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Cover definition: When is a set a subset of it's cover's union instead of being equal? |
I am trying to find a graph (ideally the smallest) that demonstrates why the following 3 necessary conditions are not sufficient for a graph to be traceable. In other words, what is the smallest untraceable graph $G$ satisfying the following 3 conditions?
1. $G$ is connected.
2. $G$ has at most two vertices of ... |
What is the smallest untraceable graph satisfying some necessary conditions for traceability? |
Let $(R, m_R, \kappa_R) $ a discrete valuation ring with quotient field $K$, and $K \subset L$ a finite field extension. Let $ B$ be the integral closure of $R$ in $L$, which is in general not a dvr, let denote by $\frak{P}_1, \frak{P}_2, ..., \frak{P}_m$ it's maximal primes lying over $m_R$. Then we obtain ramificat... |
When stuying a 1D differential equation, I tried to solve the problem by finding the Green's function and then solving with the inverse Fourier transform.
To make the problem well-behaved, I used a regularisation paremter $\epsilon$. **My problem is that when $\epsilon\to 0$, the amplitude becomes constant whereas ... |
if 4D space is represented by quaternions with real component being time and imaginary components describing 3D space what will be area element in this 3D space? Will it be negative? And will space metric be [+---]? Will volume be imaginary? |
$\lim_{k\to\infty}\frac{x_{k+1}}{x_k}=1^+$ and $\sum_k x_k$ converges? |
Suppose $S$ is a submodule of the $R$-module $M$, then it is not necessarily possible to decompose $M$ as
$$ M = S \oplus S^c . $$
However, if we suppose that $M$ is finitely generated and Noetherian, and $S$ is a submodule, can we always find a submodule $S'$ that has a trivial intersection with $S$, such that... |
Given a submodule, can we find another submodule such that their direct sum forms a finite index subgroup? |
The question came in Chemistry exam. [![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/65jgb.png
Does anyone know what does notation $[A, B]$ stands for? Thanks. |
What does this notation $[A, B]$ means for matrices? |
Let $(R, m_R, \kappa_R) $ a discrete valuation ring with quotient field $K$, and $K \subset L$ a finite field extension. Let $ B$ be the integral closure of $R$ in $L$, (which in general is not a dvr), let denote by $\frak{P}_1, \frak{P}_2, ..., \frak{P}_m$ it's maximal primes lying over $m_R$. Then we obtain ramific... |
The remark mentioned in the title states that if $L/K$ is a Galois extension, an $E$ is an intermediate field such that $E/K$ is a Galois extension, then any $K$-automorphism of $L$ restricts to a $K$-automorphism of $E$.
This should be a consequence of Theorem 4(i), section 3.5, which says that an algebraic extensi... |
I have the following coupled non-linear ODE system, which describes a biological system:
$$
\begin{cases}
\dfrac{dp}{dt} = -\gamma p f,\\
\\
\dfrac{df}{dt} = \gamma p f,\\
\\
\dfrac{dT}{dt} = \left( 1 - \dfrac{k}{T} \right) \left\{ b (1 - T) \dfrac{r^n}{f^n + r^n} - m \dfrac{f^n}{f^n + r^n} \right\},
\end{cas... |
Can an ODE system never converge to its stable equilibrium in the long run? |
I'm trying to understand a result in optimal transport. Specifically, consider the Distributionally Robust Optimization problem given by
$$\sup \left\{\int f d\mu : d_{c}(\mu, \nu) \leq \delta\right\}$$ where $\delta>0$, $f$ is integrable and lower semicontinuous, measures $\mu, \nu$ live in a Polish space $P(S)$ and ... |
Let $P_{m}^{\left(a,b\right)}\left(s\right)$ the $m$-th [Jacobi polinomial][1]. An old result of Bateman shows that $$
\left(1-x\right)^{\alpha+\mu}\frac{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(x\right)}{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(1\right)}=\frac{\Gamma\left(\alpha+\mu+1\right)}{\Gamma\left(... |
When stuying a 1D differential equation, I tried to solve the problem by finding the Green's function and then solving with the inverse Fourier transform.
