the couples are seated next to each other.
**Solution:**
This is a combinatorial problem. Let's denote the total number of ways to arrange *2n* indi... |
>Let $n,k\in\mathbb{N}.$ By a $(n,k)$ magic square, we mean a $n×n$ matrix containing non-negative integer entries such that the sum of entries of any given row or column is $k.$ Note that we don't need the diagonals to add up to $k.$ Prove that the number of $(3,k)$ magic squares is ${{k+4}\choose{4}}+{{k+3}\choose{4}... |
Suppose I want to cut a regular dodecagon into $n$ congruent simply-connected pieces. For which $n$ is this possible?
I can cut it into 24 right triangles, by cutting from the center to each vertex and the midpoint of each edge; by gluing consecutive such triangles together, I can also get any factor of $24$.
If ... |
We assume that $A$ is a borel subset of $\mathbb{R}^2$ and that $\lambda_2(A) = \pi$. Prove that:
$$\int_A (x^2 + y^2) \ d \lambda_2(x,y) \geq \frac{\pi}{2}$$
---------------------------
I see that it might be useful to use polar coordinate system. That way we would get that:
$$x = rcos(\theta) \ , \ y = rs... |
I discovered in an exercise that $$\boxed{\pi=12\int_{0}^{2-\sqrt3}\frac{dt}{1+t^2}}$$since $\frac{\pi}{12}=\arctan( 2-\sqrt3)$, what we were made to demonstrate with formulas with half angles.
I tried the following
$$\forall |t|<1, \frac{1}{1+t^2}=1-t^2+t^4-t^6+...$$So $$\pi\approx12\left( \alpha-\frac{\alpha^... |
> Can you explain to me how you know $~\displaystyle u^2 \geq a \iff -\sqrt{a} \geq -u ~?$
To attack this entire subject analytically, you need a previous result, from the Field Theory of real numbers.
Suppose that $~a < b.~$ Then:
- For any $~c > 0, ~(c \times a) < (c \times b).~$
- For any $~c < 0, ~... |
Suppose an LTI discrete-time system is given by the equations
$$
x_{k+1} = Ax_k + Bu_k,\\
y_{k} = Cx_k + Du_k
$$
with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\rho(A) < 1$ and $x_0 = 0$. Assume $r_k\in\mathbb{R}^{n}$ is a bounded reference for the output sequence. Define the t... |
When does an optimal input sequence for a discrete-time system exist? |
Through self-study, I found out that there are some pathological functions that do not have subgradients. https://www.stat.cmu.edu/~siva/teaching/725/lec2.pdf
On page 2:
> Except for some very pathological functions (and only at the boundary of their domain)
subgradients always exist
What is an example of th... |
For simplicity, I assume that $f$ and $g$ are real-valued, Lebesgue measurable functions.
In [this question](https://math.stackexchange.com/questions/2490594/product-of-measurable-functions-is-measurable) the following is asked:
> Prove that if $f$
and $g$
are measurable then $fg$
is also measurable (expres... |
I am new to measure theory. I am confused by the following claim from *Measure Theory* by Donald Cohn:
> **Claim**$\quad$ *The restriction of Lebesgue measure to the $\sigma$-algebra of Borel subsets of $\mathbb{R}$ is not complete.*
In this post, I will denote the Lebesgue outer measure by $\lambda^*$.
Accordi... |
I'm working through Problem 4.16 in Armstrong's *Basic Topology*, which has the following questions:
1) Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
2) Are these two isomorphic as topological groups?
**Some preliminaries:**
Let $\mathbb{M_n}$ denote the set of $n\times n$ matrices with real entrie... |
I am new to measure theory. I am confused by the following claim from *Measure Theory* by Donald Cohn (Section 1.5 Completeness and Regularity):
> **Claim**$\quad$ *The restriction of Lebesgue measure to the $\sigma$-algebra of Borel subsets of $\mathbb{R}$ is not complete.*
In this post, I will denote the Lebesgue... |
I did a error while understanding about the polar decompositon.
I thought polar decomposition is `PU`, but it is `UP`. While trying to understand what went wrong, I realized that it is somewhat commutative for the inputs I am giving to it because of which I could not realize about this big mistake.
It would be great... |
Is polar decomposition commutative for diagonal matrices? |
>The life of a semiconductor laser at a constant power is
normally distributed with a mean of $7000$ hours and a standard
deviation of $600$ hours.
