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2 views

property of asymptotic rate of convergence of gradient decent algorithm on a quadratic form

Geven a quadratic form $f(x)=\frac{1}{2}x^TAx+x^Tb+a$, where $A$ is a symmetric positive definite matrix. We use gradient decent to compute the global min by $x_n=x_{n-1}-\triangledown f(x_{n-1})$ and ...
2
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0answers
11 views

the tangent bundle of an $m$-dimensional manifold is $2m$

It is often taken as obvious that for $m$-dimensional manifold $M$ in $\mathbb{R}^n$, $TM$ is a $2m$-dimensional manifold in $T\mathbb{R}^n=\mathbb{R}^n\times\mathbb{R}^n$. But still is there an ...
1
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1answer
3 views

Eigenvalues and vectors of hamiltonian system

I must find the eigenvalues of the following system: $\hat H = V (|\alpha \rangle \langle \beta| +|\beta\rangle\langle\alpha|)$, where $\langle \alpha|\alpha \rangle=\langle\beta|\beta\rangle=1 $, $\...
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0answers
4 views

a simple inequality on the simplex

Could anyone help me to show: For any $v=(v_1,v_2, v_3), v_i\ge 0, \sum_{i=1}^{3} v_i=1$ one has $\sum_{i=1}^{3} v_i|\bar p_i-p|\le\sum_{i=1}^{2}\sum_{j=i+1}^{3}\frac{v_i+v_j}{2} |\bar p_i-\bar p_j|$ ...
0
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0answers
4 views

$\mathrm{Supp}(\mathrm{Exc}(f))=\mathrm{Supp}(K_X-f^*K_Y)?$

Let $f:X\to Y$ be a log resolution of normal varieties, let $\mathrm{Exc}(f)$ be the locus (the complement in $X$ of the largest open where $f$ is an isomorphism). Is there an example of $f$, such ...
0
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0answers
2 views

What do two ODEs with the same Lyapunov function have in common

Assume I have two $N$-dimensional systems ofODEs $\frac{dX}{dt}=F(X)$ with $F: \mathbb{R}^N\rightarrow\mathbb{R}^N$ and $\frac{dY}{dt}=G(Y)$ with $G: \mathbb{R}^N\rightarrow\mathbb{R}^N$. Also, ...
-1
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0answers
11 views

I'm wondering if my work is right. Propositional Logic.

So this is my works, I'm wondering if I can do step 7 or can't. If there is another way to solve this, please help me. Also sorry if my english is bad.
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1answer
12 views

Relation between roots and coefficients of an equation

If $p,q,r $ are the roots of the equation $x^3 - 3px^2 + 3q^2x - r^3 = 0$ Hello! I hope everybody is doing well. Can anybody please help me with the above problem? My Solution: From Vieta’s $p+q+r ...
-2
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0answers
16 views

Please help me to interpret this question

Prove by induction, that a set with $n$ elements has $$\frac{n(n−1)(n−2)}6$$ subsets containing exactly three elements whenever $n$ is an integer greater than or equal to $3$.
-2
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2answers
22 views

How can I show $f(x) = x^3 - 6x^2 + 11x - 6$ is one-one or not?

I am stuck in showing whether the function mentioned above is one-one or not. I can clearly tell that it is not one-one by looking the graph, but I have to show it by proving that if $f(a) \ne f(b)$, ...
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0answers
12 views

Deriving a formula for a function

Let $S$ := {$(x, y) ∈ \mathbb R^2: x^2 + y^2 = 1$} be the unit circle. Let $p = (−1, 0)$ ∈ $S$. Define a map $F$ from $\mathbb R$ × {$0$} to $S$ as follows: Given $(t, 0)$, let $F((t, 0)) = (x, ...
0
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0answers
7 views

If $(\Omega, F, P)$ is a probability space show that $(\mathbb{R}, B(\mathbb{R}), \mu_X(B))$ is a probability space

If $(\Omega, F, P)$ is a probability space show that $(\mathbb{R}, B(\mathbb{R}),\mu_X(B))$ is a probability space, where $\mu_X(B)=P(X^{-1}(B))$, $\:\: X: \Omega \to R$ and $B(\mathbb{R})$ is the ...
0
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0answers
11 views

Are there techniques for quickly seeing patterns in numbers?

