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Power-Method convergence proof for not-diagonalizble Matrix

I would like to find proof of Power-Method convergence for non-diagonalizble Matrix. I have seen the only Version on Wikipedia and do not manage to find the original proof. Maybe someone knows, where ...
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Variational Lower Bound with latent SDE

In this paper https://arxiv.org/pdf/2007.06075.pdf, the authors give a formula in equations 13 and 14, for the ELBO for a specific VAE (latent variable governed by an SDE) that I have difficulty ...
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Symmetric monoidal categories and modules over the unit

Consider a symmetric monoidal category $(C, \otimes, I)$, where $I$ is the unit. Then there is a restricted Yoneda functor $$ C \rightarrow Hom(I,I)-mod $$ taking an object $X$ of $C$ to $Hom(I, X)$, ...
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Global minimization Problem for $f(x) = \frac{|| Ax+b||^2}{c^Tx+d}$

I am having trouble proving the following and I would appreciate help with it: Let $A \in \mathbb{R}^{m\times n}$ be of rank $n$, $b \in \mathbb{R}^m, c \in \mathbb{R}^n, d \in \mathbb{R},$ and $\...
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  • 355
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$h^{1,1}$ of blow-up of a surface

Let $S$ be a smooth projective variety of dimension $2$ over $\mathbb{C}$, consider the blow-up $\tilde{S}$ of $S$ along one point $x\in S$. How can I show that $h^{1,1}(\tilde{S})=1+h^{1,1}(S)$?
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A navigation problem: Is the path of the ship straight or curved?

My first post here: I’m looking for some guidance with a maths problem. A ship sets sail from England (A) to France (B) covering a distance of 20 miles at an average speed of 5mph. If the ship sails ...
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Justify the differentiation in Durrett's investment problem.

I'm trying to solve the Exercise 2.4.3.(ii) of the Durrett's book, 5th edition. Here, $V_{i}\geq0$ is a random variable over the measure space $\Omega$ with $EV_{i}^{2}<\infty$ and $EV_{i}^{-2}<\...
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Residues at this function

Okey, so I must calculate $$ \int_{|z+1|=4}\frac{z}{e^z+3}dz $$ In order to do so I've calculated the roots of the denominator and found that (for our region of integration) $e^z+3=0 \iff z_1=ln3 +\pi ...
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  • 11
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gaussian integral without the error function

here is my attempt.is it correct? $$I = \int e^{-x^2}dx$$ let $u = e^{-x^2}$ and $v=x$ then : $$I = xe^{-x^2} - \int -2x^2e^{-x^2} dx ; J = \int -x^2e^{-x^2}dx$$ here is where I'm in doubt am I alowed ...
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Discrete Holonomy: Tapp's Differential Geometry book. Exercise 6.6

Here is the problem: Let $S$ be a complete oriented regular surface and $\gamma : [a, b] \rightarrow S$ a regular curve in $S$. Let $a = t_0 < t_1 <... < t_n = b$ be a regular partition of $[...
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A inequation about convex functions

if f is a convex function, prove: $$\frac{1}{3}\int_0^1 f(x)dx-\frac{1}{2}\int_0^1 xf(x)dx \geq \frac{1}{12}$$
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$\int_0^1 \frac{1}{u+f(x)} dx= a \exp(b u)$, solve $f(x)$

$\int_0^1 \frac{1}{u+f(x)} dx= a \exp(b u)$, solve $f(x)$. I want the integral at left is equal to the form of $a \exp(b u)$. Here $a$ and $b$ are just some parameters or constants, they both can be 1 ...
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A question about ideals $(6,7+\sqrt{-5})=(1+\sqrt{-5})$.

$(6,7+\sqrt{-5})=(1+\sqrt{-5})$. Is it correct? These are ideals of the ring $\mathbb Z[\sqrt {-5}]$. Can I write this equivalence becasue $1+\sqrt{-5}=7+\sqrt {-5}-6$ and so $(1+\sqrt{-5})$ is a ...
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1 vote
1 answer
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How can one prove that if $f$ linear, then $f$ is continuous [duplicate]

Let $(V,\|.\|)$ be a normed vector space, $ f:V\to \mathbb{R} $ be linear and $ \ker(f) $ closed. Then $f$ is continuous. How does one prove this? My idea was to distinguish between two cases. Because ...
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What is the gauge theory field-configuration moduli space?

Suppose we want to consider a (global, but not necessarily purely topological) gauge QFT for a principal $G$-bundle $P \xrightarrow{\pi} M$, with just one (bosonic) gauge field and no others. Without ...
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Show $y^4 - x^5$ is irreducible

Is there a 'conceptual' way to see that $f(x,y) = y^4 - x^5$ is not the product of two power series $a(x,y)$ and $b(x,y)$ unless either $a$ or $b$ are invertible? I guess I am thinking of $\mathbb{C}[[...
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Taking multiple optimization steps on the same trajectory not well justified.