>(d) - A product contains three lasers, and the product fails if
any of the lasers fails. Assume that the lasers fail independently. What should the mean life equa... |
1. In the category of groups, the coproduct is free product, and for any $A,B$ sets: the coproduct of $F(A),F(B)$ is $F(A\sqcup B)$. Does this property hold in any other categories?
2. What is the coproduct in the category of commutative rings? Or if the definition of it is too complicated, then just what's the copr... |
[Reference image containing the statement][1]
Can someone please explain the below text and what it means in simple words? From what I understood it means that if x=m/2^n and if m is odd then x can be represented in binary in two ways? But how is that possible. I tried with 1/2 but how can 0.5 be represented as 0.1 ... |
Can someone please explain the below and if it means that if x=m/2^n and if m is odd then x can be represented in binary in two ways? |
>In right triangle $ABC$, we have $\angle ACB=90^{\circ}$, $AC=2$, and $BC=3$. Medians $AD$ and $BE$ are drawn to sides $BC$ and $AC$, respectively. $AD$ and $BE$ intersect at point $F$. Find the area of $\triangle ABF$.
<hr>
[![enter image description here][1]][1]
I aim to use the Shoelace Theorem to calculate th... |
**Context**
With some effort we can show that Ramanujan's fastest series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)... |
Suppose $u$ satisfies the equation
$$u(x)=\beta\int_0^x u(x-y)dF(y)+v(x)$$ where $0<\beta<1$ and $F(y)$ a distribution function with $F(0)=0$ and $v(x)\geq 0$ continuous for $x\geq 0$. The Laplace Stieltjes transform is $\tilde{f}(s)=\int_0^\infty e^{-sy}dF(s)$. Let $F^{*n}$ be the $n$ times convolution of $F$ with i... |
I'm having some problems solving this integral:
$$ I = \mathcal{P} \int_{-\infty}^{+\infty} \frac{1-e^{2ix}}{x^2} \ dx$$
where $\mathcal{P}$ is the Cauchy principal value. The exercise suggests to use the fact that:
$$I_* = \frac{1}{2} \operatorname{Re} \left[I\right]=\mathcal{P} \int_{-\infty}^{+\infty} \frac{\sin... |
I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the *germ field*, $\mathcal{F}_0^+$.
Let $\mathcal{F}_t^+ = \cap_{s > t} \mathcal{F}_s^0$ and $\mathcal{F}_t^0 = \sigma(B_s,s\le t)$ where $B_t$ is Brownian motion. Also let $P_x$ deno... |
Suppose I have 2 arbitrary functions $f(\theta)$ and $g(\theta)$, and I need an expression for $\frac{\partial f}{\partial \theta +\partial g}$. How would I do so? Is the above expression equivalent to $\frac{\partial f}{\partial \theta}+\frac{\partial f}{\partial g}$ ?
Any response will be appreciated. Many thanks... |
With latex:
How to show that if $y=f(x+a)$ is an even function where $f(x)=(x-6)^2\sin(\omega x)$ then $a$ must be 6?
The original question was to solve for what $\omega$s are possible. And there was a step like this: since $y=f(x+a)$ is even, so is $y=(x+a-6)^2$ and $y=\sin(\omega x+\omega a)$. I'm confused by t... |
Suppose you have a partial order $<_o$ on $R^n$, and you restrict it to a relation $<_r$, where $x <_r y \iff x <_o y \wedge ||x - y|| < r$. Now it might happen that $<_o$ happens to be the transitive closure of $<_r$. In this case I define $<_o$ to be "$r$-smooth." If $<_o$ is $r$-smooth for all $r > 0$, then I say it... |
I'm having some problems solving this integral:
$$ I = \mathcal{P} \int_{-\infty}^{+\infty} \frac{1-e^{2ix}}{x^2} \ dx$$
where $\mathcal{P}$ is the Cauchy principal value. The exercise suggests to use the fact that:
$$I_* = \frac{1}{2} \operatorname{Re} \left[I\right]=\mathcal{P} \int_{-\infty}^{+\infty} \frac{\sin... |
\int \frac{dx}{\left(x^2+9\right)^2}
How would you solve this with partial integration (without trigonometry)? |
$\int \frac{dx}{\left(x^2+9\right)^2}$
How would you solve this with partial integration (without trigonometry)? |
$$ \int \frac{dx}{\left(x^2+9\right)^2}$$
How would you solve this with partial integration (without trigonometry)? |
Suppose I have 2 arbitrary functions $f(\theta)$ and $g(\theta)$, and I need an expression for $\frac{\partial f}{\partial \theta +\partial g}$. How would I do so? Is the above expression equivalent to $\frac{\partial f}{\partial \theta}+\frac{\partial f}{\partial g}$ ?
