text:
The Ornstein-Uhlenbeck process ${X={(X_t)}_{t\in[0,\infty)}}$ on ${\mathbb{R}^n}$ is the solution of the stochastic differential equation

$dX_t=\sqrt{2}dB_t-X_tdt$

where ${{(B_t)}_{t\in[0,\infty)}}$ is a standard Brownian motion. Since the diffusion coefficient is constant and the drift is affine, it follows that ${X}$ is a Gaussian process. The computation of the mean and of the variance of ${X_t}$ conditional on ${\{X_0=x\}}$ yields

$\mathrm{Law}(X_t\mid X_0=x)=\mathcal{N}(xe^{-t},\sqrt{1-e^{-2t}}I_n).$

This shows that for any ${x}$ and conditional on ${\{X_0=x\}}$, ${X}$ converges in distribution:

$X_t\underset{t\rightarrow\infty}{\overset{d}{\longrightarrow}}\gamma_n$

where ${\gamma_n=\mathcal{N}(0,I_n)}$ has density ${(2\pi)^{-\frac{n}{2}}e^{-\frac{1}{2}|x|^2}}$. This shows also that ${\gamma_n}$ is invariant:

$X_0\sim\gamma_n\quad\Rightarrow\quad\forall t\geq0,\quad X_t\sim\gamma_n.$

Actually a stronger property holds true: the law ${\gamma_n}$ is reversible in the sense that

$X_0\sim\gamma_n\quad\Rightarrow\quad\forall t\geq0,\quad (X_0,X_t)\overset{d}{=}(X_t,X_0).$

The explicit law of the process allows computations, for instance for any ${s,t\geq0}$,

$\mathrm{Cov}(X_s,X_t)=e^{-|t-s|}(1-e^{-2\min(s,t)}).$

For any bounded and measurable ${f:\mathbb{R}^n\rightarrow\mathbb{R}}$, any ${x\in\mathbb{R}}$ and ${t\in[0,+\infty)}$, we set

$P_t(f)(x)=\mathbb{E}(f(X_t)\mid X_0=x).$

We have ${P_t(\mathbf{1}_A)(x)=\mathbb{P}(X_t\in A\mid X_0=x)}$. The family ${{(P_t)}_{t\in[0,\infty)}}$ is a semigroup of linear operators acting on continuous and bounded functions, in the sense that

$P_0=id, \quad \forall s,t\geq0, \quad P_t\circ P_s = P_{t+s}.$

These operators are Markov operators, in the sense that for any ${t\in[0,\infty)}$,

$\forall c\in\mathbb{R}, P_t(c)=c, \quad\text{and}\quad \forall f, f\geq0 \Rightarrow P_t(f)\geq0.$

The explicit law of the process provides the Mehler formula for the semigroup

$P_t(f)(x) =\int\!f(xe^{-t}+\sqrt{1-e^{-2t}}y)\gamma_n(dy)\\ =\mathbb{E}(f(xe^{-t}+\sqrt{1-e^{-2t}}Z)).$

This gives the following commutation with the gradient when ${f}$ is smooth:

$(\nabla P_t f)(x)=e^{-t}P_t(\nabla f)(x),$

where in the right hand side, ${P_t}$ acts on each coordinates of the vector ${\nabla f}$.

The infinitesimal generator is the unbounded operator in ${L^2(\gamma_n)}$ given by

$Af =\partial_{t=0^+}P_tf =\lim_{t\rightarrow0^+}\frac{P_tf-f}{t} =\Delta f-\langle x,\nabla f\rangle.$

The Chapman-Kolmogorov evolution equation writes

$\partial_tP_t=AP_t=P_tA.$

If we fix ${f}$ and write ${u_t(x)=P_t(f)(x)}$ for any ${x}$ and ${t}$ then

$u_0=f\quad\text{and}\quad \partial_t u_t=Au_t=\Delta u_t-\langle x,\nabla u_t\rangle.$

The operator ${A}$ (and ${P_t}$ for any ${t\geq0}$) is symmetric in ${L^2(\gamma_n)}$, in other words an integration by parts holds, meaning that for any ${f}$ and ${g}$,

$-\int\!fAg\,d\gamma_n =\int\!\nabla f\cdot\nabla g\,d\gamma_n.$

If ${X_0}$ has density ${f_0}$ with respect to ${\gamma_n}$ then ${X_t}$ has also a density with respect to ${\gamma_n}$ given by ${f_t=P_tf_0}$. If ${g}$ is the Lebesgue density of ${\gamma_n}$, then ${g_t=f_tg}$ is the Lebesgue density of ${X_t}$. The evolution of ${g_t}$ with respect to ${t}$ is described by the Fokker-Planck equation, dual of the Chapman-Kolmogorov equation,

$\partial_tg_t=\Delta g_t+\mathrm{div}(xg_t).$

If ${\mu}$ and ${\nu}$ are probability measures on ${\mathbb{R}^n}$ with ${\nu\ll\mu}$ then the Kullback-Leibler divergence or relative entropy of ${\nu}$ with respect to ${\mu}$ is defined by

$H(\nu\mid\mu)=\int\!f\log f\,d\mu=\int\!\log f\,d\nu \quad\text{where}\quad f=\frac{d\nu}{d\mu}.$

We take the convention ${H(\nu\mid\mu)=+\infty}$ if ${f\log f\not\in L^1(\mu)}$ or if ${\nu\not\ll\mu}$. Note that Jensen’s inequality shows that ${H(\nu\mid\mu)\geq0}$ with equality iff ${\mu=\nu}$.

In the case where ${\mu}$ is a Boltzmann-Gibbs measure with Lebesgue density ${g(x)=e^{-V(x)}}$, the quantity ${H(\nu\mid\mu)}$ becomes a Helmholtz free energy, in the sense that

$H(\nu\mid\mu)=\int\!V\,d\nu-S(\nu)$

where the first term in the right hand side is the mean energy of ${\nu}$ while the second term in the right hand side is the Boltzmann-Shannon entropy

$S(\nu)=\displaystyle\int\!fg\log(fg)\,dx.$

Suppose that the law ${\mu_0}$ of ${X_0}$ has density ${f_0}$ with respect to ${\gamma_n}$. Then the law ${\mu_t}$ of ${X_t}$ has density ${f_t=P_tf_0}$ with respect to ${\gamma_n=\mu_\infty}$. The free energy decays along the time. Namely, using the evolution equation and the integration by parts,

$\begin{array}{rcl} \frac{d}{dt}H(\mu_t\mid\gamma_n) &=&\displaystyle\int\!\partial_t(f_t\log f_t)\,d\gamma_n\\ &=&\displaystyle\int\!(1+\log f_t)Af_t\,d\gamma_n\\ &=&\displaystyle-\int\!\frac{|\nabla f_t|^2}{f_t}\,d\gamma_n\\ &=&-J(\mu_t\mid\gamma_n) \leq0. \end{array}$

This is know as the de Bruijn identity:

$\frac{d}{dt}H(\mu_t\mid\gamma_n)=-J(\mu_t\mid\gamma_n)\leq0.$

The quantity

$J(\nu\mid\mu) =\int\!\frac{|\nabla f|^2}{f}\,d\mu=\int\!|\nabla\log f|^2\,d\nu \quad\text{where}\quad f=\frac{d\nu}{d\mu}$

is the Fisher information. How it behaves along the O.-U. dynamics? Well, using commutation, two times Jensen’s inequality, and the invariance of ${\gamma_n}$, we get

$\begin{array}{rcl} J(\mu_t\mid\gamma_n) &=&\int\!\frac{|\nabla f_t|^2}{f_t}\,d\gamma_n\\ &=&e^{-2t}\int\!\frac{|(\nabla f)_t|^2}{f_t}\,d\gamma_n\\ &\leq& e^{-2t}\int\!\frac{(|\nabla f|)_t^2}{f_t}\,d\gamma_n\\ &\leq& e^{-2t}\int\!\left(\frac{|\nabla f_0|^2}{f_0}\right)_t\,d\gamma_n\\ &=& e^{-2t}J(\mu_0\mid\gamma_n), \end{array}$

in other words the Fisher information decays exponentially:

$\forall \mu_0\ll\gamma_n, \forall t\geq0,\quad J(\mu_t\mid\gamma_n)\leq e^{-2t}J(\mu_0\mid\gamma_n).$

In particular we get \begin{align*} H(\mu_0\mid\gamma_n) =-\int_0^\infty\!\frac{d}{dt}H(\mu_t\mid\gamma_n)\,dt =\int_0^\infty\!J(\mu_t\mid\gamma_n)\,dt \leq \frac{1}{2}J(\mu_0\mid\gamma_n). \end{align*} This inequality is known as a logarithmic Sobolev inequality:

$\forall \nu\ll\gamma_n,\quad H(\nu\mid\gamma_n)\leq\frac{1}{2}J(\nu\mid\gamma_n).$

This inequality is optimal in the sense that equality is achieved when ${d\nu(x)/d\gamma_n(x)=e^{ax}}$ for some ${a\in\mathbb{R}}$. Using this inequality for ${\nu=\mu_t}$ yields

$\frac{d}{dt}H(\mu_t\mid\gamma_n) =-J(\mu_t\mid\gamma_n) \leq -\frac{1}{2}H(\mu_t\mid\gamma_n).$

which gives, by Gronwall’s lemma, an exponential decay of the free energy, namely

$\forall\mu_0\ll\gamma_n, \forall t\geq0, H(\mu_t\mid\gamma_n)\leq e^{-2t}H(\mu_0\mid\gamma_n).$

Since both sides are equal for ${t=0}$, taking the derivative at time ${t=0}$ allows to recover from this exponential decay the logarithmic Sobolev inequality!

Hypercontractivity. For any ${t\in[0,\infty)}$ and any ${p\in[1,\infty]}$, Mehler’s formula shows immediately that ${P_t}$ can be extended into a linear operator on ${L^p(\gamma_n)}$. In fact ${P_t}$ is always a contraction:

$\forall p\geq1, \forall t\in[0,\infty), \forall f\in L^p(\gamma_n),\quad \Vert P_tf\Vert_p\leq\Vert f\Vert_p.$

Namely, using Jensen’s inequality and the invariance of ${\gamma_n}$,

$\begin{array}{rcl} \Vert P_tf\Vert_p^p &=&\int\!|\mathbb{E}(f(X_t)\mid X_0=x)|^p\,d\gamma_n(x)\\ &\leq& \int\!\mathbb{E}(|f(X_t)|^p\mid X_0=x)\,d\gamma_n(x)\\ &=&\int\!P_t(|f|^p)\,d\gamma_n(x)\\ &=&\int\!|f|^p\,d\gamma_n(x)\\ &=&\Vert f\Vert_p^p. \end{array}$

Since equality is achieved for constant functions, it follows that ${\Vert P_t\Vert_{p\rightarrow p}=1}$. The semigroup ${{(P_t)}_{t\in[0,\infty)}}$ is in fact hypercontractive:

$\forall p\geq1, \forall t\geq0, \forall f\in L^p(\gamma_n), \quad \Vert P_t f \Vert_{p(t)} \leq \Vert f \Vert_p,$

where ${p(t) = 1 + (p-1)e^{2t}}$, in other words ${\Vert P_t\Vert_{p\rightarrow p(t)}=1}$, and moreover this value ${p(t)}$ is critical in the sense that if ${q > p(t)}$ then ${\Vert P_t\Vert_{p\rightarrow q}=+\infty}$.