To make the problem well-behaved, I used a regularisation paremter $\epsilon$. **My problem is that when $\epsilon\to 0$, the amplitude becomes constant whereas ... |
*Salvador Dali* applied in his 1947–1949 painting "*Leda Atomica*" a concept that relates to the golden ratio or its conjugate. It is well known that Dali used mathematical formulas to elaborate his works (e.g. golden ratio, catastrophe theory, etc.), this said, we are not talking here about a layman.
[![enter image... |
I'm currently studying the cube-solving algorithms developed by Kociemba, Korf, and Thistlethwaite and I'm interested in understanding their computational complexities.
Could someone please explain or provide the Big O notation for the time complexity of these algorithms? I'm particularly interested in how the numbe... |
Timec Complexity O-Notation for Kociemba, Korf, and Thistlethwaite's Algorithms? |
Suppose that $A, B \in \mathbf R^{m \times n}$ with $m \leq n$. Let $\|\,\cdot\,\|$ denote the Frobenius norm and let $\langle \,\cdot\,, \,\cdot\,\rangle$ denote the Frobenius inner product.
Note that
$$
\|A^T A\| = \sum_{j=1}^m \sigma_j(A)^2 \geq \frac{1}{m} \left( \sum_{j=1}^m \sigma_j(A) \right)^2 = \frac{1}{m} ... |
The product of elements in the main diagonal is $100^7$. Estimate the sum of all other summands in the determinant: notice, that every other permutation contains $100$ at most five times. How many permutations are there all together? It is $7!=5040$. But $100^7>100^5\cdot 5040$, which means that all the other permutati... |
Let w = (1 1 2 -1), W = { u € R^4 | u • w = 2 } , and
V = { u - (1 0 1 1) | for all u € W }.
(a) Is W a subspace ? Explain your answer
(b) Show that V is a subspace and that T = { (-1 1 0 0) , (-2 0 1 0) , (1 0 0 1) } is a basis for V.
(c) Let T be the basis of V defined in (b)
Find [(4 -1 1 5)]^T (Tr... |
It's from Vakil's FOAG, exercise 25.1.H, with the following text:
> If a morphism $\pi: X \rightarrow Y$ satisfies the property that the natural map $\mathscr{O}_Y \rightarrow$ $\pi_* \mathscr{O}_{\mathrm{X}}$ is an isomorphism, we say that $\pi$ is $\mathscr{O}$-connected. Clearly the notion of $\mathscr{O}$ connec... |
Proper $\mathcal{O}$-connected morphism of locally Noetherian schemes is surjective? |
I'm currently studying the cube-solving algorithms developed by Kociemba, Korf, and Thistlethwaite and I'm interested in understanding their computational complexities.
Could someone please explain or provide the Big O notation for the time complexity of these algorithms?
Any insights or resources on this topic w... |
Let $w = (1, 1 ,2, -1), W = \{ u \in R^4 \ |\ u \cdot w = 2 \}$ , and
$V = \{ u - (1, 0, 1, 1)\ |\ \forall u \in W \}$.
(a) Is W a subspace ? Explain your answer
(b) Show that V is a subspace and that $T = \{ (-1, 1, 0, 0) , (-2, 0, 1, 0) , (1, 0, 0, 1) \}$ is a basis for V.
(c) Let T be the basis of V d... |
Let $w = (1, 1 ,2, -1), W = \{ u \in\Bbb R^4\mid u \cdot w = 2 \}$, and
$V = \{ u - (1, 0, 1, 1)\mid u \in W \}$.
(a) Is W a subspace? Explain your answer
(b) Show that V is a subspace and that $T = \{ (-1, 1, 0, 0) , (-2, 0, 1, 0) , (1, 0, 0, 1) \}$ is a basis for V.
(c) Let T be the basis of V defined i... |
Let $\begin{align}
w &= \left(\begin{matrix}
1 \\
1 \\
2 \\
-1
\end{matrix}\right)
\end{align}$, $W = \{ u \in \mathbb{R^4} | u \cdot w = 2 \}$ , and
$V = \{ u - \left(\begin{matrix}
1 \\
0 \\
1 \\
1
... |
**The problem states**:
Find the remainder when $x^{73}-2x^{15}+3x-1$ is divided by $(x-1)^2$.
