Noncommutative Riemann Surfaces by Embeddings in $${\mathbb{R}^{3}}$$

We introduce C-Algebras of compact Riemann surfaces as non-commutative analogues of the Poisson algebra of smooth functions on . Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.     

Global Well-Posedness of the Energy-Critical Defocusing NLS on $${\mathbb{R} \times \mathbb{T}^3}$$

We prove global well-posedness in H ¹ for the energy-critical defocusing initial-value problem \({(i\partial_t+\Delta_x)u=u|u|^2,\quad u(0)=\phi,}\) in the semiperiodic setting \({x\in\mathbb{R} \times \mathbb{T}^3}\) .     

{\mathbb{Z}}-Actions on AH Algebras and {\mathbb{Z}^2}-Actions on AF Algebras

We consider \mathbbZ{\mathbb{Z}}-actions (single automorphisms) on a unital simple AH algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the Rohlin property under some technical assumptions. Moreover, two \mathbbZ{\mathbb{Z}}-actions with the Rohlin property on such a C*-algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent. We also prove that locally approximately inner and uniformly outer \mathbbZ²{\mathbb{Z}^2}-actions on a unital simple AF algebra with a unique trace have the Rohlin property and classify them up to cocycle conjugacy employing the OrderExt group as classification invariants.     

A Centre-Stable Manifold for the Focussing Cubic NLS in $${\mathbb{R}}^{1+3}$$

Consider the focussing cubic nonlinear Schrödinger equation in \({\mathbb{R}}^3\) :

$i\psi_t+\Delta\psi = -|\psi|^2 \psi. \quad (0.1) $

It admits special solutions of the form e ^itα ?, where \(\phi \in {\mathcal{S}}({\mathbb{R}}^3)\) is a positive (? > 0) solution of

$-\Delta \phi + \alpha\phi = \phi^3. \quad (0.2)$

The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the 8-dimensional manifold that consists of functions of the form \(e^{i(v \cdot + \Gamma)} \phi(\cdot - y, \alpha)\) . We prove that any solution starting sufficiently close to a standing wave in the \(\Sigma = W^{1, 2}({\mathbb{R}}^3) \cap |x|^{-1}L^2({\mathbb{R}}^3)\) norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that \({\mathcal{N}}\) is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones [BatJon]. The proof is based on the modulation method introduced by Soffer and Weinstein for the L ²-subcritical case and adapted by Schlag to the L ²-supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in \({\mathbb{R}}^3\) for the nonselfadjoint Schrödinger operator obtained by linearizing (0.1) around a standing wave solution. All results in this paper depend on the standard spectral assumption that the Hamiltonian

$\mathcal H = \left ( \begin{array}{cc}\Delta + 2\phi(\cdot, \alpha)^2 - \alpha &;\quad \phi(\cdot, \alpha)^2 \\ -\phi(\cdot, \alpha)^2 &;\quad -\Delta - 2 \phi(\cdot, \alpha)^2 + \alpha \end{array}\right ) \quad (0.3)$

has no embedded eigenvalues in the interior of its essential spectrum \((-\infty, -\alpha) \cup (\alpha, \infty)\) .

     

Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra \mathfrakpgl(p+1|q){\mathfrak{pgl}(p+1|q)} is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of \mathfrakpgl(n|n)\not @ \mathfraksl(n|n){\mathfrak{pgl}(n|n)\not\cong \mathfrak{sl}(n|n)}), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields.     

Quantum Sphere {\mathbb{S}^4} as a Non-Levi Conjugacy Class

We construct a ${U_\hbar(\mathfrak{sp}(4))}$ -equivariant quantization of the four-dimensional complex sphere ${\mathbb{S}^4}$ regarded as a conjugacy class, Sp(4)/Sp(2) ×?Sp(2), of a simple complex group with non-Levi isotropy subgroup, through an operator realization of the quantum polynomial algebra ${\mathbb{C}_\hbar[\mathbb{S}^4]}$ on a highest weight module of ${U_\hbar(\mathfrak{sp}(4))}$ .     

Cohomology of {\mathcal {K}(2)} Acting on Linear Differential Operators on the Superspace {\mathbb{R}^{1|2}}

We compute the first differential cohomology of the Lie superalgebra ${\mathcal{K}(2)}$ of contact vector fields on the (1, 2)-dimensional real superspace with coefficients in the superspace of linear differential operators between the superspaces of weighted densities—a superisation of a result by Feigin and Fuchs. We give explicit expressions of 1-cocycles generating these spaces.     

Mutually Unbiasedness between Maximally Entangled Bases and Unextendible Maximally Entangled Systems in $\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}}$

The mutually unbiasedness between a maximally entangled basis (MEB) and an unextendible maximally entangled system (UMES) in the bipartite system \(\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}} (k>1)\) are introduced and discussed first in this paper. Then two mutually unbiased pairs of a maximally entangled basis and an unextendible maximally entangled system are constructed; lastly, explicit constructions are obtained for mutually unbiased MEB and UMES in \(\mathbb {C}^{2}\otimes \mathbb {C}^{4}\) and \(\mathbb {C}^{2}\otimes \mathbb {C}^{8}\), respectively.     

Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local $${\mathbb{C}\mathbb{P}^2}$$

The holomorphic anomaly equations describe B-model closed topological strings in Calabi–Yau geometries. Having been used to construct perturbative expansions, it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries. These yield formal nonperturbative solutions, showing integrability of the holomorphic anomaly equations at the nonperturbative level. This paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local \({\mathbb{C}\mathbb{P}^2}\) toric Calabi–Yau background, and by addressing the associated (resurgent) large-order analysis of both perturbative and multi-instanton sectors. In particular, analyzing the asymptotic growth of the perturbative free energies, one finds contributions from three different instanton actions related by \({\mathbb{Z}_3}\) symmetry, alongside another action related to the Kähler parameter. Resurgent transseries methods then compute, from the extended holomorphic anomaly equations, higher instanton sectors and it is shown that these precisely control the asymptotic behavior of the perturbative free energies, as dictated by resurgence. The asymptotic large-order growth of the one-instanton sector unveils the presence of resonance, i.e., each instanton action is necessarily joined by its symmetric contribution. The structure of different resurgence relations is extensively checked at the numerical level, both in the holomorphic limit and in the general nonholomorphic case, always showing excellent agreement with transseries data computed out of the nonperturbative holomorphic anomaly equations. The resurgence relations further imply that the string free energy displays an intricate multi-branched Borel structure, and that resonance must be properly taken into account in order to describe the full transseries solution.     

Superconductivity Near the Normal State Under the Action of Electric Currents and Induced Magnetic Fields in {\mathbb{R}^2}

We consider the linearization of the time-dependent Ginzburg-Landau system near the normal state. We assume that an electric current is applied through the sample, which captures the whole plane, inducing thereby, a magnetic field. We show that independently of the current, the normal state is always stable. Using Fourier analysis the detailed behaviour of solutions is obtained as well. Relying on semi-group theory we then obtain the spectral properties of the steady-state elliptic operator.     

Dimensional Crossover in Anisotropic Percolation on $${\mathbb {Z}}^{d+s}$$

We consider bond percolation on \({\mathbb {Z}}^d\times {\mathbb {Z}}^s\) where edges of \({\mathbb {Z}}^d\) are open with probability \(p<p_c({\mathbb {Z}}^d)\) and edges of \({\mathbb {Z}}^s\) are open with probability q, independently of all others. We obtain bounds for the critical curve in (p, q), with p close to the critical threshold \(p_c({\mathbb {Z}}^d)\). The results are related to the so-called dimensional crossover from \({\mathbb {Z}}^d\) to \({\mathbb {Z}}^{d+s}\).     

Mutually Unbiased Unextendible Maximally Entangled Bases in $\mathbb {C}^{d}\otimes \mathbb {C}^{d + 1}$

We study mutually unbiased unextendible maximally entangled bases (MUUMEBs) in bipartite stystem \(\mathbb {C}^{d}\otimes \mathbb {C}^{d + 1}\). By deriving the sufficient and necessary conditions that two MUUMEBs in \(\mathbb {C}^{3}\otimes \mathbb {C}^{4}\) need to satisfy, we first establish two pairs of MUUMEBs in \(\mathbb {C}^{3}\otimes \mathbb {C}^{4}\). Then we present the sufficient and necessary conditions that two MUUMEBs in bipartite system \(\mathbb {C}^{d}\otimes \mathbb {C}^{d + 1}\) need to satisfy, thus generalize the main results of Halqem et al. (Int. J. Theor. Phys. 54(1), 326, 2015).     

Study of ^{201} Tl\xrightarrow{\varepsilon }^{201} Hg decay properties

The intensities of the transitions to the ground-state and the first excited level of ²⁰¹Hg in ²⁰¹Tl decay and the values of the constant log ft are estimated on the basis of an analysis of the relative intensities of the mercury M ₄ N _6,7 N _6,7 Auger electron line induced by ? capture and the mercury conversion O ₃ line caused by the 1.56-keV nuclear transition.     

A Note on the Abelian Sandpile in $\pmb{\mathbb{Z}}^{d}$

We analyze the abelian sandpile model on ℤ ^d for the starting configuration of n particles in the origin and 2d−2 particles otherwise. We give a new short proof of the theorem of Fey, Levine and Peres (J. Stat. Phys. 198:143–159, 2010) that the radius of the toppled cluster of this configuration is O(n ^1/d ).     

The Cubic Dirac Equation: Small Initial Data in {{H^{\frac{1}{2}}} (\mathbb{R}^{2}})

Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space \({{H^{\frac{1}{2}}} (\mathbb{R}^{2}}\)) is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein–Gordon equation in dimension n = 2, which is captured by constructing an adapted systems of coordinate frames.