Let us give a proof. One can assume that ${f\geq0}$ since ${|P_t f|\leq P_t|f|}$ by Jensen’s inequality. Note that ${p(0)=0}$ and ${p(t)>p}$ if ${t>0}$. Set ${\alpha(t)=\log\Vert P_t f\Vert_{p(t)}}$. To lighten the notation, let us set ${f_t=P_tf}$. We have, for any ${t\geq0}$,

$\begin{array}{rcl} \alpha'(t) &=&\left(\frac{1}{p(t)}\log\int\!f_t^{p(t)}\,d\gamma_n\right)’\\ &=&-\frac{p'(t)}{p(t)^2}\log\int\!f_t^{p(t)}\,d\gamma_n +\frac{1}{p(t)}\frac{\left(\displaystyle\int\!f_t^{p(t)}\,d\gamma_n\right)’}{\displaystyle\int\!f_t^{p(t)}\,d\gamma_n}\\ &=&-\frac{p'(t)}{p(t)^2}\log\int\!(f_t)^{p(t)}\,d\gamma_n +\frac{1}{p(t)}\frac{\displaystyle\int\!\left(p'(t)\log f_t+p(t)\frac{Af_t}{f_t}\right)f_t^{p(t)}\,d\gamma_n}{\displaystyle\int\!f_t^{p(t)}\,d\gamma_n}\\ &=&-\frac{p'(t)}{p(t)^2}\log\int\!f_t^{p(t)}\,d\gamma_n +\frac{p'(t)}{p(t)^2}\frac{\displaystyle\int\!f_t^{p(t)}\log f_t^{p(t)}\,d\gamma_n}{\displaystyle\int\!f_t^{p(t)}\,d\gamma_n} +\frac{\displaystyle\int\!(Af_t)f_t^{p(t)-1}\,d\gamma_n}{\displaystyle\int\!f_t^{p(t)}\,d\gamma_n}\\ &=&\frac{p'(t)}{p(t)^2}\left(H(h_t^{p(t)}\gamma_n\mid\gamma_n)+\frac{p(t)^2}{p'(t)}\int\!(Ah_t)h_t^{p(t)-1}\,d\gamma_n\right) \end{array}$

where ${h_t=f_t/\Vert f_t\Vert_{p(t)}}$. Now the logarithmic Sobolev inequality and the integration by parts give, for any ${h\geq0}$ such that ${h^p}$ is a probability density with respect to ${\gamma_n}$,

$\begin{array}{rcl} \mathrm{H}(h^p\gamma_n\mid\gamma_n) &\leq&\frac{1}{2}\int\!\frac{|\nabla h^p|^2}{h^p}\,d\gamma_n\\ &=&\frac{p^2}{2}\int\!|\nabla h|^2h^{p-2}\,d\gamma_n\\ &=&\frac{p^2}{2(p-1)}\int\!\left<\nabla h,\nabla h^{p-1}\right>\,d\gamma_n\\ &=&-\frac{p^2}{2(p-1)}\int\!(Ah)h^{p-1}\,d\gamma_n. \end{array}$

Using this inequality for ${h=h_t}$ and ${p=p(t)}$, and using ${2(p(t)-1)=p'(t)}$, we obtain that ${\alpha'(t)\leq0}$ for any ${t\geq0}$, and as a consequence

$\log\Vert P_tf\Vert_{p(t)} = \alpha(t) \leq\alpha(0)=\log\Vert f\Vert_p.$

Finally, if now ${q>p(t)}$ then taking ${f_\lambda(x)=e^{\langle\lambda,x\rangle}}$ for some ${\lambda\in\mathbb{R}^n}$ gives

$\Vert f_\lambda\Vert_p=e^{p|\lambda|^2/2} \quad\text{and}\quad P_t f_\lambda=e^{|\lambda|^2(1-e^{-2t})/2}f_{\lambda e^{-t}}$

and therefore

$\frac{\Vert P_t f_\lambda\Vert_{q}}{\Vert f_\lambda\Vert_p} =e^{|\lambda|^2(e^{-2t}(q-1)+1-p)/2},$

a quantity which tends to ${+\infty}$ as ${|\lambda|\rightarrow\infty}$ since ${q>p(t)=1+(p-1)e^{2t}}$.

The proof shows that conversely, from the hypercontractive statement, one can extract the logarithmic Sobolev inequality by taking the derivative at ${t=0}$.

Polynomials. The set of polynomials ${\mathbb{R}[X]}$ is dense in ${L^2(\gamma_1)}$. To see it, let us take ${f\in L^2(\gamma_1)}$, then the Laplace transform ${\varphi_\mu}$ of the signed measure ${\mu(dx)=f(x)\gamma_1(dx)}$ is finite on ${\mathbb{R}}$ since for any ${\theta\in\mathbb{R}}$, by the Cauchy-Schwarz inequality,

$(\varphi_\mu(\theta))^2=\left(\int\! \exp(\theta x)\,\mu(dx)\right)^2 \leq \int\!f^2\,d\gamma_1\int\!\exp(2\theta x)\,\gamma_1(dx)<+\infty,$

and in particular, ${\varphi_\mu}$ is analytic on a neighborhood of ${0}$. Now since for any ${k\in\mathbb{N}}$,

$\varphi_\mu^{(k)}(0) =\int\!x^kf(x)\,\gamma_1(dx) =\langle P_k,f\rangle_{L^2(\gamma_1)}\quad\text{where}\quad P_{k}(x) =x^{k},$

and if ${f\perp\mathbb{R}[X_1,\ldots,X_n]}$ in ${L^2(\mathbb{R})}$, then the derivatives of any order of ${\varphi_\mu}$ vanish at ${0}$, and since ${\varphi_\mu}$ is analytic, we get ${\varphi_\mu\equiv0}$ and then ${\mu=0}$ and then ${f=0}$ in ${L^2(\gamma_n)}$.

Hermite polynomials. Hermite’s polynomials ${{(H_k)}_{k\in\mathbb{N}}}$ are the orthogonal polynomials obtained using the Gram-Schmidt algorithm in ${L^2(\gamma_1)}$ from the canonical basis of ${\mathbb{R}[X]}$. They are normalized in such a way that the coefficient of the term of highest degree in ${H_k}$ is ${1}$ for any ${k\geq0}$. We find

$H_0(x)=1,\quad H_1(x)=x,\quad H_2(x)=x^2-1,\quad\ldots$

It can be checked that Hermite’s polynomials ${{(H_k)}_{k\geq0}}$ satisfy

• Generating series: for any ${k\geq0}$ and ${x\in\mathbb{R}}$,

$H_k(x)=\partial^k_1G(0,x) \quad\text{where}\quad G(s,x)=e^{sx-\frac{1}{2}s^2}=\sum_{k=0}^\infty\frac{s^k}{k!}H_k(x);$

• Three terms recursion formula: for any ${k\geq0}$ and ${x\in\mathbb{R}}$,

$H_{k+1}(x)= xH_{k}(x) – kH_{k-1}(x);$

• Recursive differential equation: for any ${k\geq0}$ and ${x\in\mathbb{R}}$,

$H_k'(x)=kH_{k-1}(x);$

• Differential equation: for any ${k\geq0}$ and ${x\in\mathbb{R}}$,

$H_k”(x)-xH_k'(x)+kH_k(x)=0.$

Using the generating series and Plancherel’s formula, we get

$\sum_{k=0}^\infty \frac{s^{2k}}{k!^2}\Vert H_k\Vert_2^2 =\int\!G(s,x)^2\,\gamma_1(dx)=\exp(-s^2)\int\!e^{2sx}\,\gamma_1(dx) =e^{s^2} =\sum_{k=0}^\infty\frac{s^{2k}}{k!},$

which gives ${\Vert H_k\Vert_2^2=k!}$ by identifying the series coefficients. It follows that ${{(H_k/\sqrt{k!})}_{k\in\mathbb{N}}}$ is a dense orthonormal sequence in the Hilbert space ${L^2(\gamma_1)}$.

For any ${f\in L^2(\gamma_1)}$, we have

$f=\sum_{k\geq0}a_kH_k\quad\text{where}\quad k!a_k=\int\!fH_k\,d\gamma_1.$

In particular

$\Vert f\Vert_2^2=\int\!f^2\,d\gamma_1=\sum_{k\geq0}k!a_k^2.$

Note that ${a_0=\displaystyle\int\!f\,d\gamma_1=\gamma_1(f)}$ is the mean of ${f}$ under ${\gamma_1}$.

Hermite’s polynomials and Ornstein-Uhlenbeck process. Hermite’s polynomials are eigenvectors of the operators ${P_t}$ and ${A}$, namely for any ${k\in\mathbb{N}}$ and ${t\geq0}$,

$P_tH_k=e^{-kt}H_k \quad\text{and}\quad AH_k=-kH_k.$

The property for ${A}$ is immediate from the differential equation satisfied by Hermite’s polynomials. To establish the property for ${P_t}$, we note that for any ${Z\sim\gamma_1}$,

$P_t(G(s,\cdot))(x) =e^{se^{-t}x-\frac{1}{2}s^2}\mathbb{E}\bigr(e^{s\sqrt{1-e^{-2t}}Z}\bigr),$

and since the Laplace transform of ${Z}$ is given by ${\mathbb{E}(e^{\theta Y})=e^{\frac{1}{2}\theta^2}}$ we get

$P_t(G(s,\cdot))(x)=G(se^{-t},x),$

therefore, by the generating series property of Hermite’s polynomials,

$\begin{array}{rcl} P_t(H_k)(x) &=&P_t(\partial_1^kG(0,\cdot))(x)\\ &=&\partial_{s}^k P_t(G(s,\cdot))(x)_{\vert s=0}\\ &=&\partial_{s}^k G(se^{-t},x)_{\vert s=0}\\ &=&e^{-kt}\partial_{1}^k G(se^{-t},x)_{\vert s=0}\\ &=&e^{-kt}H_k(x). \end{array}$

This shows that Hermite’s polynomials are eigenvectors of ${P_t}$.

Exponential decay. If ${f=\sum_{k\geq0}a_kH_k\in L^2(\gamma_1)}$ then for any ${t\geq0}$,

$P_tf=\sum_{k\geq 0}e^{-kt}a_kH_k,$

and thus

$\Vert P_t f-\gamma_1(f)\Vert_2^2 =\sum_{k\geq 1}a_k^2 e^{-2kt}k! \leq e^{-2t}\sum_{k\geq 1}a_k^2 k! =e^{-2t}\Vert f-\gamma_1(f)\Vert_2^2.$

We have obtained the exponential decay in ${L^2(\gamma_1)}$: for any ${t\geq0}$ and ${f\in L^2(\gamma_1)}$,

$\Vert P_t f-\gamma_1(f)\Vert_2 \leq e^{-t}\Vert f-\gamma_1(f)\Vert_2.$

Using the invariance of ${\gamma_1}$, we get ${\displaystyle\int\!P_tf\,d\gamma_1=\int\!f\,d\gamma_1=\gamma_1(f)}$ and therefore

$\mathrm{Var}_{\gamma_1}(P_tf)\leq e^{-2t}\mathrm{Var}_{\gamma_1}(f),$

which is is equivalent to the Poincaré inequality with constant ${1}$ (optimal for ${H_1}$):

$\mathrm{Var}_{\gamma_1}(f)\leq-\int\!fAf\,d\gamma_1=\int\!f’^2\,d\gamma_1.$

This inequality is the linearization at ${h=1+\varepsilon f}$ of the logarithmic Sobolev inequality

$\int\!h^2\log(h^2)\,d\gamma_1 -\int\!h^2\,d\gamma_1\log\int\!h^2\,d\gamma_1 \leq -2\int\!hAh\,d\gamma_1=2\int\!h’^2\,d\gamma_1.$

The gap between the first eigenvalue ${0}$ and the second eigenvalue ${-1}$ of ${A}$ is of length ${1}$. This spectral gap produces the exponential convergence. More generally, the semigroup preserves the spectral decomposition. If ${f\perp\mathrm{Vect}\{H_1,\ldots,H_{k-1}\}}$ in ${L^2(\gamma_1)}$ then ${P_t(f)\perp\mathrm{Vect}\{H_1,\ldots,H_{k-1}\}}$ for any ${t\geq0}$ and for any ${t\geq0}$,

$\Vert P_t f-\gamma_1(f)\Vert_2 \leq e^{-k t}\Vert f-\gamma_1(f)\Vert_2.$

Dimension ${n}$. The operator ${A}$ is a sum of operators acting on one variable:

$Af =\Delta f-\langle x,\nabla f\rangle =A_1f+\cdots+A_nf \quad\text{where}\quad A_kf=\partial_k^2f-x_k\partial_kf.$

The eigenvectors of ${A}$ are products of univariate Hermite’s polynomials. Namely, for any ${k\in\mathbb{N}^n}$, if we denote, for any ${x\in\mathbb{R}^n}$,

$H_k(x)=H_{k_1}(x_1)\cdots H_{k_n}(x_n),$

then

$AH_k=(k_1+\cdots+k_n)H_k.$

Quantum harmonic oscillator. Let ${g_n}$ be the density of ${\gamma_n}$. Consider the isometry

$\Phi:f\in L^2(dx)\rightarrow \Phi(f)=g_n^{-1/2}f\in L^2(\gamma_n).$

One can define the operators ${K}$ on ${L^2(dx)}$ from the operator ${A}$ on ${L^2(\gamma_n)}$, namely

$Kf=(\Phi^{-1}\circ A\circ\Phi)(f) =g_n^{1/2}A(fg_n^{-1/2}) =\Delta f+\Bigr(\frac{n}{2}-\frac{1}{4}|x|^2\Bigr)f.$

This is the quantum harmonic oscillator, a special kind of Schrödinger operator. We have ${\partial_t Q_t=KQ_t}$ where ${{(Q_t)}_{t\in[0,\infty)}}$ is the semigroup of operators defined by

$Q_t(f) =(\Phi^{-1}\circ P_t\circ \Phi)(f) =g_n^{1/2}P_t(g_n^{-1/2}f)$

The eigenvectors of ${K}$ are Hermite’s wave functions: for any ${k\in\mathbb{N}^n}$,

$\psi_k(x)=g_n^{1/2}(x)H_k(x)=(2\pi)^{-\frac{n}{2}}e^{-\frac{1}{4}|x|^2}H_k(x).$

For instance, for ${k=(0,1,\ldots,n-1)}$, we get the wave function

$\psi(x_1,\ldots,x_n)=g_n^{1/2}(x)e^{-\frac{1}{4}|x|^2}H_0(x_1)\cdots H_{n-1}(x_n).$

