Rm SO Research Articles

The existence of multiple topologically distinct solutions to $\sigma_{2,p}$-energy

Let ${\An} \subset \R^n$ be a bounded Lipschitz domain and consider the $\sigmap$-energy functional \begin{equation*} {{\mathbb F}_{\sigmap}}[u; {\An}] := \int_{\An} \big|{\wedge}^2 \nabla u\big|^p dx, \end{equation*} with $p\in\mathopen]1, \infty]$ over the space of measure preserving maps \begin{equation*} {\mathcal A}_p(\An) =\big\{u \in W^{1,2p}\big(\An, \R^n\big) : u|_{\partial \An} = {x},\ \det \nabla u =1 \mbox{ for ${\mathcal L}^n$-a.e.\ in $\An$} \big\}. \end{equation*} In this article we address the question of multiplicity {\it versus} uniqueness for {\it extremals} and {\it strong} local minimizers of the $\sigmap$-energy funcional $\mathbb F_{\sigmap}[\cdot; {\An}]$ in ${\mathcal A}_p({\An})$. We use a topological class of maps referred to as {\it generalised} twists and examine them in connection with the Euler-Lagrange equations associated with $\sigmap$-energy functional over ${\mathcal A}_p({\An})$. Most notably, we prove the existence of a countably infinite of topologically distinct twisting solutions to the later system in all {\it even} dimensions by linking the system to a set of nonlinear isotropic ODEs on the Lie group ${\rm SO}(n)$. In sharp contrast in {\it odd} dimensions the only solution is the map $u\equiv x$. The result relies on a careful analysis of the {\it full} versus the {\it restricted} Euler-Lagrange equations. Indeed, an analysis of curl-free vector fields generated by symmetric matrix fields plays a pivotal role.

Energetics of buoyancy-generated turbulent flows with active scalar: pure buoyant plume

A framework for the budget of Reynolds stress, temperature and density variations, turbulent mass flux and turbulent heat flux is presented for buoyancy-generated turbulent flows with a focus on turbulent plumes. The dynamical interactions between the turbulence generated from the velocity and thermodynamic fluctuations in thermal and heated gas plumes is studied. A large-eddy simulation tool has been used to simulate heavier and lighter than air plumes released from a circular heated source for high Reynolds number (Re). The budget equations are developed using a Favre-averaging approach. Both heated air plumes and heated gas plumes, namely, heated ${\rm SO}_2$ and heated ${\rm CH}_4$ plumes, are included in the study. The study focuses on addressing key questions – does the turbulence kinetic energy (TKE) spectrum follow the classical inertial $-5/3$ spectrum or is there a $-3$ buoyancy regime with a different scaling law? Fundamental pathways for generation of turbulence through velocity fluctuations, and temperature and density fluctuations are discussed. Analysis of the TKE and scalar variance budget equations has demonstrated that multiple mechanisms dominate the transport of Reynolds stresses, and in particular that correlations between the density and the velocity dominates the turbulence production mechanisms. Entrainment is connected to the turbulent transport and pressure-dilation processes.

Development of ultra-pure gadolinium sulfate for the Super-Kamiokande gadolinium project

