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Adjoint of weighted composition operator lemma
Lemma: Suppose that W^_{\psi,\phi}: H^2(D) \to H^2(D) is bounded and \beta \in D. then W^{\psi,\phi}K\beta = \bar{\psi(\beta)}K_{\phi(\beta)}. Where D is the unit disk, and H^2 is the Hardy-Hilbert ... linear-algebra complex-analysis functional-analysis- 13
Family of sets closed under intersection and union of countable chains under inclusion induces a finitary closure operator
Let $\mathcal{U} \subseteq \wp\left ( X \right )$ be a family of sets that contains both $\emptyset$ and $X$, is closed under arbitrary intersections and is closed under union of countable chains ... real-analysis general-topology metric-spaces set-theory- 65
Sylverster's law of inertia
Any bilinearform $s$ is Diagonazible and if we tweek the Basis vektor's just a little and order them accordingly,say $\mathcal{B}:=(v_1,\ldots,v_n)$ we achieve that the transformation matrix takes the ... linear-algebra- 83
Integrating $f(x,y)dx$ for unknown $y$
Is there a way to express an integral such as $$\int (2-y)x^{y-2}dx$$ where $y\equiv y(x)$ we don't know what $y$ is. Is there a way to do this type of integral or do we have to know $y(x)$. calculus integration indefinite-integrals- 420
Determine the parameters a and b so that the function is continuous
I did this task with the function and I got the result that a = -1 and b = 3, can someone help me and check if it is a good result or if I got the right solution. Determine the parameters a and b so ... functions solution-verification continuity- 1
If it exists, can cross sections of a real tesseract appear to us in 3D space completely different than Schlegel diagram?
We as human beings cannot comprehend how an object looks like in spatial dimensions higher than 3, it is in fact unimaginable. Yet, in mathematics we are able to project analogues of objects from ... geometry mathematical-physics projective-geometry projective-space dimension-theory-analysis- 1
Conversion a 2D point value, back to 3D given a known Y value
I am dealing with an image processing problem, and I think I am doing something wrong, mathematically. I have a homogenous point in 3D $P=(x,y,z,1)$, and a corresponding homogenous point on the image ... linear-algebra linear-transformations image-processing- 1,019
Finding Hatcher's notation confusing
Define $f : S^1\times I\to S^1\times I$ by $f (\theta, s) = (\theta + 2πs, s)$, so $f$ restricts to the identity on the two boundary circles of $S^1 \times I$. Show that $f$ is homotopic to the ... general-topology- 41
Find limit of inverse function
Let $x$ be a function $x(y): \mathbb{R} \to \mathbb{R}$, where it is not possible to find the inverse relation $y(x)$ in a closed form. Is there a way to find the limit \begin{equation} \lim_{x \to \... real-analysis- 39
conditional probability question doubt
Consider an experiment having three possible outcomes that occur with probabilities $p_1$, $p_2$, and $p_3$, respectively. Suppose n independent trials of the experiment are conducted and let $X_i$ ... probability probability-theory random-variables conditional-probability- 3
Number of points on an elliptic curve over $F_{p^n}$
$E$ is an elliptic curve with non-split multiplicative reduction at prime $p$. I'm trying to find the number of points $E$ over $F_{p^n}$. I know that when I remove the singularity, the rest is a ... elliptic-curves- 43
Easy Differential Series Can anyone help me to continue this equation after thia step?
I have to solve this equation by using this method and im stuck at this step Please help me to solve it :* ordinary-differential-equations partial-differential-equations power-series differential-forms formal-power-series- 1
integrating over a sphere using norm of x
My question is similar to "". I have the integral as shown in the picture. My question is how do i start the ... integration spheres- 1
Impossibility of probability measure on $2^{[0,1]}$
Let $\Omega = [0, 1]$ and define a set function on the subsets $(a, b] \subset \Omega$ as $P( (a, b] ) = b-a$ Prove that no extension of $P$ from the subsets where it is defined to the power set of $\... probability probability-theory measure-theory lebesgue-measure- 49
Show that the sequence $(T_nx)$ is bounded for every x ∈ $X$.
Here is the question: Given $X$ and $Y$ are Banach spaces and $T_n$ : $X$ → $Y$ a sequence of bounded operators. Show that the sequence $(|f(T_nx)|)$ is bounded for every x ∈ $X$ and every f ∈ $Y^∗$ ... functional-analysis normed-spaces dual-spaces- 33
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