Any response will be appreciated. Many thanks... |
Suppose I have 2 arbitrary functions $f(\theta)$ and $g(\theta)$, and I need an expression for $\frac{\partial f}{\partial \theta +\partial g}$. How would I do so? Is the above expression equivalent to $\frac{\partial f}{\partial \theta}+\frac{\partial f}{\partial g}$ ?
Any response will be appreciated. Many thanks... |
If the equation $\frac{x^3+4}{{(x+1)}^3}=c,\forall x \in \Bbb R\setminus{\{-1\}}$ possess two roots, then what are the values of $c$ ?
Attempt: $c=1$ may be a solution if the quadratic equation thus formed has distinct real roots, i.e. $x^3+4={(x+1)}^3$, which gives $ x^2+x-1=0$ which has $2$ different solutions $... |
I am trying to solve the following problem but I face difficulties with the results. Any suggestion?
The line integral $\int_C y^2 dx - x dy$, where $C$ is the boundary of the square $[-1,1]\times[-1,1]$ oriented counterclockwise, can be evaluated in two ways:
a) Using the definition of the line integral:
To co... |
Just was stuck by "[A remarkable Definite Integral][1]"
From formula (3), if i put $a=1$, then
\begin{align}
\int_{-\infty}^{\infty}\frac{2e^{-(x^2+i\pi x)t}}{e^x+e^{-x}}dx=\pi e^{-\frac{\pi^2}{4}t}
\end{align}
But I couldn't proceed further. Would someone can give hints on what tricks can be used to solve t... |
**Context**
With some effort we can show that Ramanujan's fastest series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)... |
I am trying to understand parts of the authors solution given to the following question:
>The generators of $\mathrm{SO}(3)$ can be chosen as
>$$t^1=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & -i \\\ 0 & i & 0 \\ \end{pmatrix},\ t^2=\begin{pmatrix}0 & 0 & i\\ 0 & 0 & 0 \\ -i & 0 & 0 \\ \end{pmatrix},\ t^3=\begin{pmatrix}0 & ... |
I am trying to solve the following problem but I face difficulties with the results. Any suggestion?
The line integral $\int_C y^2 dx - x dy$, where $C$ is the boundary of the square $[-1,1]\times[-1,1]$ oriented counterclockwise, can be evaluated in two ways:
a) Using the definition of the line integral:
To co... |
Consider a non-empty convex set $C \in \mathbb{R}^n$. I have the following conjecture: for all $x\in C$, there exists $y \in ri(C)$ (ri means relative interior) such that $y$ is in the neighbourhood of $x$.
The conjecture has the flavour of "the boundary is thin", or maybe "$ri(C)$ is dense in $C$"? |
Is relative interior "everywhere"? |
A vector field is a pointwise assignment of vectors(arrows).
Tensors are defined as $\mathbb{R}$-valued multilinear maps, and a tensor field is a pointwise assignment of tensors.
Also, a vector space $V$ is isomorphic to it's dual space $V^*$(thus, a vector can be realized as a linear operator too).
Is a ten... |
Theorem (Rank theorem for injective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is injective, then there are maps $(U, \varphi)$ of M around p and $(V,\psi )$ of M around $F(p)$ such that... |
Theorem (Rank theorem for injective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is injective, then there are charts $(U, \varphi)$ of M around p and $(V,\psi )$ of M around $F(p)$ such th... |
We would like to minimize the quantity
$E_{in}(\vec{w})=\frac{1}{N}\sum_{i=1}^N(\vec{w}^{T}\vec{x_n}-y_n)^2$
under the constraint
$\vec{w}^T\Gamma^T\Gamma\vec{w}\leq C$ where $\Gamma$ is a matrix, $C$ is a positive constant.
Let $\vec{w}_{reg}$ be the solution to the above problem, further suppose $\vec{w}_{... |
I'm going to provide a full characterisation of the set of SVDs for a given matrix $A$, using two different (but of course ultimately equivalent) kinds of formalisms. First a standard matrix formalism, and then using dyadic notation.