Abstract This paper reports the development and detailed properties of about 13 metric tons of gadolinium sulfate octahydrate, $\rm Gd_2(\rm SO_4)_3\cdot \rm 8H_2O$, which has been dissolved into Super-Kamiokande (SK) in the summer of 2020. We evaluate the impact of radioactive impurities in $\rm Gd_2(\rm SO_4)_3\cdot \rm 8H_2O$ on diffuse supernova neutrino background searches and solar neutrino observation and confirm the need to reduce radioactive and fluorescent impurities by about three orders of magnitude from commercially available high-purity $\rm Gd_2(\rm SO_4)_3\cdot \rm 8H_2O$. In order to produce ultra-high-purity $\rm Gd_2(\rm SO_4)_3\cdot \rm 8H_2O$, we have developed a method to remove impurities from gadolinium oxide, Gd2O3, consisting of acid dissolution, solvent extraction, and pH control processes, followed by a high-purity sulfation process. All of the produced ultra-high-purity $\rm Gd_2(\rm SO_4)_3\cdot \rm 8H_2O$ is assayed by inductively coupled plasma mass spectrometry and high-purity germanium detectors to evaluate its quality. Because of the long measurement time of high-purity germanium detectors, we have employed several underground laboratories for making parallel measurements including the Laboratorio Subterráneo de Canfranc in Spain, Boulby in the UK, and Kamioka in Japan. In the first half of production, the measured batch purities were found to be consistent with the specifications. However, in the latter half, the $\rm Gd_2(\rm SO_4)_3\cdot \rm 8H_2O$ contained one order of magnitude more 228Ra than the budgeted mean contamination. This was correlated with the corresponding characteristics of the raw material Gd2O3, in which an intrinsically large contamination was present. Based on their modest impact on SK physics, they were nevertheless introduced into the detector. To reduce 228Ra for the next stage of gadolinium loading to SK, a new process has been successfully established.

Rankin–Selberg integrals for $$\mathrm{SO}_{2n+1}{\times }\mathrm{GL}_r$$ attached to newforms and oldforms

The conjectural newform theory for generic representations of $p$-adic ${\rm SO}_{2n+1}$ was formulated by P.-Y. Tsai in her thesis in which Tsai also verified the conjecture when the representations are supercuspidal. The main purpose of this work is to compute the Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ with $1\le r\le n$ attached to newforms and also oldforms under the validity of the conjecture.

Beatings of ratchet current magneto-oscillations in GaN-based grating gate structures: Manifestation of spin-orbit band splitting

We report on the study of the magnetic ratchet effect in AlGaN/GaN heterostructures superimposed with lateral superlattice formed by dual-grating gate structure. We demonstrate that irradiation of the superlattice with terahertz beam results in the dc ratchet current, which shows giant magneto-oscillations in the regime of Shubnikov de Haas oscillations. The oscillations have the same period and are in phase with the resistivity oscillations. Remarkably, their amplitude is greatly enhanced as compared to the ratchet current at zero magnetic field, and the envelope of these oscillations exhibits large beatings as a function of the magnetic field. We demonstrate that the beatings are caused by the spin-orbit splitting of the conduction band. We develop a theory which gives a good qualitative explanation of all experimental observations and allows us to extract the spin-orbit splitting constant \alpha_{\rm SO}= 7.5 \pm 1.5 meV \unicode{x212B}. We also discuss how our results are modified by plasmonic effects and show that these effects become more pronounced with decreasing the period of the gating gate structures down to sub-microns.

Crystal-field Paschen-Back effect on ruby in ultrahigh magnetic fields

Zeeman spectra of the R lines of ruby (Cr$^{3+}$: $\alpha$-Al$_{2}$O$_{3}$) were studied in ultrahigh magnetic fields up to 230 T by magneto-photoluminescence measurements. The observed Zeeman patterns exhibit nonlinear behaviors above 100 T, evidencing the breakdown of the previously reported Paschen-Back effect for ${\bf B} \perp c$ geometry. We adopted the crystal-field multiplet theory including the cubic crystal field (${\mathcal H}_{\rm cubic}$), the trigonal crystal field (${\mathcal H}_{\rm trig}$), the spin-orbit interaction (${\mathcal H}_{\rm SO}$), and the Zeeman interaction (${\mathcal H}_{\rm Z}$). It is found that the nonlinear splitting of the R lines is owing to the hybridization between the $^{2}E$ and $^{2}T_{1}$ states, which leads to the quantization of these Zeeman levels with the orbital angular momentum. Our results suggest that the exquisite energy balance among ${\mathcal H}_{\rm cubic}$, ${\mathcal H}_{\rm trig}$, ${\mathcal H}_{\rm SO}$, and ${\mathcal H}_{\rm Z}$ realized in ruby offers a unique opportunity to observe the onset of the $crystal-field$ Paschen-Back effect toward the high-field extreme.

