Edwin Franks
  • Member for 1 year, 1 month
  • Last seen more than a month ago
  • Sydney NSW, Australia
the minimum of the sum of two convex functions
Accepted answer
5 votes

Let $f(x,y)=2(x+1)^2+((x+y)+1)^2$ and $g(x,y)=(x-1)^2+((x+y)-1)^2$. Since the second partial derivatives of $f$ and $g$ are positive and both $H_f=\begin{bmatrix} 6 & 2 \\ 2 & 2 \end{bmatrix}$ ...

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Example of a basis which is not a Riesz basis?
3 votes

Let $\{v_1,v_2,\dots\}$ be a complete orthonormal basis for $\mathcal{X}$. Let $w_1=v_1$, $w_2=\frac{1}{\sqrt 2}(v_1+v_2)$ and more generally for $k=3,4,\dots$ let $w_k=\frac{1}{\sqrt k}(v_1+v_2+\...

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Derivative of Inverse of sum of matrices
Accepted answer
3 votes

$$\frac1{\Delta x}\big[(A+(x+\Delta x)B)^{-1}-(A+xB)^{-1}\big]=$$ $$\frac1{\Delta x}(A+xB)^{-1}(A+xB)\big[(A+(x+\Delta x)B)^{-1}-(A+xB)^{-1}\big] (A+(x+\Delta x)B) (A+(x+\Delta x)B)^{-1}$$ $$=-(A+xB)^{...

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inequality of integrals
2 votes

\begin{align*} \int_0^1\int_0^1 & (x-y)f^2(x)f(y)\mathrm{d}x\mathrm{d}y = \int_0^1\int_0^y(x-y)f^2(x)f(y)\mathrm{d}x\mathrm{d}y\ + \\ & \int_0^1\int_0^x(x-y)f^2(x)f(y)\mathrm{d}y\mathrm{d}...

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For which real $s$ does $\sum _{n=1}^{\infty} \frac{1}{n^s}$ converge?
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2 votes

Fix $s>1$. For $k=2,3,\dots$, one has $2^k-1=3,7,15,\dots$ and \begin{align*} \sum_{n=1}^{2^k-1}\frac{1}{n^s} & = 1 + %% \Big(\frac{1}{2^s}+ \frac{1}{3^s}\Big) + %% \Big(\frac{1}{4^s}+...

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Invariant superoperators under unitary action
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2 votes

Let $\Phi\in\mathcal{S}(\mathcal{H})$ and suppose that for any unitary conjugation operator $\mathcal{U}\in\mathcal{S}(\mathcal{H})$, $\mathcal{U}\Phi=\Phi\mathcal{U}$. Let $\{e_1,e_2,\dots,e_d\}$ be ...

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Functional not attaining its norm
2 votes

Let $\displaystyle{F(x)=\sum_{n=1}^\infty x_n}$. $|F(x)|\leq \|x\|_1<\|x\|_1+\|x\|_p$ so that $\|F\|^*\leq 1$. Let $u_k$ be defined by $u_{k,n}=1/k$ if $n\leq k$ and $u_{k,n}=0$ if $n> k$. Since ...

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Does the quotient of the dyadic and ternary rationals greater than $1$ by their powers of $3$ have the trivial topology?
1 votes

Let $[x],[y]\in X_{>1}/\langle3\rangle$ with $x,y\in X_{>1}$ the least element of their respective equivalence classes. I claim that \begin{equation} \label{eq:pseudometric} d([x],[y])=|3^{-i}...

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Unicity of the equidistant set from the boundary of an open set
Accepted answer
1 votes

The answer I believe is no. Let $V=(0,1)\times(0,1)$, let $F$ be the square with corners $(-1,-1),\ (2,-1),\ (2,2)$, and $(-1,2)$. The minimum distance between $\partial V$ and $F$ is $a=1$. However, $...

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Does a given positive measure set in $\mathbb{R}^2$ contains product of two 1 dimensional positive measure set?
1 votes

Let $A$ and $B$ be defined by \begin{align*} A & =\big\{a\in\mathbb{R}: a^2=\sum_{i=1}^\infty a_i10^{-2i}, a_i\in\{1,2\,\dots,9\}, a^2\not\in \mathbb{Q}\big\} \\ B & =\big\{b\in\mathbb{R}: b^...

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Proving a property of exponential function applied to linear operators
1 votes

\begin{align*} %% \Big\|\sum_{k=0}^{n} \binom{n}{k}& \frac{1}{n^k} B^k-e^B\Big\| = %% \Big\|\sum_{k=0}^{n} (1-\frac1n)(1-\frac2n)\cdot(1-\frac{k-1}{n}) %% \frac{B^k}{k!} -e^B\Big\|\\ %...

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Use Laplace transform to solve $f'(t) = H(t - a) \int_{0}^{t-a} dt'\; g(t') f(t-t')$
1 votes

$$f'(t)=H(t-a)\int_{0}^{t-a} g(s) f(t-s)\mathrm{d}s =\int_{0}^\infty g(s)H\big((t-a)-s\big) f(t-s)\mathrm{d}s.$$ Note that for $t<a$, $f'(t)=0$ and hence $f(t)=f(0)$ for $0\leq t\leq a$. The ...

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Uniqueness of solution of an integral equation [Updated]
1 votes

If $f(y) = c_0 + c_2 y^2 + \int f(z) e^{f(y-z)}\mathrm{d}z$ differentiating three times gives $f'''(y)=\int f'''(z)e^{f(y-z)}\mathrm{d}z$. Thus, $f'''=f'''*e^f= f'''*\underbrace{e^f \ast \cdots \ast e^...

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Prove $s<\infty\to S<\infty \to I<\infty$
Accepted answer
0 votes

To show that $s<\infty$ implies $S<\infty$ for $p>1$ set $a_n:=\mu\big(\{x:2^n<f(x)\leq2^{n+1}\}\big)$, $b_n:=\mu\big(\{x:2^n<f(x)\}\big)$, $q=p/(1-p)$ and $M_n=\sup\{a_n\bigm| m\geq n\}...

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Pareto allocation problem when the utility is decreasing in one good
0 votes

With positive consumption, the marginal utility of $x_2$ is negative for the first consumer, and positive for the second consumer, $$\frac{\partial u_{1}}{\partial x_2}<0 \text{ and } \frac{\...

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Does concavity imply that $f'(a)<\frac {f(a)}a$?
0 votes

See equation (3) page 7, $\eta(a)=a\nu'(a)$. Thus if $\nu$ is concave, $\nu''<0$ so, $$\eta'(a)=\nu'(a)+a\nu''(a)<\nu'(a)=\frac1a(a\nu'(a))=\frac{\eta(a)}{a}.$$

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Does the ratio of the $x-$values of two continuous functions $f(x), g(x) \in \mathbb{R^+}$ tell you anything about the ratio between their $y-$values?
0 votes

If $\lim_{x \to \infty}f(x) = \lim_{x \to \infty}g(x) = 0$ and \begin{equation} \lim_{x \to \infty} \frac{f(x)}{g(x)}=L_1 \text{ and } \lim_{y \to 0+} \frac{f^{-1}(y)}{g^{-1}(y)}=L_2\tag{1} \end{...

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