The TL;DR is that if $A=UDV^\dagger=\tilde U \tilde D\tilde V^\dagger$ for $U,\tild... |
Theorem (Rank theorem for injective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is injective, then there are charts $(U, \varphi)$ of M around p and $(V,\psi )$ of M around $F(p)$ such th... |
Consider a vector $\mathbf{v} = (v_1,\dots,v_d) \in \mathbb{Z}^d$, i.e., a vector of $d$ components where each component is an integer. The norm (length) of $\mathbf{v}$ is given by
$$
\left|\mathbf{v}\right|=\sqrt{v_{1}^{2}+\dots+v_{d}^{2}}.
$$
I was curious to find either an exact expression or an estimate fo... |
For given $n>1$ I'm looking for the maximum number $k$ of colors such that there exists a $k$-coloring of the edges of the complete graph on $2n$ vertices with the property that a copy of every possible $k$-colored perfect matching is present in the colored complete graph.
For example, we know that there are $n+1$ 2... |
Theorem (Rank theorem for injective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is injective, then there are charts $(U, \varphi)$ of M around p and $(V,\psi )$ of N around $F(p)$ such th... |
Is this set uncountable? $A = \{A_n \colon n \in \mathbb{N}\}$ where $A_n$ is the set $\mathbb{N}$ with the number $n$ removed from it |
I have a beginner question in Riemannian geometry. Given a manifold $\mathcal{M}$ endowed with a Riemannian metric $g$, I assume this metric is represented locally by a family of positive definite matrices $(A_q)_q$ in local coordinates.
I assume I have an inequality of the type : $A_q \geq M_q$ near every $q \in \m... |
[On page 119][1] Edwards rewrites the $\xi(s)$ function as
$$
\begin{align*}
\xi\left (\frac{1}{2} + it \right ) &= \frac{s}{2}\Pi\left ( \frac{s}{2} - 1 \right )\cdot \pi^{-s/2}\cdot (s - 1) \cdot \zeta(s)\\\\
\\\\
&= e^{\ln \Pi[(s/2) - 1]}\cdot \pi^{-s/2} \cdot \frac{s(s-1)}{2}\cdot \zeta(s)\\\\
\\\\
&= \le... |
I have the following proof in my notes. It is a particular case of the more general rank theorem.
Theorem (Rank theorem for injective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is inj... |
I'm currently working through a proof that discusses the relationship between Lévy processes in law and infinitely divisible distributions, presented in [Sato][1]. The proof involves defining a measure (7.4), and then claims that the family {$\mu_{t_0,\cdots,t_n}$} is consistent, defined in Sato as
$$
\mu_{t_1,\ldots... |
In an abelian group, is a certain inequality involving products of cardinalities of sum and differences of finite subsets (sumsets) true or not? |
I am interested in learning about the congruence relations satisfied by the coefficients of modular forms, in particular I am interested in learning more about $\tau(n)$, the coefficients of the weight $12$ modular forms $\Delta=q\prod_{n\ge 1}(1-q^n)^{24}$. I am aware that it satisfies some [remarkable congruences][1]... |
S(ε) is the area of the shape, defined by the functions y = e^x and y = x+1+ε. ε is small positive number. When ε -> 0:
S(ε) = S0 + ε^aS1 + ...., a > 0
How can I find S0, S1 and a?
