The spin-S Heisenberg antiferromagnet on the two-dimensional lattice is investigated for S=1/2 and S=1. We consider interaction at isolated dimers ($J_{\rm d}$) and interaction bonds that form the bounce lattice ($J_{\rm b}$). For $J_{\rm d}=J_{\rm b}$, the system is reduced to the maple-leaf-lattice antiferromagnet. We primarily conduct highly parallelized numerical diagonalization to examine the spin excitation gap above the ground state for various $J_{\rm b}/J_{\rm d}$ cases. For S=1/2, we report calculations for a 42-site cluster that has not been previously treated. The S=1 case is examined for the first time for clusters up to 24 sites. Regardless of whether S=1/2 or 1, we find that the system has a gapped nature for small $J_{\rm d}/J_{\rm b}$ and becomes gapless at $J_{\rm d}/J_{\rm b}\sim 1.4$. For S=1, we also find that another gapped region appears between the gapless case at $J_{\rm d}/J_{\rm b}\sim 1.4$ and the boundary of the exact-dimer phase.
In this paper, we establish local well-posedness for the Cauchy problem associated with the Korteweg-de Vries (KdV) equation on a general metric star graph. The graph comprises m + k semi-infinite edges: k negative half-lines and m positive half-lines, all joined at a common vertex. The choice of boundary conditions is compatible with the conditions determined by the semigroup theory. The crucial point in this work is to obtain the integral formula using the forcing operator method and the Fourier restriction method of Bourgain. This work extends the results obtained by Cavalcante for the specific case of the Y junction to a more general class of star graphs.
This paper studies the noncommutative singularity theory of the double $A_n$ quiver $Q_n$ (with a single loop at each vertex), with applications to algebraic geometry and representation theory. We give various intrinsic definitions of a Type A potential on $Q_n$, then via coordinate changes we (1) prove a monomialization result that expresses these potentials in a particularly nice form, (2) prove that Type A potentials precisely correspond to crepant resolutions of cAn singularities, (3) solve the Realisation Conjecture of Brown-Wemyss in this setting. For $n \leq 3$, we furthermore give a full classification of Type A potentials (without loops) up to isomorphism, and those with finite-dimensional Jacobi algebras up to derived equivalence. There are various algebraic corollaries, including to certain tame algebras of quaternion type due to Erdmann, where we describe all basic algebras in the derived equivalence class.
Given a calibration $α$ whose stabilizer acts transitively on the Grassmanian of calibrated planes, we introduce a nontrivial Lie-theoretic condition on $α$, which we call compliancy, and show that this condition holds for many interesting geometric calibrations, including Kähler, special Lagrangian, associative, coassociative, and Cayley. We determine a sufficient condition that ensures compliancy of $α$, we completely characterize compliancy in terms of properties of a natural involution determined by a calibrated plane, and we relate compliancy to the geometry of the calibrated Grassmanian. The condition that a Riemannian immersion $ι\colon L \to M$ be calibrated is a first order condition. By contrast, its extrinsic geometry, given by the second fundamental form $A$ and the induced tangent and normal connections $ abla$ on $TL$ and $D$ on $NL$, respectively, is second order information. We characterize the conditions imposed on the extrinsic geometric data $(A, abla, D)$ when the Riemannian immersion $ι\colon L \to M$ is calibrated with respect to a calibration $α$ on $M$ which is both parallel and compliant. This motivate the definition of an infinitesimally calibrated Rieman
We discuss the existence and non-existence of constant scalar curvature, as well as extremal, Sasaki metrics. We prove that the natural Sasaki-Boothby-Wang manifold over the admissible projective bundles over local products of non-negative CSC Kähler metrics, as described in https://link-springer-com.libproxy.unm.edu/article/10.1007/s00222-008-0126-x, always has a constant scalar curvature (CSC) Sasaki metric in its Sasaki-Reeb cone. Moreover, we give examples that show that the extremal Sasaki--Reeb cone, defined as the set of Sasaki--Reeb vector fields admitting a compatible extremal Sasaki metric, is not necessarily connected in the Sasaki--Reeb cone, and it can be empty even in the non-Gorenstein case. We also show by example that a non-empty extremal Sasaki--Reeb cone need not contain a (CSC) Sasaki metric which answers a question posed in https://mathscinet-ams-org.libproxy.unm.edu/mathscinet-getitem?mr=4420789. The paper also contains an appendix where we explore the existence of Kähler metrics of constant weighted scalar curvature, as defined in https://londmathsoc-onlinelibrary-wiley-com.libproxy.unm.edu/doi/full/10.1112/plms.12255, on admissible manifolds over local produ
We prove asymptotics and study sign patterns for coefficients in expansions of elements in the Habiro ring which satisfy a strange identity. As an application, we prove asymptotics and discuss positivity for the generalized Fishburn numbers which arise from the Kontsevich-Zagier series associated to the colored Jones polynomial for a family of torus knots. This extends Zagier's result on asymptotics for the Fishburn numbers.
In this paper, we build a compactification by a strictly pseudoconvex CR structure for complete and non-compact Kähler manifolds whose curvature tensor is asymptotic to that of the complex hyperbolic space.
We briefly summarize the current status of driven solid-state and cold-atom systems, and introduce articles compiled in the Focus Section in Zeitschrift für Naturforschung A, Volume 71, Issue 10 (2016)
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Branches received exceptions to Hegseth's policy that made the shot optional