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What is the Asymptotic Order of the Sum of Random Variables with non-finite Moments? |
I showed that every Gram matrix, i.e. a $n \times n$ matrix $A$ with $A_{ij} = \langle x_i,x_j\rangle$ where $x_1,...,x_n$ are vectors in an inner product vector space $V$, is Hermitian and positive semi-definite.
But how to show the converse: For every Hermitian positive semi-definite matrix there is a inner prod... |
Past research on matrix subalgebras generated by a given set of generators has one major restriction: most of the literature on the generation of algebras automatically presumes unitality of the generator set, meaning that most proofs are based on the implicit assumption of a unit element among the given matrices; this... |
To be honest I dont know how to do this problem, I only know that we should get a second order linear differention equation for the position $x$. And we have to solve the problem using theta.
I know that $X(t)= r\sin\theta$, where r would be the length of the rod but I dont know where to go from there.
Were also gi... |
[Green's theorem](https://en.wikipedia.org/wiki/Green%27s_theorem) says that:
$$
\int_C L \ dx + \int_C M \ dy = \iint_D \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \ dx \ dy
$$
If the M and N statisfy $\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} = 1$, then the formula can be use... |
If a series $\sum u_n$ converges, $u_n \to 0$ but is there any type of condition on $u_n$ that would ensure that $nu_n \to 0$?
I can think of many $u_n's$ satisfying this, all geometric series terms, p-series terms too for $p\geq 2$, so maybe there's a general condition on $u_n$ that along with $\sum u_n < \infty$ g... |
If a series $\sum u_n$ converges, can anything be said about $nu_n$? |
Suppose we want to formalise some parts of mathematics within set theory, which is itself formalised in a first-order language (in the standard way). Logic is assumed to be classical. In particular, this means that – at the level of semantics – every term must have a denotation, which in turn means that every function ... |
Suppose we have $A,B \in \operatorname{Mat}(n, \mathbb{R})$, and let $\alpha(A,B) = \text{Tr}(AB)$ and $\beta(A,B) = \text{Tr}(AB^T)$. Then is it possible to show that both $\alpha$ and $\beta$ cannot be bilinear forms? If not then what can be shown as a counter-example?
Furthermore, how can I show that only $\beta$... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satisfying... |
Consider $n$ equally sized circles $c_i$ in the plane with their centers $x_i$ in the vertices $v_i$ of a regular $n$-gon and all meeting in the center of that $n$-gon.
Examples:
- Fig. 1: circle pattern for $n=5$:
[![enter image description here][2]][2]
- Fig 2: circle pattern for $n=15$:
[![enter i... |
I have the following problem:
Consider complex Euclidean spaces H and H′
such that $dim(H) ⩽ dim(H′)$.
Let ${|v_i⟩}^{k}_{i=1} ⊂ H$ and ${|w_i⟩}^{k}_{i=1} ⊂ H′$ denote orthonormal sets of vectors.
Show that there exists an isometry $V : H → H′$such that $V |v_i⟩ = |w_i⟩$ for all i ∈ {1,. . . , k}.
I know that... |
I am reading David Zywina's ["Elliptic curves with maximal Galois action"][1]. For a number field $K$, he defines $\mathbb{Q}^{\text{cyc}} \subset \overline{K}$ to be "the" cyclotomic extension of $\mathbb{Q}$.
1. Is $\mathbb{Q}^{\text{cyc}}$ the minimal cyclotomic extension corresponding to the conductor of $K$ o... |
Why is $K \cap \mathbb{Q}^{\text{cyc}}=\mathbb{Q}$ iff $\chi_K(G_K)=\hat{\mathbb{Z}}^{\times}$? |
Suppose we have a Compound Poisson Process with intensity $\lambda$
$$
S_t = \sum_{i=1}^{N_t} X_i.
$$
We can compute the formula for the CDF as follows
\begin{align*}
F_{S_t}(x) = P(S_t \leq x)
&= \sum_{k=0}^\infty P(S_t \leq x \mid N_t = k)P(N_t = k) \\
& = \sum_{k=0}^\... |
### Showing that $(e^{\lambda B_t - \frac{1}{2}\lambda^2t})_{t\geq 0}$ is a $\mathbb{R}$-valued martingale
Let $B$ be a standard $\mathbb{R}$-valued Brownian motion and let $\lambda\in\mathbb{R}$. From $B_t-B_s$ being independent of $\mathcal{F}_s$ and $B_t-B_s \sim \mathcal{N}\left(0, t-s\right)$ follows $\math... |
Is $(e^{i \lambda B_t - \frac{1}{2}\lambda^2t})_{t\geq 0}$ a martingale? |
I am trying to extract the mathematical expression of creating a set of elements from the diagonal of a matrix, but after filtering by a vector.