[Graph of functions][1]
[1]: https://i.stack.imgur.com/Af7Bj.png |
Consider $S^1 \subseteq \mathbb{C}$ with the Euclidean metric. Let $X = \Pi_{n=1}^\infty S^1$ as a set and define
$d_\infty: X \times X \to [0, 2], d_\infty((x_n)_{n=1}^\infty, (y_n)_{n=1}^\infty) = \operatorname{sup}_{n=1}^\infty d(x_n, y_n)$
Now $(X, d_\infty)$ is a metric space. What is the fundamental group o... |
I want to find the optimal control $u^*(t)$ for the following problem:
\begin{align*}
\dot{x}_1(t) &= x_2(t), & x_1(0) &= 3 \\
\dot{x}_2(t) &= -2x_1(t) + 5u(t), & x_2(0) &= 5
\end{align*}
which minimizes:
\begin{equation}
J = \frac{1}{2} \int_{0}^{2} [x_1^2(t) + u^2(t)] \, dt
\end{equation}
**Step 1: Define ... |
As part of an assignment, we had to investigate the asymptotic behaviour as $n$ tends to $+\infty$ of the ratio
$$\frac{\Gamma(n,n|z|^2)}{\Gamma(n)},$$ where $\Gamma(s,x)$ is defined as $$\Gamma(s,x)=\int_x^{+\infty}t^{s-1}e^{-t}\mathrm{d} t.$$
In the cases $|z|>1$ and $|z|<1,$ I was able to use a steepest descent a... |
Let $X$ be a manifold. Leray's Theorem for Čech Cohomology states that if a covering $\mathfrak{U}$ is acyclic $(\check{H}^p(U_{i_0}\cap\ldots U_{i_k},F)=0$ for every finite intersections of elements in $\mathfrak{U}$ and every $p$) for a sheaf $F$, then
$$
\check{H}^q(\mathfrak{U},F)=\check{H}^q(X,F)
$$
where the ... |
From the ["Finite affine planes" section](https://en.wikipedia.org/wiki/Affine_plane_(incidence_geometry)#Finite_affine_planes) of the Wikipedia article "Affine plane (incidence geometry)", in a (finite) affine plane of order $n$:
- each line contains $n$ points,
- each point is contained in $n+1$ lines,
- ther... |
$AG(2,2)$ and $AG(3,2)$ and their resp. triangular (Fano plane-like) and tetrahedral ($PG(3,2)$-like) representations. Are my assumptions correct? |
For X ∼ Poisson(λ) how can we calculate the coverage probability, confidence set can be obtd using chi square but not the coverage probability? |
I am trying to solve a nonlinear PDE defined over the intervals $0 <= t <= 1$ and $0 <= x <= x_{max}$, which has the following form:
$4t(1-t)U_t(x,t)=U_{xx}(x,t)+2xU_x(x,t)-4t[U(x,t)^2-U(x,t)]$.
I am looking for the solution in the form:
$U(x,t)=u_0(x) + u_1(x)t + u_2(x)t^2 + ...$
The PDE is accompanied by ... |
Is there any theory for the following series expansion method of solving PDEs? |
I am reading Rudin's "Principles of Mathematical Analysis" and I am stuck at the following theorem:
> Theorem 8.14:
> If, for some $x$, there are constants $\delta > 0$ and $M < \infty$ such that: $\left| f(x + t) - f(x) \right| \leq M|t|$ for all $t \in (-\delta, \delta)$, then $\lim_{N\rightarrow\infty} s_N(f; ... |
I am reading Rudin's "Principles of Mathematical Analysis" and I am stuck at the following theorem:
> Theorem 8.14:
> If, for some $x$, there are constants $\delta > 0$ and $M < \infty$ such that: $\left| f(x + t) - f(x) \right| \leq M|t|$ for all $t \in (-\delta, \delta)$, then $\lim_{N\rightarrow\infty} s_N(f; ... |
I am trying to solve a nonlinear PDE defined over the intervals $0 \leq t \leq 1$ and $0
\leq x \leq x_{max}$, which has the following form:
$4t(1-t)U_t(x,t)=U_{xx}(x,t)+2xU_x(x,t)-4t[U(x,t)^2-U(x,t)]$.
I am looking for the solution in the form:
$U(x,t)=u_0(x) + u_1(x)t + u_2(x)t^2 + ...$
The PDE is accom... |
From the first $(675n^6+2)$ natural numbers, if five consecutive natural numbers are chosen, $E$ be the event that sum of the numbers is cube of natural number and the sum of middle three is square of a natural number, then which of the following option(s) is/are true
(A) $P(E)=\frac{2}{673}$ if $n=1$
(B) $P(E)=... |
I have a beginner question in Riemannian geometry. Given a manifold $\mathcal{M}$ endowed with a Riemannian metric $g$, I assume this metric is represented locally by a family of positive definite matrices $(A_q)_q$ in local coordinates.
I assume I have an inequality of the type : $A_q \geq M^1_q$ near every $q \in ... |
$S(ε)$ is the area of the shape, defined by the functions $y = e^x$ and $y = x+1+ε$. $ε$ is small positive number. When $ε \to 0$:
$S(ε) = S_0 + ε^aS_1 + ...., a > 0$
How can I find $S_0, S_1$ and $a$ ?
[Graph of functions][1]
[1]: https://i.stack.imgur.com/Af7Bj.png |
I have a beginner question in Riemannian geometry. Given a manifold $\mathcal{M}$ endowed with a Riemannian metric $g$, I assume this metric is represented locally by a family of positive definite matrices $(A_q)_q$ in local coordinates.