Diagram involutions and homogeneous Ricci-flat metrics

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups ${\rm SL}(n)$, ${\rm SO}(p,q)$, ${\rm Sp}(n,\mathbb R)$. Most of these metrics are shown not to be flat.

Double and dual numbers. SU(2) groups, two-component spinors and generating functions

We explicitly show that the groups of $2 \times 2$ unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to ${\rm SO}(2,1)$ or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two-component spinors. We show that with the aid of the double numbers we can find generating functions for separable solutions of the Laplace equation in the $(2 + 1)$ Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.

T -model inflation and bouncing cosmology

We examine the dynamics of a closed cosmology whose matter source is that of a conformally coupled scalar field with a broken ${\rm SO}(1,1)$ symmetry, which correspond to the $\alpha$-attractors proposed by Linde and Kallosh. Following a field redefinition, such models give rise to "T-model" inflationary potentials, whose dynamics provide both an inflationary phase and a classical bounce. We show that the universe can undergo bounces far from the regime of quantum gravity. We analyse perturbations on this background with particular attention given to the effects of a double-bouncing scenario (with rapid recollapse between bounces) on the long wavelength modes. We demonstrate that the predictions of such models agree well with observations and might explain the suppression of power in the low multiples of the CMB.

Higher order energy functionals and the Chen-Maeta conjecture

The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called $ES-r$-energy functionals $ E_r^{ES}(\varphi) = (1/2)\int_{M}\, |(d^*+d)^r (\varphi)|^2\, dV$, where $r \geq 2 $ and $ \varphi:M \to N$ is a map between two Riemannian manifolds. The initial part of this paper is a short overview on basic definitions, properties, recent developments and open problems concerning the functionals $ E_r^{ES}(\varphi)$ and other, equally interesting, higher order energy functionals $E_r(\varphi)$ which were introduced and studied in various papers by Maeta and other authors. If a critical point $\varphi$ of $E_r^{ES}(\varphi)$ (respectively, $E_r(\varphi)$) is an isometric immersion, then we say that its image is an $ES-r$-harmonic (respectively, $r$-harmonic) submanifold of $N$. We observe that minimal submanifolds are trivially both $ES-r$-harmonic and $r$-harmonic. Therefore, it is natural to say that an $ES-r$-harmonic ($r$-harmonic) submanifold is proper if it is not minimal. In the special case that the ambient space $N$ is the Euclidean space $\mathbb{R}^n$ the notions of $ES-r$-harmonic and $r$-harmonic submanifolds coincide. The Chen-Maeta conjecture is still open: it states that, for all $r \geq2$, any proper, $r$-harmonic submanifold of $\mathbb{R}^n$ is minimal. In the second part of this paper we shall focus on the study of $G = {\rm SO}(p+1) \times {\rm SO}(q+1)$-invariant submanifolds of $\mathbb{R}^n$, $n = p+q+2$. In particular, we shall obtain an explicit description of the relevant Euler-Lagrange equations in the case that $r = 3$ and we shall discuss difficulties and possible developments towards the proof of the Chen-Maeta conjecture for $3$-harmonic $G$-invariant hypersurfaces.

Characteristics of Aerosol Pollution Under Different Visibility Conditions in Winter in a Coastal Mega-City in China