**So, what I did is**,
Assume, the remainder to be linear, i.e $r(x) = ax+b$
By Eucild''s,
$x^{73}-2x^{15}+3x-1 = q(x)(x-1)^2+r(x)$
Putting x = 1,
We get, $a+b = 1$.
Then I differentiated it, and t... |
Can I avoid differentiation in this problem? |
I have the following coupled non-linear ODE system, which describes a biological system:
$$
\begin{cases}
\dfrac{dp}{dt} = -\gamma p f,\\
\\
\dfrac{df}{dt} = \gamma p f,\\
\\
\dfrac{dT}{dt} = \left( 1 - \dfrac{k}{T} \right) \left\{ b (1 - T) \dfrac{r^n}{f^n + r^n} - m \dfrac{f^n}{f^n + r^n} \right\},
\end{cas... |
Let $w = (1, 1 ,2, -1), W = \{ u \in\Bbb R^4\mid u \cdot w = 2 \}$, and
$V = \{ u - (1, 0, 1, 1)\mid u \in W \}$.
(a) Is W a subspace? Explain your answer
(b) Show that V is a subspace and that $T = \{ (-1, 1, 0, 0) , (-2, 0, 1, 0) , (1, 0, 0, 1) \}$ is a basis for V.
(c) Let T be the basis of V defined i... |
Let $w = (1, 1 ,2, -1), W = \{ u \in\Bbb R^4\mid u \cdot w = 2 \}$, and
$V = \{ u - (1, 0, 1, 1)\mid u \in W \}$.
(a) Is $W$ a subspace? Explain your answer
(b) Show that $V$ is a subspace and that $T = \{ (-1, 1, 0, 0) , (-2, 0, 1, 0) , (1, 0, 0, 1) \}$ is a basis for $V$.
(c) Let $T$ be the basis of $V$... |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
DISCLAIMER: I also made a [post][1] about it in biology stack exchange but I'm not sure whether that forum is the most appropriate. Hence I thought to share my question here as well.
While looking at [this paper][2], I came across with the following:
> Examples of reaction-diffusion systems (1)-(2), falling into ... |
Are there known characterizations of subsets of positive matrices over which the [Loewner order](https://en.wikipedia.org/wiki/Loewner_order) is total ? |
In a previous question of mine (see [here][1] and [here][2]), I suggested the following general transformation for integrals involving $\tan{\left(\frac12\csc^{-1}{(x)}\right)}$ or $\tan{\left(\frac12\sec^{-1}{(x)}\right)}$ or both.
>$$\int f\left(\tan{\left(\frac12\csc^{-1}(x)\right)}, \tan{\left(\frac12\sec^{-1}(x... |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
Trying to understand Bredon's proof (below). Please note that $G$ is an Abelian group.
So my confusion starts with $H_1(Hom(R,G_*)) \to H_0(Hom(A,G_*)) \to H_0(Hom(F,G_*))$. I understand that he is using Zig-Zag lemma:
$H_2(Hom(R,G_*)) \to H_2(Hom(A,G_*)) \to H_2(Hom(F,G_*)) \to H_1(Hom(R,G_*)) \to H_1(Hom(A,G_*... |
I encountered this integral and I want to find its symbolic solution.
Since $x^2$ is in the denominator, we have $0< a< b$ to make sure the integrand value does not explode.
I did some search on Internet: https://www.physicsforums.com/threads/integration-of-e-x-x-2-with-respect-to-x.225955/ and I saw someone sa... |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
I have a question about the fair use of vouchers.
The facts are as follows:
At a lottery retailer, tickets were purchased with a duration of several draws. The cost was borne by person 1, who had ordered the lottery ticket. However, instead of the selected numbers, the sales clerk selected the same number of rando... |
I have the below function:
$$\pi(x) = \frac{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot r_1}{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot \gamma+ r_0}- x$$
with:
- $x, s_0, s_1, r_0, r_1 \in R^+$
- $\lambda, \gamma \in [0,1]$
- $k \in \{0.25, 4... |
For practice I am trying to use the standard definition of volume for a set $X \subset \mathbb{R}^n$ given by:
$$vol(X) := \int_X dx_1 \ \ldots \ dx_n$$
to compute the area of the triangle spanned by the points $(0,0,1), (0,2,0)$ and $(1,1,1)$ in $\mathbb{R}^3$. (The result should be $1.22$.)