Recently I was watching 8 out of 10 cats does countdown (S18E01 - 26 July 2019), for which the math question there was the following numbers: if you haven't seen countdown, you are provided 6 numbers,...
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0answers
2 views

derivative of matrix exponent

I need to calculate derivative of expression $\ a'*exp(a*a')*a\ $ with respect to n-dimensional vector a. I understand, that I can express exponential as series and use SVD, $\ a'*P*exp(D)*P^(-1)*...
2
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1answer
27 views

Integral of $(dx)^2$

What is the integral of $(dx)^2$, that is, $\int{(dx)^2}$
-1
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1answer
16 views

Be G an abelian finite group. If every element of G has at maximum order two then $G \cong (Z/2Z)^n$ [duplicate]

I'm trying to prove this proposition Be G an abelian finite group. If every element of G has at maximum order two then $G \cong (Z/2Z)^n$ I tried a lot, but my ideas are not so useful. Somebody ...
1
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0answers
8 views

Proving limits using u-substitution

I'm trying to prove that u-substitution is allowed in limits. How might one prove that if $w = \displaystyle\lim_{n \to a} g\left(n\right)$ then: $ \displaystyle\lim_{n \to a} f\left(g\left(n\right)\...
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1answer
15 views

Proof by Binomial theorem

I am supposed to prove using Binomial theorem that number $$11^{10}-1$$ ends with at least with two zeroes. My solution so far: $$11^{10}-1=(10+1)^{10}-1$$ $$\sum _{i=1}^{10} \binom{10}{i}10^{10-i}1^...
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0answers
8 views

Let n be a natural number with the prime decomposition n = p1^s1 + p2^s2 + … + pk^sk [duplicate]

Prove that if n = m^2 for some natural number m then s1, s2, . . . , sk are all even.
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0answers
7 views

Subtracting from a Venn diagram

I've come across this problem recently. I'm assuming that its a Venn diagram situation, but that's all I can figure out Q: a = a' + X b = b' + X c = c' + X Using only a, b, and ...
0
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0answers
7 views

Number of possible trees given number of nodes

I have a problem that I initially thought was easy but later turned out to seem more difficult. So, given a tree within the hydra game (http://math.andrej.com/2008/02/02/the-hydra-game/), I am ...
2
votes
0answers
16 views

Inequality with conditions $\sum_{cyc}\cos(a\cos(b))\geq \sum_{cyc}\cos(a\cos(c))$

it's an inequality by me : Let $a,b,c>0$ such that $a+b+c=1$ and $a\geq b \geq c $ then we have : $$\sum_{cyc}\cos(a\cos(b))\geq \sum_{cyc}\cos(a\cos(c))$$ This inequality is very precise . ...
0
votes
1answer
29 views

$\int \frac{\sqrt{x}}{\sqrt[4]{x^3+1}}dx$

solve:$$\int \frac{\sqrt{x}}{\sqrt[4]{x^3+1}}dx$$ Unfortunately I dont know which substitution should use
0
votes
1answer
23 views

$f$ is analytic function over a unit disk.

let $f$ be an analytic function on a unit disk $D=\left \{ z \in \mathbb{C}: |z|<1 \right \}$ such that the range of the function is contained in the $\mathbb{C} - (-\infty,0]$. Then i have to ...
0
votes
2answers
46 views

Suppose $|a+b| = 2$, $|a|=2$, $|b|=1$, find $|a\cdot b|$

I know $$|a\cdot b| = |a||b|\cos(x)$$ But I don't know how to use this formula to calculate $|a\cdot b|$ given $|a+b|$.
0
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0answers
3 views

