1
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0answers
34 views

Actuarial Problem - Conditional Probability

Given the information below $$ \begin{array}{|c|c|c|c|c|} \hline \textbf{Group} & \textbf{A} & \textbf{B} & \textbf{C}& \textbf{Total} \\ \hline \text{Age }26-35 & 60 & 40 &...
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0answers
13 views

Calculate Probability of People Stoping Cable Service

The Problem: I have 100 individuals that decided to join a cable service in the first month; the first month is free and no payment is required. After the first free month, 70% of the 100 people ...
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0answers
44 views

evaluation or simplification of integrals

i need a help to evaluate or simplify as possible the following integrals , my aim is find an approximation to this integration : here $$m(x)=\int_0^x \frac 1 {\sqrt{2\pi t^3}}\exp\left(-\frac 1 {2bx}...
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0answers
14 views

Frobenius condition in infinite dimensions

My setting is this: $X$ a smooth manifold. $\phi_t:X\rightarrow X$ a flow of a vector field. $f\in \mathcal C^{\infty}(X;\mathbb R)$ an element of the space of smooth functions on $X$ and \begin{...
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0answers
33 views

PDF of $X+Y$ for $X$ and $Y$ independent with given PDFs

Consider two independent random variables $X$ and $Y$. Let $f_X(x)=1-\frac{x}{2}$ if $ 0 \leq x \leq 2$ and $0$ otherwise. Let $f_Y(y)=2-2y$ if $ 0 \leq y \leq 1$ and $0$ otherwise. Find the ...
1
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0answers
31 views

exercice: continuous semi-martingale

Let $X_t=X_0+M_t+A_t$ a continuous semi-martingale. Let $g:R->[-1,1]$ ,$C^{\infty}$ with g(x)=-1 if $x \leq 0$ and g(x)=1 if $x \geq 1$. Let $f_n:R->R$ with $f_n(0)=0$ and $f_n'(x)=g(nx)$ with $...
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0answers
84 views

Why is proving that $10$ is solitary considered very difficult?

The title says it all. We denote the sum of the divisors of $x$ by $\sigma(x)$. The ratio $I(x)=\sigma(x)/x$ is called the abundancy index of $x$. If $I(m)=I(n)$, then $\{m,n\}$ is called a ...
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0answers
30 views

A quick manual method to find $\lim_{x\rightarrow \infty}(\frac{x}{e}-x(\frac{x}{1+x})^x)$?

Can someone suggest me a quick manual method to find $\lim_{x\rightarrow \infty}(\frac{x}{e}-x(\frac{x}{1+x})^x)$ ? I'm just going on and on... P.S:I'm just in high school..keep it down to my ...
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0answers
21 views

Weak convergence of stochastic integral and its quadratic variation

Consider a sequence of stochastic processes $Z^{(n)}$ and a stochastic process $Z$ on $[0,1]$ such that $$ \int_0^\cdot Z^{(n)}_t dW_t \overset{d}{\longrightarrow} \int_0^\cdot Z_t dW_t $$ converge in ...
1
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0answers
48 views

Is $p(p + 1)$ always a friendly number for $p$ a prime number?

Let $\sigma(x)$ denote the sum of the divisors of $x$. We call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A positive integer $N$ is friendly if there exists a positive integer $M \...
1
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0answers
34 views

Is my understanding of conjugation in a group correct?

I'm reading some lecture notes and watching some videos, but the purpose of group conjugation was never really made explicit. After a good amount of time, this is what I've come up with, in the ...
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0answers
21 views

Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional modules of $U_q(g)$?

Let $g$ be a complex simple Lie algebra and $U_q(g)$ the corresponding quantum group. Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional ...
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0answers
28 views

Convergence of infinite series.

I tried finding it's convergence by converting it into exponential series using first by ratio test and then by Raabe's test and reached closer to the answer but not the exact answer. $$\frac{a+x}{1!}...
1
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0answers
21 views

Extend of cutoff function

$(M,g_t)$ is a family of Riemannian manifold ,$g_t$ evolve under Ricci flow $\partial_t g_{ij}=-2R_{ij}$. At $t=0$ ,we define $\varphi$ as below first picture . Then ,extend $\varphi=0, $ outside $...
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0answers
35 views

Absolute convergence implies convergence .

Absolute convergence implies that $ \mid \mid s_m\mid −\mid s_n \mid \mid \space = \space \mid \mid x_{m}\mid +\mid x_{m−1}\mid +…\mid x_{n+1} \mid \mid \space \leq \space ϵ $ But $\mid s_m −...
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0answers
48 views

Sum of traces over Weyl group

I'm interested in computing sums like $\sum_{\sigma \in W} tr(\sigma ^3)$ , where $W$ is the Weyl group of $SO(2n+1)$, i.e. $W = (\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$. I tried to figure out what an ...
1
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0answers
62 views

For which sequences $\{a_n\}$ is $\lim_{n \rightarrow \infty}(1+\frac{1}{a_n} )^{a_n}=e$ true?

