We describe the one-generator braces A satisfying the condition 
$A^3 = langle 0 rangle$.
We prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colourings, edge colourings and/or edge orientations.
We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-split and have no cut vertices, in a suitable topological sense. Then, we apply a result of Haken and Matveev in order to algorithmically distinguish these blocks.
We construct and study the moduli of stable hypersurfaces in toric orbifolds. Let X be a projective toric orbifold and 
$alpha in operatorname{Cl}(X)$ an ample class. The moduli space is constructed as a quotient of the linear system 
$|alpha|$ by 
$G = operatorname{Aut}(X)$. Since the group G is non-reductive in general, we use new techniques of non-reductive geometric invariant theory. Using the A-discriminant of Gelfand, Kapranov and Zelevinsky, we prove semistability for quasismooth hypersurfaces of toric orbifolds. Further, we prove the existence of a quasi-projective moduli space of quasismooth hypersurfaces in a weighted projective space when the weighted projective space satisfies a certain condition. We also discuss how to proceed when this condition is not satisfied. We prove that the automorphism group of a quasismooth hypersurface of weighted projective space is finite excluding some low degrees.
In this note, we establish a boundary maximum principle for a class of stationary pairs of varifolds satisfying a fixed contact angle condition in any compact Riemannian manifold with smooth boundary.
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