If I am not mis... |
Let A be a $n\times n$ matrix, $s$ be real number and $I$ be the $n\times n$ identity matrix. Why does $A$ have a worse condition number than $A-sI$?
I suspect it is something to do with the singular values being closer together for the matrix $A-sI$ than for $A$. I was wondering how I could mathematically prove th... |
Why does A-sI have a better condition number than A? |
Seven people are standing in two rows, with three people in the front row and four in the back row. Among them, A and B must stand next to each other, while A and C must stand separately. How many different arrangements are there?
My thought process is like this, but the result is incorrect.
First, we consider th... |
What is the correct approach to solving this permutation and combination problem? |
I am considering the following problem：
Let $ A $ be a commutative Noetherian local ring, $ I $, $ I' $ be two ideals of $ A $ satisfying that $ dim(A/I) = dim(A/I') $. Is it true that $ dim(A/I)=dim(A/I\cap I') $ always holds? Here $ dim $ means Krull dimension.
Any ideals would be thankful. |
I am considering the following problem：
Let $ A $ be a commutative Noetherian local ring, $ I $, $ I' $ be two ideals of $ A $ satisfying that $ \dim(A/I) = \dim(A/I') $. Is it true that $ \dim(A/I)=\dim(A/I\cap I') $ always holds? Here $ \dim $ means Krull dimension.
Any ideas would be thankful. |
I have difficulty understanding the solution below and have already summarized my difficulties at the end of this post
**Problem**:
A circle is circumscribed about a triangle formed by joining three points taken at random in the surface of a circumscriptible polygon of $n$ sides, find the chance that the circle l... |
Let $\mathcal{H}$ be a Hilbert space, $A:\mathcal{H}\rightarrow\mathcal{H}$ a Hilbert-Schmidt operator on it, and $H_0:\mathcal{D}(H_0)\rightarrow\mathcal{H}$ an unbounded self-adjoint operator on it. Suppose that the operator $AH_0:\mathcal{D}(H_0)\rightarrow\mathcal{H}$ is bounded and that there exists an orthonormal... |
I am trying to understand the blow-up along a submanifold as explained in Huybrechts "Complex Geometry An Introduction", p. 99, Example 2.5.2.
For background, for $m \leq n$, we see $\mathbb C^m \subset \mathbb C^n$ as $\{ z_{m+1} = \ldots = z_n = 0 \}$ and define
$$
Bl_{\mathbb C^m} (\mathbb C^n) := \{ (x,z): z_i... |
Why are local models of blow-ups compatible by gluing with transition functions in the normal bundle? |
For practice I am trying to use the standard definition of volume for a set $X \subset \mathbb{R}^n$ given by:
$$vol(X) := \int_X dx_1 \ \ldots \ dx_n$$
to compute the area of the triangle spanned by the points $(0,0,1), (0,2,0)$ and $(1,1,1)$ in $\mathbb{R}^3$, i.e. the standard area of this triangle, ***NOT*** ... |
Let $\mathcal{H}$ be a Hilbert space, $A:\mathcal{H}\rightarrow\mathcal{H}$ a Hilbert-Schmidt operator on it, and $H_0:\mathcal{D}(H_0)\rightarrow\mathcal{H}$ an unbounded self-adjoint operator on it. Suppose that the operator $AH_0:\mathcal{D}(H_0)\rightarrow\mathcal{H}$ is bounded and that there exists an orthonormal... |
I am currently trying to apply Ljusternik-Schnirelman theory to a problem in geometric calculus of variations and have the following problem:
I consider an energy functional $\mathcal{E}$ on a submanifold $\mathcal{M}_s\subset W^{1+s,2}(I,\mathbb{R}^n)$ for some open interval $I$ and $s>\frac{1}{2}$ (Sobolev-Slobodec... |
Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$. My question is about the "inverse" problem described below.
