We execute Avdek's algorithm to find many algebraically overtwisted and tight 3-manifolds by contact surgeries. In particular, we show that a contact surgery on the standard contact 3-sphere along any Legendrian positive torus knot with the maximal Thurston–Bennequin invariant yields an algebraically overtwisted and tight 3-manifold, where is a positive integer.
We prove the so-called master relation in the tautological ring of the moduli space of curves that implies polynomial properties of the Dubrovin–Zhang hierarchies associated to different versions of cohomological field theories as well as their equivalences to the corresponding double ramification hierarchies.
We prove that the Dirichlet forms associated with the unlabelled infinite Dyson Brownian motion with the inverse temperature do not have a spectral gap.
We construct qualitatively new examples of superabundant tropical curves which are non-realisable in genuses 3 and 4. These curves are in and , respectively, and have properties resembling canonical embeddings of genus 3 and 4 algebraic curves. In particular, the genus 3 example is a degree 4 planar tropical curve, and the genus 4 example is contained in the product of a tropical line and a tropical conic. They have excess dimension of deformation space equal to 1. Non-realisability follows by combining this with a dimension calculation for the corresponding space of logarithmic curves.
We show that the Morse boundary exhibits interesting examples of both the existence and non-existence of Cannon–Thurston maps for normal subgroups, in contrast with the hyperbolic case.
It is shown that semilinear parabolic evolution equations featuring Hölder continuous nonlinearities with at most linear growth possess global strong solutions for a general class of initial data. The abstract results are applied to a recent model describing front propagation in bushfires and in the context of a reaction–diffusion system.
Goldstine and Horwitz introduced the octonionic Hilbert space in 1964, sparking extensive research into octonionic linear operators. A recent discovery unveiled a significant characteristic of octonionic Hilbert spaces: an axiom, once deemed insurmountable, has been found not to be independent of other axioms. This discovery gives rise to a novel concept known as octonionic para-linear functionals. The purpose of this paper is to establish the octonionic Hahn–Banach theorem for para-linear functionals. Due to the non-associativity, the submodule generalized by an element is no longer of the form . This phenomenon makes it difficult to apply the octonionic Hahn–Banach theorem if the theorem holds only for functionals defined on octonionic submodules. In this paper, we introduce a notion of para-linear functionals on real subspaces and establish the octonionic Hahn–Banach theorem for such functionals. Then we apply it to obtain some valuable corollaries and discuss the reflexivity of Banach -modules.
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