To investigate chemical profiles and formation mechanisms of aerosol particles in winter haze events, comprehensive measurements including hourly concentrations of PM2.5 and water-soluble inorganic ions and related gas-phase precursors were conducted via an online monitoring system from January to March of 2016 in Shenzhen, a coastal mega-city in south China. In this study, high concentrations of PM2.5, NO2 and lower levels of O3 were observed during haze periods in comparison with clear days (Visibility > 15km). The major secondary ionic species were \begin{document}${\rm NH_4^ +} $\end{document} 、 \begin{document}${\rm NO_3^ -} $\end{document} and \begin{document}${\rm SO_4^{2 - }}$\end{document} -, which varied significantly on haze and clear days. The ratio of \begin{document}${\rm NO_3^ -}$\end{document} / \begin{document}${\rm SO_4^{2 - }}$\end{document} in haze days was greater than that on clear days and tended to be larger when air pollution became more serious. At the same time, compared with previous studies, it has been found that the ratio has been increasing gradually in Shenzhen, indicating that the motor vehicle exhaust emissions have a more and more important impact on air quality in Shenzhen. Sulfur oxidation rate(SOR) and nitrogen oxidation rate(NOR) was higher during the haze period than that in clean days, indicating efficient gas to particle conversion. Further analysis shows that high concentrations of sulfate might be explained by aqueous oxidation, but gas-phase reactions might dominate nitrate formation. This study also highlights that wintertime nitrate formation can be an important contributor to aerosol particles, especially during haze periods.

Ancient solutions on bundles with non-abelian structural group

We generalize the ancient solutions of the Ricci flow on certain principal ${\rm SO}(3)$ bundles over compact quaternionic K\"ahler manifolds constructed by Bakas, Kong, and Ni to certain $RP^3$ fibre bundles over a product of two compact quaternionic K\"ahler manifolds. The ancient solutions are of Type I, $\kappa$-noncollapsed, and have positive Ricci curvature. Analogous solutions on even-dimensional non-K\"ahlerian bundle spaces are also constructed.

Emergent dynamics of an orientation flocking model for multi-agent system

We study the orientation flocking for the deterministic counterpart of a stochastic agent-based model introduced by Degond, Frouvelle and Merino-Aceituno in 2017, where the orientation is defined as a \begin{document}$ {\rm SO}(3) $\end{document} matrix. Their proposed model can be reduced to the other collective dynamics models such as the Lohe matrix model and the Viscek-type model as special cases. In this work, we study the emergent dynamics of the orientation flocking model in two frameworks. First, we present sufficient conditions leading to the orientation flocking when the natural frequency matrices are identical. To be precise, we prove that all orientation matrices asymptotically converge to the common one, and the spatial position diameter remains uniformly bounded. Second, we show the emergence of orientation-locked states for non-identical natural frequency matrices, that is, the difference of any two orientation matrices tends to the definite constant matrix. On the other hand, we establish the finite-in-time stability with respect to initial data of the proposed orientation flocking model. We also present the numerical results consistent with our rigorous analysis. Our work remains valid even for dimensions greater than three.

Consistent S3 reductions of six-dimensional supergravity

We work out the consistent AdS$_3\times S^3$ truncations of the bosonic sectors of both the six-dimensional ${\cal N}=(1,1)$ and ${\cal N}=(2,0)$ supergravity theories. They result in inequivalent three-dimensional half-maximal ${\rm SO}(4)$ gauged supergravities describing 32 propagating bosonic degrees of freedom apart from the non-propagating supergravity multiplet. We present the full non-linear Kaluza-Klein reduction formulas and illustrate them by explicitly uplifting a number of AdS$_3$ vacua.

Dynamics on the space of 2-lattices in 3-space

We study the dynamics of $${{\rm SL_3}(\mathbb{R})}$$ and its subgroups on the homogeneous space X consisting of homothety classes of rank-2 discrete subgroups of $${\mathbb{R}^3}$$ . We focus on the case where the acting group is Zariski dense in either $${{\rm SL_3}(\mathbb{R})}$$ or $${{\rm SO(2,1)}(\mathbb{R})}$$ . Using techniques of Benoist and Quint we prove that for a compactly supported probability measure $${\mu}$$ on $${{\rm SL_3}(\mathbb{R})}$$ whose support generates a group which is Zariski dense in $${{\rm SL_3}(\mathbb{R})}$$ , there exists a unique $${\mu}$$ -stationary probability measure on X. When the Zariski closure is $${{\rm SO(2,1)}(\mathbb{R})}$$ we establish a certain dichotomy regarding stationary measures and discover a surprising phenomenon: The Poisson boundary can be embedded in X. The embedding is of algebraic nature and raises many natural open problems. Furthermore, motivating applications to questions in the geometry of numbers are discussed.