**Direct prob... |
Given the following Relations over the set $M := {α, β, γ}$
$R1 := {(α, α), (α, β), (β, α), (β, β), (γ, γ)}$ \
$R2 := {(α, α), (α, γ), (β, β), (β, γ), (γ, β), (γ, γ)}$
How is $R1$ transitive? \
The condition for transitivity is \
$(a,y) (y,x) \implies (a,x)$
And how is $R2$ not a partial order? \
Part... |
Suppose that $A, B \in \mathbf R^{m \times n}$ with $m \leq n$. Let $\|\,\cdot\,\|$ denote the Frobenius norm and let $\langle \,\cdot\,, \,\cdot\,\rangle$ denote the Frobenius inner product.
Note that
$$
\|A^T A\|^2 = \sum_{j=1}^m \sigma_j(A)^4 \geq \frac{1}{m} \left( \sum_{j=1}^m \sigma_j(A)^2 \right)^2 = \frac{1}... |
I have a question about the fair use of vouchers.
The facts are as follows:
At a lottery retailer, tickets were purchased (4 weeks, for the same 6 fields). The cost of € 29.5 was borne by person 1, who had ordered the lottery ticket. However, instead of the selected numbers, the sales clerk selected the same numbe... |
The direct answer to your question is "no". At least if by "stable" you mean all eigenvalues of the Jacobian at $x^*$ have negative real part.
How then do we square this with this particular example? The issue comes when you simplify the system. You are correct in saying $w=f+p$ is constant, but you have to understa... |
Given the following Relations over the set $M := \{α, β, γ\}$
$R1 := \{(α, α), (α, β), (β, α), (β, β), (γ, γ)\}$ \
$R2 := \{(α, α), (α, γ), (β, β), (β, γ), (γ, β), (γ, γ)\}$
How is $R1$ transitive? \
The condition for transitivity is \
$(a,y)\in R1 \text{ and }(y,x) \in R1 \implies (a,x) \in R1$
And ho... |
Consider a market with $d = 1$ risky asset with prices $(S_n)_{n \geq 0}$ and interest rate $r$. Suppose an investor has initial wealth $X_0 > 0$, consumes $C_n$, and holds $\theta_n$ shares over the time interval $(n - 1, n]$, where $C_n$ and $\theta_n$ are $\mathcal{F}_{n-1}$-measurable. The investor’s goal is to max... |
The direct answer to your question is "no". At least if by "stable" you mean all eigenvalues of the Jacobian at $x^*$ have negative real part. (See, e.g., Section 2.9 Theorems 1 and 2 from Differential Equations and Dynamical Systems, L. Perko 3rd edition).
How then do we square this with this particular example? Th... |
Let $X\geq 0$ be ar real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine the derivative of $$E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$ with respect to $z$. I started like this: If we have density functions $f(x)$ and $g_x(y)$ of $X$ and $Y(X)$ I calculated
\begin{align*}
E[e^{X... |
Let $X$ be a finite simplicial complex such that its geometric realisation $|X|$ is homeomorphic to a closed, connected, orientable manifold $M^n$. Suppose we are given an element $\gamma \in \pi_1(|X|)$ (where we know the fundamental group from abstract, non-implemented reasons).
My Question: Is there a sort of a... |
Plancherel's theorem states that the Fourier transform preserves the $L^2$ norm, meaning that if $f$ and $g$ are functions whose Fourier transforms are denoted by $\hat{f}$ and $\hat{g}$ respectively, then:
$$
\| f \|_{L^2} = \| \hat{f} \|_{L^2}
$$
Similarly, is it true that: if $\mu_1$ and $\mu_2$ are probabil... |
Given the following Relations over the set $M := \{α, β, γ\}$
$R1 := \{(α, α), (α, β), (β, α), (β, β), (γ, γ)\}$ \
How is $R1$ transitive? \
The condition for transitivity is \
$(a,y)\in R1 \text{ and }(y,x) \in R1 \implies (a,x) \in R1$
And how is $R2$ not a partial order? \
Partial order if thing is a... |
Given the following Relations over the set $M := \{α, β, γ\}$
$R1 := \{(α, α), (α, β), (β, α), (β, β), (γ, γ)\}$
How is $R1$ transitive? \
The condition for transitivity is \
$(a,y)\in R1 \text{ and }(y,x) \in R1 \implies (a,x) \in R1$
And how is $R2$ not a partial order? \
Partial order if thing is an... |
Given a graph $G = (V, E)$ and an order $v_1, \ldots, v_{n}$ of its vertices, a Greedy coloring is an algorithm that assigns colors to the vertices as follows: Firstly, $c(v_1) = 0$; then it traverses the order and performs the following assignment:
$$
c(v_i) = \min_{z \in \mathbb{N}_0} \text{ s.t. no previous nei... |
it is well known that every power series has a (maybe infinite) R s.t for every |z|>R the series diverges and for every |z|<R it converges. assume R is finite and denote by C the subset of the circle of radius R on which the series conveges.