Following the gradient vector to a stationary point

Consider the surface $z=-75x^4-\frac{100}{3}x^3+\frac{199}{2}x^2-3xy-\frac{9}{2}y^2+200$. If we are located at a point, say $(0,6)$, and we head in the direction of steepest ascent, is there a way to ...
0
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0answers
15 views

Numeric functions

Let $n$ and $k$ be positive integers. A function $f$:{$1, 2, 3, 4, ... kn$} --> {$1, ... 5$} is said to be good if $f (j + k) - f (j)$ is a multiple of $k$ for all $j = 1.2, ..., kn - k$ a) Prove ...
1
vote
1answer
6 views

First Success distribution PMF sum problem

I have the following problem: Each toss of a coin results in a head with probability $p$. The coin is tossed until the first head appears. Let $X$ be the total number of tosses (including the count ...
0
votes
2answers
21 views

Is R transitive if R = {(a,a), (b,a), (b,b), (b,c), (c,a), (c,b), (c,c)}

I know the rule is ((a R b) ∧ (b R c)) → a R c but since (a,b) is not in R in the first place can R still be transitive?
0
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2answers
21 views

Is it true that $\sum_{n=0}^{\infty} a_n \frac{(z_0+h)^n-z_0^n}{h} = \sum_{n=1}^{\infty} a_n \frac{(z_0+h)^n-z_0^n}{h}$ (Notice the index)

There's a proof in my textbook where the author uses the followng equality, $$ \sum_{n=0}^{\infty} a_n \frac{(z_0+h)^n-z_0^n}{h}- \sum_{n=1}^{\infty}na_n z_0^{n-1} = \sum_{n=1}^{\infty} a_n \left(\...
0
votes
0answers
10 views

Ultra-Slow Speed Process Control (is there exist Mathematical Background?)

I want to ask a question to respected mathematicians and specialists in control theory: Is there a separate direction in nonlinear control theory that develops control of super slow processes? Are ...
1
vote
0answers
8 views

Dimensionality and functional form of the natural conjugate prior to the two-parameter Normal distribution

Exponential family distributions take the following form: $$ p(x | \boldsymbol{\eta}) = h(x) \exp\left(\boldsymbol{\eta} \mathbf{T}(x) - A(\boldsymbol{\eta}) \right) $$ where $\boldsymbol{\eta}$ are ...
0
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0answers
7 views

How to show a renewal process with $m(t)=E[N(t)]=\lambda t$ is a poisson process?

Suppose here is a renewal process $N(t), t\geq 0$. $m(t)=E[N(t)]=\lambda t$. How to show that it is a Poisson process with parameter of $\lambda$? Thanks.
0
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2answers
33 views

Calculation of binomial coefficient

I just want to ask how to calculate, if I have: $$\binom{17}{8,9}$$ Thanks
0
votes
1answer
16 views

How to split a set into two disjoint subsets in a special way?

Suppose $S$ is a finite set (the number of its members is not large). The set $\Sigma=\{s_1, \ldots, s_N\}$ is a set of subsets of $S$, i. e. $s_i \in S$. Is it possible to split $S$ into disjoint ...
-2
votes
0answers
11 views

Term for functions with results between 0 and 1?

Is there a term for functions $f: x \in \mathbb{R}_{\geq 0} \to [0, 1] $ that have the following properties: Strictly monotony $f(0) = 0$ $\lim_{x \to \infty} f(x) = 1$ Functions (found here) I ...
0
votes
0answers
18 views

How to prove that cake-eating problem has no solution?

Consider the following optimization problem (called cake-eating): $$\sum\limits_{t = 0}^{\infty} u(a_t) \to \max$$ subject to $$\sum\limits_{t = 0}^{\infty} a_t \leq s, \quad s >0, \...
-1
votes
0answers
28 views

Derivative of $a^x$ [on hold]

Assume $f_a(x)=a^x$. And given that $f_a'(x)=f_a'(0)*f_a(x)$ a. Prove that: $f_a'(x)=(Log_ba)f_b'((Log_ba)x)$ Do all this without knowing that e exists, and without knowing its properties. b. ...
0
votes
2answers
20 views