A bosonic wave function is obtained by symmetrization over ${x_1,\ldots,x_n}$. A fermionic wave function is obtained by anti-symmetrization (implies nullity on the diagonal):

$\begin{array}{rcl} \psi_{\mathrm{fermions}}(x_1,\ldots,x_n) &=&g_n^{1/2}(x)\sum_{\sigma\in\Sigma_n}(-1)^{\mathrm{signature}(\sigma)}H_{\sigma(1)-1}(x_1)\cdots H_{\sigma(n)-1}(x_n)\\ &=&g_n^{1/2}(x)\det \begin{pmatrix} H_0(x_1)&\ldots&H_0(x_n)\\ \vdots &\vdots&\vdots\\ H_{n-1}(x_1)&\ldots&H_{n-1}(x_n) \end{pmatrix}\\ &=&g_n^{1/2}(x)\det \begin{pmatrix} x_1^0&\ldots&x_n^0\\ \vdots &\vdots&\vdots\\ x_1^{n-1}&\ldots&x_n^{n-1} \end{pmatrix}\\ &=&g_n^{1/2}\prod_{1\leq i<j\leq n}(x_i-x_j). \end{array}$

The Slater determinant is here proportional to a Vandermonde determinant. Now

$|\psi_{\mathrm{fermions}}(x_1,\ldots,x_n)|^2 =(2\pi)^{-\frac{n}{2}}e^{-\frac{1}{2}(x_1^2+\cdots+x_n^2)}\prod_{1\leq i<j\leq n}(x_i-x_j)^2.$

We recognize up to normalization the formula of the density of the Gaussian Unitary Ensemble (GUE) namely the density of the eigenvalues of a Gaussian ${n\times n}$ Hermitian random matrix with Lebesgue density in ${\mathbb{R}^{n+n^2-n}=\mathbb{R}^{n^2}}$ proportional to

$H\mapsto e^{-\frac{1}{2}\mathrm{Tr}(H^2)}.$

Notes. By pure provocation, we used the Cauchy-Schwarz inequality only once. We have learned the link with the GUE during a talk by Satya Majumdar.

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Syntax · Style · .


text:
# Tag Info

6

Pretty sure it's false - maybe you should check with the guys who wrote the exam. It's going to be a counterexample in $L^2([0,1])$, with $g=1$. Say $(I_n)_{n=1}^\infty$ is a sequence of disjoint intervals with $|I_n|=2^{-n}$. Define $$F_1=\chi_{I_1}$$and $$F_n=2^{n-1}\chi_{I_n}-2^{n-2}\chi_{I_{n-1}}\quad(n>1).$$ Then $\sum F_n=0$ almost everywhere. ...

5

Indeed, usually one allows non-negative functions to take the value $+\infty$, but one usually doesn't talk about integrable functions taking values in $\overline{\Bbb R}$. There are good reasons for that; I shuddered when I saw that definition $\infty-\infty=0$. I hope his main point in doing this is to point out that it's not a good idea... Anyway. If the ...

5

It's certainly true for $1<p<\infty$. In that case the weak topology on $B$ is the same as the weak* topology regarding $\ell^p$ as the dual of $\ell^{q}$. So $B$ is a compact Hausdorff space, and so Stone-Weierstrass implies that those polynomials are dense in $C(B)$. Note that you didn't specify what topology on $C(B)$ you're talking about when you ...

4

By Holder's inequality, $$\int_{|f_n-f|>\alpha} |f_n - f| d\mu \leq \int_{|f_n-f|>\alpha} |f_n| d\mu + \int_{|f_n-f|>\alpha} |f| d \mu \leq$$ $$\mu(\{\omega\in\Omega:|f_n(\omega)-f(\omega)|>\alpha\})^{1/q} \left(||f_n||_p+||f||_p\right)$$ As $n\to\infty$, $\mu(\{\omega\in\Omega:|f_n(\omega)-f(\omega)|>\alpha\})\to 0$, and ...

4

You've done the bulk of the work already. To show that $\|f_{n_k} - f \|_{L^p(E \setminus F)}$ is bounded, note that by the triangle (Minkowski's) inequality, $$\|f_{n_k}-f \|_{L^p(E \setminus f)} \leq \|f_{n_k}\|_{L^p(E)} + \|f\|_{L^p(E)}.$$ By Fatou's lemma, $$\|f\|_{L^p(E)}^p = \int_E |f|^p = \int_E \lim\limits_{k \to \infty} |f_{n_k}|^p \leq ... 3 Suppose f\in L^\infty([0,1]) and \|f\|_\infty> 0. Let E = \{x:|f(x)|=\|f\|_\infty\}. Then$$\lim_{p\to \infty} \left( \frac{\|f\|_\infty}{\|f\|_p}\right)^p = \frac{1}{m(E)}.$$(If m(E)=0, the conclusion is that the limit is \infty.) Proof: Let M= \|f\|_\infty. Then the expression equals$$\frac{M^p}{M^p\cdot m(E) + M^p\int_{[0,1]\setminus ...

3

$X_n \to X$ in $L^p$ for $p>1$ implies $X_n \to X$ in $L^1$ because $|| X||_p$ is an increasing function in $p$ (use Jensen), in particular $EX_n \to EX$. Since $x \mapsto x^p$ is continuous this also implie s$(EX_n)^p \to (EX)^p$.

2

For $0<p\le 1$ by the $c_r$ inequality $$|\mathbb{E}|X_n|^p-\mathbb{E}|X|^p|\le \mathbb{E}|X_n-X|^p\to 0$$ For $p>1$ the Minkowski inequality implies that $$|||X_n||_p-||X||_p|\le ||X_n-X||_p\to 0$$

2

As mentioned in the comments, by the Hölder inequality (and because $\mu(\Omega)=1$) we have $$\|f\|_1 = \int_{\Omega} |f(x)|\,d\mu(x) \leq \left(\int_{\Omega}|1|^2\,d\mu(x)\right)^{\frac12}\left(\int_{\Omega}|f(x)|^2\,d\mu\right)^{\frac12} = \|f\|_2.$$ Then, if $f_n \to f$ in $L^2$ we have $0\leq \|f_n-f\|_1 \leq \|f_n-f\|_2\to 0$, thus $f_n \to f$ in ...

2

Let $f_n$ be a square function with width $1/n$ and height $\sqrt{n}$. Then $f_n\rightarrow 0$ in $L^2$. Let $g_n=f_n$...

2

Let $f(x) = \frac{1}{|x|}$ and $g(x) = \chi_{|x|>1}\frac{1}{|x|}+\chi_{|x|\le 1}|x|$... Note here that technically $g(0)=0$, but at least $g>0$ almost everywhere.

2

In general, the $p$-norms of a function aren't increasing. If there is a measurable set $E$ with $1 < \mu(E) < +\infty$, then $$\lVert \chi_E\rVert_p = \mu(E)^{\frac{1}{p}}$$ is decreasing. Generally, there are $f$ for which $\lVert f\rVert_p$ is (eventually) decreasing and $f$ for which $\lVert f\rVert_p$ is (eventually) increasing. If $\mu$ is a ...

2

Your operator is $$Tf(x) = \int^\infty_{-\infty}1_{y\leq x}e^{-(x-y)}f(y)\,dy$$ which by change of variables $z=x-y$ becomes $$Tf(x) = \int^\infty_{-\infty}1_{z\geq 0}e^{-z} f(x-z)\,dz.$$ This is just the convolution operator $$f\mapsto \phi*f$$ with $$\phi(z) = 1_{z\geq 0}e^{-z}.$$ Now one of the fundamental inequalities (not hard to prove) about ...

2

$Tf=\phi*f$, where $\phi(t)=e^{-t}\chi_{(0,\infty)}(t)$. (I had $\phi$ backwards in the first version; noticed that guest's $\phi$ was different, then noticed his was right.) So $$\widehat{Tf}=\hat\phi\hat f.$$You can easily calculate $\hat\phi$; now the norm of $T$ is $||\hat\phi||_\infty$ and the spectrum of $T$ is the essential range of $\hat\phi$. (In ...

2

Since $f$ is $L_p$-integrable, $f$ is $L_1$-integrable. By Lusin's theorem, there exists a function $h$ which is continuous in $[0,1]$ such that $$\mu(\{x:|f(x)-h(x)|>\epsilon^{1/p}/(2\mu(\Omega))\})<\delta$$ Let $A=\{x:|f(x)-h(x)|>\epsilon^{1/p}/(2\mu(\Omega))\}$. Then $$\int_{\Omega} |f-h|^p\,d\mu = \int_{\Omega-A} |f-h|^p\,d\mu +\int_{A} ... 2 You are almost right. Just use the Cauchy-Schwarz inequality$$\mu(\Omega)=\int_{\Omega} 1 \,d\mu=\int_{\Omega} \left|f^{1/2}\frac{1}{f^{1/2}}\right|\,d\mu\leqslant \left(\int_{\Omega}|f|\, d\mu\right)^{\frac{1}{2}}\left(\int_{\Omega}\frac{1}{|f|}\, d\mu\right)^{\frac{1}{2}} $$2 Both results follow immediately with an application of Young's inequality Let 1 \le p,q,r \le \infty be such that$$\frac{1}{p} + \frac{1}{q} = \frac{1}{r} + 1.$$If f \in L^p and g \in L^q then f * g \in L^r and we get the following bound on the norm$$\|f * g\|_r \le \|f\|_p\|g\|_q.To answer your first question, let f \in L^1 and ... 1 Suppose f\in L_{w}^{p}. Fix \lambda>0 and define f_{0,\lambda}:=f\chi_{\left|f\right|\geq\lambda} and f_{1,\lambda}:=f-f_{0,\lambda}. I claim that f_{0,\lambda}\in L_{w}^{p_{0}} and f_{1,\lambda}\in L_{w}^{p_{1}}. Indeed, \begin{align*} ... 1 I'm guessing that the question is to show that \phi is locally integrable. This is very simple, doesn't use any fancy theorems. You have\phi=g*f,$$where g(x)=1/|x|. Now g=g_1+g_2, where g_1\in L^1(\Bbb R^3) and g_2\in L^\infty(\Bbb R^3). So g_1*f\in L^1, while g_2*f is continuous (and bounded). 1 Your function produces infinities in both endpoints of the interval. The usual approach is to split the integral using some well-behaved intermediate point, say x=1. Therefore, \int \limits _0 ^\infty = \int \limits _0 ^1 + \int \limits _1 ^\infty. Concerning the first one, make the change of variable x = \Bbb e ^{-t}, obtaining \int \limits _0 ... 1 Elements of L^q are precisely those classes of measurable functions with \| f \| _q < \infty. Now if you have such an f \in L^q, due to the inequality you have found you will also have \| f \| _p \le C \| f \| _q < \infty, so f \in L^p, therefore L^q \subset L^p. 1 If \mu(X)=1, there exists p>0 with f\in L^p(\mu), and if we agree that \log(0)=-\infty and \exp(-\infty)=0 then$$\lim_{p\to0}||f||_p=\exp\left(\int\log|f|\,d\mu\right), $$the geometric mean of |f|. Hence some people sometimes write ||f||_0 for the geometric mean. Assume f\ge0. First, Jensen's inequality with \phi(t)=e^{pt} shows that ... 1 The counterexample doesn't hold true: Because \sup_{x\in[0,1]}x^{-\frac{1}{2}}=\infty it follows f\notin L^{\infty}([0,1]), so the pairing \langle f,f\rangle is not defined.For f\in L^p,g\in L^q, p^{-1}+q^{-1}=1 the dual pairing \langle f,g\rangle is always finite, because of Hölder's inequality. 1 "Scalar product" is a misnomer. It's actually a duality, that is, a bilinear map L^p\times L^q\to\mathbb F. 1$$ |(f*g)(x)=\int f(x-y)g(y)\, dy| \leq \left( \int |f(x-y)|^p \, dy \right)^{1/p} \left( \int |g(y)|^q \, dy \right)^{1/q} = \|f\|_p \|g\|_q $$for (almost) all x. Hence \|f*g\|_\infty \leq... 1 In case it's not clear, if you assume only u\in L^1 then no, you can't say anything nice. First: Of course in that case you really want to consider the essential supremum, not the sup: If you let u be the characteristic function of the rationals then u=0 almost everywhere although \sup_{|y|<h}|u(x+y)-u(x)|=1 for every x. That's a totally dumb ... 1 The usual way to estimate L^p norms using weak estimates is by using the Layer-Cake formula (see L^p-norm of a non-negative measurable function)$$ \int |f|^q \, d\mu = \int_0^\infty q \cdot \lambda^{q-1} \cdot \mu(\{x \mid |f(x)| \geq \lambda\}) \, d\lambda.  Now, we split the integral into two parts: \begin{eqnarray*} ...

Only top voted, non community-wiki answers of a minimum length are eligible

format: plain


text:
## Snake

Points: 25 (partial)
Time limit: 1.0s
Memory limit: 250M

Author:
Problem type

In the world of snakes, Cassandra said to Snooky "hsssch-hssch" and that means Snooky have to meet casandra at her house.