I am reading the paper Proximal Policy Optimization Algorithms found at https://arxiv.org/pdf/1707.06347.pdf. In this paper they say "While it is appealing to perform multiple steps of ...
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Nonlinear Metric Space

Is there an area of mathematics that deals with a multi-dimensional space that includes some (e.g., three) dimensions with the same distance metric and a dimension with a different distance metric?
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1 answer
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Symbolab solves $0=8a-10000/a^2$ and $0=8a-\frac{10000}{a^2}$ differently. Aren't the equations the same?

Given the both formulas - in my opinion they are exactly the same: (A): (A): https://www.symbolab.com/solver/step-by-step/0%3D8a%20-%2010000%2Fa%5E%7B2%7D?or=input (B): (B): https://www.symbolab.com/...
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  • 1
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Exceptional divisor of blow-up in a non-rational point

Let $X$ be an algebraic variety over $k$ and $Spec(L) \to X$ be a point of degree $n$ on $X$. Is there any description of $Bl_{Spec(L)}X$ in terms of $L$? Is there any connection between them in the ...
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Is there $\sigma_\infty$ such that $\sigma_\infty$ is a subsequence of $\sigma_n$ for all $n \in \mathbb N$?

I have come across this "Diagonal method" in this lecture note. Theorem: Let $S$ be a non-empty set. For each $n\in \mathbb N$, let $\sigma_n$ be a sequence of elements of $S$. We denote by ...
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Euler's Integrals

Calculate this integral through Euler's integrals ( Gamma and Beta functions ): $$\int_0^\pi x \sin^p(x)\ dx$$ Сhanging variables ( $u = \sin^p x$ ) or performing such a transformation ( $\int_0^\pi x ...
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  • 1
1 vote
2 answers
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A Simple Example of a Function on $[0, 1]$

I am looking for a concrete example of a continuous function $f$ on the unit interval $[0, 1]$ such that $0\leq f(x) <1$ for each $x\in [0, 1]$, but $\int_0^1 f(x) dx = 1$. Any help is greatly ...
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Integral inequality derived from inner product.

Let $f$ be a continuous function from $[0,1]$ to $\mathbb R$ and $f>0$. I want to show that $$1\le (\int_0^1f(t)dt)(\int_0^1\dfrac{1}{f(t)}dt)$$ My idea is to use the inner product $\langle f,g \...
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(Digital Logic Design ) Combinational circuit design problem.

Design a circuit that will add either $1$ or $2$ to a $4$-bit binary number $N$. Let the inputs $N_3, N_2, N_1, N_0$ represent $N$. The input $K$ is a control signal. The circuit should have outputs $...
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Definition of static spherically symmetric spacetime as fiber bundle

I am working on a physical paper about solutions of Einstein field equations in case of static spacetimes with perfect fluid spheres and wanted to use a new definition of the spherical symmetry there. ...
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Convergence Rate of the tail of a series.

Consider a sequence $(a_k)_{k \in \mathbb{N}}$. Then the (absolute) convergence of the series $\displaystyle\sum_{k=1}^\infty a_k$ implies that that the sequence of tail series has limit zero, i.e. $$\...
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verifying markov property

Consider only paths which: when started from $a > 0$, move with constant velocity 1. when started from $a = 0$, it stays at $0$ for a time $\xi$ and then starts moving with constant velocity 1. ...
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3 votes
1 answer
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Why does the ninth successive difference of primes appear to have two distinct groups?

Was exploring successive differences of primes and noticed an interesting pattern of the histogram of counts for the sixth and ninth difference. The ninth is more pronounced, code and image below. ...
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How to find basis and dimensions of vector spaces

Let V={(a,b,c,d) an element of R4:b-2c+d=0}and w={(a,b,c,d) an element of R4:a=d, b=2c}. Find a basis and dimension of a. V b. W c. V intersection W
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Probability problem my AP statistics teacher can't solve

This is a challenge problem that my AP Stat teacher can't solve, so I am hoping that I can find an answer here. I am aware that you could use a computer to run simulations to get an approximate, but I ...
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1 vote
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Minimizing error for A in Ax=b if x and b are matrices

There was an answer give for how to solve Ax=b if x and b are known here: Solving for $A$ in $Ax = b$ But I was wondering is there a way to solve it if there is no exact solution and you want to ...
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Unitaries, or mapping one basis to another

The 2x2 matrix answer I'm getting is wrong and I have only one trial left. Please help! Let the basis $\{\mathbf{v}\}$ be the set of vectors $\left\{ \left[\begin{array}{c} \cos\theta_{1}\\ \sin\...
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-1 votes
1 answer
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How to write in mathmatics symbol when we definite intergral not at once, twice, ... at N times a function like the image. [closed]