For example:
Be X a matrix of N x N elements, v a vector of N elements, and R the result of N elements:
$A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bm... |
I have somewhat often found myself in the following situation, especially when self-studying a mathematical subject:
I'm reading a book on a certain topic, and at certain parts I don't immediately understand something, for example a proof. But instead of re-reading it I just skip over it and continue. This then lead... |
In the book, a first course in functional analysis by D.Somasundaram, it is mentioned that $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$
But the proof is given only for one side inclusion..that is.,
The spectrum of an element x in a Complex Banach Algebra is a subset of the closed disk with cent... |
Is the spectrum of an element in an algebra ,$\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$? |
Past research on matrix subalgebras generated by a given set of generators has one major restriction: most of the literature on the generation of algebras automatically presumes unitality of the generator set, meaning that most proofs are based on the implicit assumption of a unit element among the given matrices; this... |
I was reading Rotman's "An Introduction to the theory of Groups" Chapter 12 "The Word Problem". I cannot understand the key concepts of Turing Machines and undecidibilty. I want to prove understand the proof "There exists a finitely presented group which has unsolvable word problem". Can anyone point out any beginner f... |
Is the spectrum of an element in an algebra, $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$? |
I was reading Section 1.5 of *Sheaves on Manifolds* by Kashiwara and Shapira. In the proof of Propositon 1.5.6, they states that in order to prove the functor $H^0:K(\mathcal{C})\rightarrow \mathcal{C}$ is a cohomological functor, where $\mathcal{C}$ is an abelian category, it suffices to prove that $H^0(Y)\rightarrow ... |
### Showing that $(e^{\lambda B_t - \frac{1}{2}\lambda^2t})_{t\geq 0}$ is a $\mathbb{R}$-valued martingale
Let $B$ be a standard $\mathbb{R}$-valued Brownian motion and let $\lambda\in\mathbb{R}$. From $B_t-B_s$ being independent of $\mathcal{F}_s$ and $B_t-B_s \sim \mathcal{N}\left(0, t-s\right)$ follows $\math... |
Is $(e^{i \lambda B_t + \frac{1}{2}\lambda^2t})_{t\geq 0}$ a martingale? |
In his *Mathematical Methods of Classical Mechanics*, V.I. Arnold states the following theorem without proof in pages 75-76:
> Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where
$q_1$ parameterises the curve and $q_2$ is the coordinate in the orthogonal direction to the curve. Consi... |
Assume that $X\in \mathrm{PSD}(n)$ is a symmetric, positive semi-definite matrix with real entries and $C>0$ is a positive real number. If $X$ satisfies $$
\forall \alpha\in\mathbb{Z}^n\setminus\{0\},\qquad \alpha^T X \alpha \geq C,
$$ then is it true that $X$ is positive definite? That is, if $X$ is lower bounded on... |
Can a continuous function $f:\mathbb{R}^2\to\mathbb{R}$ be constant on the following hyperbolas?
$$H_a=\{(x,y)\in\mathbb{R}^2:x+1/y=a\},a\in\mathbb{R}$$
$$H_\infty=\mathbb{R}\times\{0\}$$
That is, such that $\forall a\in\mathbb{R}\cup\{\infty\}:P_1,P_2\in H_a\Rightarrow f(P_1)=f(P_2)$. I know that, if this were the ... |
Can a continuous function be constant on these hyperbolas? |
I am trying to extract the mathematical expression of creating a set of elements from the diagonal of a matrix, but after filtering by a vector.
For example:
Let $A$ be a $3 \times 3$ matrix, $\mathbf{v}$ a $3 \times 1$ vector, and $\mathbf{R}$ be the resulting $3 \times 1$ vector:
$A = \begin{bmatrix} 1 & 0 & 1... |
Lets say there is a curve whose curvature is given as k = 0.01. What i want to calculate is angle between tangent drawn at point "P" which is at a distance of "X" along the curve and Y axis (shown as θ in pic) . [![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/CTt1d.jpg
Can any relation... |
If f”(a) is defined, is f differentiable near a?