The world of snakes is modelled as a coordinate plane, Snooky's position is (0,0) , And casandra's house position is (X,Y).

The problem is in each second Snooky may move in only two ways:

• Move 2 steps up , 1 step right.
• Move 1 step up , 2 steps right.

Can you help him find if it is possible to reach casandra's house and the shortest time to achieve that?

#### Input Specification

The first line of the input will contain a number $$T$$ , number of testcases T<=10^4.

Each of the next $$T$$ lines contains two numbers X,Y , $$0 \le X,Y \le 10^8$$.

#### Output Specification

If it is possible output YES, then in the next line the shortest possible time, Else output NO.

#### Scoring

• (20 points) $$1 \le X,Y \le 1000$$.
• (80 points) No additional constraint.

#### Sample Input

    3
1 2
3 3
4 1

#### Sample Output

    YES
1
YES
2
NO


text:
This vignette discusses data.table’s reference semantics which allows to add/update/delete columns of a data.table by reference, and also combine them with i and by. It is aimed at those who are already familiar with data.table syntax, its general form, how to subset rows in i, select and compute on columns, and perform aggregations by group. If you’re not familiar with these concepts, please read the “Introduction to data.table” vignette first.

## Data

We will use the same flights data as in the “Introduction to data.table” vignette.

flights
#         year month day dep_delay arr_delay carrier origin dest air_time distance hour
#      1: 2014     1   1        14        13      AA    JFK  LAX      359     2475    9
#      2: 2014     1   1        -3        13      AA    JFK  LAX      363     2475   11
#      3: 2014     1   1         2         9      AA    JFK  LAX      351     2475   19
#      4: 2014     1   1        -8       -26      AA    LGA  PBI      157     1035    7
#      5: 2014     1   1         2         1      AA    JFK  LAX      350     2475   13
#     ---
# 253312: 2014    10  31         1       -30      UA    LGA  IAH      201     1416   14
# 253313: 2014    10  31        -5       -14      UA    EWR  IAH      189     1400    8
# 253314: 2014    10  31        -8        16      MQ    LGA  RDU       83      431   11
# 253315: 2014    10  31        -4        15      MQ    LGA  DTW       75      502   11
# 253316: 2014    10  31        -5         1      MQ    LGA  SDF      110      659    8
dim(flights)
# [1] 253316     11

## Introduction

In this vignette, we will

1. first discuss reference semantics briefly and look at the two different forms in which the := operator can be used

2. then see how we can add/update/delete columns by reference in j using the := operator and how to combine with i and by.

3. and finally we will look at using := for its side-effect and how we can avoid the side effects using copy().

## 1. Reference semantics

All the operations we have seen so far in the previous vignette resulted in a new data set. We will see how to add new column(s), update or delete existing column(s) on the original data.

### a) Background

Before we look at reference semantics, consider the data.frame shown below:

DF = data.frame(ID = c("b","b","b","a","a","c"), a = 1:6, b = 7:12, c = 13:18)
DF
#   ID a  b  c
# 1  b 1  7 13
# 2  b 2  8 14
# 3  b 3  9 15
# 4  a 4 10 16
# 5  a 5 11 17
# 6  c 6 12 18

When we did:

DF$c <- 18:13 # (1) -- replace entire column # or DF$c[DF\$ID == "b"] <- 15:13 # (2) -- subassign in column 'c'

both (1) and (2) resulted in deep copy of the entire data.frame in versions of R versions < 3.1. It copied more than once. To improve performance by avoiding these redundant copies, data.table utilised the available but unused := operator in R.

Great performance improvements were made in R v3.1 as a result of which only a shallow copy is made for (1) and not deep copy. However, for (2) still, the entire column is deep copied even in R v3.1+. This means the more columns one subassigns to in the same query, the more deep copies R does.

#### shallow vs deep copy

A shallow copy is just a copy of the vector of column pointers (corresponding to the columns in a data.frame or data.table). The actual data is not physically copied in memory.

A deep copy on the other hand copies the entire data to another location in memory.

With data.table’s := operator, absolutely no copies are made in both (1) and (2), irrespective of R version you are using. This is because := operator updates data.table columns in-place (by reference).

### b) The := operator

It can be used in j in two ways:

1. The LHS := RHS form

DT[, c("colA", "colB", ...) := list(valA, valB, ...)]

# when you have only one column to assign to you
# can drop the quotes and list(), for convenience
DT[, colA := valA]
2. The functional form

DT[, :=(colA = valA, # valA is assigned to colA
colB = valB, # valB is assigned to colB
...
)]

Note that the code above explains how := can be used. They are not working examples. We will start using them on flights data.table from the next section.

• In (a), LHS takes a character vector of column names and RHS a list of values. RHS just needs to be a list, irrespective of how its generated (e.g., using lapply(), list(), mget(), mapply() etc.). This form is usually easy to program with and is particularly useful when you don’t know the columns to assign values to in advance.

• On the other hand, (b) is handy if you would like to jot some comments down for later.

• The result is returned invisibly.

• Since := is available in j, we can combine it with i and by operations just like the aggregation operations we saw in the previous vignette.

In the two forms of := shown above, note that we don’t assign the result back to a variable. Because we don’t need to. The input data.table is modified by reference. Let’s go through examples to understand what we mean by this.

For the rest of the vignette, we will work with flights data.table.

## 2. Add/update/delete columns by reference

### a) Add columns by reference

#### – How can we add columns speed and total delay of each flight to flightsdata.table?

flights[, :=(speed = distance / (air_time/60), # speed in mph (mi/h)
delay = arr_delay + dep_delay)]   # delay in minutes
#    year month day dep_delay arr_delay carrier origin dest air_time distance hour    speed delay
# 1: 2014     1   1        14        13      AA    JFK  LAX      359     2475    9 413.6490    27
# 2: 2014     1   1        -3        13      AA    JFK  LAX      363     2475   11 409.0909    10
# 3: 2014     1   1         2         9      AA    JFK  LAX      351     2475   19 423.0769    11
# 4: 2014     1   1        -8       -26      AA    LGA  PBI      157     1035    7 395.5414   -34
# 5: 2014     1   1         2         1      AA    JFK  LAX      350     2475   13 424.2857     3
# 6: 2014     1   1         4         0      AA    EWR  LAX      339     2454   18 434.3363     4

## alternatively, using the 'LHS := RHS' form
# flights[, c("speed", "delay") := list(distance/(air_time/60), arr_delay + dep_delay)]

#### Note that

• We did not have to assign the result back to flights.

• The flights data.table now contains the two newly added columns. This is what we mean by added by reference.

• We used the functional form so that we could add comments on the side to explain what the computation does. You can also see the LHS := RHS form (commented).

### b) Update some rows of columns by reference - sub-assign by reference

Let’s take a look at all the hours available in the flights data.table:

# get all 'hours' in flights
flights[, sort(unique(hour))]
#  [1]  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

We see that there are totally 25 unique values in the data. Both 0 and 24 hours seem to be present. Let’s go ahead and replace 24 with 0.

#### – Replace those rows where hour == 24 with the value 0

# subassign by reference
flights[hour == 24L, hour := 0L]
• We can use i along with := in j the very same way as we have already seen in the “Introduction to data.table” vignette.

• Column hour is replaced with 0 only on those row indices where the condition hour == 24L specified in i evaluates to TRUE.

• := returns the result invisibly. Sometimes it might be necessary to see the result after the assignment. We can accomplish that by adding an empty [] at the end of the query as shown below:

flights[hour == 24L, hour := 0L][]
#         year month day dep_delay arr_delay carrier origin dest air_time distance hour    speed
#      1: 2014     1   1        14        13      AA    JFK  LAX      359     2475    9 413.6490
#      2: 2014     1   1        -3        13      AA    JFK  LAX      363     2475   11 409.0909
#      3: 2014     1   1         2         9      AA    JFK  LAX      351     2475   19 423.0769
#      4: 2014     1   1        -8       -26      AA    LGA  PBI      157     1035    7 395.5414
#      5: 2014     1   1         2         1      AA    JFK  LAX      350     2475   13 424.2857
#     ---
# 253312: 2014    10  31         1       -30      UA    LGA  IAH      201     1416   14 422.6866
# 253313: 2014    10  31        -5       -14      UA    EWR  IAH      189     1400    8 444.4444
# 253314: 2014    10  31        -8        16      MQ    LGA  RDU       83      431   11 311.5663
# 253315: 2014    10  31        -4        15      MQ    LGA  DTW       75      502   11 401.6000
# 253316: 2014    10  31        -5         1      MQ    LGA  SDF      110      659    8 359.4545
#         delay
#      1:    27
#      2:    10
#      3:    11
#      4:   -34
#      5:     3
#     ---
# 253312:   -29
# 253313:   -19
# 253314:     8
# 253315:    11
# 253316:    -4

Let’s look at all the hours to verify.

# check again for '24'
flights[, sort(unique(hour))]
#  [1]  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

#### Exercise:

What is the difference between flights[hour == 24L, hour := 0L] and flights[hour == 24L][, hour := 0L]? Hint: The latter needs an assignment (<-) if you would want to use the result later.

If you can’t figure it out, have a look at the Note section of ?":=".

### c) Delete column by reference

#### – Remove delay column

flights[, c("delay") := NULL]
#    year month day dep_delay arr_delay carrier origin dest air_time distance hour    speed
# 1: 2014     1   1        14        13      AA    JFK  LAX      359     2475    9 413.6490
# 2: 2014     1   1        -3        13      AA    JFK  LAX      363     2475   11 409.0909
# 3: 2014     1   1         2         9      AA    JFK  LAX      351     2475   19 423.0769
# 4: 2014     1   1        -8       -26      AA    LGA  PBI      157     1035    7 395.5414
# 5: 2014     1   1         2         1      AA    JFK  LAX      350     2475   13 424.2857
# 6: 2014     1   1         4         0      AA    EWR  LAX      339     2454   18 434.3363

## or using the functional form
# flights[, :=(delay = NULL)]
• Assigning NULL to a column deletes that column. And it happens instantly.

• We can also pass column numbers instead of names in the LHS, although it is good programming practice to use column names.

• When there is just one column to delete, we can drop the c() and double quotes and just use the column name unquoted, for convenience. That is:

flights[, delay := NULL]

is equivalent to the code above.

### d) := along with grouping using by

We have already seen the use of i along with := in Section 2b. Let’s now see how we can use := along with by.

#### – How can we add a new column which contains for each orig,dest pair the maximum speed?

flights[, max_speed := max(speed), by = .(origin, dest)]
#    year month day dep_delay arr_delay carrier origin dest air_time distance hour    speed max_speed
# 1: 2014     1   1        14        13      AA    JFK  LAX      359     2475    9 413.6490  526.5957
# 2: 2014     1   1        -3        13      AA    JFK  LAX      363     2475   11 409.0909  526.5957
# 3: 2014     1   1         2         9      AA    JFK  LAX      351     2475   19 423.0769  526.5957
# 4: 2014     1   1        -8       -26      AA    LGA  PBI      157     1035    7 395.5414  517.5000
# 5: 2014     1   1         2         1      AA    JFK  LAX      350     2475   13 424.2857  526.5957
# 6: 2014     1   1         4         0      AA    EWR  LAX      339     2454   18 434.3363  518.4507
• We add a new column max_speed using the := operator by reference.

• We provide the columns to group by the same way as shown in the Introduction to data.table vignette. For each group, max(speed) is computed, which returns a single value. That value is recycled to fit the length of the group. Once again, no copies are being made at all. flights data.table is modified in-place.

• We could have also provided by with a character vector as we saw in the Introduction to data.table vignette, e.g., by = c("origin", "dest").

### e) Multiple columns and :=

#### – How can we add two more columns computing max() of dep_delay and arr_delay for each month, using .SD?

in_cols  = c("dep_delay", "arr_delay")
out_cols = c("max_dep_delay", "max_arr_delay")
flights[, c(out_cols) := lapply(.SD, max), by = month, .SDcols = in_cols]
#    year month day dep_delay arr_delay carrier origin dest air_time distance hour    speed max_speed
# 1: 2014     1   1        14        13      AA    JFK  LAX      359     2475    9 413.6490  526.5957
# 2: 2014     1   1        -3        13      AA    JFK  LAX      363     2475   11 409.0909  526.5957
# 3: 2014     1   1         2         9      AA    JFK  LAX      351     2475   19 423.0769  526.5957
# 4: 2014     1   1        -8       -26      AA    LGA  PBI      157     1035    7 395.5414  517.5000
# 5: 2014     1   1         2         1      AA    JFK  LAX      350     2475   13 424.2857  526.5957
# 6: 2014     1   1         4         0      AA    EWR  LAX      339     2454   18 434.3363  518.4507
#    max_dep_delay max_arr_delay
# 1:           973           996
# 2:           973           996
# 3:           973           996
# 4:           973           996
# 5:           973           996
# 6:           973           996
• We use the LHS := RHS form. We store the input column names and the new columns to add in separate variables and provide them to .SDcols and for LHS (for better readability).

