Recently Andrei Teleman considered instanton moduli spaces over negative definite four-manifolds X with b2(X) ≥ 1. If b2(X) is divisble by four and b1(X) = 1 a gauge-theoretic invariant can be defined; it is a count of flat connections modulo the gauge group. Our main result shows that if such a moduli space is non-empty and the manifold admits a connected sum decomposition X ∼= X1#X2 then both...

Let n ≥ l ≥ 2 and q ≥ 2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose l-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hy...

Let n ≥ l ≥ 2 and q ≥ 2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose l-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hy...

In this paper we deal with a theoretical question raised in connection with the application of Dirichlet-Sch type inequality, obtained by Huang (Int. Math. J. 27(02):1650009, 2016), which has been already applied to obtain multiplicity results for boundary value problems in several recent papers. We also discuss a particular case of it in more detail. As an application, we deduce the least harm...

Given the solution f of the sequential fractional differential equa1 Fite type result for SFDE’s 2 tion aD α t (aD α t f) + P (t)f = 0, t ∈ [b, c], where −∞ < a < b < c < +∞, α ∈ ( 1 2 , 1 ) and P : [a,+∞) → [0, P∞], P∞ < +∞, is continuous, assume that there exist t1, t2 ∈ [b, c] such that f(t1) = (aD α t f)(t2) = 0. Then, we establish here a positive lower bound for c−a which depends solely on...

The well-known Bernstein theorem states any complete minimal surface that can be written as the graph of a function on R must be a plane. This type of result has been generalized in higher dimension and codimension under various conditions. See [2] and the reference therein for the codimension one case and [1], [3], and [6] for higher codimension case. In this note, we prove a Bernstein type re...

We obtain a Viterbo-Hofer-Zehnder type result for Hamiltonian inclusions. Let H : : IR2N -~ IR be a locally Lipschitz function and c E IR. Suppose that E := {aE IR2~ ~ ~(a-) = c} is a nonempty compact subset of IR2N and 0 ~ ~H(x) for x E E. Then for any 03B4 > 0 the Hamiltonian inclusion x E has a conservative periodic solution x(t) such that H (~(t)~ = c’ E (c ~ , c + 8) for all t.

In this paper we look for solutions of the equation δdA = f (〈A,A〉)A in R, where A is a 1-differential form and k ≥ 2. These solutions are critical points of a functional which is strongly degenerate because of the presence of the differential operator δd. We prove that, assuming a suitable convexity condition on the nonlinearity, the equation possesses infinitely many finite energy solutions.