can C be any arbitrary finite or co-finite set?
are there any interesting ... |
Given a graph $G = (V, E)$ and an order $v_1, \ldots, v_{n}$ of its vertices, a Greedy coloring is an algorithm that assigns colors to the vertices as follows: Firstly, $c(v_1) = 0$; then it traverses the order and performs the following assignment:
$$
c(v_i) = \min_{z \in \mathbb{N}_0} \text{ s.t. no previous nei... |
Let $(E,N)$ be a normed space, and $M$ be another norm. Can you find a necessary and sufficient condition on $M$ so that $\sqrt{NM}$ is also a norm on $E$?
I hink the condition is that $M$ is proportional to $N$.
Of course, the only thing to investigate is the triangle inequality: we want to prove that if for a... |
I want to understand the semantics of rights bit shifts `x>>k` in two's complement properly, in particular why do right bit shifts of size $k$ approximately divide $x$ by $2^k$ (and ideally making this "approx" precise and also handling the arithmetic left bit shift too).
So the representation of a number in two co... |
Given a graph $G = (V, E)$ and an order $v_1, \ldots, v_{n}$ of its vertices, a Greedy coloring is an algorithm that assigns colors to the vertices as follows: Firstly, $c(v_1) = 0$; then it traverses the order and performs the following assignment:
$$
c(v_i) = \min_{z \in \mathbb{N}_0} \text{ s.t. no previous nei... |
I know there are some natural ways to make a general category additive or even K-linear by using the free group or free vector space generated by a set and a direct sums, I also know that we may make a additive category a semi-abelian category via the Karoubi envelope, my question is wheter or not there exists a constr... |
The problem states:
> Find the remainder when $x^{73}-2x^{15}+3x-1$ is divided by $(x-1)^2$.
**So, what I did is**,
Assume, the remainder to be linear, i.e $r(x) = ax+b$
By Eucild''s,
$x^{73}-2x^{15}+3x-1 = q(x)(x-1)^2+r(x)$
Putting x = 1,
We get, $a+b = 1$.
Then I differentiated it, and the r... |
Let $\mathcal{H}$ be a Hilbert space, $A:\mathcal{H}\rightarrow\mathcal{H}$ a Hilbert-Schmidt operator on it, and $H_0:\mathcal{D}(H_0)\rightarrow\mathcal{H}$ an unbounded self-adjoint operator on it. Suppose that the operator $AH_0:\mathcal{D}(H_0)\rightarrow\mathcal{H}$ is bounded and that there exists an orthonormal... |
it is well known that every power series has a (maybe infinite) $R$ s.t for every $|z|>R$ the series diverges and for every $|z|<R$ it converges. assume $R$ is finite and denote by $C$ the subset of the circle of radius $R$ on which the series converges.
can $C$ be any arbitrary finite or co-finite set?
are there any... |
Suppose $b\in L^\infty(\mathbb{R})$ and $u\colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}, \quad (t,x)\mapsto u(t,x)$ is a solution to $\partial_t^2 u = \partial_x^2 u +b(x)\partial_x u,\quad (x,t)\in \mathbb{R}\times \mathbb{R}$ with initial conditions $u(0,\cdot)=u_0$ and $\partial_t u(0,\cdot)=u_1$. Do we then have... |
>$$\Gamma(x)= \int_0^ \infty e^{-t}t^{x-1}dt \ \ \ \ \ \ \ x>0. $$
Bohr and Mollerup showed that the gamma function is the only positive function $f$ defined on $(0,\infty)$ that has these properties
1. $f(x+1)=xf(x)$
2. $f(1)=1$
3. $f(x)$ is a continuous
4. $f(x)$ is log convex
But before Bo... |
Are there alternatives functions for the gamma function? |
No, this is not true, why would it be ? In particular your second distance is nothing but the $L^2$ norm of $\mu_1-\mu_2$, which is not even always defined when $\mu_1$ and $\mu_2$ are not absolutely continuous with respect to the Lebesgue measure.