My logic goes, if f”(a) exist, then f’(a) must be defined near a, thus f is differentiable near a. Is that correct? |
Is the function differentiable near a point? |
I am going through some lecture notes in commutative algebra. I am struggling with one basic example which I want to understand fully before going further. The example is the following:
We take $\mathbb{C}$-module $\mathbb{C}^n.$ We restrict scalars to $\mathbb{R}$ and obtain $\mathbb{R}^{2n}.$ Then we extend scalar... |
Can a non-constant continuous function $f:\mathbb{R}^2\to\mathbb{R}$ be constant on the following hyperbolas?
$$H_a=\{(x,y)\in\mathbb{R}^2:x+1/y=a\},a\in\mathbb{R}$$
$$H_\infty=\mathbb{R}\times\{0\}$$
That is, such that $\forall a\in\mathbb{R}\cup\{\infty\}:P_1,P_2\in H_a\Rightarrow f(P_1)=f(P_2)$. I know that, if t... |
Can a non-constant continuous function be constant on these hyperbolas? |
If $f’’(a)$ is defined, is $f$ differentiable near $a$?
My logic goes, if $f’’(a)$ exist, then $f’(a)$ must be defined near $a$, thus $f$ is differentiable near $a$. Is that correct? |
I am learning about Havel Hakimi. I know how the algorithm works, but I'm not sure what happens when there are no enough numbers. For example, (2,3,3,4). I remove 4, then I'm supposed to subtract 1 from the next 4 numbers ın the sequence but there only 3 numbers. Does that mean it's not graphical, or should I subtract ... |
Havel Hakimi Theorem -- What happens if there are no enough numbers? |
I'm looking for a pairing function $f(x,y)$ which gives unique values for every combination of integers $x,y>0$ and 0 for $x=0$ $\forall y$.<br>
Or to be more general we can also replace $0$ with constants. <br>
- We do know the max values of $x$ and $y$ can achieve
- We are allowed to shift $x, y$ values. So we al... |
Can we find any (better) pairing function $f(x,y)$ with $f(0,y)=0$? Does it help if we do know min/max values of $x,y$? |
Is this method for constructing nilpotent matrices effective? |
I only need help with the last part of the following question:
>Let $U(\rho, \tau, \lambda)$ and $V(\rho, \tau, \lambda)$ be matrix-valued functions. Consider the following system of overdetermined linear partial differential equations:
$$
\frac{\partial}{\partial \rho} \psi=U \psi, \quad \frac{\partial}{\partial ... |
Consider the following two expressions in $\tan{x}$:
$\sqrt[3]{\tan x+1}$
$\sqrt[3]{\left(\tan x\right)^{3}+\left(\tan x\right)^{2}-\tan x-1}$?
There are trigonometric identities which can be used to express a compound angle as a polynomial in $\tan{x}$, for example $\tan{3x}=\frac{3\tan{x}-\tan^{3}{x}}{1-3... |
If I start with a set of consecutive primes, let's say {2,3,5,7}. Now I assume as in euclids theorem that these are all primes. By the proof I get 2*3*5*7 + 1 = 211. 211 is prime, so I add it to the set: {2,3,5,7,211}. If the resulting number is not prime, add the prime dividers of this number which are not in the set.... |
Does euclids proof of infinite primes generate them all if you add the new prime to the list? |
[This question][1] discusses how $\mathrm{S}L(2,\mathbb{R}$) coincides with $\pm\exp(z)$ with $z\in \frak{sl}(n,\mathbb{R}$) (the real traceless matrices). Is it known what happens for $n>2$?
Namely, one can represent all invertible matrices with determinant $1$ and trace greater than or equal to $-2$. Also, just b... |
What is in the image of the exponential of $\frak{sl}(n,\mathbb{R}$)? What do you need to get all of $\mathrm{SL}(n,\mathbb{R}$)? |
If I start with a set of consecutive primes, let's say $\{2,3,5,7\}$. Consider $2\times 3\times5\times7+ 1 = 211$. Now, $211$ is prime, so I add it to the set: $\{2,3,5,7,211\}$. If the resulting number is not prime, add the prime divisors of this number which are not in the set.
Just out of curiosity, does this ge... |
Does euclid's proof of infinitude of primes generate all the primes if you keep adding the new primes to the list? |
Let $A$ be a finitely generated algebra over some field $k$, and assume that $A$ is an integral domain.