• Note that since we allow assignment by reference without quoting column names when there is only one column as explained in Section 2c, we can not do out_cols := lapply(.SD, max). That would result in adding one new column named out_col. Instead we should do either c(out_cols) or simply (out_cols). Wrapping the variable name with ( is enough to differentiate between the two cases.

• The LHS := RHS form allows us to operate on multiple columns. In the RHS, to compute the max on columns specified in .SDcols, we make use of the base function lapply() along with .SD in the same way as we have seen before in the “Introduction to data.table” vignette. It returns a list of two elements, containing the maximum value corresponding to dep_delay and arr_delay for each group.

Before moving on to the next section, let’s clean up the newly created columns speed, max_speed, max_dep_delay and max_arr_delay.

# RHS gets automatically recycled to length of LHS
flights[, c("speed", "max_speed", "max_dep_delay", "max_arr_delay") := NULL]
#    year month day dep_delay arr_delay carrier origin dest air_time distance hour
# 1: 2014     1   1        14        13      AA    JFK  LAX      359     2475    9
# 2: 2014     1   1        -3        13      AA    JFK  LAX      363     2475   11
# 3: 2014     1   1         2         9      AA    JFK  LAX      351     2475   19
# 4: 2014     1   1        -8       -26      AA    LGA  PBI      157     1035    7
# 5: 2014     1   1         2         1      AA    JFK  LAX      350     2475   13
# 6: 2014     1   1         4         0      AA    EWR  LAX      339     2454   18

## 3) := and copy()

:= modifies the input object by reference. Apart from the features we have discussed already, sometimes we might want to use the update by reference feature for its side effect. And at other times it may not be desirable to modify the original object, in which case we can use copy() function, as we will see in a moment.

### a) := for its side effect

Let’s say we would like to create a function that would return the maximum speed for each month. But at the same time, we would also like to add the column speed to flights. We could write a simple function as follows:

foo <- function(DT) {
DT[, speed := distance / (air_time/60)]
DT[, .(max_speed = max(speed)), by = month]
}
ans = foo(flights)
#    year month day dep_delay arr_delay carrier origin dest air_time distance hour    speed
# 1: 2014     1   1        14        13      AA    JFK  LAX      359     2475    9 413.6490
# 2: 2014     1   1        -3        13      AA    JFK  LAX      363     2475   11 409.0909
# 3: 2014     1   1         2         9      AA    JFK  LAX      351     2475   19 423.0769
# 4: 2014     1   1        -8       -26      AA    LGA  PBI      157     1035    7 395.5414
# 5: 2014     1   1         2         1      AA    JFK  LAX      350     2475   13 424.2857
# 6: 2014     1   1         4         0      AA    EWR  LAX      339     2454   18 434.3363
#    month max_speed
# 1:     1  535.6425
# 2:     2  535.6425
# 3:     3  549.0756
# 4:     4  585.6000
# 5:     5  544.2857
# 6:     6  608.5714
• Note that the new column speed has been added to flights data.table. This is because := performs operations by reference. Since DT (the function argument) and flights refer to the same object in memory, modifying DT also reflects on flights.

• And ans contains the maximum speed for each month.

### b) The copy() function

In the previous section, we used := for its side effect. But of course this may not be always desirable. Sometimes, we would like to pass a data.table object to a function, and might want to use the := operator, but wouldn’t want to update the original object. We can accomplish this using the function copy().

The copy() function deep copies the input object and therefore any subsequent update by reference operations performed on the copied object will not affect the original object.

There are two particular places where copy() function is essential:

1. Contrary to the situation we have seen in the previous point, we may not want the input data.table to a function to be modified by reference. As an example, let’s consider the task in the previous section, except we don’t want to modify flights by reference.

Let’s first delete the speed column we generated in the previous section.

flights[, speed := NULL]

Now, we could accomplish the task as follows:

foo <- function(DT) {
DT <- copy(DT)                              ## deep copy
DT[, speed := distance / (air_time/60)]     ## doesn't affect 'flights'
DT[, .(max_speed = max(speed)), by = month]
}
ans <- foo(flights)
#    year month day dep_delay arr_delay carrier origin dest air_time distance hour
# 1: 2014     1   1        14        13      AA    JFK  LAX      359     2475    9
# 2: 2014     1   1        -3        13      AA    JFK  LAX      363     2475   11
# 3: 2014     1   1         2         9      AA    JFK  LAX      351     2475   19
# 4: 2014     1   1        -8       -26      AA    LGA  PBI      157     1035    7
# 5: 2014     1   1         2         1      AA    JFK  LAX      350     2475   13
# 6: 2014     1   1         4         0      AA    EWR  LAX      339     2454   18
#    month max_speed
# 1:     1  535.6425
# 2:     2  535.6425
# 3:     3  549.0756
# 4:     4  585.6000
# 5:     5  544.2857
# 6:     6  608.5714
• Using copy() function did not update flights data.table by reference. It doesn’t contain the column speed.

• And ans contains the maximum speed corresponding to each month.

However we could improve this functionality further by shallow copying instead of deep copying. In fact, we would very much like to provide this functionality for v1.9.8. We will touch up on this again in the data.table design vignette.

1. When we store the column names on to a variable, e.g., DT_n = names(DT), and then add/update/delete column(s) by reference. It would also modify DT_n, unless we do copy(names(DT)).

DT = data.table(x = 1L, y = 2L)
DT_n = names(DT)
DT_n
# [1] "x" "y"

## add a new column by reference
DT[, z := 3L]

## DT_n also gets updated
DT_n
# [1] "x" "y" "z"

## use copy()
DT_n = copy(names(DT))
DT[, w := 4L]

## DT_n doesn't get updated
DT_n
# [1] "x" "y" "z"

## Summary

#### The := operator

• It is used to add/update/delete columns by reference.

• We have also seen how to use := along with i and by the same way as we have seen in the Introduction to data.table vignette. We can in the same way use keyby, chain operations together, and pass expressions to by as well all in the same way. The syntax is consistent.

• We can use := for its side effect or use copy() to not modify the original object while updating by reference.

So far we have seen a whole lot in j, and how to combine it with by and little of i. Let’s turn our attention back to i in the next vignette “Keys and fast binary search based subset” to perform blazing fast subsets by keying data.tables.


text:
# Experimental signature of topological superconductor

+ 7 like - 0 dislike
297 views

I was wondering if someone can provides some clear experimental signatures of a topological superconductors ?

I was thinking about that, because for topological insulator, one of the hallmarks is the insulating behaviour of the bulk versus the metallic behaviour at the edge. So, a good experimental signature is simply the presence of edge conducting channel(s). But for superconductors, even the non-topological (say $s$-wave for simplicity) ones already have only edge propagating modes, as elaborated by London long times ago. So, what would be the smoking gun experiment about topological superconductors ? Are we obliged to probe the gap symmetry (what I would not really like because it is more or less a symmetry check, not a topological one), or is there something more fancy ?

Any good reference is obviously warm welcome.

This post imported from StackExchange Physics at 2014-04-05 03:34 (UCT), posted by SE-user FraSchelle
retagged Apr 5, 2014
That is an excellent point about the surfaces of superconductors. Part of the definition of topological insulators is that they are robust against all types of disorder that respect certain discrete symmetries. So maybe one should think about the behavior of trivial superconductor in the presence of random $U(1)$ perturbations.

This post imported from StackExchange Physics at 2014-04-05 03:34 (UCT), posted by SE-user BebopButUnsteady
But from another perspective aren't these modes quite different? The local excitation on the surface of a topological insulator are metallic density fluctuations, whereas on a superconductor they are $U(1)$ fluctuations. Simply there is no dissipation on a superconductor, whereas there must be on the topological edge state.

This post imported from StackExchange Physics at 2014-04-05 03:34 (UCT), posted by SE-user BebopButUnsteady

+ 8 like - 0 dislike

There are different categories of topological superconductors. I’m guessing that you are referring to the time-reversal invariant (class DIII) ones, in 2D or 3D. Yes, it is possible to distinguish the surface/edge states of 3D/2D topological superconductors from the bulk. I'm not talking about designing some intricate experimental technique to separate out the contribution from the bulk and the surface; as it turns out, they don't need any separating! The bulk and surface states are distinct in a very fundamental way. That statement is true not just in topological superconductors but also in topological insulators. First allow me to make a few comments on topological insulators case.

Yes, an ideal topological insulator is insulating in the bulk and conducting on the surface/edge. Despite the existence of an extensive list of (3D) band topological insulators discovered to date, none of them are insulating in the bulk! The Kondo topological insulator SmB$_6$ is the only one with the properties of an ideal topological insulator; but the Kondo gap opens up only at cryogenic temperatures ($\approx$ 4 K). So this example is not “natural” in the sense that it doesn’t exist in a topological phase under ambient conditions.

The reason we can distinguish between the surface (in the 3D case) and bulk states of a topological insulator, even when the bulk is conducting, is because the surface forms the so-called “helical metal.” The conducting electronic states in this helical metal do not hybridize (or mix) with the conducting states in the bulk; picture this with the analogy of two immiscible liquids in a container. The following paper discusses in detail how topological surface states refuse to hybridize with the bulk

Bohm-Jung Yang, Mohammad Saeed Bahramy, and Naoto Nagaosa. “Topological protection of bound states against the hybridization.” Nature communications 4 (2013): 1524. (arXiv version)

These helical states, which form a 2D electronic system on the boundary of a 3D system, have half the degrees of freedom (DOF) of an equivalent purely 2D system; the other half (DOF) are on the other surface (look up the holographic principle). As a result, no time-reversal symmetry preserving perturbation can gap them out. In other words, they are topologically protected (due to symmetry). Another important thing to stress is that just because an insulator is topologically trivial doesn’t mean it cannot have surface states. The only difference is that these states can be destroyed.

Now, in the case of topological superconductors, the surface states are even more exotic. The surfaces of 3D time-reversal invariant topological superconductors host Majorana modes. You would observe a topologically protected Dirac cone, just like a 3D topological insulator, on the surface of a topological superconductor. The only difference is that this Dirac cone represents the dispersion of Majorana modes instead of Dirac fermions and has half the degrees of freedom due to particle-hole symmetry. There is a tonne of literature out there which proposes schemes of experimentally detecting these Majorana modes. For the case of 3D time-reversal invariant topological superconductors, however, I am familiar with only two: Scanning Tunneling Microscopy (STM) and Angle-Resolved Photoemission Spectroscopy (ARPES). In a point contact spectroscopy measurement on a topological superconductor you are supposed to see a so-called Zero-Bias Conductance Peak (ZBCP) inside the bulk gap as shown in Fig. 2 of:

Satoshi Sasaki, M. Kriener, Kouji Segawa, Keiji Yada, Yukio Tanaka, Masatoshi Sato, and Yoichi Ando. “Topological Superconductivity in Cu$_x$Bi$_2$Se$_3$.” Physical Review Letters 107, no. 21 (2011): 217001. (arXiv version)

One small disclaimer: there is still a lot of debate whether the authors of the above paper have really seen the Majorana mode or not. For example check this article which refutes the claims of the above paper:

Niv Levy, Tong Zhang, Jeonghoon Ha, Fred Sharifi, A. Alec Talin, Young Kuk, and Joseph A. Stroscio. “Experimental Evidence for $s$-Wave Pairing Symmetry in Superconducting Cu$_x$Bi$_2$Se$_3$ Single Crystals Using a Scanning Tunneling Microscope.” Physical Review Letters 110, no. 11 (2013): 117001.
(arXiv version)

Alternatively, instead of the momentum-integrated density of states in STM, you can directly “see” the Dirac cone of Majorana modes using ARPES since you are measuring the bandstructure directly. As matter of fact, the existence of first and second generation of 3D topological insulators was identified using precisely this technique. To my knowledge, no one has so far observed Majorana modes on the surface of a topological superconductor using ARPES. The main reasons are technical: cooling capabilities, energy resolution, lack of stoichiometrically stable (unlike Cu$_x$Bi$_2$Se$_3$) material candidates which can be cleaved to give atomically flat surfaces, etc.

Aside from time-reversal invariant topological superconductors, a broad list of schemes to engineer topological superconductors and detect their Majorana modes can be seen in the following (excellent) review articles:

Jason Alicea. “New directions in the pursuit of Majorana fermions in solid state systems.” Reports on Progress in Physics 75, no. 7 (2012): 076501. (arXiv version)

C. W. J. Beenakker, “Search for Majorana Fermions in Superconductors.” Annual Review of Condensed Matter Physics 4, no. 1 (2013): 113-136. (arXiv version)

Details (continued from the comments section below)

As far as the ZBCP goes, I feel that it is still more reliable using STM (compared to transport) for 3D superconductors (class DIII). For the system investigated by Kouwenhoven and others in the 1D Rashba nanowires (class D) the situation is way more ambiguous. Transport is way more finicky; it's (obviously) not as direct as STM or ARPES. The discrepancies in the above mentioned STM studies arise (in Yoichi Ando's and my opinion) because Cu$_x$Bi$_2$Se$_3$ is very hard to work with. Since Cu$_x$Bi$_2$Se$_3$ is not stoichiometrically stable, the dopant copper atoms diffuse and form “pockets” of superconducting and normal (or topological and non-topological) regions (depending on local density of Cu). Ando thinks that the two studies measured different such regions! That being said, I don't know if Cu$_x$Bi$_2$Se$_3$ is topological if the above problem didn't exist. I'm just pointing out the source of the confusion.