I assume I have an inequality of the type : $A_q \geq M^{-1}_q$ near every $q \... |
I have a beginner question in Riemannian geometry. Given a manifold $\mathcal{M}$ endowed with a Riemannian metric $g$, I assume this metric is represented locally by a family of positive definite matrices $(A_q)_q$ in local coordinates.
I assume I have an inequality of the type : $A_q \geq M^{-1}_q$ near every $q \... |
I have a beginner question in Riemannian geometry. Given a manifold $\mathcal{M}$ endowed with a Riemannian metric $g$, I assume this metric is represented locally by a family of positive definite matrices $(A_q)_q$ in local coordinates.
I assume I have an inequality of the type : $A_q \geq M^{-1}_q$ near every $q \... |
Here's some work so far.
$$\left| \frac{x+1}{x^2+2 \sqrt{x}} - \frac{1}{4} \right| \leq \frac{\left| x+1 \right|}{ \left| x^2 + 2 \sqrt{x}\right|} + \frac{1}{4} = \frac{\left|x-4\right| + 3}{\left|x^2 + 2 \sqrt x \right|} + \frac{1}{4}$$
Suppose we let $\delta_1 = 5$, so that $1 < \left|x\right| < 9$ and $3 < 2 \... |
Given $f(x) = \frac{x+1}{x^2+2 \sqrt{x}}$, How to prove $\lim_{x \rightarrow 4} f(x)$ using $\varepsilon-\delta$ definition? |
In most of the cases I have studied, I haven't seen what it means for a form "to be" on the manifold. We usually let $M$ to be a manifold, and we let $\omega\in \bigwedge ^kT^*_pM$ in the exterior algebra of the dual tangent space, we then say $\omega$ is of the form $\sum_{I}f_Idx^I$ (I is a multiindex) and we don't g... |
I was wondering how I could use the limit of a function definition to prove that $\lim_{x \rightarrow 4} f(x) = 1/4$, where $f$ is defined as $$f(x) = \frac{x+1}{x^2+2 \sqrt{x}}.$$ Here's some work so far.
$$\left| \frac{x+1}{x^2+2 \sqrt{x}} - \frac{1}{4} \right| \leq \frac{\left| x+1 \right|}{ \left| x^2 + 2 \sqrt{... |
>Surely if $X_t$ is nice and smooth, then $\alpha > \frac{1}{2}$, but then $2\alpha > 1$, and wouldn't this degenerate into saying that $\mathbb{X}$ is constant? From the known result on Holder continuous greater than 1 $\implies$ constant?
As mentioned in the comments, the rough path lift $\mathbb{X}_{s,t}$ is n... |
I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$.
Now, I solved the problem quite easily in a sketching way: the level curves of $f(x, y)$ are concave parabolas ($y = k - (x-2)^2$), and the feasible region is the up... |
From this [answer][1], it implicitly becomes evident that
$$\sum_{k=1}^\infty \frac{kn^k}{(k+n)!} =\frac{1}{(n-1)!}$$
holds for any integer $n$.
I **guess** that the following general formula holds for any $x>0$:
$$\color{blue}{\sum_{k=1}^\infty \frac{kx^k}{\Gamma(k+1+x)}=\frac{1}{\Gamma(x)}}.$$
While... |
In most of the cases I have studied, I haven't seen what it means for a form "to be" on the manifold. We usually let $M$ to be a manifold, and we let $\omega\in \bigwedge ^kT^*_pM$ in the exterior algebra of the dual tangent space, we then say $\omega$ is of the form $\sum_{I}f_Idx^I$ (I is a multiindex) and we don't g... |
Consider the well-known [coupon collector problem][1] with $n$ potential coupons. If I collect a coupon per day, I know it will take me $n \cdot H_n$ days to finally collect all of them, where $H_n$ is the $n$-th harmonic number. For example, if the coupons are the positive integers up to 12, one potential sequence of ... |
Given the following relation, which properties does it fulfill?
Is my reasoning correct or am I missing something?
(m, n) ∈ R ⇐⇒ m > n
***1.) reflexivity***
no, 1 > 1 (trivially false)
***2.) symmetry***
no, since $x > y\not\rightarrow y > x$, since $2 > 3 \not\rightarrow 3 > 2$
***3.) antisymmetry... |
Why is the projection from the standard simplex to the sphere Lipschitz continuous? |
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