By Noether normalization, we have a finite injective ring homomorphism $\varphi: k[x_1, \ldots, x_n] \hookrightarrow A$.
**Is $A$ always a free module over $k[x_1, \ldots, x_n]$?**
I know that this fails for a... |
Consider a probability distribution on the set of all ordered binary strings of length $M$. Each element may then be written as $( s_1, s_2, \cdots, s_M )$ with each $s_k \in \{ 0, 1\}$ and the probability distribution is a non-negative function $p(\{s_k\})$ which sums to $1$.
Just by parameter counting, the probab... |
Can you obtain the probability distribution on a finite discrete space through its moments? If so, how? |
I'm looking for a pairing function $f(x,y)$ which gives unique values for every combination of integers $x,y>0$ and 0 for $x=0$ $\forall y$.<br>
Or to be more general we can also replace $0$ with constants. <br>
- We do know the max values of $x$ and $y$ can achieve
- We are allowed to shift $x, y$ values. So we al... |
Let's call an *algebraic invariant* (under the action of $\text{GL}(\mathbb{R}^n)$) of a tensor of order $N$ to an algebraic combination of the components of that tensor. In other words, a homogeneous polynomial in the components of the tensor.
* For $N = 2$ algebraic invariants are well known: given a second-order t... |
As a hint, remember that big-O is *not* a tight bound. Could you make $r(n)$ grow much more rapidly than $g(n)$, such that $O\left( \frac{s(n)}{r(n)}\right)$ ends up being much smaller than $\frac{f(n)}{g(n)}$?
Hope this helps! |
I am a bit new to functional analysis and I stumbled upon this problem that confuses me.
Consider the space $X$ of bounded, non-decreasing, right-continuous functionals $F$ on $[0,1]$. Specifically, the space of cumulative distribution functions on $[0,1]$ is a subset of $X$. I have a continuous linear functional $\... |
Riesz representation theorem with Stieltjes integral? |
Image 1 is the textbook question and image 2 is the graph in question.
[1]: https://i.stack.imgur.com/DZV8Z.jpg
[2]: https://i.stack.imgur.com/5Y9XE.jpg
The answer key says that the domain is all real numbers. I thought that the domain is actually -2<x<2.
My question is, is the answer key incorrect in thi... |
Consider a sequence of holomorphic function $\{f_n\}$ on a domain $D$ with limit function $f$ . It is known that if the convergence is compact (i.e. converge uniformly to $f$
in every compact subset of $D$), $f$ is holomorphic on $D$.
Then is there an example that $f$ is holomorphic on $D$ but the convergence is n... |
Image 1 is the textbook question and image 2 is the graph in question.
[1]: https://i.stack.imgur.com/DZV8Z.jpg
[2]: https://i.stack.imgur.com/5Y9XE.jpg
The answer key says that the domain is all real numbers. I thought that the domain is actually -2<x<2.
My question is, is the answer key incorrect in t... |
Image 1 is the textbook question and image 2 is the graph in question.
[![enter image description here][1]][1]
[![enter image description here][2]][2]
The answer key says that the domain is all real numbers. I thought that the domain is actually -2<x<2.
My question is, is the answer key incorrect in this ca... |
Image 1 is the textbook question and image 2 is the graph in question.
[![enter image description here][1]][1]
[![enter image description here][2]][2]
The answer key says that the domain is all real numbers. I thought that the domain is actually -2<x<2.
My question is, is the answer key incorrect in this ca... |
Image 1 is the textbook question and image 2 is the graph in question.
The answer key says that the domain is all real numbers. I thought that the domain is actually -2<x<2.
My question is, is the answer key incorrect in this case or am I wrong? I'm new to learning math as a hobby and I don't have anyone who can ... |
How do you evaluate the following integrals?
1. $\int f df$
2. $\int f' df$
3. $\int f'' df$
Where $f(x)$ is a function of x, and $f', f''$ are the first and second derivatives with respect to x.
Regarding integral 1, I think that the answer is: $(1/2)f^2$. But I get stack with the other two. |
How to integrate a function or its derivatives with respect to itself? |
Let $(X_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$ and $\tau$ be a finite stopping time adapted to $(X_t)_{t\ge0}$. Say I run the following algorithm:
[![enter image description here][1]][1]
Say my abort criterion ensures that $\sum_{i=0}^{n-1}\tau_i$ is a samp... |
How do we get an estimator for $\operatorname E\left[\int_0^\tau f(X_t)\:{\rm d}t\right]$? |
Image 1 is the textbook question and image 2 is the graph in question.
