We introduce a family of Finsler metrics, called the
We investigate the geometry of the space of immersed closed curves equipped with reparametrizationinvariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractionalorder
 NSFPAR ID:
 10504705
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 The Journal of Geometric Analysis
 Volume:
 34
 Issue:
 7
 ISSN:
 10506926
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract Fisher–Rao metrics$$L^p$$ ${L}^{p}$ , for$$F_p$$ ${F}_{p}$ , which generalizes the classical Fisher–Rao metric$$p\in (1,\infty )$$ $p\in (1,\infty )$ , both on the space of densities$$F_2$$ ${F}_{2}$ and probability densities$${\text {Dens}}_+(M)$$ ${\text{Dens}}_{+}\left(M\right)$ . We then study their relations to the Amari–C̆encov$${\text {Prob}}(M)$$ $\text{Prob}\left(M\right)$ connections$$\alpha $$ $\alpha $ from information geometry: on$$\nabla ^{(\alpha )}$$ ${\nabla}^{\left(\alpha \right)}$ , the geodesic equations of$${\text {Dens}}_+(M)$$ ${\text{Dens}}_{+}\left(M\right)$ and$$F_p$$ ${F}_{p}$ coincide, for$$\nabla ^{(\alpha )}$$ ${\nabla}^{\left(\alpha \right)}$ . Both are pullbacks of canonical constructions on$$p = 2/(1\alpha )$$ $p=2/(1\alpha )$ , in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of$$L^p(M)$$ ${L}^{p}\left(M\right)$ geodesics as being energy minimizing curves. On$$\alpha $$ $\alpha $ , the$${\text {Prob}}(M)$$ $\text{Prob}\left(M\right)$ and$$F_p$$ ${F}_{p}$ geodesics can still be thought as pullbacks of natural operations on the unit sphere in$$\nabla ^{(\alpha )}$$ ${\nabla}^{\left(\alpha \right)}$ , but in this case they no longer coincide unless$$L^p(M)$$ ${L}^{p}\left(M\right)$ . Using this transformation, we solve the geodesic equation of the$$p=2$$ $p=2$ connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of$$\alpha $$ $\alpha $ , and study their relation to$$F_p$$ ${F}_{p}$ .$$\nabla ^{(\alpha )}$$ ${\nabla}^{\left(\alpha \right)}$ 
Abstract We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive
meson muoproduction at COMPASS using 160 GeV/$$\rho ^0$$ ${\rho}^{0}$c polarised and$$ \mu ^{+}$$ ${\mu}^{+}$ beams impinging on a liquid hydrogen target. The measurement covers the kinematic range 5.0 GeV/$$ \mu ^{}$$ ${\mu}^{}$$$c^2$$ ${c}^{2}$ 17.0 GeV/$$< W<$$ $<W<$ , 1.0 (GeV/$$c^2$$ ${c}^{2}$c )$$^2$$ ${}^{2}$ 10.0 (GeV/$$< Q^2<$$ $<{Q}^{2}<$c ) and 0.01 (GeV/$$^2$$ ${}^{2}$c )$$^2$$ ${}^{2}$ 0.5 (GeV/$$< p_{\textrm{T}}^2<$$ $<{p}_{\text{T}}^{2}<$c ) . Here,$$^2$$ ${}^{2}$W denotes the mass of the final hadronic system, the virtuality of the exchanged photon, and$$Q^2$$ ${Q}^{2}$ the transverse momentum of the$$p_{\textrm{T}}$$ ${p}_{\text{T}}$ meson with respect to the virtualphoton direction. The measured nonzero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons ($$\rho ^0$$ ${\rho}^{0}$ ) indicate a violation of$$\gamma ^*_T \rightarrow V^{ }_L$$ ${\gamma}_{T}^{\ast}\to {V}_{L}^{}$s channel helicity conservation. Additionally, we observe a dominant contribution of naturalparityexchange transitions and a very small contribution of unnaturalparityexchange transitions, which is compatible with zero within experimental uncertainties. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow one to evaluate in a modeldependent way the role of parton helicityflip GPDs in exclusive production.$$\rho ^0$$ ${\rho}^{0}$ 
Abstract A wellknown open problem of Meir and Moser asks if the squares of sidelength 1/
n for can be packed perfectly into a rectangle of area$$n\ge 2$$ $n\ge 2$ . In this paper we show that for any$$\sum _{n=2}^\infty n^{2}=\pi ^2/61$$ ${\sum}_{n=2}^{\infty}{n}^{2}={\pi}^{2}/61$ , and any$$1/2 $1/2<t<1$ that is sufficiently large depending on$$n_0$$ ${n}_{0}$t , the squares of sidelength for$$n^{t}$$ ${n}^{t}$ can be packed perfectly into a square of area$$n\ge n_0$$ $n\ge {n}_{0}$ . This was previously known (if one packs a rectangle instead of a square) for$$\sum _{n=n_0}^\infty n^{2t}$$ ${\sum}_{n={n}_{0}}^{\infty}{n}^{2t}$ (in which case one can take$$1/2 $1/2<t\le 2/3$ ).$$n_0=1$$ ${n}_{0}=1$ 