Few comments on ARPES: this is the real deal! You can indeed have a smoking gun detection using ARPES; after all you see the Majorana mode dispersion directly. The reason people haven't seen this yet, as I mentioned, is mainly technical. First of all, the state-of-the-art ARPES systems can go to into the $\approx$ 10 K range. Most topological superconductor candidates have $T_c$’s way below that (e.g. half-Heusler compounds and heavy-fermion superconductors). Some (laser-based) systems can go down to 1.5 K. It might seem that Cu$_x$Bi$_2$Se$_3$ (with $T_c = 3.8$ K) might be perfect. But it's not enough to go just below $T_c$. You have to go way below in temperature to see the (superconducting) gap features clearly (i.e. when the gap is big enough); the gap size roughly follows a power law like $\approx 1/T^2$ below $T_c$. Also the gaps of many candidate materials are very small (compared to topological insulator band gaps). Conventionally synchrotron-based ARPES is the state-of-the-art tool for doing ARPES experiments. However, recently laser-based ARPES is becoming more commonplace in labs. Apart from the compactness and significantly less resource consumption (both monetary and operating manpower) laser-based ARPES is becoming popular because it’s “clean.” Check out this image to see what I mean by that:

The synchrotron-based ARPES systems have more “juice” (high photon energies in the tens of eV) as opposed to laser-based ones (6-7 eV), but they have very poor resolution. Good resolution is what we really need if we want to see the features of the Majorana modes inside the tiny superconducting gap.

Few comments on potential topological superconductor candidates: in addition to being very good topological insulator candidates, some of the half-Heusler compounds become superconducting at very low temperatures (1-2 K). For example LaPtBi, YPtBi, and LuPtBi have $T_c$’s of 0.9 K, 0.77 K, 1 K respectively. Unfortunately the samples of these materials that have been grown so far are “rock-like” (as opposed to crystalline) and it’s very hard to cleave them get an atomically flat surface to do ARPES on. After all the Majorana modes live on the surface!

Long story short, ARPES can give us smoking gun signatures Majorana modes. I’m sure there are other clever (indirect) ways of getting smoking gun signatures as well.

This post imported from StackExchange Physics at 2014-04-05 03:34 (UCT), posted by SE-user NanoPhys
answered Jul 13, 2013 by (360 points)
Thanks for your answer, and for the review of the topological insulator experimental issue. I'm not feeling that the ZBCP gives a smoking gun experiment: there are increasing evidences that it tells us nothing really special about topology mechanism (ZBCP can be due to disorder, bad contact, Kondo mechanism, gap symmetry, ...). But your answer suggests that a better experimental evidence is still a research issue. Post-Scriptum: I edited your answer such that the arXiv links to the papers you kindly suggested are attached. Thanks again.

This post imported from StackExchange Physics at 2014-04-05 03:34 (UCT), posted by SE-user FraSchelle
Thanks for the arXiv links Oaoa. First of all, I did't realize it was you who asked the question. Being a little more concise with my answer would have been wise since you already seem to have a good background! I remember you were the second person to answer the Majorana fermions question I answered previously! Anyways, I have editted my answer to add a small "details" section. Please see my comments there.

This post imported from StackExchange Physics at 2014-04-05 03:34 (UCT), posted by SE-user NanoPhys
Thanks for the useful precision about ARPES technique, and thanks for insisting on this. That's indeed a great technique for the surface effects, hence for the bulk-boundary transition. Looking forward for some experimental result in the next years then. By the way, I think question/answer should be useful for more than one people. To this respect, your answer is very good for a broad audience, and that's good :-). Thanks again.

This post imported from StackExchange Physics at 2014-04-05 03:34 (UCT), posted by SE-user FraSchelle
Awesome answer, learned a lot, thanks.

This post imported from StackExchange Physics at 2014-04-05 03:34 (UCT), posted by SE-user Mathusalem

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverf$\varnothing$owThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.


text:
### 0664

Motion artifact quantification and localization for whole-body MRI
Thomas Kuestner1,2, Marvin Jandt2, Annika Liebgott2,3, Lukas Mauch2, Petros Martirosian1, Fabian Bamberg3, Konstantin Nikolaou3, Sergios Gatidis3, Bin Yang2, and Fritz Schick1

1Section on Experimental Radiology, University Hospital of Tuebingen, Tuebingen, Germany, 2Institute of Signal Processing and System Theory, University of Stuttgart, Stuttgart, Germany, 3Department of Radiology, University Hospital of Tuebingen, Tuebingen, Germany

### Synopsis

Motion is still one of the major extrinsic factors degrading image quality. Automated detection of these artifacts is of interest, (i) if suitable prospective or retrospective correction techniques are not available/applicable, (ii) if human experts who judge the achieved quality are not present, or (iii) if a manual quality analysis of large databases from epidemiological cohort studies is impracticable. A convolutional neural network assesses and localizes the motion artifacts. This work extends the previously published method by proposing a general architecture for a whole-body scenario with varying contrast weightings. High accuracies of >90% were achieved in a volunteer study.

### Introduction

MRI offers a broad variety of imaging applications with targeted and flexible sequence and reconstruction parametrization. Varying acquisition conditions and long examination times make MRI susceptible to imaging artifacts, amongst which motion is one of the major extrinsic factors deteriorating image quality.

Images recorded for clinical diagnosis are often inspected by a human specialist to determine the achieved image quality which can be a time-demanding and cost-intensive process. If insufficient quality is determined too late, an additional examination may be even required decreasing patient comfort and throughput. Moreover, in the context of large epidemiological cohort studies such as UK Biobank1 or German National Cohort2 the amount and complexity exceed practicability for a manual image quality analysis. Thus, in order to guarantee high data quality, arising artifacts need to be detected as early as possible to seize appropriate countermeasures, as e.g. prospective3-5 or retrospective correction techniques6-9. However, these methods are not always available and/or applicable. Moreover, when a human expert is not present or for large cohort studies, the potential presence of motion artifacts demands an automatic processing for a prospective quality assurance or retrospective quality assessment.

Previously proposed approaches for automated medical image quality analysis require the existence of a reference image and/or were only focused on specific sequences and scenarios10-12. Reference-free approaches13-17 are mainly metric-based driven to evaluate the quality on a coarse level.

In none of the previous approaches a motion artifact localization and quantification was conducted. We therefore proposed an automatic and reference-free motion artifact detection by a machine-learning convolutional neural network (CNN) approach18,19. In our previous study, the architecture was kept intentionally shallow because of the limited amount of data and investigations focused only on motion in head and abdominal region for one imaging sequence. In this work, we investigate a general architecture for motion artifact detection in a whole-body scenario with two contrast weightings.

### Material and Methods

MR images were acquired on a 3T PET/MR (Biograph mMR, Siemens) from 18 healthy volunteers (3 female, 25±8y) with a T1w and T2w FSE sequence. The acquisition parameters for the respective body regions (head, abdomen, pelvis) are depicted in Tab.1. Each of the ﬁve sequences was acquired twice for every volunteer. During the first acquisition volunteers were asked to avoid movements (head, pelvis) or an end-expiratory breath-hold (T1w) respectively navigator-triggering (T2w) was conducted in the abdominal region. In the second acquisition, volunteers were asked to move their heads, hips (rigid deformation) or to breathe normal (non-rigid deformation). Images are normalized into an intensity range of 0 to 1 and partitioned into 50% overlapping patches of size 40x40x10 (APxLRxSI).

The proposed CNN architectures are depicted in Fig.2 and compared against the previously published 2D-CNN18 to output artifact probabilities p. 3D-CNN depicts the 3D extension of the 2D-CNN with three convolutional layers of N filter kernels/channels of size MxLxB with rectified linear unit (ReLU) activation, followed by a fully-connected dense layer with softmax decision.

The MNetArt is inspired by MNet20 consisting of four stages followed by a dense output layer. Each stage contains two convolutional layers with ReLU activation, an intermittent and finalizing concatenation layer. This resembles a residual path to forward feature maps in and between stages. The first three layers exhibit an additional max-pooling downsampling.

The VNetArt is inspired by VNet21 consisting of three stages followed by a dense output layer. Each stage has two convolutional layers with ReLU activation, a finalizing concatenation layer (residual path) and max-pooling downsampling.

The four architectures (2D-CNN, 3D-CNN, MNetArt, VNetArt) are trained by a leave-one-subject-out cross-validation on (i) the complete training database, (ii) the subset consisting only of T1w or T2w images, (iii) subsets leaving-out one body region. Categorical cross-entropy is minimized for given learning rate, $\ell_2$ regularization and dropout. Parameter ranges are estimated by the Baum-Haussler rule22 with a grid-search optimization. Testing was performed on the left-out subject to determine accuracy, sensitivity (true positive rate; TPR) and speciﬁcity (true negative rate; TNR).

### Results and Discussion

Fig.3 depicts exemplary subject slices overlaid with the derived and localized motion artifact probabilities. For this small moving subject a high accuracy of 98%/83%/90% in the head, abdomen and pelvis was achieved, respectively. Fig.4 illustrates superior performance of 3D processing and inclusion of residual paths. Performance is influenced by trained body regions (type of motion), but is independent of contrast weighting. Overall, VNetArt performs best for all metrics.

### Conclusion

Proposed architectures improve previously reported network by 17%. Automatic motion artifact quantification and localization is feasible in a whole-body setting of various imaging sequences with accuracies >90%.

### Acknowledgements

No acknowledgement found.

### References

[1] W. Ollier, T. Sprosen, and T. Peakman, “UK Biobank: from concept to reality,” Pharmacogenomics, vol. 6, no. 6, pp. 639-646, 2005. [2] F. Bamberg, H. U. Kauczor, S. Weckbach, C. L. Schlett, M. Forsting, S. C. Ladd, K. H. Greiser, M. A. Weber, J. Schulz-Menger, T. Niendorf, T. Pischon, S. Caspers, K. Amunts, K. Berger, R. Bulow, N. Hosten, K. Hegenscheid, T. Kroncke, J. Linseisen, M. Gunther, J. G. Hirsch, A. Kohn, T. Hendel, H. E. Wichmann, B. Schmidt, K. H. Jockel, W. Hoffmann, R. Kaaks, M. F. Reiser, and H. Volzke, “Whole-Body MR Imaging in the German National Cohort: Rationale, Design, and Technical Background,” Radiology, vol. 277, no. 1, pp. 206–20, 2015. [3] M. Zaitsev, J. Maclaren, and M. Herbst, “Motion artifacts in mri: A complex problem with many partial solutions,” J. Magn. Reson. Imaging., vol. 42, no. 4, pp. 887–901, 2015. [4] J. Maclaren, M. Herbst, O. Speck, and M. Zaitsev, “Prospective motion correction in brain imaging: a review,” Magn. Reson. Med., vol. 69, no. 3, pp. 621–36, 2013. [5] F. Godenschweger, U. Kagebein, D. Stucht, U. Yarach, A. Sciarra, R. Yakupov, F. Lusebrink, P. Schulze, and O. Speck, “Motion correction in mri of the brain,” Phys. Med. Biol., vol. 61, no. 5, pp. R32–56, 2016. [6] L. Feng, R. Grimm, K. T. Block, H. Chandarana, S. Kim, J. Xu, L. Axel, D. K. Sodickson, and R. Otazo, “Golden-angle radial sparse parallel mri: combination of compressed sensing, parallel imaging, and golden-angle radial sampling for fast and ﬂexible dynamic volumetric mri,” Magn. Reson. Med., vol. 72, no. 3, pp. 707–17, 2014. [7] J. Y. Cheng, T. Zhang, N. Ruangwattanapaisarn, M. T. Alley, M. Uecker, J. M. Pauly, M. Lustig, and S. S. Vasanawala, “Free-breathing pediatric mri with nonrigid motion correction and acceleration,” J. Magn. Reson. Imaging., vol. 42, no. 2, pp. 407–20, 2015. [8] G. Cruz, D. Atkinson, C. Buerger, T. Schaeffter, and C. Prieto, “Accelerated motion corrected three-dimensional abdominal mri using total variation regularized sense reconstruction,” Magn. Reson. Med., vol. 75, no. 4, pp. 1484–98, 2016. [9] T. Küstner, C. Würslin, M. Schwartz, P. Martirosian, S. Gatidis, C. Brendle, F. Seith, F. Schick, N.F. Schwenzer, B. Yang, and H. Schmidt, “Self-navigated 4d cartesian imaging of periodic motion in the body trunk using partial k-space compressed sensing,” Magn. Reson. Med., vol. 78, no. 2, pp. 632–644, 2017. [10] J. Oh, S. I. Woolley, T. N. Arvanitis, and J. N. Townend, “A multistage perceptual quality assessment for compressed digital angiogram images,” IEEE Trans. Med. Imaging., vol. 20, no. 12, pp. 1352–61, 2001. [11] J. Miao, D. Huo, and D. L. Wilson, “Quantitative image quality evaluation of MR images using perceptual difference models,” Med. Phys., vol. 35, no. 6, pp. 2541–53, 2008. [12] A. Ouled Zaid and B. B. Fradj, “Coronary angiogram video compression for remote browsing and archiving applications,” Comput. Med. Imaging. Graph., vol. 34, no. 8, pp. 632–41, 2010. [13] B. Mortamet, M. A. Bernstein, Jr. Jack, C. R., J. L. Gunter, C. Ward, P. J. Britson, R. Meuli, J. P. Thiran, and G. Krueger, “Automatic quality assessment in structural brain magnetic resonance imaging,” Magn. Reson. Med., vol. 62, no. 2, pp. 365–72, 2009. [14] J. P. Woodard and M. P. Carley-Spencer, “No-reference image quality metrics for structural MRI,” Neuroinformatics, vol. 4, no. 3, pp. 243–62, 2006. [15] M. D. Tisdall and M. S. Atkins, “Using human and model performance to compare MRI reconstructions,” IEEE Trans. Med. Imaging., vol. 25, no. 11, pp. 1510–7, 2006. [16] D. Atkinson, D. L. Hill, P. N. Stoyle, P. E. Summers, and S. F. Keevil, “Automatic correction of motion artifacts in magnetic resonance images using an entropy focus criterion,” IEEE Trans. Med. Imaging, vol. 16, no. 6, pp. 903–10, 1997. [17] K. P. McGee, A. Manduca, J. P. Felmlee, S. J. Riederer, and R. L. Ehman, “Image metric-based correction (autocorrection) of motion effects: analysis of image metrics,” J. Magn. Reson. Imaging, vol. 11, no. 2, pp. 174–181, 2000. [18] T. Küstner, A. Liebgott, L. Mauch, P. Martirosian, F. Bamberg, K. Nikolaou, B. Yang, F. Schick, and S. Gatidis, “Automated reference-free detection of motion artifacts in magnetic resonance images,” Magn. Reson. Mater. Phys., Biol. Med., Sep 2017. [19] T. Küstner, A. Liebgott, L. Mauch, P. Martirosian, K. Nikolaou, F. Schick, B. Yang, and S. Gatidis, „Automatic reference-free detection and quantification of MR image artifacts in human examinations due to motion,” ISMRM Proceedings, p. 1278, 2017. [20] R. Mehta and J. Sivaswamy, “M-net: A Convolutional Neural Network for deep brain structure segmentation,” in IEEE International Symposium on Biomedical Imaging (ISBI), April 2017, pp. 437–440. [21] F. Milletari, N. Navab, and S.-A. Ahmadi, “V-Net: Fully Convolutional Neural Networks for volumetric Medical Image Segmentation,” ArXiv e-prints, June 2016. [22] E.B. Baum and D. Haussler, “What Size Net Gives Valid Generalization?,” Neural Comput., vol. 1, no. 1, pp. 151–160, Mar. 1989.