[![enter image description here][1]][1]
[![enter image description here][2]][2]
The answer key says that the domain is all real numbers. I thought that the domain is actually -2<x<2.
My question is, is the answer key incorrect in this ca... |
I'm reading Emily Riehl's *Category Theory in Context* and pondering Exercise 1.2.i, where the author writes "Defining $\mathcal{C}/c$ to be $(c/\mathcal{C}^{op})^{op}$," in an effort to exhibit duality between the slice and co-slice categories. If you allow yourself complete freedom to just "reverse all the arrows" i... |
When performing the "opposite" operation in a category, can you reverse arrows that are part of an object's definition? |
[nonlinear differential equation][1]
Is it possible to obtain below result by considering z(0)=0?
[result][2]
[1]: https://i.stack.imgur.com/1a5gV.png
[2]: https://i.stack.imgur.com/t4Py6.png |
What kind of nonlinear differantial equation is this? |
Consider a sequence of holomorphic function $\{f_n\}$ on a domain $D$ with limit function $f$ . It is known that if the convergence is compact (i.e. converge uniformly to $f$
in every compact subset of $D$), $f$ is holomorphic on $D$.
Then is there an example that $f$ is holomorphic on $D$ but the convergence is n... |
Ross and Pekoz's book A Second Course in Probability gives the following question in their chapter on Stein-Chen Method:
Compute a bound on the accuracy of a normal approximation for a Poisson random variable with mean 100.
I believe this post was seeking help on the same question, but never found it https://math... |
Here is one way. Since $3x+2y+z=x+(x+y)+(x+y+z)$, the problem is transformed to find how many positive solutions for $a+b+c=2021$ such that $0<a<b<c$.
Without any restriction, there are $2020\choose 2$ positive solutions. Since 2021 is not divisible by 3, it is impossible to have $a=b=c$. Now consider the cases of ... |
What kind of nonlinear differential equation is this? |
Consider a sequence of holomorphic function $\{f_n\}$ on a domain $D$ with limit function $f$ . It is known that if the convergence is compact (i.e. converge uniformly to $f$
in every compact subset of $D$), $f$ is holomorphic on $D$.
Then is there an example that $f$ is holomorphic on $D$ but the convergence is n... |
The problem is to maximize $(x-y)^2(y-z)^2(z-x)^2$ where $x^2+y^2+z^2=1$.
Since the domain is compact and the function is continuous, it is certain that extrema must exist. Plus I figured out that the maximum is $1/2$ where $(x,y,z) = (1/\sqrt{2}, 1/\sqrt{2}, 0)$ by Lagrange multiplier.
However, the calculatio... |
The problem is to maximize $(x-y)^2(y-z)^2(z-x)^2$ over all $(x,y,z)\in \mathbb{R}^3$ satisfying $x^2+y^2+z^2=1$.
Since the domain is compact and the function is continuous, it is certain that extrema must exist. Plus I figured out that the maximum is $\frac{1}{2}$ where $(x,y,z) = (\frac{1}{\sqrt{2}}, \frac{1}{\... |
> Let $Q_c(x)=x^2+c$. Prove that if $c<\frac14$, there is a unique
> $\mu>1$ such that $Q_c$ is topologically conjugate to
> $F_\mu(x)=\mu(1-x)$ via a map of the form $h(x)=\alpha x+\beta$.
Interpretation: $h(x)$ is a linear map. $Q_c(x)$ and $F_\mu(x)$ are a quadratic maps. The claims is that if $c<\frac14$, then... |
Let $k$ be a local field, and consider the local ring of the formal power series $k[[x_1, \cdots, x_n]]$.
Consider the field of fraction $k((x_1,\cdots, x_n)):=\text{Frac}(K[[x_1, \cdots, x_n]])$.
It is not a local field, but we can define a discrete valuation for it.
