The combinatorial description via ribbon graphs of the moduli space of Riemann surfaces allows to define combinatorial cycles in a natural way. Witten and Kontsevich first conjectured that these classes are polynomials in the tautological classes. We answer affirmatively to this conjecture and find recursively all the polynomials. Moreover we lift the combinatorial classes to the Deligne-Mumfor...

We prove that many families of toric ideals stabilize up to symmetry. Our results imply Hillar-Sullivant’s Independent Set Theorem and settle affirmatively questions in work by Aschenbrenner-Hillar, Hillar-Sullivant, and Hillar-Martin del Campo. Our approach involves splitting an equivariant monomial map into a part for which we have an explicit degree bound of the kernel, and a part for which ...

where Ω is a bounded, smooth domain in R, φ1 is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and h ∈ C(Ω̄). We prove that given k ≥ 1 this problem has at least k solutions for all sufficiently large s > 0, which answers affirmatively a conjecture by Lazer and McKenna [22] for this case. The solutions found exhibit multiple concentration behavior around maxi...

We prove that any bounded linear operator on Lp[0, 1] for 1 6 p < ∞, commuting with the Volterra operator V , is not weakly supercyclic, which answers affirmatively a question raised by Léon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic flavored condition on an operator which prevents it from being weakly supercyclic and is satisfied for any operator commuting with V . ...

We prove a removal lemma for infinitely-many forbidden hypergraphs. It affirmatively settles a question on property testing raised by Alon and Shapira (2005) [2, 3]. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our proof constructs a constant-time probabilistic algorithm to edit a small number of edges. It also gives a quantitative bound in t...

In [12], the widely known linear time algorithm for computing the maximum area triangle in a convex polygon was found incorrect, and whether it can still be solved in linear time was proposed as an open problem. We resolved it affirmatively by presenting a new linear time algorithm. Our algorithm is simple and easy to implement, and it computes all the locally maximal triangles. 1998 ACM Subjec...

Let G be a finite group and/: A—*B a homomorphism of G-modules. In one form, Tate's theorem says that if, for some r and all subgroups U of G, Êr~liU, f) is a surjection, HriU, f) is an isomorphism, and Hr+1iU, f) is an injection, then HniU, f) is an isomorphism for all U and all ra. Whaples has asked if the modification of this theorem stated below is true, and this paper answers Whaples' ques...

We propose a preprocessing algorithm for the multiway cut problem that establishes its polynomial kernelizability when the difference between the parameter k and the size of the smallest isolating cut is at most log(k). To the best of our knowledge, this is the first progress towards kernelization of the multiway cut problem. We pose two open questions that, if answered affirmatively, would imp...

We construct explicit generating sets Sn and S̃n of the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), Sn) and C(Sym(n), S̃n) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of random walks on groups, card shuffling a...

Suppose that X = SpecR is an F -finite normal variety in characteristic p > 0. In this paper we show that the big test ideal τb(R) = τ̃ (R) is equal to ∑ ∆ τ(R; ∆) where the sum is over ∆ such that KX + ∆ is Q-Cartier. This affirmatively answers a question asked by various people, including Blickle, Lazarsfeld, K. Lee and K. Smith. Furthermore, we have a version of this result in the case that R...