Prove that this space of sequence is measurable

Let $\mathbb{R}^{\mathbb{N}} = \prod_{i \in \mathbb{N}} \mathbb{R}$ and suppose $\mathcal{B}^{\mathbb{N}}$ is product $\sigma$-algebra of Borel sets in $\mathbb{R}^{\mathbb{N}}$ For $m \in \mathbb{Z}$,...
1
vote
4answers
38 views

Three polynomials which have the same value for a variable

Let $P_1(x)= ax^2-bx-c$, $P_2(x)=bx^2-cx-a$, and $P_3(x)= cx^2-ax-b$ be three quadratic polynomials, where $a,b$, and $c$ are non-zero real numbers. Suppose that there exists a real number $k$ such ...
-1
votes
1answer
24 views

Show that there exists $x \in \mathcal{H}$ such that $\langle x,e_n \rangle = \alpha^n$

In my homework I have the following problem. Let $(e_n)_{n=1}^\infty$ be an orthonormal basis for a Hilbert space and $\alpha \in \mathbb{R}$. Show that there exists $x \in \mathcal{H}$ such that $\...
0
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0answers
13 views

Is it true that $\mu\left(\left\{x\in\partial K : \limsup_{r\to 0^+}\ \frac{\mu(B_r(x)\cap K^c)}{\mu(B_r(x))}>0\right\}\right)=0?$

Suppose $(X,d)$ is a complete separable metric space, $\mu$ a finite Borel measure, $K\subset X$ a compact of $(X,d)$ such that $\mu(\partial K)>0$. Is it true that $\mu\left(\left\{x\in\partial ...
1
vote
1answer
19 views

Asymptotic expansion to solution of $x - \log x = a$ for large $a$

Suppose $a \in \mathbb{R}$ is very large. Then there are two solutions to $x - \log x = a$. I was wondering what the asymptotic expansion of the larger solution to this equation is. The first term ...
0
votes
0answers
6 views

Image of a line bundle on an elliptic curve under Fourier-Mukai transform

Let $E$ be an elliptic curve with a base point $p$. What is the image of the line bundle $\mathcal{O}_E(np)$ under the Fourier-Mukai transform with a kernel given by the Poincare bundle on $E\times E$?...
1
vote
0answers
14 views

Proof of dimension of tangent space to manifold

I am struggling to understand the details of a proof in Wald's General Relativity, pp. 15-16. Claim Let $M$ be an $n$-dimensional manifold. Take some point $p\in M$ and let $V_p$ be the tangent space ...
0
votes
0answers
14 views

Problem with understanding solution of discrete mathematics task

The task says: three dice are thrown. On how many ways sum of numbers the dice will be 14? This a solution: https://imgur.com/Dze8vd6 I understand first two rows, but I don't understand how did we ...
0
votes
0answers
7 views

Combinatorial bijective proof equality

Let define $H^{m}((n);\mu)$ count the number of tuples $(\alpha,\tau_1,\ldots,\tau_r ,\beta)$ in symmetric group $S_n$ where $\alpha$ is fixed cycle of type $(n)$ and $\beta$ fixed cycle of type $\...
0
votes
1answer
9 views

Finding orthogonal of a set in functional analysis

I am self studying functional analysis from Kreyszig Functional analysis and I need help in Ex 3.3 Question no. 5 which is ------> If X = $ R^2 $ . Then find orthogonal complement of M if M is {x} , ...
0
votes
0answers
14 views

Why doesn't the Fourier partial sums of a continuous function diverge for the entire series?

I know that there exists a $2\pi$ periodic continuous function whose Fourier series has a divergent subsequence at a point. However, I heard that for any $2\pi$ periodic continuous function $f(x)$, ...
0
votes
2answers
49 views

$\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+\dots+\frac{1}{\sqrt{n}}<2\sqrt{n}$?

I'm trying to prove this by induction, but something doesn't add up. I see a solution given here, but it is actually proving that the expression is greater than $2\sqrt{n}$. I'd appreciate some ...

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