I am interested in automorphisms of $k((x... |
Is there any substantial study regarding the automorphism of the field $k((x_1,\cdots, x_n)):=\text{Frac}(k[[x_1, \cdots, x_n]])$? |
Recently I meet a very complex integral and wonder whether there is an analytic solution or an approximate expression to this integral? The expression is as follows:
$$ \mathcal{Z}^{l}_{n}(r, x)=\int_{0}^{\infty}exp(-\frac{R^2}{W^2}) I_{n}(\frac{BxR}{W^2}) J_{n+l}(rR) RdR$$
where $B$ and $W$ are constants. $J_n$ $I_n... |
Is there an analytic solution or approximate expression to this integral? |
Is it useful to define the derivative as $\lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$? |
> Let $Q_c(x)=x^2+c$. Prove that if $c<\frac14$, there is a unique
> $\mu>1$ such that $Q_c$ is topologically conjugate to
> $F_\mu(x)=\mu(1-x)$ via a map of the form $h(x)=\alpha x+\beta$.
Interpretation: $h(x)$ is a linear map. $Q_c(x)$ and $F_\mu(x)$ are a quadratic maps. The claims is that if $c<\frac14$, then... |
In this
>answer to [Is there any valid complex or just real solution to $\sin(x)^{\cos(x)} = 2$?](https://math.stackexchange.com/a/4693242/905886),
one must calculate $$\frac{d^{n-1}}{dw^{n-1}}\left.\frac{4^{-\frac{n}{\sqrt w}}}{\sqrt w}\right|_1=\sum_{m=0}^\infty\frac{\Gamma\left(\frac12-\frac m2\right)(-\ln(4)... |
In order to calculate the Fourier transform of a formula as follow:
$$F(x, y) = exp(-\frac{(x-x_0)^2 + (y-y_0)^2}{W^2})e^{-iltan^{-1}(\frac{y}{x})} $$ Where $B$ and $l$ are constants. After transform this formula into cylindrical coordinate system I meet a very complex integral and wonder whether there is an analytic ... |
**Does there exist two binary operations + and ∗ on {0,1} such that + and ∗ satisfy the same equations, but not the same quasi-equations? If not, are there such binary operations on a finite set of higher cardinality, and if so, what is the smallest possible cardinality?**<p>
(Partial) answers:<p>
<ol>
<li> No, ... |
Let $k,n\in\mathbb{N}$. Let $N(k,n)$ denote the size of the finite set
$$\{(x_1,\cdots,x_k)\in\mathbb{N}^k:x_1+2x_2+\cdots+kx_k=n\}. $$
I feel it special and important. Do $N(k,n)$ have names? Is there a table of values? |
In this
>answer to [Is there any valid complex or just real solution to $\sin(x)^{\cos(x)} = 2$?](https://math.stackexchange.com/a/4693242/905886),
one must calculate $$\frac{d^{n-1}}{dw^{n-1}}\left.\frac{4^{-\frac{n}{\sqrt w}}}{\sqrt w}\right|_1=\sum_{m=0}^\infty\frac{\Gamma\left(\frac12-\frac m2\right)(-\ln(4)... |
Let $G$ be a reductive algebraic group over the field $\mathbb C$, $B\subset G$ a Borel subgroup and $G/B$ their quotient variety.
For every $B$-module $(M,\rho)$ we can construct the associated $G$-equivariant vector bundle $G\times^BM\to G/B$, where $G\times^BM$ is the quotient of $G\times M$ by the relation $(g,... |
<h4>Question Summary (to Make Easier to Reference in the Answer)</h4>
$$
\text{If }x^2-16\sqrt{x}=12 \text{ what is the value of }f=x-2\sqrt{x}
\tag{Eq. 1}$$
<h4>Checking for Consistency by Plugging in the $f=x-2\sqrt{x}=2$</h4>
Check $ x - 2\sqrt{x}-2=0$ implies $(\sqrt{x}-1)^2=2+1=3$ so $\sqrt{x}=1+\sqrt{3... |
Antisymmetric, or skew-symmetric tensors on a subset of indices are those that get multiplied by $-1$ when any of the indices from the subset are transposed. This type of tensor is widely used in physics and mathematics.
Now imagine I have, say, a tensor of type $(3, 0)$ that gets multiplied by $e^{2\pi i\over3}$ wh... |
Does this type of tensor appear anywhere? |
How many functions such that the Image is a partition? |
In order to calculate the Fourier transform of a formula as follow:
$$F(x, y) = \exp\left(-\frac{(x-x_0)^2 + (y-y_0)^2}{W^2}\right)e^{-il\tan^{-1}(\frac{y}{x})} $$ Where $B$ and $l$ are constants. After transform this formula into cylindrical coordinate system I meet a very complex integral and wonder whether there is... |
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