I would like to know when I can say that $$\lim_{n \rightarrow \infty} \left(1+\frac{1}{a_n}\right)^{a_n} = e$$ Is it enough for sequence $a_n$ to be monotonic sequence than tends to infinity? Can ...
1
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0answers
29 views

Mean and variance on a metric space

It is my first post, so please correct me if I am not following the rules/etiquette. Assume that we are given a space $\mathcal{S}$ composed by vectors $x\in\mathbb{R}^L$, constrained by $$\sum x_i=...
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0answers
34 views

probability in rock paper and scissors

This is actually a programming Question I had faced 2 times. Q) Mark was left behind on Mars by his crew. He found species there similar to humans.He taught them the game of Rock-Paper-Scissor (one ...
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0answers
25 views

Trying to find a taylor series expansion a power series

Let $F(z) = \sum_{n=1}^{\infty} \frac{1}{n^z} $. I want to find taylor series of $F(z) $ around $z=2$. I have computed the $kth$ derivative of $F$ as $$ F^{(k)}(z) = \sum_{n=1}^{\infty} \frac{ (-1)...
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0answers
51 views

Methods to show that an ideal isn't principal in a quadratic number field?

Suppose $a,m\in\mathbb Z_{\ge2}$. Let's consider the ring $A=\mathbb Z[(1+\alpha)/2]$, where $\alpha^2=1-4a^m$, and the ideal $I=(a,(1+\alpha)/2)$, we need to show that $I^n$ is non-principal when $0&...
1
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0answers
47 views

Compact matrix integral operator bound via its kernel

Let $\mathcal{H} = L^p_{n}(0,a)$, where $p \in \mathbb{R}^+$, $p \geq 1$ and $n \in \mathbb{N}$, be the Hilbert space of vector-valued functions defined on the finite interval $(0,a) \subset \mathbb{R}...
1
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0answers
12 views

Proving a change of variables is a variable over a ring

I was reading through some solutions to some problems in Lang here, and I was confused by some of the terminology. In particular, on page 25 of the document it says that "Now we show that Y is ...
1
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0answers
37 views

Numbers $n$ whose prime factors are $2$ and $5$ if and only if $\sum_{k=1}^{rad(n)}\mu(k)k=0$?

In the literature if defined the arithmetical function $rad(n)$ as $1$ if $n=1$ and for $n>1$ by $rad(n)=\prod_{p|n}p$. It is obviously a multiplicative function. Too we know the Mobius function $\...
1
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0answers
47 views

Finding the coordinates of R such that PR+RQ is minimum

Let P=(1,1) and Q=(3,2). Find the point R on the axis such that PR+RQ is minimum Let the coordinates of R be $(h,k)$. For PR+RQ to be minimum, PRQ would have to be a straight line. But R lies on ...
1
vote
0answers
42 views

Does an equation of this type have complex solutions?

How can I show, that every quadratic equation of the type $z^2+az+b=0$ with complex coefficients $a,b$ has a solution in $\mathbb{C}$? And how can I get these solutions? At first, I tried applying ...
1
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0answers
37 views

Differentiability of certain integral

Is the function $$f(x,y,z)=\int_0^x\int_0^y\int_0^z\int_0^x\int_0^y\int_0^z\frac{dx_1 \, dy_1 \, dz_1 \, dx_2 \, dy_2 \, dz_2}{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$$ $C^2$-differentiable in all its ...
1
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0answers
22 views

Subscript manipulation in summation

I am looking at a proof of the commutativity of multiplication in a linear algebra, and I do not understand the steps taken to simplify a summation. The product is defined by $$(fg)_n=\sum_{i=0}^...
1
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0answers
21 views

Notating an arbitrarily long series of series

Is there a better notation for the following sum? \begin{equation} y = \sum\limits_{x_1}^{S_{x_0}} \sum\limits_{x_2}^{S_{x_1}} \dots \sum\limits_{x_{n}}^{S_{x_{n-1}}} f(x_n) \end{equation} The fact ...
1
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0answers
48 views

Tangent Spaces & Definition of Differentiation

What is meant by definition when we talk about the $c:(-\epsilon,\epsilon)\rightarrow M$ is that an interval on the curve? In differential geometry what is the difference between $D_{c(t)}$, $Df_{...
1
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0answers
26 views

relationship between curvature and exterior differential operator

I am reading selected topics in harmonic maps written by James Eells and Luc Lemaire. Let $\xi:V\to M$ be a vector bundle with connection $\triangledown$. In the book, author defines the curvature of ...
1
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0answers
38 views

Evaluate Integral of $\int_0^{\frac{\pi 2^n}{2}}\sin^{2+d}(u)\prod_{i=1}^{n}\cos^{2-d}(u/2^i)du$

Integral of $$\int_0^{\frac{\pi 2^n}{2}}\sin^{2+d}(u)\prod_{i=1}^{n}\cos^{2-d}(u/2^i)du$$ I've tried a number of ways to re-write this in a way that makes sense, but no luck thus far. The integrand ...
1
vote
0answers
20 views

Number of permutations of [2n] where $x_i + x_{i+1} \ne 2n+1$

As stated in title, what is the number of permutations of $[2n]$ where $x_i + x_{i+1} \ne 2n+1 \;,\forall i\in[2n-1]$. I want to use the inclusion-exclusion theorem, and consider separately ...
1
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0answers
26 views

What is the best low-rank approximation of a non-negative matrix?