Abstract We consider integral areaminimizing 2dimensional currents
in$T$ $T$ with$U\subset \mathbf {R}^{2+n}$ $U\subset {R}^{2+n}$ , where$\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$ $\partial T=Q\u301a\Gamma \u301b$ and$Q\in \mathbf {N} \setminus \{0\}$ $Q\in N\setminus \left\{0\right\}$ is sufficiently smooth. We prove that, if$\Gamma $ $\Gamma $ is a point where the density of$q\in \Gamma $ $q\in \Gamma $ is strictly below$T$ $T$ , then the current is regular at$\frac{Q+1}{2}$ $\frac{Q+1}{2}$ . The regularity is understood in the following sense: there is a neighborhood of$q$ $q$ in which$q$ $q$ consists of a finite number of regular minimal submanifolds meeting transversally at$T$ $T$ (and counted with the appropriate integer multiplicity). In view of wellknown examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for$\Gamma $ $\Gamma $ . As a corollary, if$Q=1$ $Q=1$ is a bounded uniformly convex set and$\Omega \subset \mathbf {R}^{2+n}$ $\Omega \subset {R}^{2+n}$ a smooth 1dimensional closed submanifold, then any areaminimizing current$\Gamma \subset \partial \Omega $ $\Gamma \subset \partial \Omega $ with$T$ $T$ is regular in a neighborhood of$\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$ $\partial T=Q\u301a\Gamma \u301b$ .$\Gamma $ $\Gamma $ 
Abstract The electric
E 1 and magneticM 1 dipole responses of the nucleus$$N=Z$$ $N=Z$ Mg were investigated in an inelastic photon scattering experiment. The 13.0 MeV electrons, which were used to produce the unpolarised bremsstrahlung in the entrance channel of the$$^{24}$$ ${}^{24}$ Mg($$^{24}$$ ${}^{24}$ ) reaction, were delivered by the ELBE accelerator of the HelmholtzZentrum DresdenRossendorf. The collimated bremsstrahlung photons excited one$$\gamma ,\gamma ^{\prime }$$ $\gamma ,{\gamma}^{\prime}$ , four$$J^{\pi }=1^$$ ${J}^{\pi}={1}^{}$ , and six$$J^{\pi }=1^+$$ ${J}^{\pi}={1}^{+}$ states in$$J^{\pi }=2^+$$ ${J}^{\pi}={2}^{+}$ Mg. Deexcitation$$^{24}$$ ${}^{24}$ rays were detected using the four highpurity germanium detectors of the$$\gamma $$ $\gamma $ ELBE setup, which is dedicated to nuclear resonance fluorescence experiments. In the energy region up to 13.0 MeV a total$$\gamma $$ $\gamma $ is observed, but this$$B(M1)\uparrow = 2.7(3)~\mu _N^2$$ $B\left(M1\right)\uparrow =2.7\left(3\right)\phantom{\rule{0ex}{0ex}}{\mu}_{N}^{2}$ nucleus exhibits only marginal$$N=Z$$ $N=Z$E 1 strength of less than e$$\sum B(E1)\uparrow \le 0.61 \times 10^{3}$$ $\sum B\left(E1\right)\uparrow \le 0.61\times {10}^{3}$ fm$$^2 \, $$ ${}^{2}\phantom{\rule{0ex}{0ex}}$ . The$$^2$$ ${}^{2}$ branching ratios in combination with the expected results from the Alaga rules demonstrate that$$B(\varPi 1, 1^{\pi }_i \rightarrow 2^+_1)/B(\varPi 1, 1^{\pi }_i \rightarrow 0^+_{gs})$$ $B(\Pi 1,{1}_{i}^{\pi}\to {2}_{1}^{+})/B(\Pi 1,{1}_{i}^{\pi}\to {0}_{\mathrm{gs}}^{+})$K is a good approximative quantum number for Mg. The use of the known$$^{24}$$ ${}^{24}$ strength and the measured$$\rho ^2(E0, 0^+_2 \rightarrow 0^+_{gs})$$ ${\rho}^{2}(E0,{0}_{2}^{+}\to {0}_{\mathrm{gs}}^{+})$ branching ratio of the 10.712 MeV$$B(M1, 1^+ \rightarrow 0^+_2)/B(M1, 1^+ \rightarrow 0^+_{gs})$$ $B(M1,{1}^{+}\to {0}_{2}^{+})/B(M1,{1}^{+}\to {0}_{\mathrm{gs}}^{+})$ level allows, in a twostate mixing model, an extraction of the difference$$1^+$$ ${1}^{+}$ between the prolate groundstate structure and shapecoexisting superdeformed structure built upon the 6432keV$$\varDelta \beta _2^2$$ $\Delta {\beta}_{2}^{2}$ level.$$0^+_2$$ ${0}_{2}^{+}$