If you want a counterexample, take $\mu_1 = \delta_0$ and $\mu_2 = \... |
Let's consider that we have a variable $X$ ranging from $0.1$ to $0.5$. For simplicity, let's assume that the cumulative distribution function (CDF) is given by $F(x) = x^{0.5}$. However, this interval is constrained by the limited availability of experimental data. How can one extend the support of $X$ and recalibrate... |
How can one expand the support of a cumulative distribution function (CDF)? |
Are there alternatives functions for the gamma function that was used as generalisation for the factorials? |
>$$\Gamma(x)= \int_0^ \infty e^{-t}t^{x-1}dt \ \ \ \ \ \ \ x>0. $$
Bohr and Mollerup showed that the gamma function is the only positive function $f$ defined on $(0,\infty)$ that has these properties
1. $f(x+1)=xf(x)$
2. $f(1)=1$
3. $f(x)$ is a continuous
4. $f(x)$ is log convex
But before Bo... |
If $f$ is a real valued measurable function in $[0, \infty)$ and $f \in L^{1}([0,\infty))$ then $$\lim_{x\to\infty} f(x) = 0$$
Here measurable is in the Lebesgue sense.
Can anyone give an outline of its proof? |
For unital Algebras $A$, the Spectrum of an element $x \in A$ is defined to be
$$
Sp_A(x) := \{ \lambda \in \mathbb{C}: x - \lambda * e \text{ is not invertible} \}
$$
According to my textbook, there is a way we can extend this definition to non-unital Algebras: Let $A$ be an algebra, then define $A_I := A \times... |
I'm wondering how can i make sense of a partial solution to a two-system of polynomial equations.
Let $\{f,g\}\subseteq K[x,y]$ be polynomials in a polynomial ring over the field $K$. Given the equations:
$$f(x,y)=a$$
$$g(x,y)=b$$
Every solution is contained in the affine variety $\mathcal{V}(I_0)≠\emptyset$ of th... |
Set of all finite subsets of the natural numbers is not able to create an infinite sequence? |
Let $X\geq 0$ be ar real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine the derivative of $$E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$ with respect to $z$. I started like this: If we have density functions $f(x)$ and $g_x(y)$ of $X$ and $Y(X)$ I calculated
\begin{align*}
E[e^{X... |
I'm having struggles understanding the last statement below from Qing Liu's "algebraic geometry and arithmetic curves".
Lemma 2.5.22 (pg. 73): Let $A$ be a finitely generated integral domain over $k$, and let $p$ be a prime ideal of $A$ of height $1$. Then $\dim A/p=\dim A-1.$
Proof. Let $f\in p\backslash \{0\}$.... |
If $f$ is a real valued measurable function in $[0, \infty)$ which is uniformly continuous and $f \in L^{1}([0,\infty))$ then $$\lim_{x\to\infty} f(x) = 0$$
Here measurable is in the Lebesgue sense.
Can anyone give an outline of its proof?
Edit: I forgot to mention the function has to be uniformly continuous.
Ot... |
Let $\mathcal{A}$ be a finite set and $n\in \mathbb N$. Moreover let $B:={b_1,\dots, b_l}\subset \mathcal{A}$ with $l\leq n$. I would like to compute the number of tuples $a\in\mathcal{A}^n$ s.t. the two conditions
- for every $b\in B$ there exists an $i=1,\dots,n$ with $a_i = b$
- for every $j=1,\dots, n$ it hol... |
Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)= \lfloor \frac{an+b}{cn+d} \rfloor , \forall n \in \mathbb N$$
Prove that $(c=0 , b < d , 0 < a \le d) \implies f$ is a surjective function
I tried to solve this Olympiad problem by myself. My sol... |
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