[enter image description here][1] [1]: https://i.stack.imgur.com/MzA4V.png ...
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Question about about the proof of Jensen Formula (Conway)

Let $F(z) = f(z)\prod_{k=1}^{n}\frac{r^{2}-\overline{a_{k}}z}{r(z-a_{k})}$ which is analytic on $\overline{D}(0;r)$ and has no zero in $D(0;r)$. I don't understand why $\vert F(z)\vert =\vert f(z)\...
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1 answer
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Category of divisible abelian groups is additive but not abelian

I read that the Category of divisible abelian groups is additive but not abelian but I can’t find a proof…I haven’t understand very much what a divisible group is and I don’t know how to work this… Is ...
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Can this problem adjust to a Bin Packing problem?

hope you are doing fine. I am facing some mathematical problem which I am not certain of which type is it, nor how to approach its solution. Here is a picture with a representation (simplify) of the ...
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1 vote
1 answer
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How many ways to make $n$ pairs from 2 groups of $n$ people?

I'm looking to count ways to make $n$ pairs if there are two groups of $n$ people and each pair must consist of one person from each group. My initial thought is $^nP_n = n!$, as we can line the ...
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Fast approximation for bivariate normal cumulative distribution function

The cdf of a normal distribution has some nice approximations like $$F(z) = 1/(1 + \exp(-0.07056 x^3 – 1.5976 x)),$$ see https://www.econstor.eu/bitstream/10419/188388/1/v02-i01-p114_60-313-1-PB.pdf ...
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Boundary of smooth domain difference

Let $\Omega_{1},\Omega_{2} \subset \mathbb{R^2}$ be open, connected and bounded domain with piecewise smooth boundary. Let suppose that $\Omega_{1}\cap\Omega_{2}\neq \emptyset$ has a piecewise smooth ...
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Show that $\varlimsup\limits_{n\rightarrow \infty}(\frac{1+a_{n+1}}{a_n})^{n} \geq e$ for any positive sequence $(a_n)$

"Show that $\varlimsup\limits_{n\rightarrow \infty}(\frac{1+a_{n+1}}{a_n})^{n} \geq e$ for any positive sequence $(a_n)$" I found this problem on page 148 of Zorich's Mathematical Analysis (...
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Every Cayley Digraph of $\mathbb{Z}_{n}$ is isomorphic to a Metacirculant digraph

Let $m,n$ be positive integers and $\alpha\in \mathbb{Z}_n^*$. Define $\rho, \tau:\mathbb{Z}_m\times \mathbb{Z}_n \to \mathbb{Z}_m\times \mathbb{Z}_n$ by $$\rho(i,j) = (i,j+1)$$ $$\tau(i,j) = (i+1,\...
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Minimize norm of difference, mapped into several spaces

I'm looking for a non-computational solution, in other words for an explicit formula, for the following problem. Let $Y$ be a vector space and $X$ a subspace of $Y$. Given $y\in Y$, and $n$ morphisms ...
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Calculating how many automorphisms a graph has

I am trying to figure out how many automorphisms graphs have but I believe I am going wrong when calculating for this graph. The answer I am getting is 48 but that seems too high for this graph: EDIT: ...
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-3 votes
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Vector space: How to prove $(a+b)(x+y) =ax+ay+bx+by$

How to prove $(a+b)(x+y) =ax+ay+bx+by$, where $x,y \in V$ and $a,b \in F$, where $V$ is a vector space and $F$ is the field?
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Reading about the null set and Lebesgue measure - asking for reading recommendation

I hope its allowed to ask a more "general" question here I just started learning multivariable integral, and we just learned about the null set and Lebesgue measure ( in our defention we use ...
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1 vote
1 answer
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Series representation of GCD(x, y)

While playing around with WolframAlpha, I discovered that apparently the GCD of any integers x, y is equal to the following sum: $$ x + y - xy + 2\sum_{k = 1}^{x-1} \lfloor \frac{ky}{x}\rfloor $$ I've ...
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Random truncation with poisson process

Suppose, a Poisson process is observed in fixed time intervals, e.g. we record the output of a Geiger counter during a time interval $\Delta t$ started randomly. The decay rate is much larger than ...
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  • 1
1 vote
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Prove uniform convergence of bounded sequence

Supose we have a sequence of real continuous functions on an interval $[0,b] \in \mathbb{R}$ $(f_n)_{n=0}^{\infty}$ (hence uniformly continuous). Moreover the sequence has the following properties: ...
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2 votes
2 answers
32 views

Explicit examples of formal power series which is not rational functions?

This MO question https://mathoverflow.net/questions/249541/formal-power-series-is-taylor-expansion-of-rational-function-iff-hankel-determin states that if $k$ is a field and $k[[T]]$ is the power ...
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