### Figures

Table 1: MR sequence parameters for acquiring the motion-free and motion-corrupted images.

Figure 2: Proposed CNN architectures for motion artifact detection. 3D image patches are processed to output an artifact probability p on a per-patch level. Respective feature map sizes and channels are stated for the examined input size of 40x40x10 (APxLRxSI).

Figure 3: MR image patches overlaid by color-coded artifact probability map p in T1w head, T1w abdominal and T2w pelvic images. The subject is only slightly moving and breathing very shallow. Some residual peristaltic movement in the motion-free acquisition can be seen.

Figure 4: Average test accuracy, sensitivity (TPR) and specificity (TNR) over all patches and over all cross-validation runs (wide bars) to illustrate impact (a) from complete training set, (b) of contrast weighting (T1, T2) and (c) of body regions (head, abdomen, pelvis). Standard deviations are negligibly small.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
0664


text:
## Connes' trace.. but what is the operator ?¿¿

What is the operator inside the Connes' approach to RH ??, he gave a trace but never says what operator is or gives or write down any operator or dynamicals system for this operator

i am talking about the paper

http://www.math.osu.edu/lectures/connes/zeta.pdf

shouldn't he prove that his traceor that the functional determinant of his operator is just the Riemann Zeta function on the critical line ??.


text:
# Leo has Rs 12.50 to spend on his weekly supply of sweets, crisps and apples. A bag of crisps costs Rs 0.65 , a bag of sweets costs Rs. 0.85 and one apple costs Rs. 0.50. The total number of packets of crisps, sweets and apples consumed in a week must be at least seven and he eats at least twice as many packets of sweets as crisps. His new healthy diet also means that the total number of packets of sweets and crisps must not exceed one-third of the number of apples. If s,c, and a denote the number of packets of sweets, crisps and apples respectively, which one of the following represents one of the constraints defining the feasible region? Which of the following represents one of the constraints in the question.   Option 1) $3s-3c+a\geq 0$ Option 2) $17s+10a+13c\leq 250$ Option 3) $3c+3s+a\leq 0$ Option 4) $a+c+s\leq 7$

According to question $c+s+a\geq 7\\ c+s\leq \frac{a}{3}\\ S=2c\\ and \: 0.65c+0.85s+0.50a\leq 12.50\\ 13c+17s+10a\leq 250$

Option 1)

$3s-3c+a\geq 0$

This solution is incorrect

Option 2)

$17s+10a+13c\leq 250$

This solution is correct

Option 3)

$3c+3s+a\leq 0$

This solution is incorrect

Option 4)

$a+c+s\leq 7$

This solution is incorrect

### Preparation Products

##### Knockout BITSAT 2020

It is an exhaustive preparation module made exclusively for cracking BITSAT..

₹ 4999/- ₹ 1999/-
##### Knockout BITSAT 2021

It is an exhaustive preparation module made exclusively for cracking BITSAT..

₹ 4999/- ₹ 2999/-
##### Knockout BITSAT-JEE Main 2021

An exhaustive E-learning program for the complete preparation of JEE Main and Bitsat.

₹ 27999/- ₹ 16999/-


text:
### Money Bags

Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

### Little Man

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

### Sizing Them Up

Can you put these shapes in order of size? Start with the smallest.

# Stop the Clock

##### Age 5 to 7 Challenge Level:

This is a game for two players. You can use the interactivity below, or you could print off a page of blank clock faces in Word or as a pdf.

Set the time on the clock to 6 o'clock to start the game.
Decide who will go first (player 1) and who will go second (player 2).
Take it in turns to choose to move the hands of the clock on by $\frac{1}{2}$ hour or by 1 hour. For example, player 1 could choose $\frac{1}{2}$ hour, so the clock hands move to 6.30, then player 2 might choose 1 hour, moving the clock hands to 7.30... etc.
The winner is the player who moves the hands exactly onto 12 o'clock.

Can you work out a winning strategy so that you can always beat your opponent?


text:
Search:

# VC dimension examples

Recall that to prove the VC-dim of this class is some value {$d$}, we need to show two things:

1. There exists a set {$S$} of size {$d$} shattered by {$\mathcal{H}$}.
2. There does not exist a set {$S$} of size {$d+1$} shattered by {$\mathcal{H}$}.

## Decision trees

• What is the VC dimension of a 3-node decision tree in one dimension?

(:ans:) Each node deﬁnes a new splitting threshold on the data. Three thresholds deﬁne at most 4 regions on the axis, so the VC dimension is 4. (:ansend:)

• What about a 7-node tree in one dimension?

(:ans:) Using the same reasoning as in the 3-node case, VC dimension in this case is 8. (:ansend:)

## Origin-centered circles and spheres

• What is the VC dimension of an origin-centered circle (2D) {$f(\mathbf{x}) = sign(a \mathbf{x}^\top\mathbf{x} + b)$}?

(:ans:) The VC dimension is 2. With any set of three points, they will be at some radii {$r_1 \leq r_2 \leq r_3$} from the origin, and no function {$f$} will be able to label the points at {$r_1$} and {$r_3$} with {$+1$} while labeling the point at {$r_2$} with {$-1$}. (The opposite labeling also cannot be achieved.) Thus, no set of three points is shatterable by origin-centered circles, making the VC dimension just 2. (:ansend:)

• What about an origin-centered sphere (3D)?

(:ans:) The VC dimension in this case is also 2. The same reasoning as for the 2D case applies. (:ansend:)

## Semi-circles

Suppose we were interested in using semi-circles as classifiers. That is, classifiers shaped like so:

Points inside the semi-circle are labeled {$+1$} and those outside are labeled {$-1$}. Any rotated or scaled versions of the above drawing are also valid semi-circles. For example:

• Which of the following conﬁgurations of 4 points can a semi-circle classifier shatter?

(:ans:) (a) and (b) are shatterable. For ©, the central point cannot be labeled {$-1$} while the other three points are labeled {$+1$}. (Keep in mind that for the hypothesis class we’re considering, the interior of the semi-circle must take on label {$+1$}, not {$-1$}.) (:ansend:)

• What about these conﬁgurations of 5 points?

(:ans:) Neither of the above sets are shatterable. For both sets, one labeling semicircles cannot achieve is shown below: (:ansend:)

## Axis-aligned hyperplanes

• What is the VC dimension of an axis-aligned hyperplane in two dimensions? Keep in mind that the formal deﬁnition of an axis-aligned hyperplane in two dimensions is {$f(\mathbf{x}) = sign(ax_c + b)$}, where {$c$} is either 1 or 2. That is, if the dimensions are {$x_1$} and {$x_2$}, then the hyperplane can be any line that is parallel to the {$x_1$} axis or the {$x_2$} axis. Hint: VC dimension is at most 3, since we know the VC dimension of general (non-axis-aligned) hyperplanes in {$n$} dimensions is {$n + 1$}.

(:ans:) The VC dimension is 3. An example of a shatterable set of three points is (0,0), (1,2), and (2,1). Invoking the upper bound of {$n + 1$}, 3 is the max size for a shatterable set. (:ansend:)

• What about an axis-aligned hyperplane in three dimensions?

(:ans:) The VC dimension is 4. An example of a shatterable set of four points is (0,0,−1), (2,1,0), (−1,2,1) and (1,−1,2). Again invoking the upper bound of {$n + 1$}, 4 is the max size for a shatterable set. (:ansend:)

## Monotone boolean conjunctions of literals

For this example, we’ll consider binary classification as in many of the previous examples, but instead of {$y \in \{-1,+1\}$} we’ll use {$y \in \{0,1\}$} because it will make notation simpler here. Let’s look at the class of monotone boolean conjunctions of literals over {$\{ 0 , 1\}^m$}. (Monotone means the conjunctions do not contain negations.) In other words, the inputs are binary vectors of length {$m$}: {$x = x_1 x_2 x_3 \dots x_m$}, and classifiers take the form of either constants (like a conjunction of 0 literals):

{$h(x) = 1$} {$h(x) = 0$}

or else they have the form:

{$conjunction, h(x) = \bigwedge_{i \in S} x_i ,$}

where {${} S \subseteq \{1, \dots, m\}$} indicates which subset of the features are used. For example, the hypothesis {$h(x) = x_1 \wedge x_3$} returns {$+1$} if the first feature is {$1$} and the third feature is {$1$}, and {$0$} otherwise.

Prove the following claim: {$VC(\mathcal{H})$} is {$m$}.

Proof:

Let’s show (1) first, that {$VC(\mathcal{H}) \ge m$}. We can do this by finding a set of points in {$m$} dimensions such that all possible labelings of these points can be achieved by monotone boolean conjunctions of literals. Consider arranging the {$m$} points {$x^{(1)}, \ldots, x^{(m)}$} such that:

{$x^{(j)}_i = \left\{ \begin{array}{cc} 0 & \;\textrm{if}\; j = i \\ 1 & \;\textrm{otherwise} \end{array} \right.$}

First, note that we can achieve the all 1s labeling for this set of points just by using the constant classifier {$h(x) = 1$}. To achieve a labeling where just one point {$x^{(i)}$} is labeled zero, we can use the classifier {$h(x) = x_i$}. (There are {$m$} such labelings, one for each of {$i = 1, \ldots, m$}.) To achieve a labeling where just two points {$x^{(i)}$} and {$x^{(j)}$} are labeled zero, we can use the classifier {$h(x) = x_i \wedge x_j$}. Extrapolating, to achieve the labelings with {$k \leq m$} zeros we can use the classifiers that are conjunctions of {$k$} literals. Therefore, we have shown that {$\mathcal{H}$} can give any labeling to these {$m$} points, meaning {$VC(\mathcal{H}) \ge m$}.