I have a non-negative matrix $X$ and I compute its Singular Value Decomposition: $$X = U \Sigma V^T$$ then, I take the lower rank approximization: $$X_k = U_k \Sigma_k V^T_k$$ where $k < rank(X)$, $...
1
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0answers
27 views

Linear Algebra - Field of 2 elements

I'm working on the following problem: $F = \mathbb{F}^2$ is the field of two elements. Let $U \subset F^4$ be the subspace generated by $(1,1,1,1),(1,1,0,0),(0,1,1,0)$ and $V \subset F^4$ be ...
1
vote
0answers
32 views

Can a non-trivial factor of a strong Fermat-pseudoprime always be found efficiently?

Suppose, $N$ is a composite Fermat-pseudoprime to base $a$ : $$a^{N-1}\equiv 1\ (\ mod \ N)$$ If $N$ is NOT strong Fermat-pseudoprime to base $a$, a non-trivial factor of $N$ can be found ...
1
vote
0answers
20 views

How to Find the Range and the set notation

For appropriate sets A and B , determine the range of a function f : A → B that assigns (a) to each integer the sum of that integer and its negative. (b) to each pair of ...
1
vote
0answers
23 views

Geometrical interpretation of the Triple Product Rule

Most of us who have done multivariable calculus are familiar with the rule $$ \left( \frac{\partial x}{\partial y} \right)_{z} \left( \frac{\partial y}{\partial z} \right)_{x} \left( \frac{\partial z}{...
1
vote
0answers
36 views

Geometric Structures of a fixed area.

Lets $M_A$ be the space of metrics of area $A$ on a two dimensional surface $S$, and let $D_0$ be the group of area-preserving diffeomorphisms whose right action on $M_A$ is given by pullback. The ...
1
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0answers
21 views

Does Linnik's approximation to Goldbach's problem also work for the power of 3, 5, 7, etc ?

Linnik proved in 1951 the existence of a constant K such that every sufficiently large even number is the sum of two primes and at most K powers of 2. Roger Heath-Brown and Jan-Christoph Schlage-...
1
vote
0answers
24 views

How to upper-bound $\sum_{i=1}^n \frac{s_i}{i + \sqrt{s_i}}$ and $\sum_{i=1}^n \frac{s_i}{i + {s_i}}$?

Any ideas how to upper bound $A$, $B$, $C$ and/or $D$: $$ B = \sum_{i=1}^n \frac{s_i}{i + {s_i}} $$ $$ C = \sum_{i=1}^n \frac{1}{i + \sqrt{s_i}} $$ $$ D = \sum_{i=1}^n \frac{1}{i + {s_i}} $$ as a ...
1
vote
0answers
78 views

Area Between Polynomials as a Similarity Measure?

I'm trying to find a conceptually easy way to calculate a "similarity" number between two polynomials. The answer to this question compares the roots of each polynomial, but it isn't obvious to me how ...
1
vote
0answers
20 views

Finding a diffeomorphism to put an ODE in a different form.

Suppose that you are given a function $F:\mathbb{R} \rightarrow \mathbb{R}$ (that is as smooth as you need to assume, I don't really care too much right now about regularity) such that $F(a) = 0, \; a&...
1
vote
0answers
20 views

Algorithm for calculating the nearest Euclidean matrix with constraints

In the area of symmetrizing the atom positions of a molecule, one runs typically into the so called Euclidean distance matrix problem. After symmetrization a distance matrix $D$ which is symmetric ($...
1
vote
0answers
23 views

Akra-Bazzi method - constructive proof

As I was familiarizing myself with different methods of computing complexities of recurrences, I stumbled upon the Akra-Bazzi method. Seeing such a beautiful result literally made my day. I was able ...
1
vote
0answers
17 views

necessary and sufficient conditions for a function to “have a tangent vector at almost all points”

If I have a nice curve $z(t)$ and a reparametrization $t(s)$, what does it mean for $z(t(s))$ to have a tangent vector at all s, (or more specifically, at almost all s)
1
vote
0answers
23 views

An apparent contradiction in the simple Lie algebra $E_8$

The following is the Dynkin diagram for simple Lie algebra $E_8$ My question is the following: It is clear that $e_i+e_j$ for $i \neq j$ is a positive root. Let $\alpha _8$ be the fundamental ...
1
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0answers
5 views

Stochastic withdrawals from finitely-lived stock

Suppose an energy source has n quanta of energy in storage, all of which are available now (t=0) until t = T, at which time the energy source disappears (or is no longer available). Suppose there are ...
1
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0answers
25 views

What can I learn from the differentials of a series?

I apologize in advance for my language - I am not in any sort of mathematical community so I do not know the proper terms for the things I am talking about. I have two series’ each containing an ...
1
vote
0answers
14 views

Probability of k nodes at a certain level.

As this is homework I do not wan't a complete solution, just pointers where to look :) given there are $n$ persons at the bottom of a building, level 0. what is the probability that $k$ persons ...

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