Let N ⊂ Rr be a lattice, and let deg : N → C be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on deg, the data (N, deg) determines a function f : H → C that is a holomorphic modular form of weight r for the congruence subgroup Γ1(l). Moreover, by considering all possible pairs (N, deg), we obtain a natural subring T (l) o...

Abstract. We analyze the behavior of the trace of the resolvent of an elliptic cone differential operator as the spectral parameter tends to infinity. The resolvent splits into two components, one associated with the minimal extension of the operator, and another, of finite rank, depending on the particular choice of domain. We give a full asymptotic expansion of the first component and expand ...

Barrier arguments are used to prove regularity of boundary points for a large class of uniformly elliptic operators when the domain satisfies a geometric condition. The condition is that the exterior of the domain contains a suitable lower dimensional set. Although necessary and sufficient conditions are known for the regularity of boundary points for the Dirichlet problem for a uniformly ellip...

We investigate which elliptic PDEs have the property that every viscosity supersolution belongs to a Sobolev space $$W^{1,q}_{loc}(\Omega )$$ , $$\Omega \subseteq \mathbb {R}^n$$ . The asymptotic cone of operator’s sublevel set seems be essential. It turns out much can said if we know how is related dominative p-Laplacian, with $$p = \frac{n-1}{n}q + 1$$ show that, in certain sense, this minima...

We study the inequality −Δu−μ|x|2u≥(|x|−α⁎up)uq in an unbounded cone CΩρ⊂RN (N≥2) generated by a subdomain Ω of unit sphere SN−1⊂RN, p,q,ρ>0, μ∈R and 0≤α<N. In above, |x|−α⁎up denotes standard convolution operator CΩρ. discuss existence nonexistence positive solutions terms N,p,q,α,μ Ω. Extensions to systems inequalities are also investigated.

The author develops the theory of pseudo differential equations and boundary value problems in nonsmooth domains. A model pseudo differential equation in a special cone is reduced to a certain integral equation. AMS Subject Classifications: 35S15, 45F05.

‎By the Mordell‎- ‎Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎This paper studies the rank of the family Epq:y2=x3-pqx of elliptic curves‎, ‎where p and q are distinct primes‎. ‎We give infinite families of elliptic curves of the form y2=x3-pqx with rank two‎, ‎three and four‎, ‎assuming a conjecture of Schinzel ...

Abstract. The locus of genus-two curves with n marked Weierstrass points has codimension n inside the moduli space of genus-two curves with n marked points, for n ≤ 6. It is well known that the class of the closure of the divisor obtained for n = 1 spans an extremal ray of the cone of effective divisor classes. We generalize this result for all n: we show that the class of the closure of the lo...

‎In a (t,n)-threshold secret sharing scheme‎, ‎a secret s is distributed among n participants such that any group of t or more participants can reconstruct the secret together‎, ‎but no group of fewer than t participants can do‎. In this paper, we propose a verifiable (t,n)-threshold multi-secret sharing scheme based on Shao and Cao‎, ‎and the intractability of the elliptic curve discrete logar...