Now let’s show (2), that {$VC(\mathcal{H}) \leq m$}. We can do this using a proof by contradiction. Suppose {$\mathcal{H}$} could shatter {$m+1$} points. Then for each point {$x^{(i)}$} there would be some conjunction {$c^{(i)}$} that labels this point 0 and all other points 1. Further, in this conjunction there must be some literal {$x^{(i)}_{f(i)}$} such that:

{$x^{(i)}_{f(i)} = 0\;\;\;\;$}

where the function {$f$} here just indicates a mapping from the datapoint index to the index of a zero literal within that datapoint. Since there are {$m+1$} points but only {$m$} literals, we have that one literal must occur at least twice in the set {$\{x_{f(1)}, \ldots, x_{f(m+1)}\}$}. Without loss of generality, suppose {$x_{f(1)} = x_{f(2)}$}. Then {$x^{(1)}_{f(1)}$} is 0 (corresponding to the point {$x^{(1)}$}) and {$x^{(2)}_{f(1)}$} is 0 (corresponding to the point {$x^{(2)}$}). But then {$c^{(1)}$} is 0 for both {$x^{(1)}$} and {$x^{(2)}$}. This is a contradiction, because our initial assumption was that {$c^{(1)}$} is only 0 for {$x^{(1)}$}. Thus, we have shown that {$\mthcal{H}$} cannot shatter any set of {$m+1$} points, so {$VC(H) \leq m$}.

References: Exact VC-Dimension of Boolean Monomials

## Upper bounding VC dimension

Suppose we have a ﬁnite hypothesis space {$\mathcal{H}$} with {$|\mathcal{H}|$} unique classifiers. Knowing nothing else about {$\mathcal{H}$}, what is the tightest upper bound you can give on its VC dimension? Hint: Write an expression in terms of {$|\mathcal{H}|$}, as in {$VC(\mathcal{H}) \leq f(|\mathcal{H}|)$}, where {$f$} is the function you need to supply.

(:ans:) To shatter {$m$} points requires {$2^m$} labellings, so {$|\mathcal{H}| \geq 2^m$} would be necessary. Solving for {$m$} in this expression, we arrive at the upper bound {$VC(\mathcal{H}) = m \leq log_2 |\mathcal{H}|$}. (:ansend:)

## Hyperplanes (revisited)

During the lecture we mensioned that “in general, we can show the VC dimension of hyperplanes in {$m$} dimensions is {$m+1$}”.

Proof: Let {$\mathcal H$} be the set of hyperplanes in {$m$} dimensions. First, we show that there exists a set {$S$} of {$m+1$} points {$\in \mathbb R^m$} shattered by {$\mathcal H$}.

Suppose {$S = \{x_1, \hdots, x_m, x_{m+1}\}$}, where {$x_i$} is a point {$\in \mathbb R^m$}, and our hyperplane is represented as {$y=w^{\top}x+w_0$}. Let {$\ y_1, \hdots, y_m, y_{m+1}$} be any set of labels assigned to the {$m+1$} points and construct the following linear system:

{$w^{\top}x_1+w_0=y_1$} {$w^{\top}x_2+w_0=y_2$} {$\hdots$} {$w^{\top}x_{m+1}+w_0=y_{m+1}$}.

Notice that the above linear system as having {$m+1$} variables {$\ w_0, \hdots, w_m$} and {$m+1$} equations, hence it must have a solution as long as {$S$} satisfies the condition that {$(1, x_1), \hdots, (1, x_m), (1, x_{m+1})$} are linearly independent.

Hence by choosing {$S$} s.t. the matrix {$\left( \begin{array}{cc} 1 & x_1 \\ \vdots & \vdots \\ 1 & x_{m+1} \end{array} \right)$} is full-rank, we can always solve for a {$m$}-dimensional hyperplane with bias term that separates the {$m+1$} points in {$S$}. Since {{$\ y_1, \hdots, y_m, y_{m+1}$}} can be any set of labels, the {$m$}-dimensional hyperplane shatters the {$m+1$} points in {$S$}.

Secondly, we show that there exists no set {$S'$} of {$m+2$} points can be shattered by {$\mathcal{H}$}.

Suppose to the contrary that {$S'=\{x_1, \hdots, x_{m+1}, x_{m+2}\}$} can be shattered. This implies that there exist {$2^{m+2}$} weight vectors {$w^{(1)}, \hdots, w^{(2^{m+2})}$} such that the matrix of inner products denoted by {$z_{i,j}=x_i^{\top}w^{(j)}$} has columns with all possible combination of signs (note here {$x_i$} contains the constant feature and {$w^{(j)}$} contains the bias term). We use {$A$} to denote this matrix and

{$A=\left( \begin{array}{ccc} z_{1,1} & \hdots & z_{1,2^{m+2}} \\ \vdots & \hdots & \vdots \\ z_{m+2,1} & \hdots & x_{m+2, 2^{m+2}} \end{array} \right)\end{array} \right)\ s.t.\ sign(A)=\left( \begin{array}{ccccc} - & - & \hdots & - & + \\ - & \dot & \hdots & \dot & + \\ \vdots & \vdots & \hdots & \vdots & \vdots \\ - & + & \hdots & + & + \end{array} \right)$}.

Then the rows of {$A$} are linearly independent because there exists a column of {$A$} with the same signs which does not sum to zero hence there are no constants {$c_1, c_2, \hdots, c_{m+2}$} such that {$\sum_{i=1}^{m+2}c_iz_{i, :}=0$} . However, notice that row {$i$} of {$A$} can be written as {$x_i^{\top}W$} where {$W=[w^{(1)}, \hdots, w^{(2^{m+2})}]$}. By linear algebra knowledge we know that {$m+2$} vectors {$\in \mathbb R^{m+1}$}, are always linearly dependent (i.e. {$x_1^{\top}, \hdots, {$x_{m+2}^{\top}$} are linearly dependent). Hence {$x_1^{\top}W, \hdots, {$x_{m+2}^{\top}W$} should also be linearly dependent, which results in a contradiction.

This contradiction proves that there are no {$m+2$} points {$\in \mathbb R^m$} which can be shattered by hyperplanes in {$m$} dimension. Thus, the VC dimension of {$\mathcal{H}$} is {$m+1$}.

Note: the argument can also be proved by induction, which is left as an exercise.

# Perceptron convergence

First, take a look at the lecture notes for the convergence proof.

Then, check out this perceptron applet to gain some intuition about the algorithm. If you try it for separable data, you should notice that it works well in the sense that it converges relatively quickly on a separating line. However, keep in mind that this line will likely not have the nice max-margin property we get from SVMs. For example, the line below is the one the perceptron algorithm chooses for this separable dataset:

If you try running the perceptron applet for non-separable data, it should become apparent that some additional machinery is necessary in order to make the algorithm converge. That is, you will notice that the perceptron algorithm will oscillate back and forth between several different classifiers when the data is non-separable. For instance, for the example data shown below, the algorithm oscillates back and forth between classifiers like these two:

In practice, there are two ways we can deal with this:

1. Choose a small learning rate: Instead of updating {$\mathbf{w}_{t+1} = \mathbf{w}_t + y_t\mathbf{x}_t$} we could use a more conservative update formula with a learning rate parameter {$\eta \in (0, 1]$} so that {$\mathbf{w}_{t+1} = \mathbf{w}_t + \eta y_t\mathbf{x}_t$}. The smaller the value of {$\eta$}, the smaller the impact of a misclassified point on {$\mathbf{w}$}. While this won’t entirely eliminate the oscillating behavior, it will make the oscillations smaller. That is, the classifiers oscillated between will be more similar to each other.
2. A more complete solution to the problem is the averaged perceptron algorithm. This is a modification to the perceptron algorithm that stores an averaged weight vector and makes final predictions based on this averaged vector. That is, we let each weight be the average of its values across all iterations. To implement this, you can simply use an auxiliary vector {$\mathbf{s}$} to store a running sum of {$\mathbf{w_t}$} and in the end divide {$\mathbf{s}$} by the total number of perceptron iterations run, {$T$}.


text:
# The J.D. Crawford Prize

Print

## Full Crawford Prize Guidelines and Previous Winners

John David Crawford (1954-1998) received his doctorate in Physics from the University of California at Berkeley in 1983 and was a Professor in Physics at the University of Pittsburgh from 1990. His research centered on collisionless plasmas and on pattern formation (particularly in the Faraday surface wave experiment) where he made fundamental contributions. Dr. Crawford served as co-organizer of the 1995 Dynamical Systems SIAG meeting; the SIAG/DS honors his memory with the establishment of this prize.

#### Prize Endowments

We would like to ask for your help in contributing to a lasting financial base for the prizes. It is our goal to raise an endowment of at least \$10,000 for each prize.

Contributions for the Crawford Prize endowment may be sent to:

        The J. D. Crawford Prize Fund
Attn:  Michelle Montgomery
Society for Industrial and Applied Mathematics
3600 University City Science Center


text:
# zbMATH — the first resource for mathematics

Outgoing solutions of $$(-\Delta + q - k^ 2)u=f$$ in an exterior domain. (English) Zbl 0201.13202

Full Text:
##### References:
 [1] Reichardt, H, Ausstrahlungsbedingungen für die wellengleichung, (), 41-53 · Zbl 0100.09501 [2] Werner, P, Randwertprobleme der mathematischen akustik, Arch. ration. mech. anal., 10, 29-66, (1962) · Zbl 0105.07405 [3] Werner, P, Beugunsprobleme der mathematischen akustik, Arch. ration. mech. anal., 12, 155-184, (1963) · Zbl 0112.20303 [4] Mizohata, S, Sur l’analyticité de la fonction spectrale de l’opérateur δ relatif au problème extérieur, (), 352-357 · Zbl 0122.33802 [5] Wilcox, C, The analytic continuation of the resolvent kernel in scattering theory, () [6] Steinberg, S, Meromorphic families of compact operators, Arch. ration. mech. anal., 31, 372-380, (1968) · Zbl 0167.43002 [7] \scN. Shenk and D. Thoe, Eigenfunction expansions and scattering theory for perturbations of −Δ, J. Math. Anal. Appl. (to be published). · Zbl 0254.47017 [8] Shenk, N, Eigenfunction expansions and scattering theory for the wave equation in an exterior region, Arch. ration. mech. anal., 21, 120-150, (1966) · Zbl 0135.15602 [9] Thoe, D, Eigenfunction expansions associated with schroedinger operators in Rn, n ⩾ 4, Arch. ration. mech. anal., 26, 335-356, (1967) · Zbl 0168.12501 [10] \scN. Shenk and D. Thoe, Resonant states and poles of the scattering matrix for perturbations of −Δ, J. Math. Anal. Appl. (to be published). · Zbl 0229.35072 [11] Dolph, C.L; McLeod, J.B; Thoe, D, The analytic continuation of the resolvent kernel and scattering operator associated with the schroedinger operator, J. math. anal. appl., 16, 311-332, (1966) · Zbl 0148.35904 [12] Lax, P.D; Phillips, R.S, Scattering theory, (1967), Academic Press New York · Zbl 0214.12002 [13] Schwartze, G, Über die 1., 2., and 3. äussere randwertaufgabe der schwingungsgleichung δF + k2F = 0, Math. nachr., 28, 337-363, (1965) · Zbl 0142.37502 [14] Ikebe, T, On the eigenfunction expansion connected with the exterior problem for the schroedinger equation, Jap. J. math., 36, 33-55, (1967) · Zbl 0164.13603 [15] Ikebe, T, Scattering for the schroedinger operator in an exterior domain, J. math. Kyoto univ., 7, 93-112, (1967) · Zbl 0153.42503 [16] Wilcox, C, Spherical means and radiation conditions, Arch. ration. mech. anal., 3, 113-148, (1959) · Zbl 0087.09603 [17] Watson, G.N, Theory of Bessel functions, (1966), Cambridge Univ. Press London/New York · Zbl 0174.36202 [18] () [19] Miranda, C, Equazioni alle derivate parziali di tipo ellittico, (1955), Springer Pub New York · Zbl 0065.08503 [20] Sovolev, S.L, Applications of functional analysis in mathematical physics, (1963), Amer. Math. Soc Providence, R.I [21] Kato, T, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. pure appl. math., 12, 403-425, (1959) · Zbl 0091.09502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
LinearRecursion IMO Challenging
1981

Problem - 2325
Find the maximal value of $m^2+n^2$ if $m$ and $n$ are integers between $1$ and $1981$ satisfying $(n^2-mn-m^2)^2=1$.

The solution for this problem is available for $0.99. You can also purchase a pass for all available solutions for$99.


text:
Shannons Expansion Theorem Help

Discussion in 'Homework Help' started by testing12, Oct 10, 2011.

Jan 30, 2011
80
2
[LEFT][COLOR=black]Can someone please help me to understand how the function in 6.5 is realized. I do not understand how they obtain (w2 OR w3) term, are there some intermediate steps I am missing? [/COLOR]
[COLOR=#000000]Thnak you. [/COLOR]

[IMG]http://img841.imageshack.us/img841/3572/unledpsk.jpg[/IMG][/LEFT]

2. Maen Member

Sep 16, 2011
12
4
$f(w_1,w_2,w_3)=\overline{w_1}w_3+w_2\overline{w_3}
f(w_1,w_2,w_3)=\overline{w_1}\cdot f(w_1=0,w2,w3)+w_1\cdot f(w_1=1,w2,w3)$

Does sthis help?

testing12 likes this.