Conformally Covariant Bi-differential Operators for Differential Forms

The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $ C^\infty(\mathbb R)$. They are covariant for the (diagonal) action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb R$ by $\mathbb R^n,$ the group ${\rm SL}(2,\mathbb R)$ by the group ${\rm SO}_0(1,n+1)$ viewed as the conformal group of $\mathbb R^n,$ and functions by differential forms.

Pullback formulae for nearly holomorphic Saito–Kurokawa lifts

We give explicit pullback formulae for nearly holomorphic Saito-Kurokawa lifts restrict to product of upper half-plane against with product of elliptic modular forms. We generalize the formula of Ichino to modular forms of higher level and free the restriction on weights. The explicit formulae provide non-trivial examples for the refined Gan-Gross-Prasad conjecture for $({\rm SO}_5,{\rm SO}_4)$ in the non-tempered cases. As an application, we obtain Deligne's conjecture for critical values of certain automorphic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$. We also expect to apply our pullback formulae to construct two-variables $p$-adic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$ in the future.

Parallelisms of $$\mathrm{PG}(3,\mathbb R)$$ PG ( 3 , R ) admitting a 3-dimensional group

Betten and Riesinger constructed Parallelisms of $\mathop{\rm PG}(3,\mathbb R)$ with automorphism group $\mathop{\rm SO}(3,\mathbb R)$ by applying the reducible $\mathop{\rm SO}(3,\mathbb R)$-action to a rotational Betten spread. This was generalized by the present author so as to include oriented parallelisms (i.e., parallelisms of oriented lines). In this way, a much larger class of examples was produced. Here we show that, apart from Clifford parallelism, these are the only topological parallelisms admitting an automorphism group of dimension 3 or larger. In particular, we show that a topological parallelism admitting the irreducible action of $\mathop{\rm SO}(3,\mathbb R)$ is Clifford.

Geodesics on ${\bf SO}(n)$ and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem

We address questions on existence, multiplicity as well as qualitative features including rotational symmetry for certain classes of geometrically motivated maps serving as solutions to the nonlinear system $$ \begin{cases} -\text{\rm div}[ F'(|x|,|\nabla u|^2) \nabla u] = F'(|x|,|\nabla u|^2) |\nabla u|^2 u &\text{in } \mathbb{X}^n,\\ |u| = 1 &\text{in } \mathbb{X}^n ,\\ u = \varphi &\text{on } \partial \mathbb{X}^n. \end{cases} $$% Here $\varphi \in \mathscr{C}^\infty(\partial {\mathbb{X}}^n, \mathbb S}^{n-1})$ is a suitable boundary map, $F'$ is the derivative of $F$ with respect to the second argument, $u \in W^{1,p}(\mathbb{X}^n, \mathbb S}^{n-1})$ for a fixed $1< p< \infty$ and ${\mathbb{X}}^n=\{x \in \mathbb R^n : a< |x|< b\}$ is a generalised annulus. Of particular interest are spherical twists and whirls, where following \cite{Taheri2012}, a spherical twist refers to a rotationally symmetric map of the form $u\colon x \mapsto \rom{Q}(|x|)x|x|^{-1}$ with $\rom{Q}$ some suitable path in $\mathscr{C}([a, b], {\rm SO}(n))$ and a whirl has a similar but more complex structure with only $2$-plane symmetries. We establish the existence of an infinite family of such solutions and illustrate an interesting discrepancy between odd and even dimensions.

Analytic torsion of arithmetic quotients of the symmetric space $${\rm SL} (n,\mathbb{R})/ {\rm SO}(n)$$ SL ( n , R ) / SO ( n )

In this paper we define a regularized version of the analytic torsion for arithmetic quotients of the symmetric space $${{\rm SL}(n,\mathbb{R})/ {\rm SO}(n)}$$ . The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat operator.