It is very easy to see that every smooth projective curve can be embedded in P3. Eisenbud and Harris, in conversation, asked whether the same is true if P3 is replaced by an arbitrary smooth rational projective 3-fold X and Eisenbud suggested starting with the case where X is toric. In this note we answer the toric question, affirmatively. For simplicity, we work over the field of complex numbers.

Recent work of [Dasgupta-Kumar-Sarlós, STOC 2010] gave a sparse Johnson-Lindenstrauss transform and left as a main open question whether their construction could be efficiently derandomized. We answer their question affirmatively by giving an alternative proof of their result requiring only bounded independence hash functions. Furthermore, the sparsity bound obtained in our proof is improved. T...

We prove that the trace of the nth power of the antipode of a Hopf algebra with the Chevalley property is a gauge invariant, for each integer n. As a consequence, the order of the antipode, and its square, are invariant under Drinfeld twists. The invariance of the order of the antipode is closely related to a question of Shimizu on the pivotal covers of finite tensor categories, which we affirm...

Combinatorial Ricci flow on an ideally triangulated compact 3-manifold with boundary was introduced by Luo as a 3-dimensional analogue of Chow-Luo's combinatorial surface and conjectured to find algorithmically the hyperbolic metric totally geodesic boundary. In this paper, we prove Luo's conjecture affirmatively extending through singularities if admits such metric.

In this paper, we prove that the tautological algebra in cohomology of the moduli space Mg of smooth projective curves of genus g is generated by the first [g/3] Mumford-Morita-Miller classes. This solves a part of Faber’s conjecture [5] concerning the structure of the tautological algebra affirmatively. More precisely, for any k we express the k-th Mumford-Morita-Miller class ek as an explicit...

Riemann’s theorems on the rearrangement of absolutely convergent and conditionally convergent series of real numbers are analysed within Bishop-style constructive mathematics. The constructive proof that every rearrangement of an absolutely convergent series has the same sum is relatively straightforward; but the proof that a conditionally convergent series can be rearranged to converge to what...

In this paper we consider the linear symmetric cone programming (SCP). At a KarushKuhn-Tucker (KKT) point of SCP, we present the important equivalent conditions for the nonsingularity of Clarke’s generalized Jacobian of the KKT nonsmooth system, such as primal and dual constraint nondegeneracy, the strong regularity, and the nonsingularity of the B-subdifferential of the KKT system. This affirm...

Let (M, ω) be a compact connected symplectic manifold of dimension 2n equipped with a symplectic circle action. In this paper we show that if the fixed point set is non-empty and isolated then the symplectic circle action must be Hamiltonian. This extends the results of Tolman–Weitsman and McDuff, and proves their conjecture affirmatively. The main strategy is to use a variant of the Euler numb...

Abstract. An eventual property of a matrix M ∈ C is a property that holds for all powers M, k ≥ k0, for some positive integer k0. Eventually positive and eventually nonnegative matrices have been studied extensively, and some results are known for eventually r-cyclic matrices. This paper introduces and establishs properties of eventually reducible matrices, establishs properties of the eigenstr...

We utilize the Legendre-Fenchel transform and weak geodesics for plurisubharmonic functions to construct a weight function that can be used in Berndtsson-Lempert method, give an Ohsawa-Takegoshi extension type of result. Theorem 4.1 0.1 \cite{OT2017} (Theorem \ref{Theorem A} B} below) follow as two special cases this result, thus answering affirmatively question posed by Ohsawa remark \cite{OT2...